Benefits of time dilation / length contraction pairing? [Archive]

JesseM

Mar15-09, 10:45 PM

Ok, I accept that I may be confused. Let's look at kinematics first.

Say I do a simple experiment with a small car designed to move at a set speed (I don't what that speed is). I run it past two posts (so and si) and measure the time.

To work out the speed, I use the equation I mentioned before.

How do I work out the time t?

This is my suggestion. I have a stop watch, I start it when the car passes the first post and I stop it when it passes the second post. The value on the watch is then t, which shows me the result in "ticks" each of which will probably be 1ms or 10ms long.
Is the watch riding in the car or at rest relative to the posts on the ground? If at rest on the ground, then it must be far from at least one of the posts when the car passes it...how do you ensure that the watch is stopped "when" it passes the post it's not next to? Let's say the watch is next to the first post, so you start it when the car passes...do you stop it when you see the light from the car passing the second post, or do you prearrange things so that it will stop simultaneously with the event of the car passing the second post, relative to the rest frame of the watch and posts? If the latter, then this isn't a novel suggestion, it's the time interval that's always used when calculating speed=distance/time.
Therefore, t= (number of ticks on my stopwatch).

Not (time between each tick on my stopwatch).
The time in speed=distance/time is always the number of ticks on your stopwatch, not "time between each tick on your stopwatch". We'd only be interested in the "time between each tick on your stopwatch" if we knew the number of ticks in the watch's frame and then wanted to figure out the time for the watch to elapse this number of ticks in a different frame where the watch was in motion.
Then I notice there was a separation between me and post si and cleverly work out that that means that I don't get to see the car pass the post instantaneously, that there are now simultaneity issues, and therefore I shouldn't really be using gallilean equations, but rather lorentz based ones.

How do I work out the speed now?

I can use the value on my stopwatch, plus the knowledge that the information about the car passing si travelled to me at c (approximately, because I am not in a vaccuum).

Still, my value of t is in terms of ticks of my stopwatch, t = (number of ticks of my stopwatch before I receive information about the car passing si minus the number of ticks of my stopwatch which elapsed while that information was in transit).
Yes, then in this case you are really calculating the time on your watch between the car passing the first post and the time on your watch that is simultaneous with the car passing the second post in the watch's frame...as I said this is the time interval you'd always want to use when calculating speed=distance/time for the car in the ground frame.
Then, I decide to get more tricky. Because I have been told that for things in motion relative to me time dilates, I want to see some empirical evidence for it.

I put a video camera on post si and a stopwatch on the car along with a mechanism which starts both my stopwatch and the stopwatch on the car as the car passes post so. I call the car "Prime". I call myself "Unprime". I call the result I calculate from my stopwatch t and the result captured on the video camera as the car passes t'.

I think that the result of my empirical experiment will be that time does not dilate, but rather contracts, since the car's t' will be less than my t.
But that's just because you have "primed" and "unprimed" backwards from the normal convention. The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame). This is consistent with the convention for length contraction, where the unprimed L represents the distance between two ends of an object in the object's own rest frame, and the primed L' represents the distance between the ends of the same object in the frame where the object is in moving.

neopolitan

Mar15-09, 10:50 PM

Neopolitan,

There are four equations, one for time each dimension. It's hard to see what you mean.
I don't think most people are 'continually confused'.

If you are talking about:

t'=\gamma t

L_{x}'=L_{x} / \gamma

L_{y}'=L_{y} / \gamma

L_{z}'=L_{z} / \gamma

Then you are confused. I don't think you mean that though.

If you are talking about:

t'=\gamma t

L'=L / \gamma

t'=\gamma (t-x.\frac{v}{c^{2}}

x'=\gamma (x-vt)

then fair enough.

The pair I am talking about is:

t'=\gamma t

L'=L / \gamma

And the (to me more intuitive) pair is:

\Delta t'=\Delta t / \gamma

\Delta x'=\Delta x / \gamma

With the explanation that lengths contract in a frame in motion (no different to existing situation) and that clocks slow down in a frame in motion. If you like, you could then talk about how this is equivalent to time intervals dilating, with each tick taking longer in a frame in motion.

Then, when people like chrisc come along (and myself, many moons ago), they won't get into trouble for doing the obvious calculation x' / t' and ending up with v / \gamma^{2} or worse ... c / \gamma^{2}.

Hopefully this also answers Jesse's question.

neopolitan

Mar15-09, 11:41 PM

But that's just because you have "primed" and "unprimed" backwards from the normal convention. The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame). This is consistent with the convention for length contraction, where the unprimed L represents the distance between two ends of an object in the object's own rest frame, and the primed L' represents the distance between the ends of the same object in the frame where the object is in moving.

I know this is the convention. What is the benefit of that convention?

Note that I clearly specified thing so that I would have one primed frame (that of the car) and one unprimed frame (mine) and values from the primed frame would be primed and values from the unprimed frame would be unprimed. Is that not consistent?

The convention is to not really talk about a single frame, but to have a length for which both ends are simultaneous and clock for which ticks are colocated.

The thing that got me thinking about this most recently is chrisc's concern which I think was about light clocks (whether his issue was or was not about light clocks is immaterial either way).

Consider a single light clock. This has both a length (between a tick mirror and a tock mirror) \Delta x and consecutive ticks \Delta t. Lie it down on a carriage of some sort so that the vector \Delta x is parallel with the direction of the carriage's potential motion.

The only speed the carriage can have for which the conditions behind time dilation and length contraction can be simultaneously applied to the dimensions of the light clock (I'm only talking little 's' version of simultaneous here, not the strict definition) is zero. Only when the carriage is at rest (relative to are observer) are the ends of the light clock simultaneous (relative to the observer) and consecutive ticks are colocal (relative to the observer).

You can only apply time dilation and length contraction to the light clock in a non-trivial way by mixing frames. And if you try to use those, to work out the speed of the photon in the light clock, you end up with a non-sensical result - because of the frame mixing.

I can see that it is useful to be able to work out the amount by which you would need to slow down a clock which is at rest relative to you to match a clock which is in motion relative to you. But I still don't see the huge benefit associated with the pairing of time dilation and length contraction.

Is it mere orthodoxy? Is it historical? Or is there a real concrete advantage?

cheers,

neopolitan

JesseM

Mar16-09, 12:32 AM

I know this is the convention. What is the benefit of that convention?
That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.
Note that I clearly specified thing so that I would have one primed frame (that of the car) and one unprimed frame (mine) and values from the primed frame would be primed and values from the unprimed frame would be unprimed. Is that not consistent?
Consistent with what? I thought we were talking about consistency between notation in the length contraction and time dilation equations, I don't know what it would even mean to ask if the time dilation equation alone is "consistent". You're certainly free to refer to the frame of the clock as the primed frame if you like, but then to be consistent you should also use primed to refer to the rest frame of the object whose length you're talking about in the length contraction equation.
The thing that got me thinking about this most recently is chrisc's concern which I think was about light clocks (whether his issue was or was not about light clocks is immaterial either way).

Consider a single light clock. This has both a length (between a tick mirror and a tock mirror) \Delta x and consecutive ticks \Delta t. Lie it down on a carriage of some sort so that the vector \Delta x is parallel with the direction of the carriage's potential motion.

The only speed the carriage can have for which the conditions behind time dilation and length contraction can be simultaneously applied to the dimensions of the light clock (I'm only talking little 's' version of simultaneous here, not the strict definition) is zero. Only when the carriage is at rest (relative to are observer) are the ends of the light clock simultaneous (relative to the observer) and consecutive ticks are colocal (relative to the observer).

You can only apply time dilation and length contraction to the light clock in a non-trivial way by mixing frames. And if you try to use those, to work out the speed of the photon in the light clock, you end up with a non-sensical result - because of the frame mixing.
I don't know what you mean by "frame mixing"--don't the time dilation and length contraction equations by definition involve two different frames, one labeled primed and one labeled unprimed? But it's still consistent in the sense that if you use unprimed to refer to the distance between the two mirrors in the clock's rest frame, then unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame, and then you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving.
I can see that it is useful to be able to work out the amount by which you would need to slow down a clock which is at rest relative to you to match a clock which is in motion relative to you.
I don't understand, the time dilation equation doesn't say anything about slowing down a clock at rest relative to you, it tells you how much time t' will have elapsed in your frame (i.e. how much time elapses on normal unslowed clocks at rest relative to you) when a moving clock ticks forward by some amount t.
But I still don't see the huge benefit associated with the pairing of time dilation and length contraction.
Length contraction and time dilation are both just useful for solving basic problems without using the full Lorentz transformation (for example, if a ship is traveling to a location 12 light year away in the Earth's frame at 0.6c relative to the Earth and you want to know what the ship's clock will read when it gets there, you can figure it out either using the time t' it takes in the Earth's frame and then applying the time dilation equation, or using the distance L' between Earth and the destination in the ship's frame, and then use time = distance/speed in that frame). And if you want to write these equations next to each other, it would be confusing if you didn't use the same convention for which notation you use for the rest frame of the clock/ruler you're talking about.

neopolitan

Mar16-09, 01:52 AM

I don't know what you mean by "frame mixing"--don't the time dilation and length contraction equations by definition involve two different frames, one labeled primed and one labeled unprimed? But it's still consistent in the sense that if you use unprimed to refer to the distance between the two mirrors in the clock's rest frame, then unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame, and then you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving.

So, cutting and pasting your words to avoid mistakes:

using "unprimed to refer to the distance between the two mirrors in the clock's rest frame" (L) and noting that "unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame" (t) and keeping in mind that this is a light clock where we are using a photon, then c = L / t.

Using your logic, you use the length contraction equation to get "the distance between mirrors" and time dilation to get "the time between ticks in the frame". How far does the photon get in how much time? That would be the speed of light: c = L' / t' = c/\gamma^2 ?

cheers,

neopolitan

JesseM

Mar16-09, 02:05 AM

using "unprimed to refer to the distance between the two mirrors in the clock's rest frame" (L) and noting that "unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame" (t) and keeping in mind that this is a light clock where we are using a photon, then c = L / t.

Using your logic, you use the length contraction equation to get "the distance between mirrors" and time dilation to get "the time between ticks in the frame". How far does the photon get in how much time? That would be the speed of light: c = L' / t' = c/\gamma^2 ?
No, because in the primed frame where the light clock is moving, the distance the light travels to get from the left mirror to the right mirror is not equal to the distance between the left and right mirror at a single instant, since both mirrors are moving in this frame. If the whole structure is going from left to right at speed v, and the light is moving at speed c in both directions, then as the light goes from left to right, the distance between the light pulse and the right mirror is shrinking at a "closing speed" of (c - v), while as the light goes from right to left, the distance between the light pulse and the left mirror is shrinking at a closing speed of (c + v). So, the time in this frame for the light to go from left mirror to right and back to left is L'/(c-v) + L'/(c+v) = 2cL'/(c^2 - v^2). So if t' is the time for the light to go from left to right and back in the frame where the light clock is moving, then t' = 2cL'/(c^2 - v^2). So, plugging in L' = L*\sqrt{1 - v^2/c^2} and t' = t/\sqrt{1 - v^2/c^2} gives:

t/\sqrt{1 - v^2/c^2} = 2cL*\sqrt{1 - v^2/c^2}/(c^2 - v^2)
multiplying both sides by \sqrt{1 - v^2/c^2} gives:
t = 2cL*(1 - v^2/c^2)/(c^2 - v^2)
And since (1 - v^2/c^2) = (1/c^2)*(c^2 - v^2) this simplifies to:
t = 2cL/c^2 = 2L/c, which is exactly what we'd expect to be true in the unprimed frame where the light clock is at rest and the distance between the mirrors is L. Of course you could also reverse this algebra to show that, since the two-way time in the light-clock rest frame is t=2L/c, the two-way time in the frame where the light clock is moving must be t'=2cL'/(c^2 - v^2).

neopolitan

Mar16-09, 03:15 AM

Yes, Jesse, I know that.

You are obliged to have two way travel to make sense of it.

And you still have the issue with redefining what t means.

This was your benefit of time dilation:

That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.

One primed frame (either moving or not moving, I don't care which). The first authoritive document on SR I read talked about K and K'. One primed frame, one unprimed frame.

Values in the primed frame are primed. Values in the unprimed frame are unprimed.

Why is this so difficult?

Time dilation applies in a clock's rest frame where consecutive ticks are colocal.

Length contraction applies in a length's rest frame where the ends of the length are simultaneous (along with the bits in the middle).

This is on wikipedia (http://en.wikipedia.org/wiki/Special_relativity#Time_dilation_and_length_contra ction), and it has been there for a long time:

Writing the Lorentz transformation and its inverse in terms of coordinate differences we get

\Delta t' = \gamma . (\Delta t - \frac{v.\Delta x}{c^{2}})

\Delta x' = \gamma . (\Delta x - v \Delta t)

and

\Delta t = \gamma . (\Delta t' - \frac{v.\Delta x'}{c^{2}})

\Delta x = \gamma . (\Delta x' - v \Delta t')

Suppose we have a clock at rest in the unprimed system S. Two consecutive ticks of this clock are then characterized by Δx = 0. If we want to know the relation between the times between these ticks as measured in both systems, we can use the first equation and find:

\Delta t' = \gamma . \Delta t for events satisfying \Delta x = 0

This shows that the time Δt' between the two ticks as seen in the 'moving' frame S' is larger than the time Δt between these ticks as measured in the rest frame of the clock. This phenomenon is called time dilation.

Similarly, suppose we have a measuring rod at rest in the unprimed system. In this system, the length of this rod is written as Δx. If we want to find the length of this rod as measured in the 'moving' system S', we must make sure to measure the distances x' to the end points of the rod simultaneously in the primed frame S'. In other words, the measurement is characterized by Δt' = 0, which we can combine with the fourth equation to find the relation between the lengths Δx and Δx':

\Delta x' = \frac{\Delta x}{\gamma} for events satisfying \Delta t = 0

This shows that the length Δx' of the rod as measured in the 'moving' frame S' is shorter than the length Δx in its own rest frame. This phenomenon is called length contraction or Lorentz contraction.

In that text I see the words, I can understand the words, but I see that different versions of the Lorentz transformation are used. It is equivalent to either mixing frames or redefining what t means.

I am not saying it is wrong. It is clearly the preferred way of doing things. I am just not convinced that there is any real benefit in doing things that way.

cheers,

neopolitan

JesseM

Mar16-09, 03:46 AM

Yes, Jesse, I know that.
Then what were you asking? Why did you say "using your logic ... the speed of light: c = L' / t' " if you knew perfectly well "my logic" would not give me that equation?
You are obliged to have two way travel to make sense of it.
You can make sense of one-way travel too if you keep in mind that stopwatches situated at the position of each mirror which are synchronized in the mirror's frame will be out-of-sync by vL/c^2 in the frame where the mirror is moving at speed v. In this case, in the frame where the mirror is moving, it takes a time of t' = L'/(c-v) for the light to get from the back mirror to the front mirror. So, in this time, the stopwatch at the front mirror has ticked forward from its starting time by t = L*(1 - v^2/c^2)/(c-v). But in the primed frame the front stopwatch started vL/c^2 behind the back watch, so the time on the front watch when the light reaches it is only ahead of the time on the back watch when the light left it by [L*(1 - v^2/c^2)/(c-v)] - [vL/c^2], which can be rewritten as [Lc^2*(1 - v^2/c^2) - vL*(c-v)]/[c^2*(c-v)] = [L*(c^2 - v^2) - vLc + Lv^2]/[c^2*(c - v)] = [Lc*(c-v)]/[c^2*(c-v)] = Lc/c^2 = L/c. And of course, if we consider things in the light clock's rest frame, it makes perfect sense that if we have synchronized stopwatches next to each mirror, the time on the stopwatch next to the front mirror when the light reaches it will be greater than the time on the stopwatch next to the back mirror when the light reaches it by L/c.
And you still have the issue with redefining what t means.
Where did I redefine it?
This was your benefit of time dilation:
That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.

One primed frame (either moving or not moving, I don't care which). The first authoritive document on SR I read talked about K and K'. One primed frame, one unprimed frame.

Values in the primed frame are primed. Values in the unprimed frame are unprimed.

Why is this so difficult?
Why is what so difficult? I have never said anything that contradicted that, have I?
Time dilation applies in a clock's rest frame where consecutive ticks are colocal.
It's not clear what you mean by "applies in". Obviously in the clock's own rest frame, its time is not dilated! It's in the frame where the clock is moving that its ticks are dilated relative to ticks of coordinate time in that frame.
Length contraction applies in a length's rest frame where the ends of the length are simultaneous (along with the bits in the middle).
Same confusion about your use of "applies in".
This is on wikipedia (http://en.wikipedia.org/wiki/Special_relativity#Time_dilation_and_length_contra ction), and it has been there for a long time:

In that text I see the words, I can understand the words, but I see that different versions of the Lorentz transformation are used. It is equivalent to either mixing frames or redefining what t means.
What different versions? Can you please be specific about what particular equations on that wikipedia page you think are using contradictory definitions or are mixing frames? Everything looks consistent to me, they are using \Delta t to refer to a time interval on a clock at rest in the unprimed frame, and \Delta x to refer to the distance between ends of an object at rest in the unprimed frame. And of course, x and t refer to the coordinates of particular events in the unprimed frame (which is a slightly different convention than I was using, but theirs is probably more clear as I didn't use a notation for time intervals that was clearly distinct from notation for time coordinates, although I did use 'L' rather than 'x' for spatial intervals).
I am not saying it is wrong. It is clearly the preferred way of doing things. I am just not convinced that there is any real benefit in doing things that way.
Doing things that way as opposed to what other way? You really need to be way more specific about what exactly you're objecting to and what alternative you think would make more sense, I can't make heads or tails of what you're complaining about.

neopolitan

Mar16-09, 05:08 AM

It's not clear what you mean by "applies in". Obviously in the clock's own rest frame, its time is not dilated! It's in the frame where the clock is moving that its ticks are dilated relative to ticks of coordinate time in that frame.

Time dilation t' = \gamma t applies in the clock's own rest frame where a rest frame is unprimed. The prime is an indication that something different is happening. Same with length contraction, it applies in the frame in the length's own rest frame where the rest frame is unprimed.

What I am saying is that the frames are not the same. You can see that in the wikipedia article quite clearly.

What different versions? Can you please be specific about what particular equations on that wikipedia page you think are using contradictory definitions or are mixing frames? Everything looks consistent to me, they are using \Delta t to refer to a time interval on a clock at rest in the unprimed frame, and \Delta x to refer to the distance between ends of an object at rest in the unprimed frame. And of course, x and t refer to the coordinates of particular events in the unprimed frame (which is a slightly different convention than I was using, but theirs is probably more clear as I didn't use a notation for time intervals that was clearly distinct from notation for time coordinates, although I did use 'L' rather than 'x' for spatial intervals).

The second pair of equations in the article are the "inverse in terms of coordinate differences".
The first pair of equations give an observer in the unprimed frame the distance and time difference between two events from the perspective of an observer in the primed frame, using the measurements that the observer in the primed frame would use (contracted distances, longer periods between ticks in the primed frame).

The second pair of equations give an observer in the primed frame the distance and time difference between two events from the perspective of an observer in the unprimed frame, using the measurements that the observer in the unprimed frame would use (contracted distances, longer periods between ticks in the unprimed frame).

Because the Lorentz boosts like the Gallilean boosts use a version t such that x/t makes sense, where you could stand there counting ticks and the number of ticks is your value of t, the authors of the wikipedia article had to change frames between the derivations of length contraction and time dilation.

The authors mix frames to get around the redefinition of t problem.

Either t is redefined (from number of ticks to time between ticks) while still using the same sort of notation, or you've got frame mixing.

That's the source of a lot of confusion. And I still can't see what the benefits justifying this confusion are.

cheers,

neopolitan

BTW I usually don't mean "you" personally, but when I said you were obliged to use two way travel to make sense, I meant you personally. You can make sense of it with a one way trip (impersonal "you"). I am assuming that you know that, but also realise that one may have to complicate the issue further by bringing in simultaneity type issues. Anyway, the distance travelled between leaving one mirror and hitting the other mirror divided by the time taken to move between those mirrors will give you the speed of light in all frames. I do hope we don't disagree on that?

When I wrote L'/t' I was meaning (in context) the primed observer's length divided by the primed observer's time. Your equations didn't use the same definitions. Your quoting of me left out, quite inconveniently, the words "the time between ticks in the frame" where it was clear that "the frame" was the frame where the light clock is moving since I was responding to you where you said "you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving".

Then you went and used "the time between ticks in the resting frame". So yes, I know what you did. You changed definitions midstream again.

It reminds me of the scene in A Fish Called Wanda, where Kevin Kline is told to raise his other hand so he does, while putting down the first hand. "Don't change definitions." Okay, I'll mix frames then! Tada, same answer! So I have to say, don't mix frames and don't change definitions and then there is no confusion. Or, alternatively, do either but make clear that you are changing definitions or mixing frames and explain why it helps to do so.

It's that last little bit I am after.

JesseM

Mar16-09, 05:37 AM

Time dilation t' = \gamma t applies in the clock's own rest frame where a rest frame is unprimed.
You are just repeating yourself without giving any answer to my question of what "applies in" means. Do you agree that in the clock's rest frame, its ticks are not dilated relative to coordinate time in that frame? Do you agree that the clock's ticks are not "dilated" in any objective frame-independent sense? If so, what do you mean by "time dilation applies in the clock's own rest frame"?
The prime is an indication that something different is happening.
Again, a totally vague and incomprehensible statement if you don't give more specifics. What is different from what, exactly? Primed and unprimed are just arbitrary conventions for identifying two different frames, but each frame's view of the other is totally symmetrical--a clock at rest in the primed frame would be dilated in the unprimed frame just as a clock at rest in the unprimed frame is dilated in the primed frame.
What I am saying is that the frames are not the same.
The laws of physics are the same in each frame, even if the behavior of specific objects is different in different frames.
You can see that in the wikipedia article quite clearly.
Where?
The second pair of equations in the article are the "inverse in terms of coordinate differences".
And when you say they are "not the same", are you referring to the fact that the first pair of equations have minus signs while the second pair have plus signs? If so, this is only because we have defined "v" in a specific way--it is the velocity of the origin of the primed frame along the x-axis of the unprimed frame (so a positive value for v means the origin is moving in the +x direction, a negative value means it's moving in the -x direction). If you instead define v' to be the velocity of the origin of the unprimed frame along the x' axis of the primed frame, then the second pair of equations would have a minus sign instead:
\Delta t = \gamma(\Delta t' - v' \Delta x' /c^2)
\Delta x = \gamma(\Delta x' - v' \Delta t')
The first pair of equations give an observer in the unprimed frame the distance and time difference between two events from the perspective of an observer in the primed frame, using the measurements that the observer in the primed frame would use (contracted distances, longer periods between ticks in the primed frame).
The notion of what is determined by an "observer" in a given frame usually is just a cute way of talking about measurements in that frame and that frame alone, so it's totally confusing to talk about an observer in one frame learning the values of measurements made in a different frame. Let's just drop the talk of "observers", OK? The first pair of equations tells us the distance and time intervals between two events in the primed frame if we already know the distance and time intervals between the same two events in the unprimed frame.
The second pair of equations give an observer in the primed frame the distance and time difference between two events from the perspective of an observer in the unprimed frame, using the measurements that the observer in the unprimed frame would use (contracted distances, longer periods between ticks in the unprimed frame).
The second pair of equations tells us the distance and time intervals between two events in the unprimed frame if we already know the distance and time intervals between the same two events in the primed frame. Agreed?
Because the Lorentz boosts like the Gallilean boosts use a version t such that x/t makes sense, where you could stand there counting ticks and the number of ticks is your value of t, the authors of the wikipedia article had to change frames between the derivations of length contraction and time dilation.
Where? I asked you to refer to the specific steps and equations you have a problem with when making complaints, you aren't doing so--if you refuse to do so I will have to bow out of this conversation. I see no way in which your claim makes sense, since in the time dilation and length contraction equations they consistently use \Delta t to refer to the time between events on a clock at rest in the unprimed frame, and \Delta x to refer to the distance between events on either end of an object which is at rest in the unprimed frame.
Either t is redefined (from number of ticks to time between ticks) while still using the same sort of notation, or you've got frame mixing.
\Delta t always refers to the number of ticks of coordinate time in the unprimed frame between two events (and for events which occur at the same position in the unprimed frame, this is equal to the number of ticks between the events on a clock at rest at that position), if you think it's ever used differently you're confused about something.

neopolitan

Mar16-09, 05:55 AM

And when you say they are "not the same", are you referring to the fact that the first pair of equations have minus signs while the second pair have plus signs? If so, this is only because we have defined "v" in a specific way--it is the velocity of the origin of the primed frame along the x-axis of the unprimed frame (so a positive value for v means the origin is moving in the +x direction, a negative value means it's moving in the -x direction). If you instead define v' to be the velocity of the origin of the unprimed frame along the x' axis of the primed frame, then the second pair of equations would have a minus sign instead:
\Delta t = \gamma(\Delta t' - v' \Delta x' /c^2)
\Delta x = \gamma(\Delta x' - v' \Delta t')

Actually no, I was referring to the primes. Good try though.

The notion of what is determined by an "observer" in a given frame usually is just a cute way of talking about measurements in that frame and that frame alone, so it's totally confusing to talk about an observer in one frame learning the values of measurements made in a different frame. Let's just drop the talk of "observers", OK? The first pair of equations tells us the distance and time intervals between two events in the primed frame if we already know the distance and time intervals between the same two events in the unprimed frame.

The second pair of equations tells us the distance and time intervals between two events in the unprimed frame if we already know the distance and time intervals between the same two events in the primed frame. Agreed?

Agreed

\Delta t always refers to the number of ticks of coordinate time in the unprimed frame between two events (and for events which occur at the same position in the unprimed frame, this is equal to the number of ticks between the events on a clock at rest at that position), if you think it's ever used differently you're confused about something.

Ok, we are getting somewhere. According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. \Delta t is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

cheers,

neopolitan

neopolitan

Mar16-09, 05:59 AM

You are just repeating yourself without giving any answer to my question of what "applies in" means. Do you agree that in the clock's rest frame, its ticks are not dilated relative to coordinate time in that frame? Do you agree that the clock's ticks are not "dilated" in any objective frame-independent sense? If so, what do you mean by "time dilation applies in the clock's own rest frame"?

I mean, that for time dilation the primed frame is the non-rest frame. It is the intent, is it not?

The same with length contraction. The primed frame is the non-rest frame. Perhaps you thought I was saying something more complicated. I wasn't.

cheers,

neopolitan

Ich

Mar16-09, 08:28 AM

I'd guess you're lost in definitions:
Time dilation: t'=t*gamma - valid for objects at rest in S
Length contraction: L'=L/gamma - length of an object at rest in S (the ends measured simultaneously in S')

What you want to have:
c = l/T with
T = time between emission and absorption of a photon
l = distance between emission and absorption of a photon

Neither T nor l are defined like t or L. The simpler formulas do not apply here.
Use the doppler formula instead:
T' = T \sqrt{\frac{1-v}{1+v}}
l' = l \sqrt{\frac{1-v}{1+v}}

to see why you have to use a different formula, draw spacetime diagrams and apply the Lorentz trasformation, from which all those special cases are derived.
Space and time are not seperate in SR, so you can't simply define one length and one time interval and apply both to situations that do not match the definition.

JesseM

Mar16-09, 11:40 AM

Ok, we are getting somewhere. According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. \Delta t is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

Why are you specifying which frame "you" are at rest in here? That would seem totally irrelevant to what you're asking, nothing would change if you just said "how many ticks will occur in the primed frame between two events if:

1. \Delta t is the number of ticks in the unprimed frame between those two events,
2. a clock at rest in the primed frame has a speed of v relative to the unprimed frame"

Anyway, the answer to whether it's more ticks or less ticks than the unprimed frame depends on what events you choose. If you choose two events on the worldline of a clock at rest in the unprimed frame, then it'll be more ticks in the primed frame than the unprimed frame because the clock is slowed down in the primed frame, so its ticks are extended by the amount given by the time dilation equation. If you choose two events on the worldline of a clock at rest in the primed frame, then it'll be less ticks in the primed frame because now the clock is slowed down in the unprimed frame. And if you pick a pair of events that don't occur at the same spatial location in either frame, you have to use the more general equation:

\Delta t' = \gamma (\Delta t - v \Delta x/c^2)
I mean, that for time dilation the primed frame is the non-rest frame. It is the intent, is it not?

The same with length contraction. The primed frame is the non-rest frame. Perhaps you thought I was saying something more complicated. I wasn't.
I've discussed this with you in the past, but when you talk about "the" rest frame in a given problem, you aren't really making much sense; you seem to have gotten in your head that when analyzing a particular problem we are supposed to pick one frame to label as "the rest frame" (or 'the observer's frame', which may be related to your insertion of an irrelevant observer above), but no such convention exists. The phrase "rest frame" is generally used only in the context of talking about some specific object; for example, if clock A is at rest in the unprimed frame and clock B is at rest in the primed frame, then the unprimed frame is "the rest frame of clock A" and the primed frame is "the rest frame of clock B".

Bob S

Mar16-09, 11:58 AM

neopolitan

Mar16-09, 07:52 PM

A few years ago, experimenters at Brookhaven National Laboratory put a beam of muons (mass = 105.658 MeV) into a magnetic ring and stored them at a gamma of 29.3. The measured lifetime in the ring was about 64.4 microseconds, a factor of 29.3 longer than their measured lifetime at rest (2.2 microseconds). The dilated lifetime gave the experimenters more time to make accurate QED measurements on the muons. During the 64.4 microseconds, the muons traveled about beta*gamma*c*tau meters, where c and tau are the speed of light, and tau is the lifetime at rest. So time dilation works.

I'm not denying that it works. I don't have a problem with the principle of time dilation. It's pretty much a notation issue.

However, you do give an example of what use time dilation has for which I thank you.

The lifetime at rest is 2.2 microseconds. The "contracted time" for the muons in magnetic ring was ... 2.2 microseconds, yes? If there was a clock stored in that magnetic ring at a gamma of 29.3 it would tick off 2.2 microseconds while clocks not stored in that magnetic ring would tick off 64.4 microseconds. If that clock had a rest length of L in the direction of motion (all sorts of problems here since the clock would have to rotate to keep that length in the direction of motion which is circular, but it's hypothetical), then that length in motion would be L/gamma.

L_{in. the. ring}=L_{at. rest. in. the. laboratory} / \gamma

\Delta t_{in. the. ring}=t_{at. rest. in. the. laboratory} / \gamma

I see no problem with using the inverse of "contracted time", or time dilation, to work out that if muons are stored at a gamma of 29.3 relative to the laboratory, then those muons will last 29.3 times longer. But that equation is:

\Delta t_{at. rest. in. the. laboratory}=\gamma . t_{in. the. ring}

It seems a little odd, under these circumstances, to call t_{in. the. ring} a "rest frame" ( or the rest frame of one clock).

cheers,

neopolitan

neopolitan

Mar16-09, 07:58 PM

Jesse,

Let me repeat the question. I will highlight something for you, hopefully it will answer the question you had before:

According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. \Delta t is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

cheers,

neopolitan

JesseM

Mar16-09, 08:19 PM

Jesse,

Let me repeat the question. I will highlight something for you, hopefully it will answer the question you had before:
No, it doesn't. You are asking how many ticks will occur between two specific events "in the primed frame" (your words), so all that matters is the difference in time-coordinate between these events in the primed frame, the fact that you have defined "yourself" to be at rest in the unprimed frame is irrelevant to the problem as you've stated it. If you instead wanted to know how many ticks occur between these events in your rest frame, then the issue of which frame you were at rest in would be relevant, but that isn't what you asked.

Mentz114

Mar16-09, 08:20 PM

neopolitan:
It seems a little odd, under these circumstances, to call t_{in. the. ring} a "rest frame" ( or the rest frame of one clock).
Why ? The muon is at rest in it's own frame.

neopolitan

Mar16-09, 08:24 PM

I've discussed this with you in the past, but when you talk about "the" rest frame in a given problem, you aren't really making much sense; you seem to have gotten in your head that when analyzing a particular problem we are supposed to pick one frame to label as "the rest frame" (or 'the observer's frame', which may be related to your insertion of an irrelevant observer above), but no such convention exists. The phrase "rest frame" is generally used only in the context of talking about some specific object; for example, if clock A is at rest in the unprimed frame and clock B is at rest in the primed frame, then the unprimed frame is "the rest frame of clock A" and the primed frame is "the rest frame of clock B".

Jesse,

You have an equation, right, t' = \gamma t. One t is primed, one t is not primed. The equation is discussing the effects of motion on time intervals with the underlying assumption that one value applies to a clock in one frame and one value applies another clock in another frame and so long as \gamma is not equal to 1, the clock, and therefore the frames, are not at rest relative to each other. Look at the equation. It is taking the perspective of one clock in it's rest frame. I know you can swap the perspective over, from clock A's perspective to clock B's perspective, if you like - but still t will be the time interval for the clock whose perspective we are examining, in other words the frame in which the clock whose perspective we are examining is at rest. In terms of the equation, we could call the unprimed frame "the rest frame". But it is context.

Take it out of context (ie don't have an equation, just compare the frames) and yes, you can't justify referring to either frame as "the rest frame". It is writing the equation that tags one of the frames as "the rest frame" (in context). I've not talked about the frames at all without referring to a specific equation.

cheers,

neopolitan

JesseM

Mar16-09, 08:26 PM

However, you do give an example of what use time dilation has for which I thank you.

The lifetime at rest is 2.2 microseconds. The "contracted time" for the muons in magnetic ring was ... 2.2 microseconds, yes? If there was a clock stored in that magnetic ring at a gamma of 29.3 it would tick off 2.2 microseconds while clocks not stored in that magnetic ring would tick off 64.4 microseconds.
Yes, if by "stored in that magnetic ring" you mean "traveling along with a muon in the ring which has been accelerated to a relativistic velocity relative to the lab"
\Delta t_{at. rest. in. the. laboratory}=\gamma . t_{in. the. ring}

It seems a little odd, under these circumstances, to call t_{in. the. ring} a "rest frame" ( or the rest frame of one clock).
Why do you find it odd? Doesn't t_{in. the. ring} refer to the time interval in the coordinate system where the clock's position-coordinate is constant over time? That is all that "the rest frame of an object" means", the frame in which it has constant position coordinate (and is therefore at rest in that frame). You seem to be confused about the meaning of the term "rest frame", although I don't really understand what you think it means.

neopolitan

Mar16-09, 08:29 PM

neopolitan:

Why ? The muon is at rest in it's own frame.

The muon is stored in a ring, it is tracing out a circular trajectory. There is no single rest frame for the stored muon over 2.2 milliseconds.

JesseM

Mar16-09, 08:31 PM

Jesse,

You have an equation, right, t' = \gamma t. One t is primed, one t is not primed. The equation is discussing the effects of motion on time intervals with the underlying assumption that one value applies to a clock in one frame and one value applies another clock in another frame and so long as \gamma is not equal to 1, the clock, and therefore the frames, are not at rest relative to each other. Look at the equation. It is taking the perspective of one clock in it's rest frame.
I don't know what it means to "take the perspective" of a frame when your equation expresses a relation between two frames. t represents the time-interval between two events on the worldline of a clock as measured in the coordinates of the clock's rest frame, and t' represents the time-interval between the same two events as measured in the coordinates of a frame where the clock is moving at speed v (which we can imagine as the rest frame of some external 'observer' although this is not really necessary). The equation relates the time intervals of the two different frames--why do you think it's "taking the perspective" of one of them?

JesseM

Mar16-09, 08:37 PM

The muon is stored in a ring, it is tracing out a circular trajectory. There is no single rest frame for the stored muon over 2.2 milliseconds.
In that case you can't really use the time dilation equation without additional justification, since it's normally meant to tell you the time dilation of an inertial clock as measured in a clock where it's moving--as it happens the math still works for the muon since the speed is constant, but the full equation for time elapsed on an accelerating clock with speed v(t) in an inertial frame would be \int \sqrt{1 - v(t)^2/c^2} \, dt.

neopolitan

Mar16-09, 08:43 PM

No, it doesn't. You are asking how many ticks will occur between two specific events "in the primed frame" (your words), so all that matters is the difference in time-coordinate between these events in the primed frame, the fact that you have defined "yourself" to be at rest in the unprimed frame is irrelevant to the problem as you've stated it. If you instead wanted to know how many ticks occur between these events in your rest frame, then the issue of which frame you were at rest in would be relevant, but that isn't what you asked.

Jesse,

Let me repeat the question again. I will highlight something else for your attention.

According to me, how many ticks will I calculate to occur in the primed frame between two events if:
1. \Delta t is the number of ticks in the unprimed frame between those two events,
2. a clock in the primed frame is has a speed of v relative to me,
3. I am at rest with the clock in the unprimed frame, and
4. I remember to take into account the inertial motion of the other frame (thus adding or deleting some ticks as required)?

More ticks or less ticks than the unprimed frame?

I know the time interval between the two events in the unprimed frame. I know the speed, relative to me, of a clock in the primed frame (a clock which is at rest in the primed frame, if you prefer). I want to know how many ticks occur in the primed frame between the two events, taking into account the motion of the clock relative to those events. I think that is clear above.

What you could have justifiably called me on is that I said "speed" when I should have said "velocity" (direction and speed) and that I assumed but did not specify that I knew all about the events (not only when but also where they take place, in the unprimed frame). Mea culpa.

If I specify that I knw the velocity and that I know all about the events (time and location, in the unprimed frame), does that make things easier?

cheers,

neopolitan

neopolitan

Mar16-09, 08:45 PM

In that case you can't really use the time dilation equation without additional justification, since it's normally meant to tell you the time dilation of an inertial clock as measured in a clock where it's moving--as it happens the math still works for the muon since the speed is constant, but the full equation for time elapsed on an accelerating clock with speed v(t) in an inertial frame would be \int \sqrt{1 - v(t)^2/c^2} \, dt.

True, blindingly irrelevant given the immediate context, but true.

neopolitan

Mar16-09, 08:57 PM

I don't know what it means to "take the perspective" of a frame when your equation expresses a relation between two frames. t represents the time-interval between two events on the worldline of a clock as measured in the coordinates of the clock's rest frame, and t' represents the time-interval between the same two events as measured in the coordinates of a frame where the clock is moving at speed v (which we can imagine as the rest frame of some external 'observer' although this is not really necessary). The equation relates the time intervals of the two different frames--why do you think it's "taking the perspective" of one of them?

Context Jesse. Context. Do you grasp the concept?

You have an equation, right, t' = \gamma t. One t is primed, one t is not primed. The equation is discussing the effects of motion on time intervals with the underlying assumption that one value applies to a clock in one frame and one value applies another clock in another frame and so long as \gamma is not equal to 1, the clock, and therefore the frames, are not at rest relative to each other. Look at the equation. It is taking the perspective of one clock in it's rest frame. I know you can swap the perspective over, from clock A's perspective to clock B's perspective, if you like - but still t will be the time interval for the clock whose perspective we are examining, in other words the frame in which the clock whose perspective we are examining is at rest. In terms of the equation, we could call the unprimed frame "the rest frame". But it is context.

One clock in one frame. Another clock in another frame.

You are comparing them. But you can't compare them from outside because they are equivalent, neither has precedence, neither is privileged.

What you can say is that, according to one, the other runs slow. And it doesn't matter which clock you pick as "one", they both run slow according to the other.

When I say "according to one" above, I mean the same as "from the perspective of one". They are meant to be equivalent statements. If you don't like one variant, replace it in your mind with the other.

cheers,

neopolitan

JesseM

Mar16-09, 09:15 PM

Jesse,

Let me repeat the question again. I will highlight something else for your attention.
OK, I see that because of the way you described it, we only know the clock at rest in the primed frame has speed v in the unprimed frame by virtue of the fact that it has speed v relative to you, and you are at rest in the unprimed frame. But as I said, you could easily cut out the middleman and remove the observer from the statement of the problem, shortening the statement of the problem to the two assumptions I gave in post #28, the physical content would be exactly the same. More importantly, the "according to me" phrase at the beginning is superfluous; as long as your assumptions can be used to infer that the primed frame is moving at v relative to the unprimed frame, then the answer to the question "how many ticks will I calculate to occur in the primed frame between two events ... more ticks or less ticks than the unprimed frame?" will be the same regardless of who is doing the calculating, it will only depend on which events you pick in the unprimed frame. I thought your "according to me" might suggest a misunderstanding about that fact, that somehow the answer could be observer-dependent, which is why I said the observer was irrelevant.
I know the time interval between the two events in the unprimed frame. I know the speed, relative to me, of a clock in the primed frame (a clock which is at rest in the primed frame, if you prefer).
Are the two events supposed to be events on the worldline of that clock (so they occur at the same position in the primed frame), or two events on the worldline of an object at rest in the unprimed frame, or are they completely arbitrary?
I want to know how many ticks occur in the primed frame between the two events, taking into account the motion of the clock relative to those events. I think that is clear above.
As I said in post #28, in general for arbitrary events the answer would be \Delta t' = \gamma (\Delta t - v \Delta x /c^2). If you specify that the events occur at the same position in one of the two frames, then an answer in terms of the time dilation equation can be given.

JesseM

Mar16-09, 09:41 PM

Context Jesse. Context. Do you grasp the concept?
Yes, I understand the concept of "context", you just aren't making your notion of the context particularly clear.
You have an equation, right, . One t is primed, one t is not primed. The equation is discussing the effects of motion on time intervals with the underlying assumption that one value applies to a clock in one frame and one value applies another clock in another frame and so long as is not equal to 1, the clock, and therefore the frames, are not at rest relative to each other. Look at the equation. It is taking the perspective of one clock in it's rest frame. I know you can swap the perspective over, from clock A's perspective to clock B's perspective, if you like - but still t will be the time interval for the clock whose perspective we are examining, in other words the frame in which the clock whose perspective we are examining is at rest. In terms of the equation, we could call the unprimed frame "the rest frame". But it is context.

One clock in one frame. Another clock in another frame.

You are comparing them. But you can't compare them from outside because they are equivalent, neither has precedence, neither is privileged.
First of all, I'm not really comparing the measurements of two physical clocks, I'm comparing the time intervals between a pair of events as defined in two different frames (if you're dealing with a pair of events that don't happen at the same location in one frame, you need two different clocks which are at rest and 'synchronized' in that frame to measure the time interval between them). But that's easy enough to fix, we can just rewrite your two sentences above as A time interval in one frame. Another time interval in another frame. But with this modification I can't make sense of your subsequent statement "you can't compare them from outside because they are equivalent, neither has precedence, neither is privileged." Of course, neither is privileged physically, but they're just two numbers, why can't I compare them? If in one frame the time-interval between events A and B is 5 seconds, and in a second frame the time-interval between the same events is 10 seconds, then clearly I can say "the time-interval between these events is twice as large in the second frame as it is in the first frame". Have I somehow "privileged" one of these time-intervals in making this statement? If so, which one?
What you can say is that, according to one, the other runs slow. And it doesn't matter which clock you pick as "one", they both run slow according to the other.
OK, so your argument does depend critically on the notion that you want to compare the rate that two physical clocks are ticking rather than comparing the time-intervals between a specific pair of events? If so, then in this case I agree that we can't say which is ticking faster or slower without first picking a frame. But the time dilation equation as it's normally written is about time-intervals, not instantaneous rates of ticking. Maybe a failure to realize this could be the source of much of your confusion? If we have an unprimed clock moving at velocity v relative to the primed clock, then the normal time dilation equation \Delta t' = \Delta t * \gamma can be written in words like this:

"time interval between two events on worldline of unprimed clock as measured in primed frame = time interval between two events on worldline of unprimed clock as measured in unprimed frame (which is of the same as the time interval between those events as measured by the unprimed clock itself, since it's at rest in the unprimed frame) * gamma"

On the other hand, if we wanted to talk about clock rates in the primed frame, then we could use d\tau' /dt' to represent the rate the primed clock is ticking relative to the primed frame's coordinate time (this would just be equal to 1 of course), and d\tau / dt' to represent the rate the unprimed clock is ticking relative to the primed frame's coordinate time (this would be less than 1), in which case the equation would be:

d\tau' / dt' = (d\tau / dt' )/ \gamma

So in this equation, we are dividing by gamma rather than multiplying by it as in the standard time dilation equation. And in this equation we are clearly looking at things "from the perspective" of a particular frame, namely the primed frame where the unprimed clock is moving. This equation could be written in words like this:

"Rate that primed clock is ticking in primed frame = rate that unprimed clock is ticking in primed frame divided by gamma"

Does this distinction between clock rates and time intervals help at all?

JesseM

Mar16-09, 09:49 PM

True, blindingly irrelevant given the immediate context, but true.
The relevance was to your statement "It seems a little odd, under these circumstances, to call t_{in. the. ring} a "rest frame" ( or the rest frame of one clock)."--I didn't understand that you were specifically saying it was odd because the muon was moving non-inertially, I thought you considered it odd because the muon wasn't at rest in the lab observer's frame or something. If your argument there specifically depended on the notion that we were accelerating the muon rather than talking about a muon moving at high velocity in a straight line (like the muons from cosmic ray showers), then the simple answer is that we don't call t_{in. the. ring} the time in the "rest frame", my earlier statement about the unprimed referring to time in the clock's rest frame was meant to refer specifically to the usual time dilation equation, which is defined to relate time-intervals measured on one inertial clock compared with the time-intervals in an inertial frame where the clock is moving. You can come up with an identical-looking equation that applies to the special case of a clock that's moving non-inertially as viewed from the perspective of an inertial frame where its speed is constant, but this is not really "the time dilation equation", and in this equation t_{in. the. ring} would have to refer to the proper time on the non-inertial clock between two events on its worldline, compared with the coordinate time interval between those same two events in the inertial frame where the clock's speed is constant.

neopolitan

Mar16-09, 10:37 PM

Yes, I understand the concept of "context", you just aren't making your notion of the context particularly clear.

First of all, I'm not really comparing the measurements of two physical clocks, I'm comparing the time intervals between a pair of events as defined in two different frames (if you're dealing with a pair of events that don't happen at the same location in one frame, you need two different clocks which are at rest and 'synchronized' in that frame to measure the time interval between them). But that's easy enough to fix, we can just rewrite your two sentences above as A time interval in one frame. Another time interval in another frame. But with this modification I can't make sense of your subsequent statement "you can't compare them from outside because they are equivalent, neither has precedence, neither is privileged." Of course, neither is privileged physically, but they're just two numbers, why can't I compare them? If in one frame the time-interval between events A and B is 5 seconds, and in a second frame the time-interval between the same events is 10 seconds, then clearly I can say "the time-interval between these events is twice as large in the second frame as it is in the first frame". Have I somehow "privileged" one of these time-intervals in making this statement? If so, which one?

OK, so your argument does depend critically on the notion that you want to compare the rate that two physical clocks are ticking rather than comparing the time-intervals between a specific pair of events? If so, then in this case I agree that we can't say which is ticking faster or slower without first picking a frame. But the time dilation equation as it's normally written is about time-intervals, not instantaneous rates of ticking. Maybe a failure to realize this could be the source of much of your confusion? If we have an unprimed clock moving at velocity v relative to the primed clock, then the normal time dilation equation \Delta t' = \Delta t * \gamma can be written in words like this:

"time interval between two events on worldline of unprimed clock as measured in primed frame = time interval between two events on worldline of unprimed clock as measured in unprimed frame (which is of the same as the time interval between those events as measured by the unprimed clock itself, since it's at rest in the unprimed frame) * gamma"

On the other hand, if we wanted to talk about clock rates in the primed frame, then we could use d\tau' /dt' to represent the rate the primed clock is ticking relative to the primed frame's coordinate time (this would just be equal to 1 of course), and d\tau / dt' to represent the rate the unprimed clock is ticking relative to the primed frame's coordinate time (this would be less than 1), in which case the equation would be:

d\tau' / dt' = (d\tau / dt' )/ \gamma

So in this equation, we are dividing by gamma rather than multiplying by it as in the standard time dilation equation. And in this equation we are clearly looking at things "from the perspective" of a particular frame, namely the primed frame where the unprimed clock is moving. This equation could be written in words like this:

"Rate that primed clock is ticking in primed frame = rate that unprimed clock is ticking in primed frame divided by gamma"

Does this distinction between clock rates and time intervals help at all?

Ok, your words do sort of encapsulate the issue.

Getting back to the original concern about confusion that the pairing of length contraction and time dilation:

if you pick L and \Delta t, such that L/\Delta t= c,

and you had a length (with rest length L) and a clock (with a rest tick-tick time of \Delta t) in a frame in motion, the equations for working out what the speed of light is in that frame in motion using the measurements in that frame in motion are not:

\Delta t' = \Delta t * \gamma
L' = L / \gamma

but something closer to (but not quite):

d\tau' / dt' = (d\tau / dt' )/ \gamma
L' = L / \gamma

cheers,

neopolitan

JesseM

Mar16-09, 11:17 PM

if you pick L and \Delta t, such that L/\Delta t= c,

and you had a length (with rest length L) and a clock (with a rest tick-tick time of \Delta t) in a frame in motion, the equations for working out what the speed of light is in that frame in motion using the measurements in that frame in motion are not:

\Delta t' = \Delta t * \gamma
L' = L / \gamma
You can figure out what the speed of light in the second frame is using these equations along with the equation for the relativity of simultaneity, as I showed in post #22 (and if you're talking about two-way speed of light you don't have to worry about simultaneity, as I showed in #20).
but something closer to (but not quite):

d\tau' / dt' = (d\tau / dt' )/ \gamma
L' = L / \gamma
I don't really see how the second pair of equations would be more helpful in calculating the speed of light in the primed frame than the first pair was, you'd still have to worry about simultaneity along with the fact that the object with length L' is moving in this frame. Also, if you know the \Delta x and \Delta t between the event of the light being sent and the event of it being received, the mathematically simplest thing to do is to calculate:

\Delta x' = \gamma (\Delta x - v \Delta t)
\Delta t' = \gamma (\Delta t - v \Delta x /c^2)

And in this case it will be true that \frac{\Delta x'}{\Delta t'} = c.

neopolitan

Mar16-09, 11:46 PM

I don't really see how the second pair of equations would be more helpful in calculating the speed of light in the primed frame than the first pair was ...<snip>

Because I know that in the frame in motion relative to me lengths contract and clocks slow down. I know that relative to that frame in motion relative to me, I am the one in motion, with contracted lengths and slow clocks.

Using my contracted lengths and slow clocks (relative to the frame which is in motion relative to me), I should be able to calculate that the speed of light to be invariant. Similarly, I expect that anyone in motion relative to me will calculate, using their contracted lengths and slow clocks (relative to me), that the speed of light is invariant.

I'm not really trying to work out what photons are doing relative to me (which is what your work in posts #20 and #22 are about). I'm trying to work out that the relativistically affected lengths and time, in another frame, still give you an invariant speed of light.

Why? Because that is where people tend to get confused, doing just that using lc and td - and these are the people who care enough to try to work it out, the world is full of people who really don't care. (See this thread (http://www.physicsforums.com/showthread.php?t=289509) and this thread (http://www.physicsforums.com/showthread.php?t=298145) for recent examples.)

cheers,

neopolitan

whybother

Mar16-09, 11:49 PM

"Can you think of any benefits?"
"Benefits? No, he'll die."
"I get that, but have there ever been any fundraisers?"

JesseM

Mar17-09, 12:00 AM

Because I know that in the frame in motion relative to me lengths contract and clocks slow down. I know that relative to that frame in motion relative to me, I am the one in motion, with contracted lengths and slow clocks.
And you don't think the time dilation equation tells you that? A slow clock should naturally take longer to tick out a certain number of seconds, no?
I'm not really trying to work out what photons are doing relative to me (which is what your work in posts #20 and #22 are about). I'm trying to work out that the relativistically affected lengths and time, in another frame, still give you an invariant speed of light.
Don't really understand the distinction you're making here. My posts only made use of photons indirectly--I was picking two different clock readings which both lay on the path of the photon in the unprimed frame where the rulers and clocks were at rest, and then from then on I didn't say anything about photons, I just figured out the distance and time between these same two clock readings in the coordinates of the primed frame, based on the idea that the clocks were slowed down (and out-of-sync in the case of one-way speed) and the rulers were shrunk, and they were all moving at speed v in this frame. How would you use "relativistically affected lengths and times" to show "an invariant speed of light" if you weren't even allowed to select events on the worldlines of clocks in the primed frame such that the distance between the clocks in that frame divided by the difference in the clocks' readings at those events should be c?
Why? Because that is where people tend to get confused, doing just that using lc and td
Sorry, what do lc and td refer to?
- and these are the people who care enough to try to work it out, the world is full of people who really don't care. (See this thread (http://www.physicsforums.com/showthread.php?t=289509) and this thread (http://www.physicsforums.com/showthread.php?t=298145) for recent examples.)
Those are long threads, can you give some specific quotes which you think are examples of the type of confusion you're trying to describe? (I still don't understand quite what the confusion is that you're talking about--is it just the confusion between clock rates and time intervals, or something else?)

neopolitan

Mar17-09, 01:23 AM

Sorry, what do lc and td refer to?

Those are long threads, can you give some specific quotes which you think are examples of the type of confusion you're trying to describe? (I still don't understand quite what the confusion is that you're talking about--is it just the confusion between clock rates and time intervals, or something else?)

lc = length contraction (or Lorentz contraction)
td = time dilation

The threads are examples. One of them you posted to. I'm not going to pick statements from either and post here out of context.

As to confusion about clock rates and time intervals, yes. Not quite the words I would use, but pretty much yes. I do struggle for terminology which you would understand and which is still what I mean. Perhaps "displayed time" and "time interval" - displayed time is what is on the display of your clock, time interval is the period between ticks.

I know you didn't want an observer, but I need one. Because I need someone to look at the clock face, take a measurement and calculate the speed of light. Then, if that observer is in motion (in a primed frame), and looks at the clock face (in the primed frame), takes a measurement (in the primed frame) and calculates the speed of light again (using a primed time value and a primed length), that speed of light won't have changed.

All I am saying is that to do that, that observer (in the primed frame) won't be using what is to an unprimed frame (ie one notionally at rest) a dilated time interval and a contracted length, but rather a contracted displayed time value (t/gamma) and a contracted length (but in the frame in which those measurements are actually made, neither the displayed value nor the length will be contracted, they will be normal).

cheers,

neopolitan

JesseM

Mar17-09, 02:20 AM

The threads are examples. One of them you posted to. I'm not going to pick statements from either and post here out of context.
Can you give me a post # from one of those threads where you think someone is getting confused by the time interval/rate of ticking distinction, at least? If that is indeed the confusion you're talking about?
All I am saying is that to do that, that observer (in the primed frame) won't be using what is to an unprimed frame (ie one notionally at rest) a dilated time interval and a contracted length, but rather a contracted displayed time value (t/gamma) and a contracted length (but in the frame in which those measurements are actually made, neither the displayed value nor the length will be contracted, they will be normal).
Why do you think that, if you haven't even done the calculation yourself?

I realize in retrospect that I did refer to the photon in my post #22 a bit more than just calculating the clock readings in the unprimed frame. But let me give an altered calculation which doesn't. Say in the unprimed frame we have two clocks at either end of a rod at rest in this frame with length L, the clock on the left reads some time T when the photon leaves it, and the clock on the right reads T + L/c when the photon reaches it. Now I'll show that the distance/time between these two events must be c in the primed frame too, using only the two clock readings and the length of the rod and its velocity, along with the time dilation, length contraction and relativity of simultaneity equations (no further reference to a photon). If the rod is moving to the right with velocity v in the primed frame, then using the relativity of simultaneity equation, we know that at the "same moment" that the left clock reads T ('same moment' according to the primed frame's definition of simultaneity), the right clock must read T - vL/c^2. So, by the time the right clock reads T + L/c, it has ticked forward by (T + L/c) - (T - vL/c^2) = L/c + vL/c^2 = cL/c^2 + vL/c^2 = (c+v)*L/c^2. So, this must be the time interval in the unprimed frame between the event of the right clock reading (T - vL/c^2) and the event of the right clock reading (T + L/c), so we can use the time dilation equation with \Delta t = (c+v)*L/c^2 to conclude that the time interval between these events is \Delta t' = (c+v)*L/c^2 \sqrt{1 - v^2/c^2} in the primed frame. And since the event of the left clock reading T is simultaneous with the event of the right clock reading (T - vL/c^2) in the primed frame, this must also be the time interval between the event of the left clock reading T and the event of the right clock reading (T + L/c), the same two events we were considering in the unprimed frame.

Now we just have to find the spatial distance between these two events in the primed frame. Well, using the length contraction equation we know that the right clock was initially a distance of L * \sqrt{1 - v^2/c^2} from the left clock at the moment the left clock read T. We also know that the time between these events in the primed frame was \Delta t' = (c+v)*L/c^2 \sqrt{1 - v^2/c^2}, and the right clock was moving at velocity v to the right the whole time, so by the time of the second event (the right clock reading T + L/c) the right clock will have moved an additional distance of v times that time interval, or v*(c+v)*L/c^2 \sqrt{1 - v^2/c^2}. So, adding that additional distance to the distance of L * \sqrt{1 - v^2/c^2} that the right clock was from the left clock initially when the left clock read T, the total distance \Delta x' between these two events in the primed frame must be [L * \sqrt{1 - v^2/c^2}] + [v*(c+v)*L/c^2 \sqrt{1 - v^2/c^2}], or [L*c^2*(1 - v^2/c^2) + v*(c+v)*L]/[c^2 \sqrt{1 - v^2/c^2}]. So, dividing this \Delta x' by \Delta t' = (c+v)*L/c^2 \sqrt{1 - v^2/c^2} which we found earlier gives:
[L*c^2*(1 - v^2/c^2) + v*(c+v)*L]/[(c+v)*L]
or
[L*(c+v)*(c-v) + v*(c+v)*L]/[(c+v)*L]
or
(c-v) + v = c. So, that completes the demonstration that \Delta x' / \Delta t' in the primed frame for the event of the left clock reading T and the event of the right clock reading T + L/c must also be equal to c. And as I said initially, nowhere did I have to talk about photons except at the very beginning when finding that these two clock-readings would both lie on the path of a photon moving at c in the unprimed frame.

Can you show how you would demonstrate that distance/time in the primed frame between these same two events would be c, using your alternate equation for the instantaneous rate that clocks at rest in the unprimed frame will be seen to be ticking in the primed frame, as opposed to the time dilation equation I used? If you're claiming that the demonstration would somehow be a lot simpler using this equation, then you need to show how this simpler demonstration would actually work in order to convince me of this; I don't believe it would actually make things any simpler.

neopolitan

Mar17-09, 03:02 AM

JesseM

Mar17-09, 03:47 AM

Here you go Jesse.

I have an experimental apparatus. Using that apparatus myself (no other frame, just me), I can measure a distance, L and the measured time it takes a photon to travel that distance, t (the specifics of the measurement is immaterial). I specifically made this apparatus so that L / t = c.

Then I give it to my buddy, I put him on a carriage with a velocity of v. According to his measurements, the distance I measured to be L would be L and the measured time it takes a photon to travel that distance would be t.

However, I know that the ruler he took with him has shrunk, relative to me. So really what he is measuring to be L, is really a shorter version of L, which I call L'.
L' represents the length of the apparatus in your frame, yes?
Then, because I know that I set up my apparatus so that c = L/t, and the speed of light is invariant, I can therefore work out that t' = L' /c.

If I work out that L' = L / gamma, then I must work out that t' = t /gamma.
I thought you were supposed to be deriving the fact that the speed of light is invariant, but here you seem to simply assume it. Also, what does t' represent, physically? Is it the time-interval in your (primed) frame between the event of your buddy measuring the light passing the left end of his apparatus and the event of his measuring the right end of the apparatus? If so, you can't assume the distance between these events is L' in your frame even though that's the length of the apparatus in your frame, because of course his apparatus is moving in your frame.

If t' does not represent a time interval in your frame between two specific events, then you have to either specify in clear physical terms what it does represent or your argument is totally incoherent, which is what it appears to be right now. I can't see how the equation c = L'/t' would make any sense unless it's interpreted as (distance between two events on the worldline of a light beam in the primed frame)/(time between the same two events in the primed frame). That's what speed always means in physics, (change in position)/(change in time), where the "change" is between two events on the worldline of the object whose speed you're measuring. Like I said, if you have some other clear physical definition of what t' in your equation represents if not the time-interval in your frame between two specific events, by all means present it, but I suspect you're just playing with symbols without having really thought through what they are supposed to represent physically.

neopolitan

Mar17-09, 04:01 AM

I thought you were supposed to be deriving the fact that the speed of light is invariant, but here you seem to simply assume it.

I would have started with L' = L / gamma and t' = t / gamma.

If the apparatus is such that L/t = c, then:

L' / t' = (L / gamma) / (t / gamma) = L/t = c

Are you happier with that?

cheers,

neopolitan

JesseM

Mar17-09, 04:26 AM

I would have started with L' = L / gamma and t' = t / gamma.

If the apparatus is such that L/t = c, then:

L' / t' = (L / gamma) / (t / gamma) = L/t = c

Are you happier with that?
No, because you haven't defined the physical meaning of t' in any way that leads me to think L'/t' can be understood as the "speed" in any frame. As an analogy, I could define L'=L*pi/(the gravitational constant), and t'=t*pi/(the gravitational constant), and then it would be true mathematically that if L/t=c then L'/t'=c as well, but this would have no physical interpretation in terms of L'/t' representing distance/time in some inertial frame (and therefore would have nothing to do with the idea that the speed of light is frame-invariant), it would just be a meaningless math game.

Mentz114

Mar17-09, 07:44 AM

The muon is stored in a ring, it is tracing out a circular trajectory. There is no single rest frame for the stored muon over 2.2 milliseconds.

Rubbish. You are are really missing something here.

Dmitry67

Mar17-09, 09:00 AM

neopolitan

Mar17-09, 10:33 AM

No, because you haven't defined the physical meaning of t' in any way that leads me to think L'/t' can be understood as the "speed" in any frame. As an analogy, I could define L'=L*pi/(the gravitational constant), and t'=t*pi/(the gravitational constant), and then it would be true mathematically that if L/t=c then L'/t'=c as well, but this would have no physical interpretation in terms of L'/t' representing distance/time in some inertial frame (and therefore would have nothing to do with the idea that the speed of light is frame-invariant), it would just be a meaningless math game.

I haven't defined either L or L' in the post you are responding to either.

I have to admit hovering my cursor over a section of text and wondering if I would have to define t' for you.

Here goes:

My buddy is measuring a length which is at an angle in spacetime due to his motion relative to me. In my frame, the extremities of the length that he is measuring are not simultaneous and not as far apart as he measures them to be. The overall effect is that his lengths are contracted, according to me. So my apparatus, at rest with me, has a greater length than his apparatus.

My buddy is also measuring a time interval, between two events which are colocated in his frame. His clock runs slow, so that while his clock measures off t, the "real" time elapsed (according to me) is a greater period. So on my clock, at rest with me, I have a displayed time which is greater than his displayed time.

Since I was calling my length and displayed time L and t respectively, that makes his smaller length L' and his lesser displayed time t'.

So, if L/t = c then L'/t'=c.

(Little reminder here, t' here is not derived from time dilation. It is the time displayed in my buddy's frame where t is the time displayed in my frame.)

------------------------------------

You may have a problem with "displayed time". I do understand that.

However, think about how we measure the average speed of a horse running on a race track. We press a button on the stopwatch when the race starts. We press it again when the horse passes the finish line. We know the length of the course (L) and the time it took the horse to run that distance (t) because we see the time displayed on the stopwatch.

Working out the average speed (which you seemed to have difficult coping with) is relatively simple. Take the length measured and divide by the displayed time.

neopolitan

Mar17-09, 10:45 AM

Rubbish. You are are really missing something here.

So the muons in question had a single inertial rest frame? Is that what I am missing?

I'd be curious to know how they managed to keep that up for 64.4 ms while at a gamma factor of 29.3 (about 99.9% of the speed of light). Was their apparatus 20,000km long? I am pretty sure I would have noticed it when visiting New York.

Mentz114

Mar17-09, 12:39 PM

So the muons in question had a single inertial rest frame? Is that what I am missing?

I'd be curious to know how they managed to keep that up for 64.4 ms while at a gamma factor of 29.3 (about 99.9% of the speed of light). Was their apparatus 20,000km long? I am pretty sure I would have noticed it when visiting New York.
You don't know the meaning of 'rest frame'. Forget the muons for a momemnt and suppose I was on space ship accelerating away from you - are you saying I don't have a rest frame ?
It may not be inertial but I've still got one.

Your attempt at sarcasm is pathetic.

JesseM

Mar17-09, 01:09 PM

My buddy is measuring a length which is at an angle in spacetime due to his motion relative to me. In my frame, the extremities of the length that he is measuring are not simultaneous and not as far apart as he measures them to be. The overall effect is that his lengths are contracted, according to me. So my apparatus, at rest with me, has a greater length than his apparatus.
So are you just talking about the standard notion of "length" in the standard length contraction equation, where L' represents the distance between ends of his apparatus at a single moment in your primed frame?
My buddy is also measuring a time interval, between two events which are colocated in his frame.
If he's measuring a time between events that are colocated in this frame, what does this have to do with measuring the speed of light? Why did you say that L/t=c? Is he measuring the two-way speed of light using a clock at one end of his apparatus and a mirror at the other?
His clock runs slow, so that while his clock measures off t, the "real" time elapsed (according to me) is a greater period.
The time elapsed in your frame between the same two events that were colocated in his frame? But if the time elapsed in your frame \Delta t' is greater than the time elapsed in his frame \Delta t, that would imply you should be using the standard time dilation equation \Delta t' = \Delta t * \gamma, so what were you talking about when you said "If I work out that L' = L / gamma, then I must work out that t' = t /gamma"? It was the meaning of t' and t' in that equation that I wanted a physical definition of, of course I know the physical definition of \Delta t and \Delta t' in the normal time dilation equation.
Since I was calling my length and displayed time L and t respectively that makes his smaller length L' and his lesser displayed time t'.
Aarrgh, you never specified that L and t represented your frame, and throughout this thread I've been using the convention (and you've been quoting from things like the wikipedia page which use the same convention) that the rest length of the ruler whose length is shrunk in the length contraction equation, and the time in the rest frame of the clock whose time is dilated in the time dilation equation, is the unprimed frame, which is the reverse of what you're doing above. I even asked "L' represents the length of the apparatus in your frame, yes?" back in post #53 and you never corrected me (keep in mind that in your example, your buddy was the one carrying 'the apparatus' along with him, not you). And then in this very post you say above "His clock runs slow, so that while his clock measures off t ..." (unprimed), so now you seem to be contradicting yourself when you say your time is t and he has "lesser displayed time t' ".

In any case, if the length in his frame is L', is this supposed to be the length of something at rest in his frame, like the apparatus he's carrying? If so, it makes no sense for you to say "his smaller length L' " since of course the rest length of an object is greater than its contracted length in the frame of an observer (like you) who sees it in motion. If you wanted L' to represent the rest length of his apparatus in his frame, and L to represent the length of his apparatus in your frame, the equation would have to be L' = L*gamma, not L'=L/gamma like you wrote (note that if you want to use a weird convention where primed is the rest length, then it would make more sense to have the equation giving unprimed as a function of primed, i.e. L = L' / gamma, since the length contraction and time dilation equations are always written to give you time and distance in the frame where the ruler/clock are moving as a function of time and distance in the frame where they're at rest. I really wish we could just agree to use the standard convention that unprimed represents the rest frame of the ruler and clock though, it would lead to much less confusion when quoting sources like wikipedia, and it's the convention I've been using throughout the entire thread which you never objected).

Also, if t' represents the time in his frame between events that are colocated in his frame, and t represents the time in your frame between these same events, and you said yourself that his time t' would be smaller (because his clock has a 'lesser displayed time'), then there is nothing nonstandard about your equation "t' = t / gamma"--this is just a reshuffling of the standard time dilation equation, which if we write it in words so we avoid primed and unprimed confusion, would be (time interval between events on clock's worldline in frame where clock is in motion) = (time interval between events on clock's worldline in frame where clock is at rest) * gamma, so obviously dividing both sides by gamma will give (time interval between events on clock's worldline in frame where clock is at rest) = (time interval between events on clock's worldline in frame where clock is in motion) / gamma, which is what you seem to be expressing with your equation t' = t / gamma.

BTW, maybe to avoid any further possible confusion about notation, we should use notation like \Delta t_{buddy} and L_{buddy} to express times and lengths measured in your buddy's frame, and \Delta t_{neo} and L_{neo} to express times and lengths in your frame? In this case if your buddy says the length of his apparatus is L_{buddy}, then the length of his (moving) apparatus in your frame is given by:

L_{neo} = L_{buddy} / \gamma

And if your buddy says the time between two events which are colocal in his frame is t_{buddy}, then the time between those same two events in your frame is:

t_{neo} = t_{buddy} * \gamma

Which of these equations are you disagreeing with, if any?
So, if L/t = c then L'/t'=c.
Why does L/t=c? Again, if we're not talking about the distance and time between two events on the worldline of a lightbeam, that equation makes little sense.
You may have a problem with "displayed time". I do understand that.
If we're dealing with two events which are colocal in your buddy's frame and your buddy has a watch at the position of these events, then on his clock "displayed time" is of course exactly equal to the time interval between the events in his frame, and that's exactly what's in the standard time dilation equation. So are you saying the time in your frame where your buddy is moving is also "displayed time"? But if you're talking about events which are colocal in his frame, in your frame these events are at different positions, so if you use a single stopwatch you have to worry about light-speed delays.
However, think about how we measure the average speed of a horse running on a race track. We press a button on the stopwatch when the race starts. We press it again when the horse passes the finish line.
Yes, that works fine in a horce-race because the delay between the events of the horse departing/finishing and the events of our seeing these things happen is totally negligible, since the time for light to get from the horse's position to our eyes is miniscule compared to the time of the race. If we were measuring a horse traveling at close to the speed of light, though, then if I'm standing at a fixed position with a stopwatch while the horse races by me, then the time between my seeing the horse begin and end might be significantly different than the time interval between the horse actually beginning and ending in my rest frame, because of these delays for light to reach my eyes. Of course if I know the distance the horse was from me when it began and ended I can compensate for this (for example, if I see the horse cross the finish line when my watch reads 8 seconds but I know the finish line is at a distance of 3 light-seconds from me, then I can say the event 'really' occurred simultaneously with my clock reading 5 seconds in my frame), but this requires some more calculation than just looking at my watch and noting "displayed time" directly. The other option, of course, would be the one Einstein imagined where there are multiple clocks which are "synchronized" in my frame, so I could note the displayed time on the clock at the starting line when the horse starts, then note the displayed time on the clock at the finish line when the horse crosses it, and the second displayed time minus the first displayed time would be the time interval in my frame.
We know the length of the course (L) and the time it took the horse to run that distance (t) because we see the time displayed on the stopwatch.
But this seems to contradict what you said earlier about your buddy measuring the time "between two events which are colocated in his frame".

JesseM

Mar17-09, 04:23 PM

Shouldnt muon decays be additionally slowed down because of very strong centrifugal forces affecting the muon? (GR time dilation)?
Centrifugal forces are "fictitious" forces that are only introduced for bookkeeping purposes when you use accelerating (non-inertial) reference frames (see the wikipedia page on fictitious forces (http://en.wikipedia.org/wiki/Fictitious_force), along with the illustrated discussion here (http://hyperphysics.phy-astr.gsu.edu/Hbase/corf.html), especially the last box), accelerating observers can experience them in totally flat (uncurved) spacetime, and as long as you stick to inertial frames in flat spacetime you can analyze everything about the behavior of such accelerating observers within the context of SR using only SR time dilation to calculate their aging (though as this section (http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html) of the twin paradox (http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html) page says, even in flat spacetime you can have 'pseudo-gravitational' time dilation if you use a non-inertial frame in flat spacetime).

DaleSpam

Mar17-09, 04:28 PM

JesseM

Mar17-09, 04:51 PM

I have not been following this conversation very closely, but I do find it confusing that the length contraction formula (http://en.wikipedia.org/wiki/Length_contraction) is given as:
L' = L/γ

whereas the time dilation formula (http://en.wikipedia.org/wiki/Time_dilation) is given as:
Δt' = γΔt

when the 1-1 d Lorentz transform looks the same for time and space (in units where c=1):
t' = γ(t-vx)
x' = γ(x-vt)
You can see visually why this is if you look at the diagram I drew up for neopolitan in a previous discussion, which he posted in post #5--basically the two equations are telling you somewhat different things if you think in terms of a spacetime diagram, and it is possible to come up with a "spatial analogue for the time dilation equation" which looks like the time dilation equation but with L substituted for \Delta t (in this equation you are talking about the spatial distance in the primed frame between two events which occur at the same time but at a spatial separation of L in the unprimed frame, which is analogous to how time dilation tells you the time interval in the primed frame between two events which occur at the same position but at a temporal separation of \Delta t in the unprimed frame...note that this is not what the standard length contraction equation tells you), and it's also possible to come up with a "temporal analogue for the length contraction equation" which looks like the length contraction equation but with \Delta t substituted for L (this is more difficult to state in words, but it's basically the time-interval in the primed frame between two surfaces of constant t in the unprimed frame which have a temporal distance of \Delta t in the unprimed frame, which is analogous to how length contraction tells you the spatial distance in the primed frame between two worldlines of constant position in the unprimed frame which have a spatial separation of L in the unprimed frame).

DaleSpam

Mar17-09, 05:26 PM

Yes, I understand, and I also know it is correct as defined. I just find it confusing, especially when I was first learning.

Because of my initial confusion I made it a general rule to never ever use either the time dilation or length contraction formula, and I have continued to stick to that rule since then. Instead I always use the Lorentz transform directly in order to avoid making a mistake in which frame is doing what.

Rasalhague

Mar17-09, 08:28 PM

I’m new to SR and have also been struggling with this apparent asymmetry between the equations for time dilation and length contraction. I found the diagram http://www.physicsforums.com/attachment.php?attachmentid=17992&d=1237127533 very helpful, and Jesse’s explanations, e.g. #64, of the "temporal analogue to length contraction", and "spatial analogue to time dilation".

If I try putting this in my own words, could you tell me if I've got it right?

As I understand it, the time dilation formula takes as its input the time between two events in a frame where there’s no space between them, i.e. two events colocal in the unprimed frame. When you plug into it this time and the speed the primed frame is moving relative to the unprimed, the formula t’ = t * gamma tells you the time in the primed frame between the two events.

The length contraction formula takes as its input the distance between two events that are simultaneous in the unprimed frame. When you plug into it this distance and the speed the primed frame is moving relative to the unprimed frame, the formula L’ = L/gamma tells you the distance in the primed frame between one of these events *and a third event*, which third event is simultaneous in the primed frame with the first, and in the unprimed frame is colocal with the second.

More generally, I guess I’ve been trying to understand what exactly the asymetry is, and where it comes from: whether a physical difference between time and space, an accident of the kind of questions asked in textbooks, or of the way the idea is expressed, or if there’s something about our relationship to time and space that makes us want to ask a different question of each--that is, something that makes this pair of not-directly-analogous questions more useful to us than any other combination of the four possible questions illustrated in the diagram. Does that make any sense?

neopolitan

Mar17-09, 08:55 PM

Note that I too have said that time dilation is correct as defined.

I have also said that the type of time used in time dilation is a different type of time to the one we usually use (one is the time between ticks, the other is the number of ticks displayed).

Jesse,

In post #51 you posted: Can you show how you would demonstrate that distance/time in the primed frame between these same two events would be c, using your alternate equation for the instantaneous rate that clocks at rest in the unprimed frame will be seen to be ticking in the primed frame, as opposed to the time dilation equation I used? If you're claiming that the demonstration would somehow be a lot simpler using this equation, then you need to show how this simpler demonstration would actually work in order to convince me of this; I don't believe it would actually make things any simpler.

I accept that there has been some dancing around, which makes things confusing. But the bit I was trying to do, the bit I was focussed on was "using your alternate equation". I ignored the words following it, because you were taking something out of context from post #45: but something closer to (but not quite):

d\tau' / dt' = (d\tau / dt' )/ \gamma
L' = L / \gamma

I specifically said "closer to (but not quite)" because it is not an instantaneous rate that I am talking about. So what you demanded of me was unreasonable.

The alternate equation I was using is this:

t' = t / \gamma

You want to know in which frame the events are colocated. Really, they don't need to be colocated in any frame.

Perhaps if I rephrase it will make it easier. I want to know the relativistic effects on my buddy's measurements of time and space. I can use that to calculate what the speed of light is in his frame (and I am most interested in that speed when light moves parallel to the direction of his motion). I want to use the same sort of dimensions that we usually use to calculate a speed (a length which can be traversed and a readout of time elapsed).

What effect does putting my buddy into motion have on his dimensions?

The answer is that his lengths contract and the readouts of time elapsed are reduced - according to me. (And my lengths and my readouts of time elapsed are reduced - according to him.)

Are we agreed on this? (I agree that the period between his ticks is greater than the period between my ticks, according to me.)

cheers,

neopolitan

neopolitan

Mar17-09, 09:02 PM

I regret posting exactly when I did. Rasalhague asks a fine question in the post before mine.

JesseM

Mar17-09, 10:53 PM

Note that I too have said that time dilation is correct as defined.

I have also said that the type of time used in time dilation is a different type of time to the one we usually use (one is the time between ticks, the other is the number of ticks displayed).
This distinction seems meaningless--unless you are postulating some notion of absolute time, the only way we can talk about "time between ticks" is by looking at the "number of ticks displayed" on a clock (or pair of synchronized clocks) with some smaller interval between ticks; for example, if the time between ticks is 1 second, put another clock next to it that ticks once every millisecond, and find that the second clock elapses 1000 ticks between each tick of the first clock. The time between a clock's ticks is just like any other time interval, we measure it using the difference in reading between the end of the interval and the beginning (or more abstractly, we can define it using the difference between the time coordinates between the events of two successive ticks).
I accept that there has been some dancing around, which makes things confusing. But the bit I was trying to do, the bit I was focussed on was "using your alternate equation". I ignored the words following it, because you were taking something out of context from post #45
When I said "using your alternate equation" I really meant "using whatever alternate equation you wish to define", not specifically using the alternate equation I gave in post #45.
The alternate equation I was using is this:

t' = t / \gamma

You want to know in which frame the events are colocated. Really, they don't need to be colocated in any frame.
Fine, but regardless of what events you choose, you still haven't given any coherent definition of what t and t' mean in terms of coordinates assigned to the events, or in terms of actual physical measurements involving those events. Does t represent the coordinate time interval between two events in the unprimed frame, and t' represent the coordinate time interval between the same two events in the unprimed frame? If not, what do t and t' mean? You can't write down equations and say they have any relevance to physics if you can't even define the physical meaning of the variables in the equations!
Perhaps if I rephrase it will make it easier. I want to know the relativistic effects on my buddy's measurements of time and space. I can use that to calculate what the speed of light is in his frame (and I am most interested in that speed when light moves parallel to the direction of his motion). I want to use the same sort of dimensions that we usually use to calculate a speed (a length which can be traversed and a readout of time elapsed).

What effect does putting my buddy into motion have on his dimensions?

The answer is that his lengths contract and the readouts of time elapsed are reduced - according to me. (And my lengths and my readouts of time elapsed are reduced - according to him.)

Are we agreed on this?
Your words are too vague and would only make sense with specific types of elaborations. Does "his lengths contract ... according to me" mean that the length you assign to an object at rest in his frame is smaller than the length he assigns to that same object? In that case I would agree. Does "his lengths contract ... according to me" mean that the length you assign to an object at rest in your frame is smaller than the length he assigns to that same object? If so, the statement is wrong. Does "his lengths contract ... according to me" mean that the spatial distance between two arbitrary events as measured by you is smaller than the spatial distance between the same two events as measured by him? If so, this would be true in some cases but not in others.

And what does "readouts of time elapsed are reduced ... according to me" mean? Does it mean that if you consider a time interval from some time t'_0 to another time t'_1 in your frame, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs at time t'_1 in your frame) - (his clock's readout at the event on the clock's worldline that occurs at time t'_0 in your frame) is smaller than the difference t'_1 - t'_0 which represents the size of the time interval in your frame? In that case I would agree. Does it mean that if you consider two arbitrary events A and B (like two events on the worldline of a photon), with A happening at t'_0 and B happening at t'_1 in your frame, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs simultaneously with B in your frame) - (his clock's readout at the event on the clock's worldline that occurs simultaneously with A in your frame) is smaller than the time interval between A and B in your frame? If so, this is exactly equivalent to the first one, so I'd agree with this too. But does it mean that if you consider the same two events A and B, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs simultaneously with B in his frame) - (his clock's readout at the event on the clock's worldline that occurs simultaneously with A in his frame) is smaller than the time interval between A and B in your frame? If so this is not true in general. And if the two events are events on the path of a photon that occur on either end of a measuring-rod of length L in his frame, then it is that last difference in his clock's readouts that you divide L by to get c.
I regret posting exactly when I did. Rasalhague asks a fine question in the post before mine.
Don't worry, I intend to reply to it.

Rasalhague

Mar17-09, 11:06 PM

The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame). This is consistent with the convention for length contraction, where the unprimed L represents the distance between two ends of an object in the object's own rest frame, and the primed L' represents the distance between the ends of the same object in the frame where the object is in moving.

As someone new to SR, the use of primed and unprimed coordinates has been a source of some confusion for me. I think that’s partly because, to begin with, I didn’t know which details of a textbook explanation were going to turn out to be the significant ones, and which were accidental to a particular author’s presentation. At first, I got the impression that the convention was simply that primed frame = rest frame. But then I encountered examples such as: “Consider a rod at rest in frame S’ with one end at x’2 and the other end at x’2 [...]” (Tipler & Mosca: Physics for Scientists and Engineers, 5e, extended version, p. 1274). From this, I guessed that the desciding factor in whether to call a frame primed or unprimed was the direction it was moving, a primed frame being one that moves in the positive x direction with respect to the unprimed frame, hence the signs used in versions of the full Lorentz transformation such as this:

x' = gamma (x – ut)
t' = gamma (t – ux/c^2)

x = gamma (x' + ut')
t = gamma (t' + ux'/c^2)

I think that’s the traditional choice of signs, isn’t it? But I don’t know yet how rigid that convention is. The practice of using primed coordinates to represent a frame moving in the positive x direction agrees with their use in discussions of Euclidian rotation for a frame rotated in the positive (counterclockwise) direction. At least, that’s how Euclidian rotation was introduced in the first books where I met it. But in Spacetime Physics, Taylor and Wheeler do it the other way around, using primed coordinates for a negative (clockwise) rotation. I suppose, looking on the bright side, the advantage to having a variety of presentation methods is that it, eventually, you can see by comparing them which details are the physically significant ones, and which a matter of convention. But to begin with it’s a lot of information to take in.

JesseM

Mar17-09, 11:12 PM

If I try putting this in my own words, could you tell me if I've got it right?

As I understand it, the time dilation formula takes as its input the time between two events in a frame where there’s no space between them, i.e. two events colocal in the unprimed frame. When you plug into it this time and the speed the primed frame is moving relative to the unprimed, the formula t’ = t * gamma tells you the time in the primed frame between the two events.

The length contraction formula takes as its input the distance between two events that are simultaneous in the unprimed frame. When you plug into it this distance and the speed the primed frame is moving relative to the unprimed frame, the formula L’ = L/gamma tells you the distance in the primed frame between one of these events *and a third event*, which third event is simultaneous in the primed frame with the first, and in the unprimed frame is colocal with the second.
Yes, these look right to me. But to make the physical meaning of the second one a little more clear, you might point out that if the first event occurs along the worldline of the left end of an object at rest in the unprimed frame, and the second event occurs along the worldline of the same object's right end (so if the distance between the events is L in the unprimed frame, this must be the length of the object in the unprimed frame), that means that the third event also occurs along the worldline of the object's right end, so since the first and third event are simultaneous in the primed frame, L' must be the length of the object in the primed frame. The idea is that the "length" of an object in any frame is defined as the distance between its two ends at a single instant in that frame.
More generally, I guess I’ve been trying to understand what exactly the asymetry is, and where it comes from: whether a physical difference between time and space, an accident of the kind of questions asked in textbooks, or of the way the idea is expressed, or if there’s something about our relationship to time and space that makes us want to ask a different question of each--that is, something that makes this pair of not-directly-analogous questions more useful to us than any other combination of the four possible questions illustrated in the diagram. Does that make any sense?
I guess I would say the usefulness of these two equation is that in physics it is typical to calculate dynamics by taking as "initial conditions" a spatial arrangement of objects at a single moment in time (along with the instantaneous velocities at that time), and then use the dynamical equations of physics to evolve that initial state forward through time. So, it's useful to know the set of spatial coordinates an object occupies at a single instant which is where the length contraction equation comes in handy, and it's useful to know how much a clock will have moved forward if you evolve the initial state forward by some particular amount of coordinate time, which is where the time dilation equation comes in handy. The "spatial analogue of time dilation" and the "temporal analogue of length contraction" equations would come in more handy if we took as our "initial state" a surface of constant x rather than a surface of constant t, and then evolved this state forward by increasing the x coordinate and looking at how things changed in successive surfaces of constant x. But this points to a real difference between how the laws of physics treat time and space; in a deterministic universe the laws of physics do allow you to determine what the physical state will be in later surfaces of constant t if you know the physical state at an earlier surface of constant t, but they don't allow you to determine the state of a surface of constant x if all you know is the state of some other surface of constant x. I guess this is also related to the fact that in SR the worldlines are timelike, so if we assume no worldline has a start or end (no worldlines starting or ending at singularities as in GR, and no particle worldlines ending because the particle annihilates with another particle as in quantum theory), then a given worldline will pierce every surface of constant t precisely once, while a worldline can have no intersections with a surface of constant x, or one pointlike intersection, or multiple pointlike intersections, or an infinite collection of points on that worldline can lie on a single surface of constant x.

JesseM

Mar18-09, 12:08 AM

As someone new to SR, the use of primed and unprimed coordinates has been a source of some confusion for me. I think that’s partly because, to begin with, I didn’t know which details of a textbook explanation were going to turn out to be the significant ones, and which were accidental to a particular author’s presentation. At first, I got the impression that the convention was simply that primed frame = rest frame. But then I encountered examples such as: “Consider a rod at rest in frame S’ with one end at x’2 and the other end at x’2 [...]” (Tipler & Mosca: Physics for Scientists and Engineers, 5e, extended version, p. 1274). From this, I guessed that the desciding factor in whether to call a frame primed or unprimed was the direction it was moving, a primed frame being one that moves in the positive x direction with respect to the unprimed frame, hence the signs used in versions of the full Lorentz transformation such as this:

x' = gamma (x – ut)
t' = gamma (t – ux/c^2)

x = gamma (x' + ut')
t = gamma (t' + ux'/c^2)

I think that’s the traditional choice of signs, isn’t it? But I don’t know yet how rigid that convention is. The practice of using primed coordinates to represent a frame moving in the positive x direction agrees with their use in discussions of Euclidian rotation for a frame rotated in the positive (counterclockwise) direction. At least, that’s how Euclidian rotation was introduced in the first books where I met it. But in Spacetime Physics, Taylor and Wheeler do it the other way around, using primed coordinates for a negative (clockwise) rotation. I suppose, looking on the bright side, the advantage to having a variety of presentation methods is that it, eventually, you can see by comparing them which details are the physically significant ones, and which a matter of convention. But to begin with it’s a lot of information to take in.
There isn't really any absolute convention about primed and unprimed, they're just ways of differentiating two distinct frames, although most authors seem to follow the conventions that you said you were most used to above. But as long as you understand the physical relations of the two frames that's all that really matters. For example, if someone writes \Delta t' = \Delta t * \gamma (the most common form I've seen), then you know \Delta t must refer to two events which occur at the same position in the unprimed frame, like two readings on a clock at rest in the primed frame; if someone instead wrote \Delta t = \Delta t' * \gamma, then unless they just made a mistake, you'd know they intended to refer to the time intervals between two events which occur at the same position in the primed frame. Likewise, if someone wrote the following as the Lorentz transformation:

x = gamma (x' - vt')
t = gamma (t' - vx'/c^2)

Then although this is different from how they're usually written, you can infer that this is just a situation where it's assumed the origin of the unprimed frame is moving at velocity v along the x' axis of the primed frame.

neopolitan

Mar18-09, 12:38 AM

Remember a while back I talked about an apparatus I had. I have it and it is at rest relative to me.

Associated with this apparatus are a length and a time measurement. I called these L and t.

I give these to my buddy, and he sets off on a carriage with a speed of v (in a direction that is convenient so that the length I measured as L is parallel to the direction of motion).

My buddy will, if he checks, find a length and time measurement of L and t.

But while my buddy in motion measures L and t, I will work out that, because he is motion, the length is contracted. I call that L', because I already have an L (unprimed is my frame so I making primed my buddy's frame). I will also work out that, because he is in motion, my buddy's clock will have slowed down. What reads on his clock will less than what I read on mine. If confuses you, and god knows it confuses me, because you have to step back a bit from the intial t I had. So, let's do it another way.

Say I have two sets of the apparatus. I keep one, and give the other to my buddy. I know they are identical. I ask him to measure it lengthwise, he gets L and I get L. But if I compare my length to his length (and I can do this with lasers and time measurements in my frame), I will find that he is "confused". His length is actually L'=L / \gamma. (And yes, I know if he does the same thing, he will find that I am "confused".)

Time is a little more complex to describe, but equivalent to using lasers and time measurements in my frame. Using a very high quality telescope, I keep track of my buddy's apparatus, most specifically the clock. I note down two times on his clock, t'_{o} and t'_{i} along with the times that I make them (my times, my frame, unprimed). I have to take into account how long it took each of those displayed times on his clock to get to me.

I will find that \Delta t' = \Delta t / \gamma (<- this is my equation, this is not time dilation!)

Now I know that when \Delta t has elapsed in my frame, \Delta t' elapses in his frame. It is not just ticks on clocks, or the time interval between two events - his time dimension is affected. And it is affected in the same way as his spatial dimension is affected. So any speed in his frame will be calculated using contracted length divided by shortened time which will give you the same result as using unaffected length divided by unaffected time. Picking appropriate values of L and \Delta t:

L / \Delta t = c = L' / \Delta t'

Does that help?

cheers,

neopolitan

Rasalhague

Mar18-09, 01:02 AM

JesseM

Mar18-09, 02:23 AM

But while my buddy in motion measures L and t, I will work out that, because he is motion, the length is contracted. I call that L', because I already have an L (unprimed is my frame so I making primed my buddy's frame).
You seem to be confused about what "unprimed is my frame so I am making primed my buddy's frame" means. The length of your buddy's apparatus is contracted when measured in your frame, his apparatus is not contracted in his own frame, so it's totally wrong to call the contracted length L' if you just said your frame is unprimed! If your frame is unprimed, then any variable that refers to how something appears in your frame--like the coordinate distance between either ends of an apparatus at single instant of time in your frame--must be unprimed, regardless of whether the physical object that you're measuring is at rest in your frame or not. Remember, physical objects aren't "in" one frame or another, different frames are just different ways of assigning coordinates to events associated with any object in the universe. And it's true that, as you say, "you already have an L" if you previously defined L to be the length of the same apparatus in your frame when it was at rest relative to you, but that just mean you need some different unprimed symbol to refer to the length of the apparatus in your frame once you've given it to your buddy and it's at rest relative to him, like L_{cbb} where "cbb" stands for "carried by buddy".

Perhaps this confusion about what quantities should be primed and what quantities should be unprimed is related to your (so far unexplained) belief that there is something "inconsistent" about the way the standard time dilation and length contraction equations are written?

And even if I changed your statement above to "I will work out that, because he is motion, the length is contracted. I call that L_{cbb}, because I already have an L", your statement would still be too vague, for exactly the same reason as the statement in your last post was too vague (I offered several possible clarifications so you could pick which one you meant, or offer a different clarification). If L_{cbb} refers to the length of the apparatus in your frame when it's being carried by your buddy, and L'_{cbb} refers to the length your buddy measures the apparatus to be using his own ruler (which is equal to L, the length you measured the apparatus to be using your ruler before you gave it to your buddy, when it was still at rest relative to you), then these will be related by the equation L_{cbb} = L'_{cbb} / \gamma, which is just the length contraction equation with slightly different notation. If on the other hand what your buddy "measures" is a distance of L' between two events using his apparatus, then the distance you measure between the same two events will not necessarily be L'/gamma, in fact it could even end up being larger than L'. So you really need to be specific about precisely what is being measured like I keep asking.
I will also work out that, because he is in motion, my buddy's clock will have slowed down. What reads on his clock will less than what I read on mine.
It is meaningless to compare two compare two clock readings unless A) the clocks are located at the same position at the moment you do the comparison, or B) you have specified which frame's definition of simultaneity you're using. Do you disagree? If not, which one are you talking about here? If it's B, and if you're using your own frame's definition of simultaneity, and if the clocks initially read the same time at some earlier moment in your frame, then I agree that at the later moment his clock will read less than yours. But again, you really need to be way more specific or you'll end up using inconsistent definitions in different statements and end up with conclusions that don't make any sense, as seems to be true of your "t' = t/gamma has to be true in order that L/t=c and L'/t'=c" argument.
Say I have two sets of the apparatus. I keep one, and give the other to my buddy. I know they are identical. I ask him to measure it lengthwise, he gets L and I get L.
"It" is too vague since you have two sets, but I assume you mean "I ask him to measure the apparatus at rest relative to him, while I measure the apparatus at rest relative to me, his value L'_{cbb} is exactly equal to my value L." Correct?
But if I compare my length to his length (and I can do this with lasers and time measurements in my frame), I will find that he is "confused". His length is actually L'=L / \gamma. (And yes, I know if he does the same thing, he will find that I am "confused".)

Time is a little more complex to describe, but equivalent to using lasers and time measurements in my frame. Using a very high quality telescope, I keep track of my buddy's apparatus, most specifically the clock. I note down two times on his clock, t'_{o} and t'_{i} along with the times that I make them (my times, my frame, unprimed). I have to take into account how long it took each of those displayed times on his clock to get to me.
And when you "take into account how long it took", you are using your frame's measurement of the distance that his clock was from yours when it read each of these two times, and assuming that the light from his clock travels at c in your frame, and subtracting distance/c from the time on your clock when you actually saw these readings, is that correct? For example, if when your clock reads 10 seconds you look through your telescope and see his clock reading 6 seconds, and at this moment you see his clock is next to a mark that's 2-light seconds away from you on your ruler, then you'd say his clock "really" read 6 seconds at the moment your clock read 8 seconds, correct?

If that is what you mean by "take into account"--and please actually tell me yes or no if it's what you meant--then note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading t'_{o} and t'_{i}, using your own frame's definition of simultaneity. So note that although you didn't really respond to my list of possible clarifications, it appears that your meaning is exactly identical to the first one I offered, which I'll put in bold (in the original comment I was using unprimed to refer to the buddy's frame and primed to refer your frame, but since you appear to want to reverse that convention by making times on your buddy's clock primed, I'll change the quote to reflect the idea that times in your frame are unprimed and times in your buddy's are primed):
And what does "readouts of time elapsed are reduced ... according to me" mean? Does it mean that if you consider a time interval from some time t_0 to another time t_1 in your frame, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs at time t_1 in your frame) - (his clock's readout at the event on the clock's worldline that occurs at time t_0 in your frame) is smaller than the difference t_1 - t_0 which represents the size of the time interval in your frame? In that case I would agree. Does it mean that if you consider two arbitrary events A and B (like two events on the worldline of a photon), with A happening at t_0 and B happening at t_1 in your frame, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs simultaneously with B in your frame) - (his clock's readout at the event on the clock's worldline that occurs simultaneously with A in your frame) is smaller than the time interval between A and B in your frame? If so, this is exactly equivalent to the first one, so I'd agree with this too. But does it mean that if you consider the same two events A and B, then for a clock at rest in his frame, the difference (his clock's readout at the event on the clock's worldline that occurs simultaneously with B in his frame) - (his clock's readout at the event on the clock's worldline that occurs simultaneously with A in his frame) is smaller than the time interval between A and B in your frame? If so this is not true in general. And if the two events are events on the path of a photon that occur on either end of a measuring-rod of length L in his frame, then it is that last difference in his clock's readouts that you divide L by to get c.
I will find that \Delta t' = \Delta t / \gamma (<- this is my equation, this is not time dilation!)
So, if according to your frame's definition of simultaneity, your clock's reading t_ 0 is simultaneous with your buddy's clock reading some time t'_0, and according to your frame's definition of simultaneity your clock's reading t_1 is simultaneous with your buddy's clock reading some time t'_1, and if \Delta t' = t'_1 - t'_0 and \Delta t = t_1 - t_0, then we get the equation \Delta t' = \Delta t / \gamma. Is that what you mean? If so, then yes, I agree, and as I said this is exactly equivalent to the statement from my earlier post that I bolded above. But in this case you are simply confused if you think this is any different from the standard time dilation equation--it only looks different because you've reversed the meaning of primed and unprimed from the usual convention and then divided both sides by gamma! Normally, if we want to take two events on the worldline of a clock (in this case your buddy's) and then figure out the time interval between these events in a frame where the clock is moving (in this case yours--of course, figuring out the time interval between these events in your frame is exactly equivalent to figuring out which readings on your clock these two events are simultaneous with in your frame and then finding the difference between the two readings on your clock), the usual convention is to call the first frame unprimed and the second frame primed, in which case we get the time dilation equation \Delta t' = \Delta t * \gamma. You have simply adopted the opposite convention, calling the first frame primed and the second frame unprimed, so the time dilation equation would just have to be rewritten as \Delta t = \Delta t' * \gamma using this convention. And of course, if we now divide both sides by gamma, we get back the equation you offered, \Delta t' = \Delta t / \gamma. You can see that this is just a trivial reshuffling of the usual time dilation equation, not anything novel.
Now I know that when \Delta t has elapsed in my frame, \Delta t' elapses in his frame.
No you don't, not for an arbitrary pair of events! Say you pick two events A and B which don't occur on the worldline of his clock (they may be two events on the worldline of a light beam for example), but such that according to his frame's definition of simultaneity, A is simultaneous with t'_0 and B is simultaneous with t'_1. Then would you agree that the time interval between these events in his frame is \Delta t' = t'_1 - t'_0? And we also know that the time interval in your frame between the event of his clock reading t'_1 and the event of his clock reading t'_0 is related to this by \Delta t = \Delta t' * \gamma. But that doesn't mean the time interval in your frame between A and B is \Delta t = \Delta t' * \gamma!!! This is because although it's true that his frame's definition of simultaneity says that A is simultaneous with his clock reading t'_0 and B is simultaneous with his clock reading t'_1, your frame uses a different definition of simultaneity, so according to your frame's definition of simultaneity A may not be simultaneous with his clock reading t'_0 and B may not be simultaneous with his clock reading t'_1, so knowing the time-interval in your frame between his clock reading t'_0 and his clock reading t'_1 tells us nothing about the time interval in your frame between A and B.

Do you understand and agree with all this? Please tell me yes or no.
It is not just ticks on clocks, or the time interval between two events - his time dimension is affected. And it is affected in the same way as his spatial dimension is affected. So any speed in his frame will be calculated using contracted length divided by shortened time which will give you the same result as using unaffected length divided by unaffected time. Picking appropriate values of L and \Delta t:

L / \Delta t = c = L' / \Delta t'
Nope, you still are unable or unwilling to define what you are actually supposed to be measuring the length of and time-intervals between, "appropriate values" is hopelessly vague. Do L and L' represent the distance between a single pair of events on the worldline of a photon, as measured in each frame? Or are you measuring two separate photons with two separate apparatuses, so L is the distance between one pair of events as measured in your frame and L' is the distance between another pair as measured in your buddy's frame? Or is it something else entirely? And how about \Delta t and \Delta t', are you going with the definition I suggested earlier where \Delta t' is the difference between two clock readings t'_1 and t'_0 on your buddy's clock, and \Delta t is the difference between two clock readings t_1 and t_0 on your clock, where you have picked the readings so that according to your frame's definition of simultaneity t_1 is simultaneous with t'_1 and t_0 is simultaneous with t'_0? If not, can you be specific about what events you are taking "deltas" between? And if so, are any of these events on the clocks' worldlines supposed to be simultaneous with events on the worldline of a photon in some frame?

JesseM

Mar18-09, 02:37 AM

Wow, thanks for your answer Jesse. That's given me lots to think about! In everyday life we're more used to regarding future time as being what's unpredictable, so it's curious to think of "the state of a surface of constant x" as being more fundamentally impossible to deduce "if all you know is the state of some other surface of constant x". But suppose that, in a deterministic universe, you knew everything about a surface of constant x to some arbitrary degree of precision. You'd know the positions of particles in that surface and be able to say something about other surfaces of constant x by examining the forces operating on the particles in your surface.
That's an interesting point, but consider the fact that it's quite possible to have a surface of constant x such that of the particles that cross it (and many particles may never cross a particular surface of constant x at all), each one crosses it only at a single point in spacetime. In this case you'd only know the instantaneous velocity of each particle at its crossing point, but I don't see how this would allow you to deduce the force unless you also knew the instantaneous acceleration (and keep in mind that in deterministic theories like classical electromagnetism, merely knowing the position and instantaneous velocity of each particle in a surface of constant t, along with the direction and magnitude of force field vectors in space in that surface, is sufficient to allow you to predict what will happen at later times, you don't need to know the instantaneous accelerations). Also consider that in principle it would be possible to have a surface of constant x where no particles crossed it at any point, even though particles did exist in that universe--I suppose you could still be told the direction and magnitude of force field vectors in this otherwise empty surface, since force fields like the electromagnetic field are imagined to fill all of space, but I don't think this would allow you to deduce the complete history of every particle in the universe (it's possible I could be wrong about this since I haven't actually seen any discussions of this question, though).
Meanwhile, less philosophically, just to check I've understood: is the case where a worldline has multiple pointlike intersections with a surface of constant x only possible for a particle for which there's no inertial frame in which the particle can be said to be at rest (i.e. its worldline isn't a straight line)? (The other two cases you mention--no intersections, one pointlike intersection--being possible for a particle which can be described as being at rest in some inertial frame.)
That's right, assuming of course that we're talking about the x-coordinate of an inertial reference frame.

neopolitan

Mar18-09, 06:04 AM

"Picking appropriate values of L and \Delta t" was too vague. The rest of what you were saying was akin to "You can't park four tanks on the rubber dingy you're designing".

Here's what I mean about picking appropriate values, pick any value of L, any value you like - in the real world you probably want a really big value, but this is hypothetical world, so it is not so important.

Then pick the value of \Delta t so that L/\Delta t = c. If you haven't picked a really big value of L, then /Delta t will be pretty damn small so that it will be challenging to take two readings t_{o} and t_{i} where t_{i} - t_{o} = \Delta t - but we are in hypothetical world.

We have no argument about length contraction. But do you deny that when I use my readings from my buddy's clock, and take into account the motion that I know he has, that I will get a /Delta t' which is shorter than mine?

Do you deny that the extent to which it is shorter is the same as the extent to which L' is shorter than L (where these are given by standard length contraction)?

I've described the events, they aren't simultaneous (and in fact, I don't care about simultaneity, I know the time readings on my buddy's clock are not simultaneous with the time readings on mine, the only thing I bother with, or need to bother with, is the extra time the second reading takes to get to me because he has moved during the time). Any discussion of simultaneity in this scenario is a distraction. If you must have some simultaneity, then try thinking that my seeing the time on my buddy's clock is simultaneous with my reading of the time on my clock, but even that is not necessary since I could use a splitframe camera and look at the results afterwards.

The bottom line, from you Jesse, is "there is no other way to do it" when the question is "what is the benefit with time dilation". It seems you truly think there is no other option. You have a very long winded way to say it, but I don't think there is any other way to interpret your approach to the original question. And yes, I haven't forgotten the original question.

cheers,

neopolitan

JesseM

Mar18-09, 06:42 AM

Here's what I mean about picking appropriate values, pick any value of L, any value you like - in the real world you probably want a really big value, but this is hypothetical world, so it is not so important.

Then pick the value of \Delta t so that L/\Delta t = c. If you haven't picked a really big value of L, then \Delta t will be pretty damn small so that it will be challenging to take two readings t_{o} and t_{i} where t_{i} - t_{o} = \Delta t - but we are in hypothetical world.
Are you just picking a value of \Delta t out of thin air, with no connection to anything physical (so you could just as easily pick a \Delta t such that L/\Delta t = 5c or any number you wish), or is it supposed to represent the time interval between some specific pair of events, like t_{o} representing the time a photon passes next to one end of an object of length L which is at rest in your frame, and t_{i} representing the time that photon passes next to the other end of the same object?
We have no argument about length contraction. But do you deny that when I use my readings from my buddy's clock, and take into account the motion that I know he has, that I will get a /Delta t' which is shorter than mine?
What does "use my readings from my buddy's clock, and take into account the motion that I know he has" mean? This is something I specifically asked you about in my last response to you (post 75):
And when you "take into account how long it took", you are using your frame's measurement of the distance that his clock was from yours when it read each of these two times, and assuming that the light from his clock travels at c in your frame, and subtracting distance/c from the time on your clock when you actually saw these readings, is that correct? For example, if when your clock reads 10 seconds you look through your telescope and see his clock reading 6 seconds, and at this moment you see his clock is next to a mark that's 2-light seconds away from you on your ruler, then you'd say his clock "really" read 6 seconds at the moment your clock read 8 seconds, correct?

If that is what you mean by "take into account"--and please actually tell me yes or no if it's what you meant--then note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading t'_{o} and t'_{i}, using your own frame's definition of simultaneity.
Can you please answer this question? And if the answer is "no, that's not what I meant" could you please tell me exactly what you do mean, preferably giving some sort of numerical example? For example, suppose your buddy's clock is moving at 0.6c in your frame and is initially right next to your, and when they are next to each other both your clock and his read 0 seconds; then later when your clock reads 16 seconds, you look through your telescope and see his clock reading 8 seconds, and you also see that when his clock reads 8 seconds it's next to a mark on your ruler that's 6 light-seconds away from you (so in your frame the event must have 'really' happened 6 seconds before you saw it through your telescope). How would the phrase "use my readings from my buddy's clock, and take into account the motion that I know he has" apply to this specific example. What would \Delta t' be, and what would \Delta t be?
I've described the events
Have you? Where? Are the events just the two readings on your buddy's clock?
they aren't simultaneous (and in fact, I don't care about simultaneity, I know the time readings on my buddy's clock are not simultaneous with the time readings on mine, the only thing I bother with, or need to bother with, is the extra time the second reading takes to get to me because he has moved during the time). Any discussion of simultaneity in this scenario is a distraction.
But if by "taking into account" the light delay, you mean taking the time on your clock (t=16 seconds in my example above) when you saw a reading on your buddy's clock (t'=8 seconds in the example) and then subtracting the distance/c that your buddy's clock was from you in your frame when it showed that reading (6 light-seconds/c = 6 seconds in the example) to get an earlier time on your clock (t=16-6=10 seconds), then this is physically equivalent (meaning you'll get the same answer for what the two clock readings would be) to just asking for the time on your clock that was simultaneous in your frame with the reading you saw on your buddy's clock (i.e. in the example, the reading of t'=8 seconds on your buddy's clock is simultaneous in your frame with the reading of t=10 seconds on your clock). If this is indeed what you meant by "taking into account", then do you agree that this is physically equivalent to my statement about simultaneity? Please answer yes or no.
The bottom line, from you Jesse, is "there is no other way to do it" when the question is "what is the benefit with time dilation".
No it isn't, because I don't even know what the physical meaning of the "it" that you want to do actually is, your posts are just too unclear for me to judge them right or wrong. It will help if you give clear yes-or-no answers to questions about what you're saying when I ask for them.

neopolitan

Mar18-09, 07:08 AM

JesseM,

You fragment too much. It leads, inexorably, to loss of context. That's why I am not responding to your fragmenting.

Look back in previous posts and I explained what I meant about taking readings on my buddy's clock. I made mention of a telescope.

But you must have overlooked it in your apparent excitement to demolish any discussion (and I mean that, "discussion", not argument because to demolish an argument you have to make an effort to understand).

You make an effort to understand, pose an unfragmented question which indicates that you have made the slightest effort to understand, rather than attack, and I will answer it.

cheers,

neopolitan

Rasalhague

Mar18-09, 10:29 AM

it's also possible to come up with a "temporal analogue for the length contraction equation" which looks like the length contraction equation but with \Delta t substituted for L (this is more difficult to state in words, but it's basically the time-interval in the primed frame between two surfaces of constant t in the unprimed frame which have a temporal distance of \Delta t in the unprimed frame, which is analogous to how length contraction tells you the spatial distance in the primed frame between two worldlines of constant position in the unprimed frame which have a spatial separation of L in the unprimed frame).

Suppose Alice and Bob are each wearing a watch. Bob, moving in the positive x direction in Alice's rest frame, at 0.6c, passes Alice and they synchronise watches. Some time later, Alice looks at her watch and wonders, "What time does Bob's watch say at the moment which in Bob's rest frame is simultaneous with me asking this question?" The answer is given by t_{B} = \gamma t_{A}, the standard time dilation formula. If Alice's watch says 4, Bob's will say 5. "How about that," thinks Alice. "To Bob, for whom my watch is moving, it's running slow."

Now suppose that Alice wonders, "What time does Bob's watch say at the moment which in my rest frame is simultaneous with me asking this question?" The answer to this is given by t_{B} = \frac{t_{A}}{\gamma}. If Alice's watch says 5, Bob's will say 4. "Fancy," thinks Alice. "From my perspective, Bob's watch, which is moving relative to me, is running slow."

And, of course, Bob can ask the equivalent questions about the time on Alice's watch with identical results by virtue of the fact that the two frames don't agree on which events are simultaneous (except for those that happen in the same place, such as their synchronisation).

But isn't Alice's second question none other than this exotic "temporal analogue for the length contraction equation"? She wants to know "the time-interval in the primed frame" (the time shown by Bob's watch, indicating a time interval along Bob's worldline) "between two surfaces of constant t in the unprimed frame" (one being the one which Alice and Bob's worldlines intersected when they synchronised watches, the other being Alice's present when she looks at her watch) "which have a temporal distance of \Delta t in the unprimed frame" (the time shown by Alice's watch when she looks at it and makes her query).

Is Alice's second question in any way less natural than the first, or a less useful thing to ask of time than of space? I'm puzzled as to how it can be, if it is, as Jesse said, "just a trivial reshuffling of the usual time dilation equation"?

JesseM

Mar18-09, 02:12 PM

You make an effort to understand, pose an unfragmented question which indicates that you have made the slightest effort to understand, rather than attack, and I will answer it.
I have made an effort to understand, and in fact the questions above are pretty clearly requests to nail down the meaning of your statements by asking if they match up with the precise definitions that I have suggested, for example:
And when you "take into account how long it took", you are using your frame's measurement of the distance that his clock was from yours when it read each of these two times, and assuming that the light from his clock travels at c in your frame, and subtracting distance/c from the time on your clock when you actually saw these readings, is that correct? For example, if when your clock reads 10 seconds you look through your telescope and see his clock reading 6 seconds, and at this moment you see his clock is next to a mark that's 2-light seconds away from you on your ruler, then you'd say his clock "really" read 6 seconds at the moment your clock read 8 seconds, correct?

If that is what you mean by "take into account"--and please actually tell me yes or no if it's what you meant--then note that this is exactly the same as asking what times on your clock were simultaneous with his clock reading t'_{o} and t'_{i}, using your own frame's definition of simultaneity.

Can you please answer this question? And if the answer is "no, that's not what I meant" could you please tell me exactly what you do mean, preferably giving some sort of numerical example? For example, suppose your buddy's clock is moving at 0.6c in your frame and is initially right next to your, and when they are next to each other both your clock and his read 0 seconds; then later when your clock reads 16 seconds, you look through your telescope and see his clock reading 8 seconds, and you also see that when his clock reads 8 seconds it's next to a mark on your ruler that's 6 light-seconds away from you (so in your frame the event must have 'really' happened 6 seconds before you saw it through your telescope). How would the phrase "use my readings from my buddy's clock, and take into account the motion that I know he has" apply to this specific example. What would \Delta t' be, and what would \Delta t be?
If you see this line of questioning as simply an "attack" rather than an attempt to actually understand in precise terms the meaning of your phrase "take into account how long it took" (and thereby to figure out the precise physical relationship between the two intervals \Delta t and \Delta t' which appear in your equation L/\Delta t = c = L'/\Delta t'), then I suppose that means you are simply too mistrusting of my motives to ever be interested in the process of actual communication with me (and 'communication' necessarily requires a willingness to clarify what the other person doesn't understand, especially in a discussion of physics where precise definitions are needed), in which case I take it there is basically nothing I could do other than nodding my head and agreeing with all your statements (even when I don't really understand what they mean) that would make you want to continue the discussion.

JesseM

Mar18-09, 05:23 PM

Suppose Alice and Bob are each wearing a watch. Bob, moving in the positive x direction in Alice's rest frame, at 0.6c, passes Alice and they synchronise watches. Some time later, Alice looks at her watch and wonders, "What time does Bob's watch say at the moment which in Bob's rest frame is simultaneous with me asking this question?" The answer is given by t_{B} = \gamma t_{A}, the standard time dilation formula. If Alice's watch says 4, Bob's will say 5. "How about that," thinks Alice. "To Bob, for whom my watch is moving, it's running slow."

Now suppose that Alice wonders, "What time does Bob's watch say at the moment which in my rest frame is simultaneous with me asking this question?" The answer to this is given by t_{B} = \frac{t_{A}}{\gamma}. If Alice's watch says 5, Bob's will say 4. "Fancy," thinks Alice. "From my perspective, Bob's watch, which is moving relative to me, is running slow."
Here you can use the time dilation formula too. If the time dilation formula is written \Delta t' = \Delta t * \gamma, then that' formula is comparing the amount of time that's elapsed on a clock (whose rest frame is labeled the unprimed frame) with the amount of time that's elapsed in a frame where the clock is moving, with "time elapsed" in that frame being based on that frame's definition of simultaneity (and with this second frame being labeled primed). So in your second example, the clock is Bob's and the second frame whose definition of simultaneity you're using is Alice's, so you can just treat Bob's frame as the unprimed frame in the standard time dilation equation and Alice's question will be the same as asking for the time elapsed in the primed frame, meaning you're just substituting t_A and t_B into the time dilation equation giving t_A = t_B * \gamma. Of course, if you wish to divide both sides by gamma, you can get back the formula t_{B} = \frac{t_{A}}{\gamma} you wrote above, but this is just a reshuffling of the time dilation equation.

But you do raise a great point which I hadn't thought of before, which is that the "temporal analogue of the length contraction equation" can always be used to calculate things that you'd normally use the time dilation equation to calculate, provided you reverse the meaning of which frame is primed and which frame is unprimed. Let me give a numerical example similar to yours. Suppose Bob is moving away from Alice at 0.6c and that both their clocks read 0 when they crossed paths as you suggested. But instead of starting Bob's time interval when his clock reads 0 as in your example, suppose we were interested in the time interval on Bob's clock that started with the event of his clock reading t_{B1} = 8 seconds, and ended with his clock reading t_{B2} = 12 seconds, so the length of the interval in Bob's frame is \Delta t_B = (t_{B2} - t_{B1}) = 4 seconds. If Alice wanted to know the time interval \Delta t_A between these same two events in her frame, which is equivalent to wanting to know the time interval between the event t_{A1} on her clock which is simultaneous in her frame with t_{B1} (in this case t_{A1} = 10 seconds) and the event t_{A2} on her clock which is simultaneous in her frame with t_{B2} (in this case t_{A2} = 15 seconds), then she would plug these two different time intervals into the time dilation equation \Delta t' = \Delta t * \gamma, treating Bob's frame as unprimed and her frame as primed, which gives \Delta t_A = \Delta t_B * \gamma. If she wanted to reverse this and figure out the time interval \Delta t_B on Bob's clock between two events on t_{B1} and t_{B2} on his clock's worldline that are simultaneous in her frame with two events on her clock's worldline t_{A1} and t_{A2} that are the beginning and end of a time interval t_A (in the example above she would start with times 10 seconds and 15 seconds on her clock and then try to figure out how much time had elapsed on Bob's clock between these moments in her frame), she'd just divide the time dilation equation by gamma so it gives \Delta t_B as a function of \Delta t_A, i.e. \Delta t_B = \frac{\Delta t_A}{\gamma}.

On the other hand, the "temporal analogue of length contraction" \Delta t' = \Delta t / \gamma would be telling her something conceptually different, assuming she continues to treat Bob's frame as unprimed and her frame as primed. Basically, it would be saying "if you use t_{A1} to label the time on Alice's clock that's simultaneous in Bob's frame with Bob's clock reading t_{B1}, and you use t_{A2} to label the time on Alice's clock that's simultaneous in Bob's frame with Bob's clock reading t_{B2}, then the time interval on Alice's clock (t_{A2} - t_{A1}) is related to the time interval on Bob's clock (t_{B2} - t_{B1}) by the formula (t_{A2} - t_{A1}) = (t_{B2} - t_{B1}) / \gamma. If we use the same numbers for t_{B1} and t_{B2} on Bob's clock as before, namely t_{B1} = 8 seconds and t_{B2} = 12 seconds, then in this case we'd have t_{A1} = 8*0.8 = 6.4 seconds (I just multiplied 8 by 0.8 because I know both clocks read 0 when they were next to each other and Alice's clock is moving at 0.6c in Bob's frame, so the standard time dilation equation tells me her clock is slowed by a factor of 0.8 in his frame) and t_{A2} = 12*0.8 = 9.6 seconds. So the equation (t_{A2} - t_{A1}) = (t_{B2} - t_{B1}) / \gamma does work here, since (t_{A2} - t_{A1} = 9.6 - 6.4 = 3.2, t_{B2} - t_{B1} is still 4 seconds, and gamma is still 0.8. But you can see that the time interval in Alice's frame we're talking about now (3.2 seconds) is different than the time interval in Alice's frame we were talking about when we were using the usual time dilation equation (5 seconds). But, that's only because we were treating Alice's frame as the primed frame in both equations! If we reverse the labels and treat Bob's frame as primed and Alice's frame as unprimed, then the standard time dilation equation \Delta t' = \Delta t * \gamma does tell you that when 3.2 seconds have elapsed on Alice's clock, 4 seconds of time have passed in Bob's frame (or equivalently, if you look at the readings on Bob's clock that are simultaneous in Bob's frame with the two readings on Alice's clock, the difference between these two readings on Bob's clock is 4 seconds).

So I guess if you take the time dilation equation and divide both sides by gamma to solve for the interval in the primed frame, this is really just equivalent to taking the "temporal equivalent of length contraction" equation and reversing which frame we call primed and which we call unprimed. To me there's still a little bit of a conceptual difference though, in the sense that normally I think of these equations as relating a clock time-interval to a coordinate time-interval, with unprimed normally being the clock time-interval. For instance, when I read the time dilation equation \Delta t' = \Delta t * \gamma, I find it most natural to think that \Delta t represents the difference between two clock-readings on a clock at rest in the unprimed frame, and then \Delta t' represents the difference between the coordinate times of these two readings in the primed frame. Of course, because a clock at rest in the primed frame will keep time with coordinate time in that frame, this is equivalent to imagining there's also a clock at rest in the primed frame, and saying \Delta t' represents the difference between two readings on the primed clock that are simultaneous in the primed frame with the two readings on the unprimed clock that were mentioned earlier. The first way of stating it just makes the usefulness of the time dilation equation more intuitive to me; as I said before, physics is all about setting up a spacetime coordinate system and then using equations to figure out how the state of objects in space changes as the time-coordinate increases.

neopolitan

Mar18-09, 07:58 PM

JesseM responding to Rasalhague:
But you do raise a great point which I hadn't thought of before, which is that the "temporal analogue of the length contraction equation" can always be used to calculate things that you'd normally use the time dilation equation to calculate, provided you reverse the meaning of which frame is primed and which frame is unprimed.

<snip>

The first way of stating it just makes the usefulness of the time dilation equation more intuitive to me; as I said before, physics is all about setting up a spacetime coordinate system and then using equations to figure out how the state of objects in space changes as the time-coordinate increases.

Thanks, I think you've given an answer my original question. I think you have said this:

There is another way of approaching the relativistic effects other than time dilation-length contraction. That is to use "temporal analogue of the length contraction equation"-length contraction. However, using time dilation is more intuitive to you - and possibly also for the majority of people. That said, there is nothing inherently wrong with using a "temporal analogue of the length contraction equation" (although one must note that a different prime convention is required).

Rasalhague has shown me that instead of:

What exactly is the greater utility of time dilation and length contraction equations which prevents the use of two contraction equations ...?

I should have asked:

What exactly is the greater utility of length contraction and time dilation equations which prevents the use of a length contraction equation and a temporal equivalent of the length contraction equation ...?

For that I thank you Rasalhague.

cheers,

neopolitan

JesseM

Mar18-09, 08:46 PM

Thanks, I think you've given an answer my original question. I think you have said this:

There is another way of approaching the relativistic effects other than time dilation-length contraction. That is to use "temporal analogue of the length contraction equation"-length contraction. However, using time dilation is more intuitive to you - and possibly also for the majority of people. That said, there is nothing inherently wrong with using a "temporal analogue of the length contraction equation" (although one must note that a different prime convention is required).
Yes, although I hadn't actually realized that the "temporal equivalent of length contraction" equation could be used to answer exactly the same physical questions as the time dilation equation until Rasalhague asked that. And it was the specificity of the way he asked the question that made me realize this--he was asking about particular events on the worldlines of two physical clocks and stating in which frame an event on one clock's worldline was supposed to be simultaneous with a corresponding event on the other clock's worldline. You say that you "should have asked" this question:
What exactly is the greater utility of length contraction and time dilation equations which prevents the use of a length contraction equation and a temporal equivalent of the length contraction equation ...?
But such a broadly-worded question would almost certainly not have led me to the same realization. This illustrates why I keep asking you to answer specifics about what you are saying, and I don't really understand why you are unwilling to grant these requests--is it that you don't like my attitude, is it that your ideas are mostly intuitive so you're not sure what the answers should be yourself, or something else? I really think a willingness to delve into specifics could allow us to make progress on things like the meaning of "L/\Delta t = c = L'/\Delta t'" which I have been unable to make sense of so far, just as the specifics of Rasalhague's question allowed for progress on the issue of the uses of the "temporal analogue of length contraction" equation, so I hope that even if you decide you are not interested in continuing this line of discussion for whatever reason, you will at least consider that there may be a lesson here about the value of engaging with nitty gritty details when talking physics.

neopolitan

Mar18-09, 11:08 PM

We were probably arguing at cross purposes, frustratingly enough for both of us. I thought you had taken that "temporal equivalent of length contraction" thing onboard a long time ago (it is in your diagram after all). So I was totally confused as to where you were coming from.

Since I thought you had understood the point and were still arguing it, it felt as if you were just trying to play games. That may have been a form of "tranference" (psychological term, relating to ascribing apparent motives of one person to another), since in real life I had a rather difficult person at work doing what I thought you were doing - playing dominance games through irrational argument.

I take your point about specifics. You may see that I have tried to be specific with figures in another thread.

May I ask why you had not come to the understanding that you just came to, when it seems that both Rasalhague and I did? This is not a hidden "you must be stupid" insult. I find you annoying, as you surely find me, but I don't find you stupid. What I am trying to do is see if you can identify, from the vantage point of someone who has only just came to this understanding, what prevents people from coming to this understanding naturally. Is there a block of some kind? If so, is it pedagogical or psychological?

(Clarification follows: I am distinguishing here between pedagogical and psychological, with a definition of "pedagogical" relating to how subjects are taught and "psychological" relating to the different ways in which people think and learn. Specific examples: "whole language" is a pedagogical method for teaching kids to read, moving away from phonics and instead recognition of whole words. As for "psychological", I am a visual, pattern identifying person which means that having a graph in front of me is more useful than a page of numbers. My visual, pattern identifying nature may lead me to link together all things that look the same (like all things with primes against them get grouped).)

This is the sort of discussion I really wanted back when I started the thread. Perhaps you might understand why I found the 80 or so posts in between frustrating, even if they were my own fault.

cheers,

neopolitan

JesseM

Mar19-09, 04:12 AM

May I ask why you had not come to the understanding that you just came to, when it seems that both Rasalhague and I did? This is not a hidden "you must be stupid" insult. I find you annoying, as you surely find me, but I don't find you stupid. What I am trying to do is see if you can identify, from the vantage point of someone who has only just came to this understanding, what prevents people from coming to this understanding naturally. Is there a block of some kind? If so, is it pedagogical or psychological?
Sure, it basically comes from the way I had drawn it in that diagram I gave you, which was the first time I had even thought about the issue of a "temporal analogue for length contraction" (let's call it the TAFLC equation for short). Note that if we write the standard time dilation equation as \Delta t' = \Delta t * \gamma, I have no problem with the idea that you can divide both sides by gamma to get \Delta t = \Delta t' / \gamma (call this the 'reversed time dilation equation'), which I think of conceptually as telling us the time elapsed on a clock at rest in the unprimed frame between two events on its worldline which we know are separated by a time-interval of \Delta t' in the primed frame (I said basically the same thing about reshuffling the time dilation equation in post #61, the paragraph beginning with "Also"). But although this "reversed time dilation equation" looks exactly like the TAFLC equation \Delta t' = \Delta t / \gamma except for the switch between primed and unprimed, I was mistakenly thinking that the physical meaning of \Delta t' and \Delta t in the TAFLC equation was totally different from either of the terms in the reversed time dilation equation. Again, the reason was how it was drawn in my diagram--I was thinking that \Delta t' represented some weird notion of the temporal distance in the primed frame between two surfaces of simultaneity from the unprimed frame that crossed through readings on the worldline of the clock at rest in the unprimed frame which have a separation of \Delta t. Superficially the notion of taking the temporal distance in the primed frame between two surfaces of constant t in the unprimed frame seems pretty weird and disconnected from anything physical (at least it did to me), an idea created only because it was analogous with taking the spatial distance in the primed frame between two worldlines of constant x in the unprimed frame, which is what length contraction is about.

What I had failed to realize was that if we imagine a physical clock at rest in the primed frame, then the "temporal distance between surfaces of constant t from the unprimed frame" just represents the difference \Delta t' between the clock's readings at the two points where its worldline intersects these surfaces of constant t from the unprimed frame, and that if we then shift our perspective back to the unprimed frame, \Delta t is now just the coordinate time between two readings on the primed clock, so now this is exactly like how I'd conceptualize the physical meaning of the terms in the reversed time dilation equation except with the roles of primed and unprimed reversed. So, this is one or two mental steps from what the TAFLC seemed to mean based on my diagram, and I didn't see the connection until I started working through a numerical example in response to Rasalhague's question. Also, it didn't help that I was used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame, rather than normally thinking in terms of a clock at rest in the primed frame too. I was aware intellectually of the fact that the coordinate time in the primed frame between two events A and B could be rephrased in terms of readings on a physical clock at rest in the primed frame, specifically the difference between the reading that was simultaneous with A and the reading that was simultaneous with B according to the prime frame's definition of simultaneity. But that seemed like a more complicated way of thinking about the physical meaning of \Delta t' (you can see it took me longer to write it out) so I usually just thought of it in terms of coordinate time.

neopolitan

Mar19-09, 11:04 AM

Sure, it basically comes from the way I had drawn it in that diagram I gave you, which was the first time I had even thought about the issue of a "temporal analogue for length contraction" (let's call it the TAFLC equation for short). Note that if we write the standard time dilation equation as \Delta t' = \Delta t * \gamma, I have no problem with the idea that you can divide both sides by gamma to get \Delta t = \Delta t' / \gamma (call this the 'reversed time dilation equation'), which I think of conceptually as telling us the time elapsed on a clock at rest in the unprimed frame between two events on its worldline which we know are separated by a time-interval of \Delta t' in the primed frame (I said basically the same thing about reshuffling the time dilation equation in post #61, the paragraph beginning with "Also"). But although this "reversed time dilation equation" looks exactly like the TAFLC equation \Delta t' = \Delta t / \gamma except for the switch between primed and unprimed, I was mistakenly thinking that the physical meaning of \Delta t' and \Delta t in the TAFLC equation was totally different from either of the terms in the reversed time dilation equation. Again, the reason was how it was drawn in my diagram--I was thinking that \Delta t' represented some weird notion of the temporal distance in the primed frame between two surfaces of simultaneity from the unprimed frame that crossed through readings on the worldline of the clock at rest in the unprimed frame which have a separation of \Delta t. Superficially the notion of taking the temporal distance in the primed frame between two surfaces of constant t in the unprimed frame seems pretty weird and disconnected from anything physical (at least it did to me), an idea created only because it was analogous with taking the spatial distance in the primed frame between two worldlines of constant x in the unprimed frame, which is what length contraction is about.

What I had failed to realize was that if we imagine a physical clock at rest in the primed frame, then the "temporal distance between surfaces of constant t from the unprimed frame" just represents the difference \Delta t' between the clock's readings at the two points where its worldline intersects these surfaces of constant t from the unprimed frame, and that if we then shift our perspective back to the unprimed frame, \Delta t is now just the coordinate time between two readings on the primed clock, so now this is exactly like how I'd conceptualize the physical meaning of the terms in the reversed time dilation equation except with the roles of primed and unprimed reversed. So, this is one or two mental steps from what the TAFLC seemed to mean based on my diagram, and I didn't see the connection until I started working through a numerical example in response to Rasalhague's question. Also, it didn't help that I was used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame, rather than normally thinking in terms of a clock at rest in the primed frame too. I was aware intellectually of the fact that the coordinate time in the primed frame between two events A and B could be rephrased in terms of readings on a physical clock at rest in the primed frame, specifically the difference between the reading that was simultaneous with A and the reading that was simultaneous with B according to the prime frame's definition of simultaneity. But that seemed like a more complicated way of thinking about the physical meaning of \Delta t' (you can see it took me longer to write it out) so I usually just thought of it in terms of coordinate time.

Thanks for that. With some things going on the background it took me some time to digest.

There is something which I find curious. It is a criticism of the pedagogy not of you nor of what time dilation is actually representing.

Note that you are "used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame". It's quite a complex thing to internalise. When being taught, or trying to teach oneself, it is going to be a real uphill struggle to grasp that particular nature of the standard time dilation equation.

I certainly struggled with it and it was not helped that I have "back to fundamentals" sort of approach to mathematics which I applied to SR by reading a translation of Einstein's 1905 paper (I use the one at fourmilab.ch (http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf)). I noted, and agonised over the fact that one of the standard equations is shown in mathematical form (length contraction) but the other is only given in words (on page 10) and this directly follows what was to me the far more intuitive equation - a form of "TAFLC" with \tau instead of t'.

I do think there is a source of confusion there. I'd be willing to accept that it just me, but it seems there are many people with some problem or another with SR, which seems really odd. Why SR? Some are kooks for sure, but there are many people who seem to be otherwise able to maintain perfectly normal lives apart from an intuitive feeling that something is just not quite right about SR.

I won't say his name, and sadly he has probably passed away with cancer by now, but a professor in a city not far from where I lived up until recently was the lead lecturer for relativity at his university. He generously gave me many hours of his time to discuss my concerns and proposed solutions, and admitted that really, he didn't fully understand it. He did not stop me or explain that my concerns about time dilation were invalid, because he had never intuitively grasped the principles the standard way either. There is a fellow in southern Europe, another physics professor, albeit in a different field who expressed stronger views than I during our discussions that there was something amiss with SR. (I don't think SR is wrong but I do think it could be taught better.) A quantum physics professor in southern England also felt I was onto something with my arguments.

If professors of physics don't grasp SR properly, what chance do the average visitors to these forums have?

To make sure I am not presenting a biased account, I should clarify that at least four professors I corresponded with gave clear indications that they grasped SR well enough as taught (at least enough as to not be intrigued by my concerns), but sadly they had no time to go into it in depth with me. I had learned SR at university, read up on it, even going back to the original documents (Einstein and Feynman, Feynman because the light clock is sort of his). I had my uneasy feeling despite all this, and being told to go learn SR (again!) didn't really help.

Anyways, I've cast away a lot of my original stuff because I can now see that I was looking at the same thing as standard SR from a different perspective (I've not cast away everything, but I may in time cast away even the little that remains) and my deep-seated concerns that time dilation could actually be wrong were not justified.

However, if this feeling of there being something not quite right (which in my case were, as I said, deep-seated and may be equally concerning to others) is due to something as harmless as a pedagogical/psychological issue where some people intuitively think the way you do and others intuitively think another, yet both ways of thinking are completely valid, being just slightly different perspectives on the same thing, then it seems that there is some scope for improvement on how SR is taught.

I really do think that your suggestion a long time ago, when we had the discussion in which the diagram I posted here was central, was a good one.

You said that you would show your new students a similar diagram and explain that time dilation is not a TAFLC, and is not meant to be. I think you could go a little further and explain the physical significance of the actual TAFLC, and how it relates to length contraction so that c is invariant. That way, you would catch the people like me who feel that TAFLC is a useful equation and gently guide them towards a proper understanding of time dilation. At the same time, you would catch people like you, who go many years without grasping that there is any significance to a TAFLC.

Does that sound unreasonable?

cheers,

neopolitan

(I'm trying to get a lot into as few words as possible, it is late and it has been a long day. Sorry if there is anything which is hard to follow.)

JesseM

Mar19-09, 11:03 PM

Thanks for that. With some things going on the background it took me some time to digest.

There is something which I find curious. It is a criticism of the pedagogy not of you nor of what time dilation is actually representing.

Note that you are "used to conceptualizing the standard and reversed time dilation equations as relating a clock time-interval on a clock at rest in the unprimed frame with a coordinate time-interval in the primed frame". It's quite a complex thing to internalise. When being taught, or trying to teach oneself, it is going to be a real uphill struggle to grasp that particular nature of the standard time dilation equation.
But as I said to Rasalhague, part of why this may seem "natural" and not really that complex if you've already spent some time studying other theories of physics like Newtonian mechanics or QM is that when solving physics problems, the usual convention is to pick some initial conditions, representing a frozen instant in which you can visualize the arrangement of all the parts of the system in space at that instant, and then evolve them forward in (coordinate) time using the dynamical equations. Once you've picked the frame, the coordinate time of that frame is something that you just get used to thinking of as "time" for the sake of solving that problem, almost like the absolute time of Newton, although in the background of your mind you know that it's frame dependent. Put it this way--if everyone still believed in absolute time and space (and therefore absolute velocity), and we knew there was a time dilation effect where clocks in motion relative to absolute space "ran slow" in an absolute sense, then would you still find it complex or difficult to internalize the notion of a time dilation equation that tells you how much real time goes by when a certain amount of ticks go by on a moving clock? And wouldn't this be pretty much analogous to wanting to know how much real space is taken up by a moving ruler whose marks indicate it's a certain length, but which is shrunk in absolute terms because it's moving? And in this case if you treat the frame corresponding to absolute space and time as the primed frame, and the frame of the moving clock and moving ruler as the unprimed frame, you get the usual equations \Delta t' = \Delta t * \gamma and \Delta x' = \Delta x / \gamma. Of course you could also ask how many ticks go by on the moving clock given a certain amount of real time has passed, or what the rest length of a ruler is given that it's a certain length in real space, in this case you'd have to divide both sides of the first by gamma to get what I called the "reversed time dilation equation" \Delta t = \Delta t' /\gamma, and multiply both sides of the second by gamma to get \Delta x = \Delta x' * \gamma. On the other hand, if you're thinking in terms of absolute space and time and treating the absolute frame as the primed one, then the meaning of the TAFLC equation \Delta t' = \Delta t /\gamma seems less intuitive to me; I guess it would come out to something like "given that two events on the worldline of a clock at rest in absolute space are separated by a coordinate time of \Delta t in a frame moving at velocity v relative to absolute space, how much real time (or clock time, given that the clock is not slowed-down) passes between those two events?"

Anyway, if you can see my point that the time dilation and length contraction equation seem fairly "natural" in a universe with absolute space and time, then maybe you can see why, once a physics student has gotten used to the idea of picking a coordinate system and then taking that system's space and time coordinates for granted for the purposes of actual calculating the dynamical behavior of physical systems, then it might seem equally natural to ask how much coordinate time goes by when a certain amount of ticks go by on a moving clock (moving relative to that coordinate system), or how much coordinate space is taken up by a moving ruler. That's the best way I can think of to explain why the equations make intuitive sense to me, but obviously it's subjective so not everyone would have the same intuitions.
I do think there is a source of confusion there. I'd be willing to accept that it just me, but it seems there are many people with some problem or another with SR, which seems really odd. Why SR? Some are kooks for sure, but there are many people who seem to be otherwise able to maintain perfectly normal lives apart from an intuitive feeling that something is just not quite right about SR.
I've always thought that the main reason so many people have a problem with SR is because of the relativity of simultaneity, and what that might be taken to imply about the lack of any "objective" or "true" present, and therefore the lack of a real flow of time. In my experience--and I have seen a few physicists say the same thing--whenever people claim they have found a paradox in SR, the majority of the time it seems to come down to a failure to consider the relativity of simultaneity.
I won't say his name, and sadly he has probably passed away with cancer by now, but a professor in a city not far from where I lived up until recently was the lead lecturer for relativity at his university. He generously gave me many hours of his time to discuss my concerns and proposed solutions, and admitted that really, he didn't fully understand it. He did not stop me or explain that my concerns about time dilation were invalid, because he had never intuitively grasped the principles the standard way either.
Is it possible that, like DaleSpam said above, he just found it simpler to use the full Lorentz transform to approach any problem which compares different frames? Since all the other distinct equations like the time dilation equation, the length contraction equation, the relativity of simultaneity equation, and the velocity addition equation are all derived from the Lorentz transform, I can see the appeal of just using that one set of equations rather than bothering to keep track of a bunch of different ones dealing with different quantities and concepts in SR.
There is a fellow in southern Europe, another physics professor, albeit in a different field who expressed stronger views than I during our discussions that there was something amiss with SR. (I don't think SR is wrong but I do think it could be taught better.) A quantum physics professor in southern England also felt I was onto something with my arguments.
I have actually heard a few people working in quantum gravity who speculate that perhaps such a theory will restore a "true" flow of time and an objective present, Lee Smolin comes to mind for example. But I don't think this is a very common view.

I really do think that your suggestion a long time ago, when we had the discussion in which the diagram I posted here was central, was a good one.

You said that you would show your new students a similar diagram and explain that time dilation is not a TAFLC, and is not meant to be.
Yes, if I was ever actually in a position to be teaching a class on SR, I'd be sure to do that! I don't want people reading this to get the impression that I'm a professor or anything... ;)
I think you could go a little further and explain the physical significance of the actual TAFLC, and how it relates to length contraction so that c is invariant.
But when you say "how it relates to length contraction so that c is invariant", are you referring to the L/\Delta t = c = L'/\Delta t' argument? As I said that doesn't really make sense to me, because even if we assume that \Delta t and \Delta t' have the meaning given to them in the TAFLC equation, and L and L' have the usual meaning from the length contraction equation, I still don't see how it would make sense physically that L/\Delta t and L/\Delta t' could represent the "speed" of a single photon in two different frames if speed is given its usual interpretation as the distance covered in a certain interval of time. We can try going back to discussing this point if you want, or not if you don't want to get into it. As for the physical significance of the TAFLC, if we write it as \Delta t' = \Delta t / \gamma I guess I would basically write it out as ""given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of \Delta t in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

By the way, I'm about to go on a trip for a few days, so I probably won't be able to continue the discussion until next week sometime.

neopolitan

Mar20-09, 12:44 AM

But when you say "how it relates to length contraction so that c is invariant", are you referring to the L/\Delta t = c = L'/\Delta t' argument? As I said that doesn't really make sense to me, because even if we assume that \Delta t and \Delta t' have the meaning given to them in the TAFLC equation, and L and L' have the usual meaning from the length contraction equation, I still don't see how it would make sense physically that L/\Delta t and L/\Delta t' could represent the "speed" of a single photon in two different frames if speed is given its usual interpretation as the distance covered in a certain interval of time. We can try going back to discussing this point if you want, or not if you don't want to get into it. As for the physical significance of the TAFLC, if we write it as \Delta t' = \Delta t / \gamma I guess I would basically write it out as ""given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of \Delta t in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

That definition is correct, although I would imagine a new student would need to be eased into it.

I can see why you can't make sense of L/t = c = L'/t'.

You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame.

I wasn't doing that. I was saying that any time inverval in the primed frame between two events which are colocal in the primed frame, will be shorter in the unprimed frame than two analogous (but not the same) events in the unprimed frame. The half-life of one muon in the primed frame (viewed from the primed frame) will be the same as the half-life of a totally different muon in the unprimed frame (viewed from the unprimed frame. (Yes, I know half-lives are statistical, but using a gross misrepresentation here might still be instructive.)

What I am saying is that the half-life of the muon in the primed frame (viewed from the primed frame) will be less than the half-life of the muon in the primed frame (viewed from the unprimed frame).

In the example BobS raised earlier in the thread, a muon at a gamma of 29.3 had a measured life time of 64.4ms as opposed to a normal (gamma of 1) life time of 2.2ms.

In the experiment he refers to, I would call the measured lifetime t and I could use the gamma to calculate what the life time in muon's "rest frame" was (quotation marks because "rest frame" is a bit of a misnomer under the circumstances). I'd prime the rest frame of the muon and leave the laboratory rest frame unprimed. That would give me:

t' = t/gamma = 64.4ms / 29.3 = 2.2ms

If I had a different experiement, using light clocks, this is how I would be doing it.

At rest in the laboratory, my light clock has a tick time of 2.2ms. That makes the distance between mirrors ct/2 = 330km (giving a L = 660km, the total distance a photon travels between ticks).

Conceptually, put the light clock at a gamma factor of 29.3 (in reality, this would prove difficult).

I will measure, in the laboratory, that the time between ticks of the light clock is now 64.4ms.

This 64.6ms is the t which is equivalent to the t from the muon example. It is not equivalent to the t which I used in ct/2 = 330km (that t was 2.2ms).

What I do know is that, in the laboratory's frame, the photon in the light clock has not travelled 330km in 64.4ms. As you showed before (using time dilation) the photon has to travel much further from one mirror to the other mirror in one direction and a bit less in the other direction.

So the distance travelled between ticks (in the laboratory) is not the same L as before but rather ct where t = 64.4ms ... eg, 19320km.

This L, divided by this t = 19320km/64.4ms = 300000 km/s

The distance travelled in the rest frame of the light clock is the old L (330km) and the time a photon takes to travel between them and back again is the old t (2.2ms).

This L, divived by this t = 660km/2.2ms = 300000 km/s

If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, travelling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3). The test clock will not have ticked. In the rest frame of the test clock, the test clock's photon will only have travelled a distance of 22.5 km.

This time and this distance are t' and L'.

L'/t' = 22.5km/75μs = 300000 km/s

I hope this helps.

cheers,

neopolitan

Rasalhague

Mar20-09, 02:51 AM

Anyway, if you can see my point that the time dilation and length contraction equation seem fairly "natural" in a universe with absolute space and time, then maybe you can see why, once a physics student has gotten used to the idea of picking a coordinate system and then taking that system's space and time coordinates for granted for the purposes of actual calculating the dynamical behavior of physical systems, then it might seem equally natural to ask how much coordinate time goes by when a certain amount of ticks go by on a moving clock (moving relative to that coordinate system), or how much coordinate space is taken up by a moving ruler. That's the best way I can think of to explain why the equations make intuitive sense to me, but obviously it's subjective so not everyone would have the same intuitions.

Maybe this is part of what confuses a novice like me with less experience of physics in general to draw on. Namely that, having been taught that there is no absolute space and time, we're then tacitly invited to pretend there is “for the sake of solving the problem”. But as a beginner, that leaves you wondering *how much* of the your intuition you're allowed to hang onto in this particular exercise. And unless it’s made explicit, you just can't tell, because the one thing you've been warned is that you can't trust your intuition when it comes to relativity. So I worry that I might make mistakes by being lulled by such a natural-seeming way of conceptualising it. Or, to put it another way, the "natural" way of treating one frame as preferential, for the sake of convenience, can sometimes feel to me as if it's bringing swarming after it all those apparent paradoxes that disappear only when you abandon certain intuitions, such as--especially--absolute simultaneity. But maybe when I'm more familiar with SR, that won't be so much of an issue.

I suppose "time dilation" and "length contraction" being just a shorthand for the full Lorentz transformation, of use in a special cases, the thing to be learnt is what those special cases are, and (on a more philosophical or abstract level) why a different special case is thus highlighted for time from the special case thus highlighted for space. Regarding which, I've found this a fascinating discussion.

I've always thought that the main reason so many people have a problem with SR is because of the relativity of simultaneity, and what that might be taken to imply about the lack of any "objective" or "true" present, and therefore the lack of a real flow of time. In my experience--and I have seen a few physicists say the same thing--whenever people claim they have found a paradox in SR, the majority of the time it seems to come down to a failure to consider the relativity of simultaneity.

Definitely! I certainly found that when I learnt that simultaneity was relative too--although it's such a fiendishly counterintuitive concept--that was the moment when some of these bizarre ideas first started to fall into place. They're still very hard for me to understood, but they no longer feels an affront to reason! The other technique that often clears things up for me is to break the problem down and think of it in terms of events. That often helps to root out the false assumptions lurking in my brain.

As for the physical significance of the TAFLC, if we write it as \Delta t' = \Delta t / \gamma I guess I would basically write it out as "given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of \Delta t in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

As in my example, I found it helpful to give the notional observers names, and make their circumstances perfectly symmetrical. That seemed to be the only way I could start get my head around which parts of the problem were significant features of spacetime geometry, and which were accidental details of the example. In the descriptions I'd encountered, I often found myself struggling to keep track over whether a particular author was using the primed/unprimed convention to represent some unique feature of a particular frame. Some introductions use unprimed as you describe, but others use it according to some other convention, or arbitrarily. And of course, where the problem is more elaborate and involves converting back and forth between frames, or where the direction of movement is significant, it's less obvious which frame is the more natural choice to be called unprimed.

JesseM

Mar20-09, 04:42 AM

That definition is correct, although I would imagine a new student would need to be eased into it.

I can see why you can't make sense of L/t = c = L'/t'.

You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame.

I wasn't doing that. I was saying that any time inverval in the primed frame between two events which are colocal in the primed frame, will be shorter in the unprimed frame than two analogous (but not the same) events in the unprimed frame. The half-life of one muon in the primed frame (viewed from the primed frame) will be the same as the half-life of a totally different muon in the unprimed frame (viewed from the unprimed frame. (Yes, I know half-lives are statistical, but using a gross misrepresentation here might still be instructive.)

What I am saying is that the half-life of the muon in the primed frame (viewed from the primed frame) will be less than the half-life of the muon in the primed frame (viewed from the unprimed frame).
OK, agree with you so far.
In the example BobS raised earlier in the thread, a muon at a gamma of 29.3 had a measured life time of 64.4ms as opposed to a normal (gamma of 1) life time of 2.2ms.

In the experiment he refers to, I would call the measured lifetime t
OK, as long as you are aware that here you are using the reverse of the "normal" convention, which is to use the unprimed frame for the rest frame of the "clock" (in this case the natural clock provided by the muon's decay) and the primed frame the frame where we are measuring the time interval between events on the worldline of a moving clock. If you want to reverse this and call the muon's rest frame the primed frame, then the "normal" time dilation equation would be written as \Delta t = \Delta t' * \gamma, the "reversed time dilation equation" would be written as \Delta t' = \Delta t / \gamma, and the TAFLC would be written as \Delta t = \Delta t' / \gamma.
and I could use the gamma to calculate what the life time in muon's "rest frame" was (quotation marks because "rest frame" is a bit of a misnomer under the circumstances). I'd prime the rest frame of the muon and leave the laboratory rest frame unprimed. That would give me:

t' = t/gamma = 64.4ms / 29.3 = 2.2ms
Yes. But just to be clear about terminology, do you agree that this is not the TAFLC, but just the reversed version of the regular time dilation equation?
If I had a different experiement, using light clocks, this is how I would be doing it.

At rest in the laboratory, my light clock has a tick time of 2.2ms. That makes the distance between mirrors ct/2 = 330km (giving a L = 660km, the total distance a photon travels between ticks).

Conceptually, put the light clock at a gamma factor of 29.3 (in reality, this would prove difficult).

I will measure, in the laboratory, that the time between ticks of the light clock is now 64.4ms.

This 64.6ms is the t which is equivalent to the t from the muon example. It is not equivalent to the t which I used in ct/2 = 330km (that t was 2.2ms).

What I do know is that, in the laboratory's frame, the photon in the light clock has not travelled 330km in 64.4ms. As you showed before (using time dilation) the photon has to travel much further from one mirror to the other mirror in one direction and a bit less in the other direction.

So the distance travelled between ticks (in the laboratory) is not the same L as before but rather ct where t = 64.4ms ... eg, 19320km.

This L, divided by this t = 19320km/64.4ms = 300000 km/s
Yes.
The distance travelled in the rest frame of the light clock is the old L (330km) and the time a photon takes to travel between them and back again is the old t (2.2ms).

This L, divived by this t = 660km/2.2ms = 300000 km/s
For clarity we can call this distance in the light clock rest frame L' = 660 km and this time t' = 2.2 ms so it maps to your L/t = c = L'/t', correct? In this case, do you agree that t and t' are related not by the TAFLC but by the standard time dilation equation (written with your unusual convention of labeling the clock rest frame as the primed frame) t = t' * gamma? And do you also agree that L and L' are related not by the length contraction equation but by an equation which looks like the "spatial analogue of time dilation" (although I'm not sure L and L' can be assigned the same physical meaning) L = L' * gamma?

As long as you agree with this stuff I have no problem with the L/t = c = L'/t' argument, but I thought you had been saying that the TAFLC was the equation that was useful in understanding the invariance of c, not the time dilation equation. I guess if you want to say that the equation L = L' * gamma is useful for understanding the invariance of the speed of light that would have some truth, although I think this only works when you're talking about the two-way speed away from some fixed point in the clock's frame and back, and as I said I don't know if the physical meaning of L and L' here can be mapped to the "spatial analogue of time dilation" equation even though it looks the same.

Finally, you said earlier: "You specifically want to measure a time interval between two events in the primed frame and then compare that to a time inverval in the unprimed frame. I wasn't doing that." It seems to me you are doing that, with the two events being 1) the event of the photon leaving the bottom mirror of the light clock which is moving in the lab frame, and 2) the event of the photon returning to the bottom mirror of that same clock. The time between these events is t' = 2.2 ms in the clock rest frame and t = 64.4 ms in the lab frame. The part I had not understood was that you were not using L and L' to represent the distance between these events in the two frames, but rather the total distance covered by the photon in each frame between these two events; this would be identical to the distance between the events if the events were on a single straight photon worldline, but since you are talking about the two-way speed of light rather than the one-way speed of light, the photons are reflected so their worldlines aren't straight.
If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, travelling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3).
No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

neopolitan

Mar20-09, 05:26 AM

There still seems to be some confusion.

We talk about L as it has to be a ruler or a rod or a length. They are convenient devices, but L could be a distance between two randomly selected points in a rest frame (I want to say my rest frame, but it can be any rest frame).

We talk about t as if it has to be attached to events, like ticking of a clock, or formally defined events. But t could be the time interval between two randomly selected times.

We can imagine putting two pins on a map and measuring the distance. We have difficulty putting two pins in time and measuring the temporal distance. But I take TAFLC as being for measuring between these two pins in time, in the same was a LC is for measuring between two pins on the map. We take a different perspective on them by putting us and pins into different inertial frames.

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be. If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

I'm deliberately using a different approach.

cheers,

neopolitan

neopolitan

Mar20-09, 05:44 AM

No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.

Simultaneity issues here. I was only taking the lab's perspective, one perspective at a time. But yes, you can reverse it around, and due to relativity of simultaneity, the muon will decay in its own frame while the lab clock reads 75 microsecond (but since we can't be muons anymore than we can be photons, it makes sense to use the lab perspective. I don't think that particle physicists would report that the muon decays after 75 microseconds on the lab clock when read from the muon's own frame in which a clock, if it could be accelerated to a gamma of 29.3, would read 2.2 ms).

Mentz114

Mar20-09, 08:00 AM

(but since we can't be muons anymore than we can be photons, it makes sense to use the lab perspective.
Muons have mass, so we can 'be muons'. I thought you might like to know this.

neopolitan

Mar20-09, 09:56 AM

Muons have mass, so we can 'be muons'. I thought you might like to know this.

Hm, if you were a muon, you wouldn't be one for long.

However, I don't think there is much difference between "I can't be a northern polka dotted, orange bellied, bearded unicorn" and "I can't be a ballet lady". I think there are a few good reasons why I can't be a muon (even though a muon has mass). Equally, I don't think that not having mass is the only thing preventing me from being a photon.

Perhaps I missed a key lecture at uni.

(There's another feeble attempt at sarcasm :smile:)

PS Have you got a thing about muons? It's just that you have only popped your head in to make comment about them. If they are off limits or something, just let me know and I will use another example.

JesseM

Mar20-09, 11:48 AM

Damn, I edited this post when I meant to reply to it to add a small comment, hence erasing everything else but the small new comment...I'll have to try to reconstruct it.

Rasalhague

Mar20-09, 12:04 PM

If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be. If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.

But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.

That's a very neat summary. It brings out very clearly where the symmetry lies (between time and space), and where the difference lies (between (1) frames in which a timelike separation has no space component, or frames in which a spacelike separation has no time component, and (2) other frames in which the separation, timelike or spacelike, has a mixture of time and space coordinates). Finger's crossed I've got the terminology corrent there...

Rasalhague

Mar20-09, 12:10 PM

That's a very neat summary. It brings out very clearly where the symmetry lies (between time and space), and where the difference lies (between (1) frames in which a timelike separation has no space component, or frames in which a spacelike separation has no time component, and (2) other frames in which the separation, timelike or spacelike, has a mixture of time and space coordinates). Finger's crossed I've got the terminology corrent there...

Ah, I just read Jesse's reply after I posted this. I see the point about it being the minimum separation. Taking that into account, it does still seem a satisfying way of looking at it.

Rasalhague

Mar20-09, 12:14 PM

"given that two events on the worldline of a clock at rest in the primed frame are separated by a coordinate time of \Delta t in the unprimed frame, how much clock time (or coordinate time in the primed frame where the clock is at rest) passes between those two events?" Can you think of a more intuitive way to express the physical significance?

Is this equivalent to saying: "Two events are separated by a timelike interval \Delta \tau. In frame S[tex], this separation has a time component [tex]\Delta t > \Delta \tau. Given the value of \Delta t, how can we calculate \Delta \tau? Answer: \Delta \tau = \Delta t / \gamma. The inverse question being: "Given the value of \Delta \tau, how can we calculate \Delta t? Answer: \Delta t' = \Delta \tau * \gamma.

Alternatively:

Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame, moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t_{a1} = t_{b1}, and t_{a2} = t_{b2}?

\Delta t' = \Delta t / \gamma.

As opposed to time dilation:

Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame S', moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t'_{a1} = t'_{b1}, and t'_{a2} = t'_{b2}?

\Delta t' = \Delta t * \gamma.

So would it be fair to say that there really is no fundamental or physical difference between "reverse time dilation" and "temporal analogue of length contraction" ("time contraction")? They ask the same question, only with different names given to the frames. If the problem you're working on only involves one question, or if it only involves asking one type of question of one frame, and the other type of question of the other frame, then you can avoid ever having to use the form \Delta t' = \Delta t / \gamma, and instead always use \Delta t = \Delta t' / \gamma. But if you want to ask both types of question in both directions, then you'd have to use \Delta t' = \Delta t / \gamma, wouldn't you? Or else swap over the labels you've given to the frames as the occasion demands.

Rasalhague

Mar20-09, 12:30 PM

The garbled text in my previous post should have read:

Is this equivalent to saying: "Two events are separated by a timelike interval \Delta \tau. In frame S, this separation has a time component \Delta t > \Delta \tau. Given the value of \Delta t, how can we calculate \Delta \tau? Answer: \Delta \tau = \Delta t / \gamma. The inverse question being: "Given the value of \Delta \tau, how can we calculate \Delta t? Answer: \Delta t' = \Delta \tau * \gamma.

Rasalhague

Mar20-09, 12:52 PM

Actually, writing it out in these terms and then thinking about how I'd write out the TAFLC equation in words makes me realize that the question of whether there's really any difference between the TAFLC equation and the reversed time dilation equation is actually rather subtle. If you look at my diagram (http://www.physicsforums.com/attachment.php?attachmentid=17992&d=1237127533), you see that the TAFLC isn't really giving you the time-interval between any pair of pink events at all, since none of the events are at the position of the top of the double-headed arrow that I use to represent the delta-t' of the TAFLC equation; it's only if you were to draw a new pair of events that are colocated in the primed frame, at the top and bottom of that double-headed arrow on that diagram, that the TAFLC would tell you the same think about the time between those new events in both frames that the reversed time dilation equation tells you about the time between the colocated events in the primed frame. So I guess what that would mean conceptually is that if you choose your pair of events at the start, then the time dilation equation + reversed time dilation equation tell you everything you need about the relation between the time intervals connecting those specific events in your two frames (one of which must be the frame where they're colocated). In this context the TAFLC equation is actually not telling you about the time-interval between those specific events in either frame, although you could of course draw in some new events such that the times delta-t and delta-t' in the TAFLC equation had the same meaning for that new pair of events that the times delta-t and delta-t' in the time dilation (and reversed time dilation) equation have for the original pair of events. But then if you want to talk about the time between the new pair, why not just start over and have them be the starting events? I guess conceptually what I would say is that to use any of these time equations you should always be clear on what two events you're interested in at the start, and once you've picked them then it's the time dilation and reversed time dilation equation that tell you the relation between the time-intervals in both frames, while the TAFLC is telling you something more abstract about the time in the non-colocated frame between planes of simultaneity from the colocated frame that pass through both events.

But couldn't you look at the conventional time dilation equation in a similar way? In each case you want to know something about the timing of two events. You specify something about the events which you want information about (which other events they have to be simultaneous with, and according to whose definition of simultaneity), in both cases without knowing exactly which events you're looking for, and the equations tell you. It could well be that I'm missing the subtlety though. I need to read these posts more carefully and think this over.

JesseM

Mar20-09, 01:21 PM

OK, here's the recreation of the last post I accidentally edited away: There still seems to be some confusion.

We talk about L as it has to be a ruler or a rod or a length. They are convenient devices, but L could be a distance between two randomly selected points in a rest frame (I want to say my rest frame, but it can be any rest frame).

We talk about t as if it has to be attached to events, like ticking of a clock, or formally defined events. But t could be the time interval between two randomly selected times.

We can imagine putting two pins on a map and measuring the distance. We have difficulty putting two pins in time and measuring the temporal distance. But I take TAFLC as being for measuring between these two pins in time, in the same was a LC is for measuring between two pins on the map. We take a different perspective on them by putting us and pins into different inertial frames.
But look at my diagram (http://www.physicsforums.com/attachment.php?attachmentid=17992&d=1237127533) again. If the pink dots are the pins, with two colocated in the unprimed frame and two simultaneous in the unprimed frame, then it is actually the time dilation equation that compares the time in the two frames between the events that are colocated in the unprimed frame, and the "spatial analogue for time dilation" (SAFTD) equation that compares the distances in the two frames between the events that are simultaneous in the unprimed frame. The TAFLC equation doesn't tell you the time between any pair of pink events in the diagram, although you could invent a new pair of events such that it would--these new events would have to be colocated in the primed frame.
If I get myself an inertial frame where two time pins are in the same position, then they will be as far apart in time as they can be.
Actually that's backwards, the time between events is minimized in the frame where they're at the same position. Suppose I have been moving inertially my whole life, and one event is the event of my birth while the other is the event of my turning 30. The time between these events is 30 years in the frame where I am at rest and they occur at the same location, but in a frame where I am moving there is a greater time between the events because I am aging more slowly.
If I get myself an inertial frame where the two length pins are simultaneous, then they will be as far apart in length as they can be.
That's not quite correct either. If you want to analyze length contraction in terms of just two events rather than three (in the case of three, #1 would be an event on the worldline of the object's left end, #2 would be an event on the worldline of the object's right end that's simultaneous with #1 in the object's rest frame, and #3 would be an event on the worldline of the right end that's simultaneous with #1 in the frame where the object is moving), then you have to pick two events on the worldline of either end of the object that are simultaneous in the frame where the object is moving, but non-simultaneous in the object's rest frame (since both ends of the object have a constant position in the object's rest frame, events on either end will still be separated by the rest length L even if they aren't simultaneous). The distance between these events will be greater in the object's rest frame where they're non-simultaneous (because rest length is greater than moving length), so they aren't at a maximal separation in the frame where the events are simultaneous. In fact it turns out that events will actually have a minimal spatial distance in the frame where they are simultaneous, you can see this by considering the more general equation for the separation between events in two arbitrary frames:

\Delta x' = \gamma (\Delta x - v \Delta t)

If you choose the unprimed frame to be the one where they're simultaneous, then \Delta t = 0 so you're left with \Delta x' = \gamma * \Delta x, which shows that the distance is always greater in the non-simultaneous frame.

Aside from these caveats, I agree with the idea that you can define the meaning of the two frames in equations like time dilation by first picking two events and then making clear which is supposed to be the frame where they are colocated (if they are timelike-separated) or which is supposed to be the frame where they are simultaneous (if spacelike-separated). Writing it out in words, the standard time dilation equation would be:

(time between events in frame where they are not colocated) = (time between events in frame where they are colocal) * gamma

Likewise, the reversed time dilation equation would be:

(time between events in frame where they are colocal) = (time between events in frame where they are not colocated) / gamma

Thinking about writing it in words, it may seem a bit subtle to say what the difference is between the TAFLC equation and the reversed time dilation equation. As I said, if you look at my diagram (http://www.physicsforums.com/attachment.php?attachmentid=17992&d=1237127533) you see that the double-headed arrow representing the dt' in the TAFLC does not have any of the three pink events at the top end of it; you would have to invent a new pair of events at either end of this double-headed arrow in order to phrase the TAFLC in terms of time intervals between events, and in that case you would write it exactly like the reversed time dilation equation above, except with the understanding that you were now referring to that new pair of events. So the way I would conceptualize this situation is to say that in order to talk about any of these equations, you first have to specify a single pair of events you want to talk about, and then in terms of those specific events the time dilation and reversed time dilation equations tell you everything you want to know about the time interval between the events in two frames (one of which is the one where they're colocated), whereas in terms of those events the TAFLC is telling you something more abstract about the time-interval (in the frame where the events are not colocated) between surfaces of simultaneity from the the frame where the events are colocated. Of course you could start with a new pair of events so that the time interval given by the TAFLC applied to the previous events is just the time interval between the new events in the frame where they're colocated, but then you're really talking about the reversed time dilation equation for these new pair of events, not the TAFLC for them.
But, assuming all the pins are in the same frame (ie they share a frame in which the time pins have zero length separation and the length pins have zero time separation), then from any other frame: t' = t/gamma and L'=L/gamma where t and L are the maximum time and length separations for the respective pins.
As I said above, t and L should be the minimum time and distance separation for the pins, there is no upper limit on their separations (there is an upper limit on the length of a physical object when viewed in different frames, but the concept of the length of an object in different frames is quite different from the concept of the spatial distance between a pair of events in different frames). And if the unprimed frame is the one where the time pins are colocated and the space pins are simultaneous, then the equations above are incorrect, they should be t' = t*gamma and L' = L*gamma, representing the standard time dilation equation along with the SAFTD equation. Do you disagree?

JesseM

Mar20-09, 01:25 PM

Ah, I just read Jesse's reply after I posted this. I see the point about it being the minimum separation. Taking that into account, it does still seem a satisfying way of looking at it.
Also see the points I made in the re-created version of that post (the original of which I accidentally deleted) about the differences between the concept of the length of a physical object in different frames vs. the concept of the distance between a pair of events in different frames. Even though the length of an object is maximized in its rest frame, the distance between a pair of events is minimized in the frame where they are simultaneous.

Mentz114

Mar20-09, 02:12 PM

JesseM

Mar20-09, 02:27 PM

Is this equivalent to saying: "Two events are separated by a timelike interval \Delta \tau.
OK, that would be equivalent to the proper time along the worldline of an inertial object that goes from one event to the other, which of course is the same as the coordinate time between the events in that object's rest frame, where the events occur at the same coordinate position.
In frame S, this separation has a time component \Delta t > \Delta \tau. Given the value of \Delta t, how can we calculate \Delta \tau? Answer: \Delta \tau = \Delta t / \gamma. The inverse question being: "Given the value of \Delta \tau, how can we calculate \Delta t? Answer: \Delta t = \Delta \tau * \gamma.
Yes, although your "inverse question" corresponds to the normal time dilation equation (with the most common notation being to use a primed t' where you've used an unprimed t, and an unprimed t where you've used \tau), whereas your first question corresponds to what I've called the "reversed time dilation equation" (where you just divide both sides of the normal time dilation equation by gamma).
Alternatively:

Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame, moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t_{a1} = t_{b1}, and t_{a2} = t_{b2}?

\Delta t' = \Delta t / \gamma.
Since you wrote t_{a1} = t_{b1} rather than t'_{a1} = t'_{b1}, I take it you want these events to be simultaneous in the unprimed frame rather than the primed frame? If so, then if we want to conceptualize this in terms of the coordinate time in two frames between a single pair of events as in neopolitan's formulation, then we're really talking about the second pair of events E_{b1} and E_{b2} here; we know the time between them in the unprimed frame, and want to know the time between them in the primed frame where they are colocated. So, this would indeed be the "reversed time dilation equation" you have above, but it would be the opposite of the usual convention about primed and unprimed (the usual convention being that the frame in which the two events are colocated would be the unprimed one).
As opposed to time dilation:

Given two events E_{a1} and E_{a2}, colocal in some frame S, with (time) interval \Delta t, what is the (time) interval \Delta t' in some other frame S', moving at constant velocity u relative to S, between two events E_{b1} and E_{b2}, colocal in S', if t'_{a1} = t'_{b1}, and t'_{a2} = t'_{b2}?

\Delta t' = \Delta t * \gamma.
Yes, although if we think in terms of a single pair of events as before, here you've reversed the convention about which frame is the one where they're colocated.
So would it be fair to say that there really is no fundamental or physical difference between "reverse time dilation" and "temporal analogue of length contraction" ("time contraction")? They ask the same question, only with different names given to the frames.
I don't think so--as I said in my post to neopolitan, if you think in terms of starting with a pair of events and then asking various questions about time-intervals involving those specific events, then the TAFLC equation is really asking something more like "in the frame where the events are not colocated, what is the temporal separation between two surfaces of simultaneity from the frame where they are colocated, given that each surface passes through one of the two events?" But this point about starting with a single pair of events brings me to your next post where you were responding to a similar comment from the post I accidentally deleted:
But couldn't you look at the conventional time dilation equation in a similar way? In each case you want to know something about the timing of two events. You specify something about the events which you want information about (which other events they have to be simultaneous with, and according to whose definition of simultaneity), in both cases without knowing exactly which events you're looking for, and the equations tell you. It could well be that I'm missing the subtlety though. I need to read these posts more carefully and think this over.
I think you always have to know what the events are physically, like particular readings on a physical clock, or any other observed events you like, and are then interested in saying various things relating to how different coordinate systems view them, like the difference in coordinate time between the events or which readings on a different physical clock are simultaneous with these events in a particular frame (and what the difference is between the two readings on that clock). I suppose you can ask questions in such a way that you don't know both events in advance, like "which reading on this clock occurs at a time interval of \Delta t after the clock reading 0 in my frame", but for the question to be well-defined it must uniquely determine the events in question even if you don't know them until you do some calculations.
If the problem you're working on only involves one question, or if it only involves asking one type of question of one frame, and the other type of question of the other frame, then you can avoid ever having to use the form \Delta t' = \Delta t / \gamma, and instead always use \Delta t = \Delta t' / \gamma. But if you want to ask both types of question in both directions, then you'd have to use \Delta t' = \Delta t / \gamma, wouldn't you? Or else swap over the labels you've given to the frames as the occasion demands.
But what do you mean by "both types of questions"? What events are you asking questions about? If you're asking about more than a single pair of events then in that case I'd agree you might use both of those equations to talk about time intervals between events, but since you're no longer talking about a single pair of events you'd have to have some different notation to distinguish between verbal formulations like "time-interval in the unprimed frame between events A and B" and "time-interval in the unprimed frame between events C and D"--perhaps you could use \Delta t_{AB} and \Delta t_{AC} in this case. Then if A and B are colocated in the primed frame while C and D are colocated in the unprimed frame, you might write \Delta t'_{AB} = \Delta t_{AB} / \gamma along with \Delta t_{CD} = \Delta t'_{CD} / \gamma, but I would refer to the first as "the reversed time dilation equation for events A and B" and the second as "the reversed time dilation equation for events C and D", in words they would both come out to:

(time between specified events in frame where they are colocated) = (time between specified events in frame where they are not colocated) / gamma

neopolitan

Mar20-09, 07:31 PM

neopolitan;
no, I don't have a thing about muons. I did not introduce the subject so your comment makes no sense. You give the impression that muons can't have a frame of reference, in which you are wrong. I'm trying to shine some light here into your fog of misunderstanding, and you respond with insults and sarcasm.

I enjoyed your little biog about talking to people ( Professors even ) about your doubts and problems with relativity. I hope you get cured soon because it's costing some people an awful lot of effort.

M

Actually, if you read the text around the comment I made about not being able to be muon, you will see that it was made in the context of a decision about which frame to use. Most readers would be able to interpret from that that I did realise that the muon had a frame of reference. The laboratory frame is a sensible frame. It's not the only frame.

I accept that I may have misunderstandings, but shining light on the blindingly obvious it not helping anyone.

You clearly don't understand the message behind my story about speaking to various people about some "doubts and problems".

As to being cured of my curiosity, did you never have it, or were you cured? (:smile:)

Mentz, I know you are curious, I know you think I am obsessing on an unimportant detail. But equally I think you were obsessing on an unimportant detail regarding the muons. It wasn't even me who introduced them. It was BobS. I just thought he raised an interesting and useful real world example.

cheers,

neopolitan

neopolitan

Mar20-09, 11:25 PM

bernhard.rothenstein

Mar21-09, 03:15 AM

There have been more than a few threads where there clearly is confusion about the use of time dilation and length contraction.

People initially think that:

1. in an frame which is in motion relative to themselves, time dilates and lengths contract; and
2. velocities in a frame which is in motion relative to themselves are contracted lengths divided by dilated time.

I admit that it stumped me for a long time, because of what I see as inconsistent use of primes and for me a much more useful pair of equations would have a more consistent use of primes, similar to the Lorentz transformations.

I was told during a long discussion that time dilation and length contraction are used, even though they pertain to different frames, because they have greater utility. I took that at face value, but now I wonder again.

What exactly is the greater utility of time dilation and length contraction equations which prevents the use of two contraction equations which would do away with the confusion I mentioned above?

(And by the way, introducing arguments that t in time dilation is the period between tick and tock doesn't really help, because this is more indicative of the confusion since we use clocks everyday to measure the time between events in terms of the number of ticks and tocks rather than in terms of the duration of pause between each tick and tock. Reinterpreting how we use time to make the equation work is not indicative of any greater utility.)

If it is a purely historical thing, then I would be far happier with it if that little tidbit were taught at the same time as the equations are introduced. But it isn't.

There is also the potential argument that they are only useful right at the beginning of one's odyssey into relativity, so it doesn't really matter. Sure, ok, then it doesn't matter if you use a more intuitive pairing does it?

Bottom line: what is so great with time dilation?

cheers,

neopolitan
Is
(T0/T)(L/L0=1 an important consequence?
I think that all we discuss there is a conswequence of the standard clock synchronization and of the measurement procedures. In general we ca have length contraction, length dilation and no disrtion at all.

Rasalhague

Mar21-09, 03:45 AM

Yes, although your "inverse question" corresponds to the normal time dilation equation (with the most common notation being to use a primed t' where you've used an unprimed t, and an unprimed t where you've used ), whereas your first question corresponds to what I've called the "reversed time dilation equation" (where you just divide both sides of the normal time dilation equation by gamma).

Given that this was my attempt at paraphrasing your definition of TAFLC and its inverse (hence the choice of primed and unprimed), I guess that shows I’m still having trouble separating these concepts of TAFLC and inverse time dilation.

(time between events in frame where they are not colocated) = (time between events in frame where they are colocal) * gamma

And (space between events in a frame where they are not simultaneous) = (space between events in a frame where they are simultaneous) * gamma. What does this tell us about the length of an object: if I measure a moving object, this is how long it would be if measured in its rest frame?

(time between events in frame where they are colocal) = (time between events in frame where they are not colocated) / gamma

And (space between events in a frame where they are simultaneous) = (space between events in a frame where they are not simultaneous) / gamma. This being length contraction.

...the concept of the length of an object in different frames is quite different from the concept of the spatial distance between a pair of events in different frames).

I wonder if this is the crucial factor in how the apparent asymmetry comes about between time dilation and length contraction? When the concepts are introduced, in a way that makes one seem somehow parallel to the other, it’s so easy to jump to that conclusion. So would it be correct to say that the ends of an object aren’t events, but that each end of an object occupying some specific location at some specific time does comprise an event (a different event in the case of each end)?

So the way I would conceptualize this situation is to say that in order to talk about any of these equations, you first have to specify a single pair of events you want to talk about, and then in terms of those specific events the time dilation and reversed time dilation equations tell you everything you want to know about the time interval between the events in two frames (one of which is the one where they're colocated), whereas in terms of those events the TAFLC is telling you something more abstract about the time-interval (in the frame where the events are not colocated) between surfaces of simultaneity from the frame where the events are colocated. Of course you could start with a new pair of events so that the time interval given by the TAFLC applied to the previous events is just the time interval between the new events in the frame where they're colocated, but then you're really talking about the reversed time dilation equation for these new pair of events, not the TAFLC for them.

But if we think of, say, the time dilation equation as a function f(t) = t * \gamma which takes as its input some time, and gives as its output some other time, this function has an inverse f^{-1}(t) = t / \gamma, the inverse being also a function over t, the real valued set of all possible time intervals, we can conceptualise both functions as abstract entities, without specifying any particular events until we actually want to calculate something about particular events. In the abstract, they’re functions that tell you something about *any* pair of events. As such, until the events are specified one way or the other--aside from matters of frame-labelling convention--aren’t TAFLC and reverse TD equivalent? And when we do want to specify a pair of events, what’s the difference between performing the same mathematical operation on the same values whether you call it “start[ing] with a new pair of events” or letting the equation tell you about a new pair of events, since, in the latter way of conceptualising it, the events would still be specified uniquely by the question, wouldn't they? (Namely the equation chosen and the value plugged into it.)

But what do you mean by "both types of questions"? What events are you asking questions about?

I meant questions of the type answered by the traditional time dilation equation (or equivalently, I assumed, reverse TAFLC) versus questions of the type answered by reverse time dilation (or equivalenty, I assumed, TAFLC), regardless of how the frames are labelled. Of course, I could well be mistaken to assume that equivalence.

(1) “In Alice’s rest frame, what time on Bob’s watch is simultaneous with Alice’s 4?” Answer: t_{B} = t_{A} / \gamma = 3.2. What do we call this: time contraction, temporal analogue of length contraction, reverse time dilation?

(2) “In Bob’s rest frame, what time on Bob’s watch is simultaneous with Alice’s 4?” Answer t_{B} * \gamma = 5. Time dilation, right? Or is it reversed TAFLC?

If you're asking about more than a single pair of events then in that case I'd agree you might use both of those equations to talk about time intervals between events...

Yes, I can see that if you input the same (nonzero) value into these two equations, you’d be talking about more than a single pair of events.

...but since you're no longer talking about a single pair of events you'd have to have some different notation to distinguish between verbal formulations like "time-interval in the unprimed frame between events A and B" and "time-interval in the unprimed frame between events C and D"--perhaps you could use \Delta t_{AB} and \Delta t_{AC} in this case. Then if A and B are colocated in the primed frame while C and D are colocated in the unprimed frame, you might write \Delta t’_{AB} = \Delta t_{AB} / \gamma along with \Delta t_{CD} = \Delta t’_{CD} / \gamma, but I would refer to the first as "the reversed time dilation equation for events A and B" and the second as "the reversed time dilation equation for events C and D", in words they would both come out to:

(time between specified events in frame where they are colocated) = (time between specified events in frame where they are not colocated) / gamma

So what, if anything, in this situation would you describe as TAFLC? Thanks for your patience, by the way, and sorry if I'm repeating myself or demanding answers to questions you've already answered in detail. Perhaps it'll become clearer to me once I've solved some more problems and got a bit more experience of the sort of questions these concepts are used to deal with, and when I've looked more at time dilation and length contraction in the wider context of the Lorentz transformation and spacetime geometry.

neopolitan

Mar21-09, 05:29 AM

Is
(T0/T)(L/L0=1 an important consequence?
I think that all we discuss there is a conswequence of the standard clock synchronization and of the measurement procedures. In general we ca have length contraction, length dilation and no disrtion at all.

It depends a little on what Lo and To are.

I am tempted to think (using standard pairing, time dilation and length contraction):

T = To * gamma
L = Lo / gamma

so:

To / T = 1 / gamma
L / Lo = 1 / gamma

so:

(To / T)(L / Lo) = 1 /(gamma)2

Which is partly why I question it.

Rearranging (To / T)(L / Lo) = 1 gives you:

(To / Lo)(L / T) = 1

or

L / T = Lo / To

Which I think is an important consequence. In much later posts we are nearing a resolution ... maybe :)

For me that discussion could revolve, conceptually, around what a photon does travelling along between two events (but I stress that it doesn't have to). In one frame, it could be said that that photon travels L in time T (so L/T=c). In another frame, it could be said that that same photon travels Lo or L' in time To or T' (so that Lo/To=c or L'/T'=c).

cheers,

neopolitan

bernhard.rothenstein

Mar21-09, 08:17 AM

Mentz114

Mar21-09, 10:35 AM

\mathbf{t'}=\mathbf{t}\cosh(\beta)+\mathbf{x}\sinh (\beta)
\mathbf{x'}=\mathbf{x}\cosh(\beta)+\mathbf{t}\sinh (\beta)

What more needs to be said ?

neopolitan

Mar21-09, 08:57 PM

\mathbf{t'}=\mathbf{t}\cosh(\beta)+\mathbf{x}\sinh (\beta)
\mathbf{x'}=\mathbf{x}\cosh(\beta)+\mathbf{t}\sinh (\beta)

What more needs to be said ?

Mentz old boy,

You are clearly extremely intelligent, very highly educated and totally untroubled by curiosity not to mention modest. Most of the rest of us would need more than those equations during our years of education even you are able to deduce all that needs to known from them.

Would you replace time dilation and length contraction with those equations? Do you suggest that presenting the new student with those equations would inform them or are you just planning to bludgeon them into conformity?

Since you seem to have said all that needs to be said, I do hope you don't plan to say any more. I am happy for you to leave to my rhetorical questions unaddressed.

cheers,

neopolitan

bernhard.rothenstein

Mar22-09, 03:36 AM

\mathbf{t'}=\mathbf{t}\cosh(\beta)+\mathbf{x}\sinh (\beta)
\mathbf{x'}=\mathbf{x}\cosh(\beta)+\mathbf{t}\sinh (\beta)

What more needs to be said ?
I think that in order to help the learner there are a lot of thinks which should be mentioned.
1. Length contraction is obtained from the Lorentz transformations if in one of the involved inertial frames a simultaneous detection of the moving rod is performed. Recent papers have shown that the same result could be obtained without imposing the mentioned condition.
2. Time dilation is obtained if in one of the involved inertial frame a proper time interval is measured.
3. Time dilation and length contraction could be derived from thought experiments and that makes the beauty of teching relativity to beginners.
4. If the clocks of the involved inertial frames are standard synchronized there is no time dilation without length contraction.
Kind regards

Mentz114

Mar22-09, 04:14 PM

Mentz old boy,

You are clearly extremely intelligent, very highly educated and totally untroubled by curiosity not to mention modest. Most of the rest of us would need more than those equations during our years of education even you are able to deduce all that needs to known from them.

Would you replace time dilation and length contraction with those equations? Do you suggest that presenting the new student with those equations would inform them or are you just planning to bludgeon them into conformity?

Since you seem to have said all that needs to be said, I do hope you don't plan to say any more. I am happy for you to leave to my rhetorical questions unaddressed.

cheers,

neopolitan
Thanks.

Would you replace time dilation and length contraction with those equations?
Those equations are length contraction and time dilation.
Do you suggest that presenting the new student with those equations would inform them or are you just planning to bludgeon them into conformity?
This remark first presupposes something then makes a damning inference. Ungentlemanly and very rude.

I am happy for you to leave to my rhetorical questions unaddressed.
Please look up the meaning of 'rhetorical'. Surely you wanted someone to respond.

Please, cut out the personal stuff, ironic or not.

Mentz114

Mar22-09, 04:23 PM

I think that in order to help the learner there are a lot of things which should be mentioned.
1. Length contraction is obtained from the Lorentz transformations if in one of the involved inertial frames a simultaneous detection of the moving rod is performed. Recent papers have shown that the same result could be obtained without imposing the mentioned condition.
2. Time dilation is obtained if in one of the involved inertial frame a proper time interval is measured.
3. Time dilation and length contraction could be derived from thought experiments and that makes the beauty of teching relativity to beginners.
4. If the clocks of the involved inertial frames are standard synchronized there is no time dilation without length contraction.
Kind regards
Bernhard,
I'm sure you're a dedicated and earnest teacher of the subject, but do beginners have to go into SR as deeply as you enjoy going ?

M

neopolitan

Mar22-09, 11:20 PM

Mentz,

The equations you provided would not help the new student to SR to understand the physical significance of the standard time dilation and length contraction equations that they are normally presented with.

I am pretty sure that they would confuse. It seems to have confused either you or the author of this site on hyperbolic functions (http://hubpages.com/hub/Hyperbolic-Functions).

On his site, time dilation is given by cosh u (probably cosh \beta of your equation set, but since you did not define \beta, I don't know).

In the same vein, length contraction (he calls it spatial contraction) is given by sech u (again probably sech \beta).

He shows you graphically what u is in his equations (the area between the asymptote and the x axis). He also clarifies that sech u is the reciprocal of cosh u.

That is slightly more helpful.

I expect that the equation pair you gave really represents the Lorentz Transformations, but in a format which is far less intuitively comprehensible to the new student. I suspect that the equation pair requires you to make reference to the function under which the area \beta is found, namely S2 = x2 - (ct)2 and that where you have written t, you should have written (ct).

But all of this is extraneous to what we were discussing.

cheers,

neopolitan

bernhard.rothenstein

Mar23-09, 06:40 AM

There have been more than a few threads where there clearly is confusion about the use of time dilation and length contraction.

People initially think that:

1. in an frame which is in motion relative to themselves, time dilates and lengths contract; and
2. velocities in a frame which is in motion relative to themselves are contracted lengths divided by dilated time.

I admit that it stumped me for a long time, because of what I see as inconsistent use of primes and for me a much more useful pair of equations would have a more consistent use of primes, similar to the Lorentz transformations.

I was told during a long discussion that time dilation and length contraction are used, even though they pertain to different frames, because they have greater utility. I took that at face value, but now I wonder again.

What exactly is the greater utility of time dilation and length contraction equations which prevents the use of two contraction equations which would do away with the confusion I mentioned above?

(And by the way, introducing arguments that t in time dilation is the period between tick and tock doesn't really help, because this is more indicative of the confusion since we use clocks everyday to measure the time between events in terms of the number of ticks and tocks rather than in terms of the duration of pause between each tick and tock. Reinterpreting how we use time to make the equation work is not indicative of any greater utility.)

If it is a purely historical thing, then I would be far happier with it if that little tidbit were taught at the same time as the equations are introduced. But it isn't.

There is also the potential argument that they are only useful right at the beginning of one's odyssey into relativity, so it doesn't really matter. Sure, ok, then it doesn't matter if you use a more intuitive pairing does it?

Bottom line: what is so great with time dilation?

cheers,

neopolitan
What is so great with time dilation???
1. In teaching it can be derived from the two postulates and from Pythagoras' throrem.
2. It leads directly to length contraction.
3. Length contraction leads directly to the Lorentz transformations.
4. Lorentz transformation lead directly to the formulas that account for all the formulas we encounter in special relativity theory.
Is there to say when it is about its benefits?

neopolitan

Mar23-09, 07:12 AM

What is so great with time dilation???
1. In teaching it can be derived from the two postulates and from Pythagoras' throrem.
2. It leads directly to length contraction.
3. Length contraction leads directly to the Lorentz transformations.
4. Lorentz transformation lead directly to the formulas that account for all the formulas we encounter in special relativity theory.
Is there to say when it is about its benefits?

I think you are talking about the light clock? or something similar? Someone else stated recently on a thread hereabouts that the light clock derivation has weaknesses. I would think anything similar has similar weaknesses.

The confusion I see comes after getting the students to get shown how time dilation is derived but no clarification is given along the lines that you can't take a contracted length and a dilated time to get a speed which the postulates you started with said was invariant.

Most students won't think more deeply than is required to pass the test and so will learn very little.

Others will instinctively grasp what has not been clarified.

Some, perhaps only a few, will be left with a vague unease because if L and t are such that L/t=c then L'/t' is not c.

I do think that we can derive the Lorentz transformations without even stopping at length contraction and time dilation. Lorentz seemed to and you can go directly from Galilean boosts to Lorentz transformations without having previously derived length contraction or time dilation, you just remove the assumption of instantaneous information transfer and use the first postulate. The second postulate falls out as a consequence.

I'd be happy to dispense with time dilation and length contraction altogether, and just go with Lorentz transformations, as Mentz possibly meant in an earlier post. But this is not the standard approach. Additionally, I would clarify just what it is that the Lorentz transformations can tell you, because if you just plug in t=0 into the spatial transformation, you end up with "length dilation" and that hardly matches with the contraction we expect.

cheers,

neopolitan

bernhard.rothenstein

Mar23-09, 11:00 AM

I think you are talking about the light clock? or something similar? Someone else stated recently on a thread hereabouts that the light clock derivation has weaknesses. I would think anything similar has similar weaknesses.

The confusion I see comes after getting the students to get shown how time dilation is derived but no clarification is given along the lines that you can't take a contracted length and a dilated time to get a speed which the postulates you started with said was invariant.

Most students won't think more deeply than is required to pass the test and so will learn very little.

Others will instinctively grasp what has not been clarified.

Some, perhaps only a few, will be left with a vague unease because if L and t are such that L/t=c then L'/t' is not c.

I do think that we can derive the Lorentz transformations without even stopping at length contraction and time dilation. Lorentz seemed to and you can go directly from Galilean boosts to Lorentz transformations without having previously derived length contraction or time dilation, you just remove the assumption of instantaneous information transfer and use the first postulate. The second postulate falls out as a consequence.

I'd be happy to dispense with time dilation and length contraction altogether, and just go with Lorentz transformations, as Mentz possibly meant in an earlier post. But this is not the standard approach. Additionally, I would clarify just what it is that the Lorentz transformations can tell you, because if you just plug in t=0 into the spatial transformation, you end up with "length dilation" and that hardly matches with the contraction we expect.

cheers,

neopolitan
Thanks for your answer. As an old teacher of physics I have studied the different ways in which the Lorentz transformations could be derived.
1. I learned a lot from Paul Kard [1] who derives first the formula that accounts for the length contraction, which leads him to the formula that accounts for the Doppler shift which leads to the addition law of relativistic velocities and derives the formula that accounts for the time dilation from the Doppler shift formula. I knew all that from Kard's original papers in Russian.
[1] Leo Karlov, "Paul Kard and the Lorentz-free special relativity," Phys.Educ. 24, 165 (1989)
2. Kalotas and Lee [2] convinced me that the Doppler shift formula could be derived from the formula that accounts for the "Police Radar" an experiment performed in a single inertial reference frame, involving a single clock and so no clock synchronization. The formula that accounts for the Doppler shift is derived by simple injection of the first postulate. He also shows that the Lorentz transformations could be derived from the Doppler shift formula.
[2] T.M. Kalotas and A.R. Lee, "A "two line" derivation of the relativistic longitudinal Doppler formula," Am.J.Phys. 58, 187 (1990)
3. Asher Peres [3] taught me that the basic formulas of relativistic kinematics could be derived from Einstein's postulate: "All the physical laws are the same for all inertial observers,in particular the speed of light is the same" in the following order: radar echo, time dilation, additions of velocities, the Doppler Effect and optical aberration. He does not derive the Lorentz transformations even if starting with one of the basic formulas mentioned above could lead to them.
[3] Asher Peres, "Relativistic telemetry," Am.J.Phys. 55, 516 (1987)

When I started learning English from BBC, Professor Grammar told me that English is a very flexible language. I would say that Special Relativity is a very flexible chapter of physics. We can start with Einstein's postulte, derive the equation that accounts for one of the effects mentioned above and it leads us to the Lorentz transformations.
I would highly appreaciate the criticism of the approaches presented above. My students enjoyed them.
Kind regards and thanks for giving me the opprtunity to discuss about the teaching of special relativity.

Ich

Mar23-09, 11:43 AM

I'd be happy to dispense with time dilation and length contraction altogether, and just go with Lorentz transformations, as Mentz possibly meant in an earlier post.
Then why don't you? IMHO, TD and LC are tools for professionals to shortcut calculations, but they're bound to mislead beginners. Students begin to think in those mechanical, ether-like terms instead of appreciating the interdependence of space and time. It needs enormous knowlegde and mental discipline to get calculations right when working with these tools - not when they're applicable, but when you have to find out whether they are or not and in which direction.
But this is not the standard approach.
Yes, the standard approach is to teach LET and tell the students that, nevertheless, there is no absolute frame. At least that was what I experienced in school. It's a time-saving approach, but a dead end.
Additionally, I would clarify just what it is that the Lorentz transformations can tell you, because if you just plug in t=0 into the spatial transformation, you end up with "length dilation" and that hardly matches with the contraction we expect.

Well, but it is exactly what you did expect: the x-basis transforms like the t-basis, therefore, in the decomposition of a null-vector, the ratio of the t- and x-component stays the same, namely c.
Length contraction is something completely different and not applicable. You can use the Lorentz-transforms to find out how length contraction is defined to see this.

DaleSpam

Mar23-09, 04:14 PM

I find myself in the (for me) very odd position of reconciler or mediator or some similar "kum-ba-ya" campfire nonsense. I am travelling and at high altitude, so I will blame it on reduced oxygen saturation.

Specifically, I agree with neopolitain, Mentz114, and JesseM (why can't you all just get along) even though you all disagree with each other. I agree with neopolitain that the standard equations are confusing. Because of that confusion and the possibility of error I follow Mentz114's approach of only using the Lorentz transform equations. Instead, I would use JesseM's spacetime diagrams to geometrically demonstrate the idea of time dilation and length contraction to a new student without ever using any formulas other than the Lorentz transform.

Don't worry, I am sure that this momentary lapse into reconciliation and agreement will pass as soon as I can get back down into more breathable atmosphere.

Mentz114

Mar23-09, 10:09 PM

Dalespam,
you're right. I wave the flag of truce. SR is too important to fall out over.

Before I go I just want to emphasize the bigger picture. What SR shows is that all physical effects ( that is, those agreed on by all observers) must be based on the proper interval, which leads to the requirement that correct physical laws must be covariantly expressed. In curved space-time this is still true and leads to the conclusion that only scalar contractions of tensors can be physical effects ( or do I mean observables ?). This is probably the most important thing so far to happen in physics.

neopolitan

Mar23-09, 11:35 PM

Wise words from DaleSpam and Mentz. I too will rein myself in.

I have privately tried to clarify what my concern is to Mentz. Hopefully that has helped sort things out.

I am aware that these forums, particularly the relativity forum, are constantly under a form of intellectual attack by people who think the whole framework of relativity is wrong somehow. There are certainly a lot of sites championing ideas which seem (at best) to be at odds with relativity and/or the standard cosmological model. I can understand that a siege mentality could result.

But not all questions are intended as attacks. Most, I suspect, are from people on the cusp of understanding SR. As I said to Mentz privately, these people are "intellectually vulnerable". Depending on circumstance, they could have their doubts and concerns addressed and continue on to be happy with SR, or they could feel that they have had their doubts and concerns minimised or ridiculed and end up turning to SR luddites or Bad Astronomers or whatever.

If we can identify what causes this confusion (and I am in the privileged position of having gone through that confusion myself), then perhaps we can identify a method of removing or reducing it.

If the forum hierarchy can clearly show that they understand where the confused student is coming from (after all, surely some of them had to go through a similar period of confusion?), then an appropriate sticky FAQ post could reduce the number of times that DaleSpam and others have to tell a new poster that he or she is mixing frames.

That post, I suggest, could contain the distilled wisdom of Jesse (his diagram and simultaneity considerations), Mentz (perhaps the student should really be using some variation of Lorentz transformations) and Dale (don't forget that one can look at this from a geometric perspective). My wisdom would be limited to the suggestion that it be explained clearly, once and for all, just where the frame mixing is taking place and why you can't use time dilation and length contraction the way it often is (mis)used.

If that saves a few students a year from the clutches of the lunatic fringe, it would surely be worth it?

cheers,

neopolitan

bernhard.rothenstein

Mar24-09, 03:41 AM

Bernhard,
I'm sure you're a dedicated and earnest teacher of the subject, but do beginners have to go into SR as deeply as you enjoy going ?

M
Thanks.
YES if we start with the Lorentz transformations in order to derive the formulas that account for the different relativistic effect.
Kind regards

bernhard.rothenstein

Mar24-09, 04:22 AM

It depends a little on what Lo and To are.

I am tempted to think (using standard pairing, time dilation and length contraction):

T = To * gamma
L = Lo / gamma

so:

To / T = 1 / gamma
L / Lo = 1 / gamma

so:

(To / T)(L / Lo) = 1 /(gamma)2

Which is partly why I question it.

Rearranging (To / T)(L / Lo) = 1 gives you:

(To / Lo)(L / T) = 1

or

L / T = Lo / To

Which I think is an important consequence. In much later posts we are nearing a resolution ... maybe :)

For me that discussion could revolve, conceptually, around what a photon does travelling along between two events (but I stress that it doesn't have to). In one frame, it could be said that that photon travels L in time T (so L/T=c). In another frame, it could be said that that same photon travels Lo or L' in time To or T' (so that Lo/To=c or L'/T'=c).

cheers,

neopolitan
I think we could breath more life into the problem.
Consider an experiment in which an observer of I located at the origin O of its rest frame measures the velocity V of the origin O' of I'. He uses a rod at rest of proper length L0 and measures the coordinate time interval T during which O' covers the distance L0 concluding that
V=L0/T (1)
In a second experiment an observer locared at the origin O' of I' measures the velocity of the measuring rod used in the previous experiment. He measures the the distorted length of the rod L and the proper time interval T0 concluding that
V=L/T0 (2)
obtaining from (1) and (2)
L/T0=L0T.
At that moment we could say that the formula that accounts for the time dilation effect is the function which best fits experimental results, length contraction being a natural consequence of the first postulate: If you move relative to me with velocity V I move relative to you with speed -V.
Quoting you

For me that discussion could revolve, conceptually, around what a photon does travelling along between two events (but I stress that it doesn't have to). In one frame, it could be said that that photon travels L in time T (so L/T=c). In another frame, it could be said that that same photon travels Lo or L' in time To or T' (so that Lo/To=c or L'/T'=c).
I would say that the photon synchronizes the clocks in the two frame leading to transformations in which L/t=L'/t' becase L and t transform via the same Doppler factor.
Kind regards

neopolitan

Mar24-09, 10:04 PM

bernhard.rothenstein

Mar25-09, 01:10 AM

Bernhard,

I think you are giving a variation of the example which JesseM gave a while back, but without specifically taking into account the simultaneity issues that your example requires, because the person taking the measurement has to either be at the origin or at the end of the distance Lo.

Because the simultaneity issues are in there, but unstated, I am not convinced that the new student will emerge unconfused.

I am a little bemused by the idea of teaching relativity via the relativistic doppler equations.
In your last post you referred to:

I am pretty sure that "L and t transform via the same Lorentz factor" would be more accurate, so long as you were talking about appropriately defined L' and t'. I'm not sure what you mean by the photon synchronising the clocks, did you mean "the photon could be used for einstein synchronisation of the clocks"? I think my meaning in the last paragraph of post #110, which should be considered in context of an earlier post #107, might have been misunderstood.

I will wait until Jesse has had a chance to respond to post #107 before trying to get to the heart of what I mean in yet another post.

cheers,

neopolitan
Thanks Neopolitan
1. I have mentioned in a previous thread that the Lorentz contraction could be derived without imposing simultaneous detection of the space coordinates of the ends of the moving rod.
2. Consider that a source of light located at the origin O of I emits a light signal at t=0 in the positive direction of the x axis. After a given time of propagation it generates the event
(x=ct;t=x/c). The same event detected from I' is characterized by the space time coordinates
x'=g(x-Vt)=gx(1-V/c)=Dx
t'=g(t-Vx/cc=gt(1-V/c)=Dt
x/t=x'/t'=c
3. Have a look at
M/ Moriconi, "Special theory of relativity through the Doppler Effect," Eur.J.Phys. 27,1400-1423 (2006)
Kind regards
Bernhard

neopolitan

Mar25-09, 09:52 AM

Thanks Neopolitan
1. I have mentioned in a previous thread that the Lorentz contraction could be derived without imposing simultaneous detection of the space coordinates of the ends of the moving rod.
2. Consider that a source of light located at the origin O of I emits a light signal at t=0 in the positive direction of the x axis. After a given time of propagation it generates the event
(x=ct;t=x/c). The same event detected from I' is characterized by the space time coordinates
x'=g(x-Vt)=gx(1-V/c)=Dx
t'=g(t-Vx/cc=gt(1-V/c)=Dt
x/t=x'/t'=c
3. Have a look at
M/ Moriconi, "Special theory of relativity through the Doppler Effect," Eur.J.Phys. 27,1400-1423 (2006)
Kind regards
Bernhard

Ok, gotcha - I think.

Because x=ct, the standard Lorentz factor multiplied by the Galilean boost (at least spatially) resolves back to the relativistic doppler factor, where

D = \sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}

Correct?

The bit I would need to think about very carefully is the underlying assumption in the Lorentz Transformation. Do we have hidden assumptions somewhere which are incompatible?

My gut feeling is that there might be. Specifically, whenever I have thought about a Lorentz transformation and indeed a Galilean boost, it has been about an event which is colocated with neither the origin of I nor the origin of I', but considered by both to have been simultaneous with the event characterised by the colocation of origins. In other words, when the origins of I and I' were colocated, then xo=x'o=0 and to=t'o=0. Later, an event is detected at the origin of I and that same event is detected at the origin of I' (not simultaneously, one photon from the event will reach one, and then another photon from the event will reach the other).

I think I might have to go into this in more detail. But at the moment, I don't quite have enough time to give it justice.

cheers,

neopolitan

bernhard.rothenstein

Mar25-09, 01:42 PM

Ok, gotcha - I think.

Because x=ct, the standard Lorentz factor multiplied by the Galilean boost (at least spatially) resolves back to the relativistic doppler factor, where

D = \sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}

Correct?

The bit I would need to think about very carefully is the underlying assumption in the Lorentz Transformation. Do we have hidden assumptions somewhere which are incompatible?

My gut feeling is that there might be. Specifically, whenever I have thought about a Lorentz transformation and indeed a Galilean boost, it has been about an event which is colocated with neither the origin of I nor the origin of I', but considered by both to have been simultaneous with the event characterised by the colocation of origins. In other words, when the origins of I and I' were colocated, then xo=x'o=0 and to=t'o=0. Later, an event is detected at the origin of I and that same event is detected at the origin of I' (not simultaneously, one photon from the event will reach one, and then another photon from the event will reach the other).

I think I might have to go into this in more detail. But at the moment, I don't quite have enough time to give it justice.

cheers,

neopolitan
That is one of the papers in which length contraction is derived without simultaneous detection of the space coordinates of the ends of the moving rod
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How to obtain the Lorentz space contraction formula for a moving rod from knowledge of the positions of its ends at different times

M Fernández Guasti et al 2009 Eur. J. Phys. 30 253-258 doi: 10.1088/0143-0807/30/2/003

In what concerns the transformation via the Doppler factor holds only in the case when I and I' are in the standard configuration and the light signal is emitted at t=t'=0 when the origins of the two frames are overlapped. Under such conditions
x'-0=D(x-0)
t'-0=D(t-0)
If the signal is emitted at a time t different from zero the straight line in I is no longer a straight line in I'.
Kind regards

JesseM

Apr8-09, 02:26 PM

If you want to use the clock in the laboratory you as your reference point, you have to do this:

While a photon in the laboratory moves between mirrors, travelling 660km in 2.2ms - what happens to a photon which is at gamma of 29.3?

If 2.2ms has elapsed in the laboratory, then a period of 2.2ms/29.3 = 75μs will have elapsed in the rest frame of the test clock (the one accelerated to gamma of 29.3).
No, 75 microseconds would represent how much time has elapsed on the test clock (if the test clock had closer mirrors so it could show time-intervals that small) in 2.2 ms of time in the lab frame. In the test clock's own frame, it's the lab clock that's running slow relative to the test clock, so when the lab clock has ticked forward 2.2 ms, the test clock has ticked forward 64.4 ms.
My fault. I was not clear about photons. It took a moment to see where you didn't agree since you seemed to be saying exactly the same as I said in my quote.

The light clock in the test frame is at gamma of 29.3 (it makes no sense to talk about a photon at gamma of 29.3).

Thinking about the light clock in the test frame, while 2.2ms has elapsed in the laboratory (one full in-laboratory tick-tick), the photon has travelled 1/29.3 of the distance it needs to travel for the clock to go through a full tick to tick sequence, which, according the laboratory, is 660km*29.3. According to the laboratory, the photon in the test frame's clock has travelled 660km in 2.2ms. According to the laboratory, the photon in the laboratory frame's clock has travelled 660km in 2.2ms. According to the laboratory, both photons have travelled 600km in 2.2ms.
Yes, all that makes sense. In the laboratory frame the photon in the lab's own light clock traveled vertically 660 km, while the photon in the moving "test" light clock traveled 660 km on a diagonal whose vertical component is only 660 km/29.3, and whose horizontal component is 2.2 ms times whatever speed the light clock is moving horizontally (the speed that gives a gamma of 29.3, which works out to 0.9994174c).
According to the test frame, what the laboratory frame "thinks" is 660km is actually 660km/29.3 and what the laboratory frame "thinks" is 2.2ms is actually 2.2ms/29.3.
Yes, although we should keep in mind that the last part only works if you're talking about a 2.2 ms time between two events which are located at the same horizontal position in the test frame, like the two events on the worldline of the photon in the test frame's own light clock. If the laboratory frame "thinks" there is a 2.2 ms period between two events which do not occur at the same horizontal position in the test frame, then the time between these same two events in the test frame will not be 2.2 ms/29.3.

(Aside: You can go through the last two paragraphs and swap the words "test" and "laboratory". The arguments would be the same. To reconcile the different views, you have to use relativity of simultaneity concepts. You shouldn't necessarily forget this next step, but at the moment, it is not necessary.)

If you want to call L'/t' LAFTD/time dilation that is fine. I do see here that that makes sense. But I also see that L'/t' length contraction/TAFLC makes equal sense. (Note that above I have not defined any primed frame or any unprimed frame.)

(660km * 29.3) / (2.2ms * 29.3) = (660km) / (2.2ms) = (660km / 29.3) / (2.2ms / 29.3) = 300000 km/s
But I don't think calling it (length contraction)/TAFLC makes sense, not unless you can justify it physically in terms of what events you're actually supposed to be measuring the distance and time between. For instance, consider your equation (660km / 29.3) / (2.2ms / 29.3). From the previous discussion, it seems this distance and time are meant to be between the following two events: 1) the event of the photon bouncing off the bottom of the test clock, and 2) the event on the photon's worldline that occurs 2.2 ms after it hits the bottom of the test clock as measured in the lab frame. In the lab frame, the spatial separation between events 1 and 2 is 660 km. Now, it's true that in the test clock's own frame, the spatial separation between these same events 1 and 2 is only (660 km / 29.3), and the time between events 1 and 2 is only (2.2 ms / 29.3). But the spatial separation here is not really obtained by either the length contraction equation (since 660 km and 660 km/29.3 don't represent the length of a single object in two different frames) or by the spatial analogue for time dilation (since we're looking at a single pair of events that are not simultaneous in either frame, whereas the SAFTD assumes the events are simultaneous in one of the two frames). Instead, the fact that the distance in one frame is equal to the distance in the other frame divided by 29.3 is really just a consequence of the fact that the two events are on the path of a photon, which must move at the same speed in both frames, and since the time between events in one frame is equal to the time in the other frame divided by 29.3, the equal speeds in both frames imply that the same must be true for the distance.

As for the fact that the time between events 1 and 2 in the test frame is equal to the time between these events in the lab frame divided by 29.3, I would say that this is obtained via the "reversed time dilation equation" where you've divided both sides of the regular time dilation equation by gamma. If the usual time dilation equation can be written in words as (time between events in frame where they're not colocated at same horizontal position) = (time between events in frame where they are colocated at same horizontal position) * gamma, then you're just dividing both sides by gamma to get (time between events in frame where they are colocated at same horizontal position) = (time between events in frame where they're not colocated at same horizontal position) / gamma. Here we know that in the lab frame where events 1 and 2 are not at the same horizontal position, the time between them is 2.2 ms, so we're dividing by gamma = 29.3 to get the time between them in the test frame where they are colocated at the same horizontal position.

On the other hand, consider your equation (660km * 29.3) / (2.2ms * 29.3). From your previous discussion, here I would imagine you are considering a different pair of events: 1b) the event of the photon hitting the bottom of the test clock, and 2b) the event of the photon hitting the top of test clock. In the test clock's own frame the spatial distance between these events is 660 km and the time between them is 2.2 ms. In the lab frame, though, the distance is 660 km * 29.3 and the time between them is 2.2 ms * 29.3. But here again the distance relation cannot be said to be an example of either length contraction or the SAFTD, but is just a consequence of the relation between the times of the events and the fact that they both lie on the worldline of a photon which moves at the same speed in both frames. As for the time relation, this is just the normal time dilation equation of the form (time between events in frame where they're not colocated at same horizontal position) = (time between events in frame where they are colocated at same horizontal position) * gamma. Here, the two events are colocated at the same horizontal position in the test frame, and the time between them in that frame is 2.2 ms; multiply this by gamma=29.3 and you get the time between the same two events in the lab frame where they are not colocated at the same horizontal position.

So, it's really important to state specifically what each of the distances and times in your equations are actually supposed to represent physically before you can decide what rules define the relation between them; as seen above, the mere fact that you're taking a time and dividing by gamma doesn't show that you're using the TAFLC, since you also do this in the reversed time dilation equation, but the two are conceptually different since if you pick a given pair of events, the reversed time dilation equation refers to the following:

(time between events in frame where they are colocated at same position along axis of motion between frames) = (time between events in frame where they are not colocated at same position along axis of motion between frames) / gamma

Whereas the TAFLC refers to the following:

(time in the non-colocated frame between two surfaces of simultaneity from the colocated frame which pass through the two events) = (time in the colocated frame between two surfaces of simultaneity from the colocated frame which pass through the two events--or equivalently, time between the events themselves in the colocated frame) / gamma

What's more, as I hadn't noticed until I thought about this problem specifically, the fact that a distance in one frame is equal to a distance in another frame multiplied or divided by gamma does not mean you are making use of either the length contraction equation or the the SAFTD equation, since in your example we were talking about the distance between a specific pair of events in two frames (so it wasn't length contraction), but the events were not simultaneous in either frame (so it wasn't the SAFTD equation).
I prefer keeping in mind that lengths which are not at rest with respect to my rest frame are contracted. So I do prefer "length contraction/TAFCL" (or if you must, you can call it "length contraction/inverse time dilation" but I don't like it, because I interpret time dilation as talking about what happens between two full ticks, not about measured time, eg numbers of ticks or number of graduations between ticks).
It's not really a matter of choice, "inverse time dilation" and "TAFCL" refer to different ideas, as my word-summary above tries to show. What's more I think you are getting yourself confused by thinking in terms of discrete "ticks" of a clock rather than continuous coordinate time--the time dilation equation is just about the coordinate time between an arbitrary pair of events in a frame where they're colocated as compared to the coordinate time between the same pair of events in a frame where they're not colocated, there's no need to consider the two events to be ticks of a single clock at rest in the frame where they're colocated, and even if you do want to think of it that way, there's no need to consider them consecutive ticks as opposed to, say, two ticks at different times which have 10,000 ticks between them. And if you think in this way, the "reversed time dilation equation" says that if you know the amount of coordinate time in frame X between some arbitrary pair of events on a clock that's moving in frame X, and you want to know how many ticks there were between these events as measured by the clock itself, then you take the original coordinate time in frame X and divide by gamma.
You might prefer to think about the fact that compared to your clock, the period between ticks of a clock in motion with respect to you is longer. (Or whatever physical definition you ascribe to time dilation, the point is that you may prefer to keep the time dilation equation whereas I prefer to keep the length contraction equation.)
Again, I don't think there's any matter of preference here--if you specify exactly what your numbers represent physically in terms of actual events or objects, then I think it's always clear what equation you're using implicitly.
There is subtle difference in approaches which might be illustrative to highlight. You are focussed very much on the relativity (which is the bit I coloured silver above, so you have to select it to read it).
I don't see how I am, perhaps you're misunderstanding me somehow. All of my above analysis is about events on the worldline of the photon in the test clock, and the distances/times between these events in both the frame of the lab and the frame of the test clock; while it's true that everything would be reversed if you instead considered events on the worldline of the photon in the lab clock as seen from both frames, that's something I haven't even been discussing.
Relativity says two things:

Something that is in motion relative to me will be length contracted and experience less time than me, relative to me.

and

The reverse is true, relative to that something.

I am really only looking at the first part, because I know the second part is true, but not terribly useful for working out the extent of that contraction and reduction of time experienced.
As I said, the distances in your equations don't actually correspond to the length of a single object in two different frames at all. And if by "something that is in motion relative to me will ... experience less time than me, relative to me" you mean "events which occur at the same position (or same horizontal position) in the frame of the 'something' in motion relative to me will have less of a time-separation in their frame than they do in my frame", then that's exactly what I was talking about in my analysis, since in both cases I was dealing with two events located at the same horizontal position in the test clock's frame, and comparing the time between them in the test clock's frame with the time between them in the lab frame (with the time always being larger in the lab frame). Never was I looking at the "reverse", if by that you mean events located at the same horizontal position in the lab frame.

neopolitan

Apr9-09, 07:38 AM

neopolitan

Apr9-09, 08:35 AM

DrGreg

Apr9-09, 09:24 AM

The observers are advised that the events were either collocated but not simultaneous or simultaneous but not collocated (assume they were told that the events happened "together" - a vague term which could mean either - and they were told in such a way that they could reasonably assume that the "togetherness" related to their own inertial frame).If both observers are told this, at least one of them is being lied to (assuming v is not zero). If we don't know which one, then there's insufficient information to answer your question.

Suppose further that the other observer (B) is moving towards the events (according to A). You can't move relative to an event. You can move towards an object that "passes through" the event. I guess you mean moving towards the location of the event in A's frame.

neopolitan

Apr9-09, 11:32 AM

If both observers are told this, at least one of them is being lied to (assuming v is not zero). If we don't know which one, then there's insufficient information to answer your question.

Suppose further that the other observer (B) is moving towards the events (according to A). You can't move relative to an event. You can move towards an object that "passes through" the event. I guess you mean moving towards the location of the event in A's frame.

I did say that the experimental controllers might have been lying altogether, they could be lying to both, it doesn't really matter. I disagree that there is insufficient information, you are just looking at it the wrong way.

Perhaps a geometric sort of person can provide an answer (I imagine that DaleSpam could probably do it).

What did "according to A" mean to you? To me it was just shorthand for saying that B is between A and where the events took place. Since A is at rest in A's frame, then in A's frame, B is moving towards where the events were (I didn't said "relative to" in this context). But anyway, yes, "moving towards the location of the event in A's frame" is right too.

With this clarified, do you have enough information?

cheers,

neopolitan

DrGreg

Apr9-09, 12:33 PM

With this clarified, do you have enough information?
No. If I don't know who's being told the truth, then I know nothing.

DrGreg

Apr9-09, 12:56 PM

Rethink.

Are A and B capable of measuring things for themselves, or do they have to rely on what they are told? If they can measure, they don't need to be told anything, they can work it out for themselves.

If A measures a length of LA and a time interval of tA, then B will measure a length and time given by

L_B = \gamma(L_A-vt_A)
t_B = \gamma(t_A - vL_A/c^2)

Furthermore, A can calculate Q = L_A^2 - c^2t_A^2 and B can calculate Q = L_B^2 - c^2t_B^2. They will both get the same answer.

If that answer is positive then there is a frame in which the events occurred simultaneously at a distance of \sqrt{Q} apart. That frame might be A, B or some other frame.

If that answer is negative then there is a frame in which the events occurred at the same place separated by a time interval of \sqrt{-Q}/c. That frame might be A, B or some other frame.

If the answer is zero, then neither statement is true in any frame.

I don't think that was the answer you were looking for, but that's the answer based on my understanding of the problem you posed.

JesseM

Apr9-09, 02:41 PM

If you don't like:

length / time = contracted length / TAFLC = SAFTD / dilated time

would you accept:

length / time = contracted length / "inverse dilated" time = "inverse contracted" length / dilated time
The problem is, length and time of what? It really is necessary to at least outline in broad terms what is supposed to be measured physically (even if it's something vague like 'the time between two events with a lightlike separation in two frames, one of which the events have the same horizontal position in'). As I pointed out in my last post, in the example with the light clocks if you pick two events on the worldline of the photon in the test clock as I did, then although it's true the distance between them will involve a gamma factor, this won't be because you were measuring the "length" of any single object in two frames, and it also won't be because you were using the SAFTD since the events were not simultaneous in either frame; rather, it's just because the distance between the events in each frame is just equal to c*(time between events in that frame), and the time between events was related by "inverse dilated time". So what you really have is something more like:

c*(time)/time = c*(inverse dilated time)/(inverse dilated time)

JesseM

Apr9-09, 03:00 PM

How about this:

You have two observers, not at rest with respect to each other (they separate at v).

Two events are observed, at a distance along the axis defined by the separation of the observers.

The observers are advised that the events were either collocated but not simultaneous or simultaneous but not collocated (assume they were told that the events happened "together" - a vague term which could mean either - and they were told in such a way that they could reasonably assume that the "togetherness" related to their own inertial frame).
"Together" in just one of the two frames, or in both? Assuming the events were at different points in spacetime, then if they were simultaneous in one frame, then they'd have a spacelike separation so they can't be colocated in either frame, and the only way they could be simultaneous in the other frame is if they occurred at the same x-coordinate in the first frame and the same x' coordinate in the second frame (x and x' defined as the axes on which the two observers are moving relative to one another, as in the usual way of writing the Lorentz transformation). On the other hand, if they were colocated in one frame then they can't be simultaneous or colocated in the other frame.
Suppose that one observer (A) notices that the separation, if spatial, matches that of a rod in possession (length L) or, if temporal, is a period of t.

Suppose further that the other observer (B) is moving towards the events (according to A).

If A and B have identical rods and identical clocks, what sort of conclusions will B come to? Results in terms of L and t would be appreciated :smile:
It seems pointless to leave it a mystery whether the events are colocated or simultaneous (and in whose frame this is true), since aside from using DrGreg's most general possible answer from post #137 in terms of the Lorentz transformation equations, the only way I can think of to answer this question is to break it down into different possibilities like:

1. Events are colocated in frame A, neither colocated nor simultaneous in B
2. Events are colocated in frame B, neither colocated nor simultaneous in A
3. Events are simultaneous in A, also simultaneous in B because both occur at same x-coordinate in A and same x'-coordinate in B
4. Events are simultaneous in A, neither colocated nor simultaneous in B
5. Events are simultaneous in B, neither colocated nor simultaneous in A

...and then answer what conclusions B would reach in each of the 5 cases. So, could you just specify which of these 5 cases applies? If not, can you explain why you want it to be mysterious?

neopolitan

Apr10-09, 05:19 AM

.... the only way I can think of to answer this question is to break it down into different possibilities like:

1. Events are colocated in frame A, neither colocated nor simultaneous in B
2. Events are colocated in frame B, neither colocated nor simultaneous in A
3. Events are simultaneous in A, also simultaneous in B because both occur at same x-coordinate in A and same x'-coordinate in B
4. Events are simultaneous in A, neither colocated nor simultaneous in B
5. Events are simultaneous in B, neither colocated nor simultaneous in A

...and then answer what conclusions B would reach in each of the 5 cases. So, could you just specify which of these 5 cases applies? If not, can you explain why you want it to be mysterious?

None of the cases.

Both A and B have been told the events happened "together".

Both A and B receive photons from the events with a temporal delay (since they both consider themselves to be at rest, there is no spatial component related to where they receive the photons).

From that they work out that the events have either a spatial "togetherness" and a temporal separation or a temporal "togetherness" and a spatial separation.

"Truth" or "reality" about the timing and locations of events is inconsequential.

I didn't ask for reality, I asked about "what sort of conclusions will B come to?"

cheers,

neopolitan

DrGreg

Apr10-09, 06:25 AM

...the only way I can think of to answer this question is to break it down into different possibilities like:

1. Events are colocated in frame A, neither colocated nor simultaneous in B
2. Events are colocated in frame B, neither colocated nor simultaneous in A
3. Events are simultaneous in A, also simultaneous in B because both occur at same x-coordinate in A and same x'-coordinate in B
4. Events are simultaneous in A, neither colocated nor simultaneous in B
5. Events are simultaneous in B, neither colocated nor simultaneous in A

...and then answer what conclusions B would reach in each of the 5 cases. So, could you just specify which of these 5 cases applies? If not, can you explain why you want it to be mysterious?
None of the cases.

Both A and B have been told the events happened "together".

Both A and B receive photons from the events with a temporal delay (since they both consider themselves to be at rest, there is no spatial component related to where they receive the photons).

From that they work out that the events have either a spatial "togetherness" and a temporal separation or a temporal "togetherness" and a spatial separation.

"Truth" or "reality" about the timing and locations of events is inconsequential.

I didn't ask for reality, I asked about "what sort of conclusions will B come to?"

cheers,

neopolitan
I really don't understand what you are talking about here. If none of the cases 1 to 5 are true, then both A and B are being lied to. The only conclusion that both A and B can come to is that they are being lied to and the experiment is a waste of time.

JesseM

Apr10-09, 06:25 AM

None of the cases.

Both A and B have been told the events happened "together".

Both A and B receive photons from the events with a temporal delay (since they both consider themselves to be at rest, there is no spatial component related to where they receive the photons).
I don't understand the phrase "no spatial component related to where they receive the photons", can you explain what you mean? What is a "spatial component", and what does it have to do with whether or not they consider themselves at rest?
From that they work out that the events have either a spatial "togetherness" and a temporal separation or a temporal "togetherness" and a spatial separation.
What method would they use to "work out" that this is true? Do the photons they receive from the events help them work this out, or are they working it out solely based on what they were told? Also, are "spatial togetherness" and "temporal togetherness" just shorthand for "occur at the same position" and "occur at the same time"? If so, do you agree that these are meaningless without reference to a specific frame--that, for example, if two events occur at the same position in one frame, that means the two events occur at different positions in a frame moving relative to the first?
"Truth" or "reality" about the timing and locations of events is inconsequential.

I didn't ask for reality, I asked about "what sort of conclusions will B come to?"
I don't understand the question. To the extent that you can make "conclusions" about a physical scenario where certain things are uncertain (like not knowing whether events are 'together in space' or 'together in time', but knowing one of the two must be true), it's only by listing various possibilities (like the 5 possibilities I mentioned) that are consistent with your knowledge and saying what you would conclude in each possible circumstance, and perhaps also by finding some broad conclusions that would hold in every possible case (like the equations DrGreg provided). Do you think any other types of conclusions can be made about a situation where all the physical details are not known? If so, please give a specific example of a situation (which can be completely unrelated to this one) where conclusions are drawn from partial information to help me understand what type of conclusions you're thinking about, giving both what is known and unknown in the example, and what specific physical conclusions you would draw in this example.

DrGreg

Apr10-09, 08:37 AM

neopolitan

I've re-read your post and let me speculate what you mean.

I suggest you are saying that each observer receives photons from the two events and measures the time interval between those receptions. If one of them knows the events were colocated in his frame, then that time is also the time interval between the events themselves. If one of them knows the events were simultaneous in her frame, then that time gives the distance between the events, after multiplying by c. Am I interpreting you correctly?

This still doesn't solve the problem. If you told us that A was being told the truth (and therefore B was being lied to), we could then answer the question as to what B would calculate based on the false assumption. Is that what you want us to do?

Note that "observers" in relativity don't, in general, actually measure distances and times in this way, because there won't be an external authority to tell you that events are colocated or simultaneous.

One way an oberver can assign a time and a position to an event is by radar. The observer sends out a radar pulse, at time t1, which is reflected from an object and an echoed pulse is received by the observer, at time t2.

The observer then assigns time and distance coordinates to the reflection event by the formulas

t = \frac {t_2 + t_1}{2}
x = c \frac {t_2 - t_1}{2}

Time intervals and distances between pairs of events are then calculated by subtraction.

neopolitan

Apr10-09, 09:01 AM

neopolitan

Apr10-09, 09:08 AM

neopolitan

I've re-read your post and let me speculate what you mean.

I suggest you are saying that each observer receives photons from the two events and measures the time interval between those receptions. If one of them knows the events were colocated in his frame, then that time is also the time interval between the events themselves. If one of them knows the events were simultaneous in her frame, then that time gives the distance between the events, after multiplying by c. Am I interpreting you correctly?

No. I'm being strict here about the term "know", neither know anything, they are told something and make calculations on that basis. But yes, that is how they would make their calculations.

This still doesn't solve the problem. If you told us that A was being told the truth (and therefore B was being lied to), we could then answer the question as to what B would calculate based on the false assumption. Is that what you want us to do?

Sort of. But I don't care who was being lied to. Two wavefronts of photons pass each observer. They've been led to believe they originated simultaneously, in their own frame, or from the same location, in their own frame.

Both could be working on a false assumption.

Note that "observers" in relativity don't, in general, actually measure distances and times in this way, because there won't be an external authority to tell you that events are colocated or simultaneous.

Exactly, but here they have someone implying that the events were simultaneous or collocated.

cheers,

neopolitan

DrGreg

Apr10-09, 10:32 AM

This is beginning to make some sense now.

OK. A (Alice) receives signals from the two events at a time tA apart by her clock. B (Bob) receives signals from the two events at a time tB apart by his clock. The two are related by the doppler factor

t_B = t_A \sqrt{\frac{c-v}{c+v}}

If either A or B believes the events were colocated in their own frame they will believe the time they measured to be the time between the events themselves.

If either A or B believes the events were simultaneous in their own frame they will believe the time they measured to be, after multiplication by c, the distance between the events themselves.

However in your original post #133 you referred to a rod in A's possession. It's not clear how your method of measurement involves the rod. So far I've needed only two clocks and v and c.

JesseM

Apr10-09, 12:30 PM

It's quite simple really.

A and B are told that two distant events happen "together", vaguely enough to be able to think that they happen together for themselves (in their own frame) but not be sure whether the events happen together spatially (collocated) or temporally (simultaneous).

Photons pass them from the events with a delay. A and B are not at rest with respect to each other, and B is between A and where the events take (or took) place. A and B separate with a speed of v.

A notes that, if indeed the events were together in A's frame, then they either happened a period of t apart, or a distance of L apart.

B will note that, if indeed the events were together in B's frame, then they either happened a period of ??? apart, or a distance of ??? apart.
OK, suppose A receives photons from the first event at t=10 and t=26. So you're saying that A will conclude that that if they were simultaneous they were 16 light-seconds apart (note that this is true only if A is also told that simultaneous events occur on different points on an axis that also crosses through A, if the axis between events does not cross through A then the distance could be different). This case would correspond to possibility #4 from from my post #139: "Events are simultaneous in A, neither colocated nor simultaneous in B". On the other hand, with these numbers A will also say that if they were colocated they must have happened 16 seconds apart, and this case corresponds to possibility #1 from post #139: "Events are colocated in frame A, neither colocated nor simultaneous in B".

Now suppose that in A's frame, B is heading towards the position(s) of the events at 0.6c. This means that if the events were actually simultaneous in A's frame (possibility #4 again), then in A's frame the farther event happens 16 light-seconds further away from B than the first, and since B is traveling towards the light from the farther event after seeing the closer event the time between the light from the farther event will hit B at 16 light-seconds/(1c + 0.6c) = 10 seconds after the light from the closer event hits him, which means according to B's own clock the light from each event is received 8 seconds apart due to the time dilation of B's clock in A's frame. On the other hand, if the events were actually colocated in A's (possibility #1), and B is headed towards the position of the events, then B will have traveled 16*0.6 = 9.6 light-seconds between the time of the two events in A's frame, so the light from the first event took an extra time of 9.6 light-seconds/(1c + 0.6c) = 6 seconds to reach B, meaning in A's frame the time between the light from each event hitting B is 16 - 6 = 10 seconds again, so the time on B's clock is 8 seconds again.

So since both possibility #1 and possibility #4 imply that B receives light from the two events 8 seconds apart, we can use this number but now suppose the events were simultaneous in B's frame and occurred along an axis that also crosses B (possibility #5 from post #139, 'Events are simultaneous in B, neither colocated nor simultaneous in A'), in which case they must have been 8 light-seconds apart in B's frame, or else the events might have been colocated in B's frame (possibility #2 from post #139, 'Events are colocated in frame B, neither colocated nor simultaneous in A') in which case they must have happened 8 seconds apart in B's frame. By the way, note that things work in reverse too; if we assume either of these possibilities is true in B's frame, where A is moving away from the events at 0.6c, we will conclude that the light from each event will hit A 20 seconds apart, so that due to time dilation A's own clock will show an interval of 16 seconds between being hit by light from each event. Also note that B's time (8 seconds) is half that of A's time (16 seconds), which fits with DrGreg's formula above if you plug in v=0.6c.

Is this the sort of thing you were asking for? If so, note that you do solve it by breaking it down into one of the five possibilities from post #139 (I didn't consider #3 because if we assume the events are simultaneous in one frame but the spatial axis between them does not contain the positions of A and B when they saw the light from each event, then it becomes impossible for A or B to say anything definite about the distance between the events in their own frame, and they will also see the light from the events coming from different angles so they'll know they weren't colocated). And if this is not what you were asking for, could you give us a numerical example and tell us what answers you would give to these problems?

It's not impossible for this to be true, if the events were simultaneous in one frame and collocated in the other - however, it would mean there is a limitation on what value v has. ( I retract this, since I suspect that the limitation on v would be v = c.)
I realize you've retracted this, but just to elaborate, it's impossible for events to be simultaneous in one frame and colocated in another (and keep in mind that only objects moving at sublight speeds have inertial rest frames, you can't have two inertial frames with a relative velocity of c). In any frame, if you calculate -c^2*\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 between two events, you get an invariant quantity, meaning if another frame calculates the same quantity for these two events using its own coordinates, although the other frame's values for the individual parts like \Delta t and \Delta x may be different, the value of the equation as a whole will be the same as in the first frame. If the value is negative, the events are said to have a "timelike separation", which implies one of the two events lies in the other event's future light cone; events which are colocated in some frame necessarily have a timelike separation. On the other hand, if the value of the equation is positive, the events are said to have a "spacelike separation", which means neither event lies in the past or future light cone of the other one; events which are simultaneous in some frame necessarily have a spacelike separation. And again, since this is an invariant quantity, it's impossible that two events could have a timelike separation in one frame but a spacelike separation in another.
You do have enough information to provide an answer.

Psychologically, it is fascinating to note that you don't seem to be able to see that, but this observation was not what I was after.
Well, it seems to me you gave a wrong or misleading answer in response to my post #139 when you said that none of the 5 cases applied; my above analysis was based on figuring out what would be true in each of the different cases, I don't see how you could do it otherwise. Perhaps you just meant that we don't have to select any single case as being the correct one since the two observers don't know which of the cases holds, but I think I made it clear in post #139 that I was talking about considering each case in turn as a possibility rather than picking one as the truth, I did say "the only way I can think of to answer this question is to break it down into different possibilities ... and then answer what conclusions B would reach in each of the 5 cases."
For Jesse, since A and B can both consider themselves to be at rest, then the photons which pass by them are at the same location when they pass by (in the relevant frame). Therefore, no spatial component, just a temporal delay.
So you're just noting that in an observer's own rest frame, the events of the light from each event hitting them happen at the same spatial position? This is tautologically true, there's no way it could be otherwise, so I was confused since your comment about "no spatial component" seemed to be saying something about the assumptions we should make in this problem, as opposed to something you'd assume in every possible SR problem (like 'light should be assumed to move at c').

neopolitan

Apr10-09, 08:50 PM

We seem to be getting somewhere. First to answer a question DrGreg posed, what about the rod? I asked for results to be given in terms of the length of the rod, L, and the time between the arrivals of photons, t - both in A's frame.

So, in JesseM's numerical attempt, t = 16 and L = 16 ls

OK, suppose A receives photons from the first event at t=10 and t=26. So you're saying that A will conclude that that if they were simultaneous they were 16 light-seconds apart (note that this is true only if A is also told that simultaneous events occur on different points on an axis that also crosses through A, if the axis between events does not cross through A then the distance could be different). This case would correspond to possibility #4 from from my post #139: "Events are simultaneous in A, neither colocated nor simultaneous in B". On the other hand, with these numbers A will also say that if they were colocated they must have happened 16 seconds apart, and this case corresponds to possibility #1 from post #139: "Events are colocated in frame A, neither colocated nor simultaneous in B".

Now suppose that in A's frame, B is heading towards the position(s) of the events at 0.6c. This means that if the events were actually simultaneous in A's frame (possibility #4 again), then in A's frame the farther event happens 16 light-seconds further away from B than the first, and since B is traveling towards the light from the farther event after seeing the closer event the time between the light from the farther event will hit B at 16 light-seconds/(1c + 0.6c) = 10 seconds after the light from the closer event hits him, which means according to B's own clock the light from each event is received 8 seconds apart due to the time dilation of B's clock in A's frame. On the other hand, if the events were actually colocated in A's (possibility #1), and B is headed towards the position of the events, then B will have traveled 16*0.6 = 9.6 light-seconds between the time of the two events in A's frame, so the light from the first event took an extra time of 9.6 light-seconds/(1c + 0.6c) = 6 seconds to reach B, meaning in A's frame the time between the light from each event hitting B is 16 - 6 = 10 seconds again, so the time on B's clock is 8 seconds again.

So since both possibility #1 and possibility #4 imply that B receives light from the two events 8 seconds apart, we can use this number but now suppose the events were simultaneous in B's frame and occurred along an axis that also crosses B (possibility #5 from post #139, 'Events are simultaneous in B, neither colocated nor simultaneous in A'), in which case they must have been 8 light-seconds apart in B's frame, or else the events might have been colocated in B's frame (possibility #2 from post #139, 'Events are colocated in frame B, neither colocated nor simultaneous in A') in which case they must have happened 8 seconds apart in B's frame. By the way, note that things work in reverse too; if we assume either of these possibilities is true in B's frame, where A is moving away from the events at 0.6c, we will conclude that the light from each event will hit A 20 seconds apart, so that due to time dilation A's own clock will show an interval of 16 seconds between being hit by light from each event. Also note that B's time (8 seconds) is half that of A's time (16 seconds), which fits with DrGreg's formula above if you plug in v=0.6c.

Is this the sort of thing you were asking for? If so, note that you do solve it by breaking it down into one of the five possibilities from post #139 (I didn't consider #3 because if we assume the events are simultaneous in one frame but the spatial axis between them does not contain the positions of A and B when they saw the light from each event, then it becomes impossible for A or B to say anything definite about the distance between the events in their own frame, and they will also see the light from the events coming from different angles so they'll know they weren't colocated). And if this is not what you were asking for, could you give us a numerical example and tell us what answers you would give to these problems?

Why not use Lorentz transformations for each of the events (in two stages, one assuming collocation, the other assuming simultaneity) and subtract the difference? (I hasten to point out that I do know the answer.)

I don't I need to show you how to do that.

Rather than wait for the answer to my rhetorical question (rhetoric as a discussion technique, not rhetorical as in not requiring an answer), I want to highlight your sentence here:

which means according to B's own clock the light from each event is received 8 seconds apart due to the time dilation of B's clock in A's frame.

This is the physical meaning of TAFLC. In so much as the length is contracted, so too is the time. That might be inverse time dilation to you, but what I see happening here is the time in B's frame is contracted.

I have hereby answered a question JesseM asked quite a while ago.

Well, it seems to me you gave a wrong or misleading answer in response to my post #139 when you said that none of the 5 cases applied; my above analysis was based on figuring out what would be true in each of the different cases, I don't see how you could do it otherwise. Perhaps you just meant that we don't have to select any single case as being the correct one since the two observers don't know which of the cases holds, but I think I made it clear in post #139 that I was talking about considering each case in turn as a possibility rather than picking one as the truth, I did say "the only way I can think of to answer this question is to break it down into different possibilities ... and then answer what conclusions B would reach in each of the 5 cases."

I thought you were trying to pick a truth. If you were trying to pick possibilities, then the answer would be "all of them". They are all possibilities. All A and B get are two photons (or wavefronts of photons, expanding in sphere from the events) with a temporal delay. They aren't really told where the photons came from so there was no "truth" to be selected.

I don't see that as five cases, I see that as one case (two photons, one temporal delay. one vague piece of advice). But I do see what you are getting at.

So you're just noting that in an observer's own rest frame, the events of the light from each event hitting them happen at the same spatial position? This is tautologically true, there's no way it could be otherwise, so I was confused since your comment about "no spatial component" seemed to be saying something about the assumptions we should make in this problem, as opposed to something you'd assume in every possible SR problem (like 'light should be assumed to move at c').

I've had to explain other things which I thought were tautological. And when I haven't, I have on occasion been berated.

cheers,

neopolitan

DrGreg

Apr10-09, 09:14 PM

JesseM

Apr10-09, 10:30 PM

Why not use Lorentz transformations for each of the events (in two stages, one assuming collocation, the other assuming simultaneity) and subtract the difference? (I hasten to point out that I do know the answer.)

I don't I need to show you how to do that.
But I already calculated that, the answer is that B will receive the light from each event 8 seconds apart according to his own clock. The result would be the same if I used the full Lorentz transform. Do you think that answer was incorrect?
Rather than wait for the answer to my rhetorical question (rhetoric as a discussion technique, not rhetorical as in not requiring an answer), I want to highlight your sentence here:
which means according to B's own clock the light from each event is received 8 seconds apart due to the time dilation of B's clock in A's frame.
This is the physical meaning of TAFLC. In so much as the length is contracted, so too is the time. That might be inverse time dilation to you, but what I see happening here is the time in B's frame is contracted.

I have hereby answered a question JesseM asked quite a while ago.
I don't think that makes sense at all. The TAFLC equation was t' = t / gamma, and here gamma = 1.25. If t' and t refer to the time between each observer seeing the events (which would be the same as what they'd conclude was the coordinate time between the events if they each postulated the events were colocated in their own frame, although they couldn't actually both be correct in postulating this), then the two times would be 16 and 8, and of course 8 is 16 divided by 2, not 16 divided by 1.25. I suppose the other alternative is that you want me to assume the events really were colocated in one frame but not the other; in this case, if they were colocated in B's frame and the time between the events was t=8 seconds in that frame, then although A would still see them 16 seconds apart, they would really have happened t'=10 seconds apart in A's frame. But if we're assuming they were really colocated in one frame and not the other, then was the whole business about their not being enough information and each observer assuming they were "together" in his own frame totally pointless?

What's more, although it's true that in this case (where the events are assumed colocated in one frame but not the other) the factor between the two times is gamma=1.25, this is just the standard time dilation equation, not the TAFLC. Remember that if you write the time dilation equation as t' = t * gamma, then the idea is that the events are colocated in the unprimed frame in this equation (and if you want the events to be colocated in the primed frame, then the standard time dilation equation would be written as t = t' * gamma). From the above calculations, you can see I assumed that the interval in the frame where they were colocated was 8 seconds, and the interval in the frame where they were not was 10 seconds, so with t=8 and t'=10, this does indeed fit the equation t' = t * 1.25. The trick with the TAFLC is that if you write it as t' = t / gamma to contrast with the standard time dilation equation t' = t * gamma, then in order for the notation to be consistent you have to be assuming the same thing about the unprimed frame being the one where the events are colocated in both cases.
I thought you were trying to pick a truth. If you were trying to pick possibilities, then the answer would be "all of them". They are all possibilities.
OK, but even if I wasn't clear enough that this was what I meant in #139, I elaborated on this point in #142:
To the extent that you can make "conclusions" about a physical scenario where certain things are uncertain (like not knowing whether events are 'together in space' or 'together in time', but knowing one of the two must be true), it's only by listing various possibilities (like the 5 possibilities I mentioned) that are consistent with your knowledge and saying what you would conclude in each possible circumstance, and perhaps also by finding some broad conclusions that would hold in every possible case (like the equations DrGreg provided).
Anyway, we seem to be clear now, although see my question above about why we even bothered to introduce this uncertainty if t and t' are only related by a gamma factor when we assume that the events were colocal in one frame and non-colocal and the other, whereas if t and t' represented the times each observer would calculate under the assumption that the events were colocal in their own frame, they would not be related by the gamma factor at all.

JesseM

Apr10-09, 10:32 PM

neopolitan

In post #146 I gave the answer

t_B = t_A \sqrt{\frac{c-v}{c+v}}

which applies if A and B are both under the delusion that the events are co-located in their own frame. If they are both under the delusion that the events are simultaneous in their own frame, the same conversion factor applies, because for each L = ct.

As far as I can tell, that is exactly what you asked for. Or have I still misunderstood the problem?

And, JesseM, do you follow my logic and agree with my conclusion?
Yes, if we assume the events are colocated in one frame, then the time interval for the other observer to see them will just be given by the relativistic Doppler shift equation, and of course if that second observer assumes they are colocated in his own frame then he'll conclude the time between the events was the same as the time between his seeing the light from the events. And as you say, it works the same if we assume simultaneity.

neopolitan

Apr11-09, 12:43 AM

JesseM,

Did you or did you not apply "time dilation" so that in B's frame, the time between arrival of photons was less than it otherwise would be?

That is, you arrived at 10s, then applied time dilation and arrived at 8 seconds.

That's all I wanted you to do. The rest of the scenario is now irrelevant.

You asked at post #131 for a physical use for TAFLC, I see that as asking "where do you see time being contracted?" Here is my answer. "Precisely where you took 10s and contracted it to 8s."

cheers,

neopolitan

JesseM

Apr11-09, 01:07 AM

JesseM,

Did you or did you not apply "time dilation" so that in B's frame, the time between arrival of photons was less than it otherwise would be?

That is, you arrived at 10s, then applied time dilation and arrived at 8 seconds.
Yes, 10 s was the time between the light from each event hitting B as measured in A's frame, 8 s was the time between the light from each event hitting B in B's own frame. Here we are dealing with two events that are colocated in B's frame (the events of the light hitting B's worldline, not the events that emitted the light in the first place).
That's all I wanted you to do. The rest of the scenario is now irrelevant.
So do you agree that the stuff about uncertainty about which frame they were "together" in was never relevant? Did you think it was relevant and then change your mind? It would be easier to understand your arguments if you would be explicit about when you have changed your mind or realized that part of a previous approach was not really important.
You asked at post #131 for a physical use for TAFLC, I see that as asking "where do you see time being contracted?" Here is my answer. "Precisely where you took 10s and contracted it to 8s."
But this is not a use of the TAFLC, it's just the regular time dilation equation (if you start with the idea that the events happened 8 s apart in the frame where they were colocated and want the time in the frame where they were not colocated), or the inverse time dilation equation obtained by dividing both sides by gamma (if you start with the idea that the events happened 10 s apart in the frame where they were not colocated and want the time in the frame where they were colocated). As I said before, the TAFLC is different:
The trick with the TAFLC is that if you write it as t' = t / gamma to contrast with the standard time dilation equation t' = t * gamma, then in order for the notation to be consistent you have to be assuming the same thing about the unprimed frame being the one where the events are colocated in both cases.

neopolitan

Apr11-09, 01:34 AM

So do you agree that the stuff about uncertainty about which frame they were "together" in was never relevant? Did you think it was relevant and then change your mind? It would be easier to understand your arguments if you would be explicit about when you have changed your mind or realized that part of a previous approach was not really important.

Try going back to the original post in this sub-thread.

Answer the question I asked there. Then you might see why the "togetherness" had some relevance.

The reason why the scenario is no longer relevant is that you came up with what I wanted you come up with another way. I thought you might grasp something from the process, but it doesn't seem like it. (Here, I should point out that what I want you to grasp may not be right, so don't take this as a personal affront.)

But this is not a use of the TAFLC, it's just the regular time dilation equation (if you start with the idea that the events happened 8 s apart in the frame where they were colocated and want the time in the frame where they were not colocated), or the inverse time dilation equation obtained by dividing both sides by gamma (if you start with the idea that the events happened 10 s apart in the frame where they were not colocated and want the time in the frame where they were colocated). As I said before, the TAFLC is different:

I feel that we could circle forever on this.

You wanted to break things up into TAFLC and "inverse time dilation". I'd prefer "temporal contraction" being the relativistic effect on time which is analagous to length contraction as well as being "inverse time dilation". I don't make such a huge separation between them, this might be because I visualise things a different way.

This visualising things a different way may be why I find it hard to credit that you had so much trouble coming to a result with the scenario I posited - where it is possibly due to the fact that the explanation I gave was more oriented towards someone who visualises things the way I do.

I'm going to withdraw, enjoy my Easter, and if I can think of another way to try to explain the physical significance of temporal contraction (or TAFLC or inverse time dilation), I'll let you know in due course.

cheers,

neopolitan

JesseM

Apr11-09, 02:02 AM

Try going back to the original post in this sub-thread.

Answer the question I asked there. Then you might see why the "togetherness" had some relevance.
You mean the question "If A and B have identical rods and identical clocks, what sort of conclusions will B come to" from post #133? I thought I already answered that. If it wasn't that, which question/post are you referring to?
I feel that we could circle forever on this.

You wanted to break things up into TAFLC and "inverse time dilation". I'd prefer "temporal contraction" being the relativistic effect on time which is analagous to length contraction as well as being "inverse time dilation". I don't make such a huge separation between them, this might be because I visualise things a different way.
I don't see how this has anything to do with visualizations, terms like "TAFLC" and "inverse time dilation" are ultimately supposed to refer to specific equations where the terms have specific physical meanings, if you aren't clear on this then any "visualizations" you come up with will be too ill-defined to have any real meaning. In the original diagram you reposted in post #5 of this thread, I specifically defined the TAFLC as t' = t / gamma, in contrast to the normal time dilation which I wrote as t' = t * gamma. What you're talking about is just dividing both sides of the normal time dilation equation by gamma, a simple algebraic operation which gives t = t' / gamma, which is different than the TAFLC above because if you want the notation to be consistent the unprimed frame has to be the one where the events separated by a time interval of t are colocated. If you just want a change in terminology from what I wrote in that diagram, and want to call the equation t = t' / gamma the "TAFLC", then this is just a semantic issue, I'm fine with calling that equation whatever you want to call it as long as you agree that it's nothing more than the ordinary time dilation equation with both sides divided by gamma. But if you think the TAFLC as I defined it, t' = t / gamma (with t the frame in which whatever events we're dealing with are colocated), is somehow applicable to the problem with the light hitting B, you have to explain what t and t' are supposed to represent physically. And if you just aren't willing to pin down what you mean by terms like "TAFLC" and "inverse time dilation" in terms of specific equations where the terms have specific meanings, then the whole discussion is pointless because pictures in physics are essentially just ways of illustrating the math (as is true of all Minkowski diagrams, for example).
This visualising things a different way may be why I find it hard to credit that you had so much trouble coming to a result with the scenario I posited - where it is possibly due to the fact that the explanation I gave was more oriented towards someone who visualises things the way I do.
When did I have trouble coming to a result? I didn't immediately give an answer because, as I think you now acknowledge, you misunderstood my question about dealing with multiple possibilities and thus gave me a misleading answer which made me unclear what you could be asking. Once this was cleared up I gave a pretty detailed numerical example.
I'm going to withdraw, enjoy my Easter, and if I can think of another way to try to explain the physical significance of temporal contraction (or TAFLC or inverse time dilation), I'll let you know in due course.
All you need to do is give the equations which are supposed to correspond to these words (preferably side-by-side with how you'd write the normal time dilation equation so I can check that the notation is consistent), and what the terms in the equations represent physically. In the meantime, happy Easter!

neopolitan

Apr14-09, 12:31 AM

JesseM

Apr14-09, 01:09 AM

Jesse,

Attached is a diagram of the sort of situation I had in mind.

Note that I don't have the events which created the photons we have been discussing (red and green arrows, at 135 degrees from the tA axis. The events could have taken place millennia ago, really all that I am concerned about is the separation between the arrivals of the photons.

The diagram is in drawn in terms of the A frame.

B is moving towards where the events took place, which means that B intercepts the photons before A (in any frame) and B is closer than A to where the events took place (in any frame).

Down in the bottom right hand corner is an indication of what A and B would take to be "togetherness" for events which could have produced the photons.

Note that initiating events that were 1) together spatially for A and 2) could have produced the photons are further apart temporally than intiating events that were 1) together spatially for B and 2) could have produced the photons. This is true in any frame.

Note also that initiating events that were 1) together temporally for A and 2) could have produced the photons are further apart spatially than intiating events that were 1) together temporally for B and 2) could have produced the photons. This is also true in any frame.

Remember in the first post in this subthread, I said that the delay of t between the arrivals of the photons meant that, to have been emitted together, they were either emitted with a delay of t from the same location (relative to A) or simultaneously with a separation of L (relative to A). These values are also marked in the bottom right hand corner.

Now the thing that concerns me is that if I were to plug your figures into my drawing, I don't think I would arrive at 16 seconds between the arrival of the photons for A and 8 seconds between the arrival of the photons for B. I'd arrive at 16s and 12.8s.
Keep in mind that if you draw 1-second ticks along A's time axis and 1-light second ticks along A's space axis, then also draw similar ticks along B's space and time axis, then when drawn from the perspective of A's frame, B's space and time axes are not merely rotated versions of A's, in A's frame the distances (in the diagram) between ticks on B's axes are greater than the distances between ticks on A's axes. You can sort of see this if you look at the diagram from post #5, where in the diagram from the unprimed perspective you can see that the distance between pink events on the light blue space axis of the unprimed frame is just 4 light-seconds, and the distance between pink events on the yellow time axis of the unprimed frame is 4 seconds; but in the diagram from the primed frame, the light blue and yellow axes are not only slanted, but the distance along these axes between the same pink events also looks stretched (and the location of the three pink events in both frames was calculated using the Lorentz transform so I know they're correct).

neopolitan

Apr14-09, 03:02 AM

You mean like this?

JesseM

Apr14-09, 03:07 AM

You mean like this?
Yup, like that (assuming the angle of the dotted lines isn't supposed to have any meaning), and same for the x-axes obviously.

neopolitan

Apr14-09, 03:56 AM

Yup, like that (assuming the angle of the dotted lines isn't supposed to have any meaning), and same for the x-axes obviously.

I thought about that shortly after I posted the drawing. The angle of the dotted lines, if they were to remain should be 135 degrees (signifying the speed at which a photon or information could reach A and assuming a c=1 sort of relationship between the x and t axes).

(The following is off-track somewhat.)

Now as I wrote it, I realised that "the speed at which a photon or information could reach A" is maybe a little misleading. It's not really that the photon is travelling to intercept A. It's more like the photon is everywhere (and everywhen) along its worldline and there is just a question of when and where the path of A through spacetime intersects that photon relative to when and where the path of B does, and vice versa.

That sort of thinking would upset Tam Hunt ... since unless you somehow shake off causality, it makes the universe highly deterministic - and predetemined at that. If a photon is everywhen along its worldline, and millions of years from now (in my frame, for want of a better frame) that photon is going to hit a descendant of mine, then does that mean that my descendent is predetermined? Note: I know that, without some very special conditions, a photon that I am colocated with will never be colocated with any decendant of mine. Less extreme though, I could trap a number of photons between two mirrors, leave them for a day and come back, then release them. Would my return to release the photons be predetermined by the abrupt end to the worldline of a number of the photons when they are absorbed by my inquisitive eyes?

cheers,

neopolitan

JesseM

Apr14-09, 04:26 AM

Now as I wrote it, I realised that "the speed at which a photon or information could reach A" is maybe a little misleading. It's not really that the photon is travelling to intercept A. It's more like the photon is everywhere (and everywhen) along its worldline and there is just a question of when and where the path of A through spacetime intersects that photon relative to when and where the path of B does, and vice versa.
But what do you mean by "the photon is everywhere (and everywhen) along its worldline"? We can't talk about the time dilation of a photon since it doesn't have any sort of internal clock, and we can't talk about what would be true in the frame of the photon since all inertial frames are sublight frames. It's tempting to think that since in the limit as an object approaches c relative to some external landmark like the galaxy, the distance between the two ends of the galaxy that the object passes approaches zero in the object's own frame, that somehow that means the distance really is zero "for the photon"...see my post #7 on this thread (http://www.physicsforums.com/showthread.php?t=227254) for a discussion of the idea. But you can't really use such a limit to define the "perspective" of a photon since certain things aren't well-defined, like what speed one photon would be traveling from the "perspective" of another photon (see my post #4 from the same thread).

neopolitan

Apr16-09, 09:28 PM

But what do you mean by "the photon is everywhere (and everywhen) along its worldline"? We can't talk about the time dilation of a photon since it doesn't have any sort of internal clock, and we can't talk about what would be true in the frame of the photon since all inertial frames are sublight frames. It's tempting to think that since in the limit as an object approaches c relative to some external landmark like the galaxy, the distance between the two ends of the galaxy that the object passes approaches zero in the object's own frame, that somehow that means the distance really is zero "for the photon"...see my post #7 on this thread (http://www.physicsforums.com/showthread.php?t=227254) for a discussion of the idea. But you can't really use such a limit to define the "perspective" of a photon since certain things aren't well-defined, like what speed one photon would be traveling from the "perspective" of another photon (see my post #4 from the same thread).

But what do you mean by "the photon is everywhere (and everywhen) along its worldline"? We can't talk about the time dilation of a photon since it doesn't have any sort of internal clock, and we can't talk about what would be true in the frame of the photon since all inertial frames are sublight frames. It's tempting to think that since in the limit as an object approaches c relative to some external landmark like the galaxy, the distance between the two ends of the galaxy that the object passes approaches zero in the object's own frame, that somehow that means the distance really is zero "for the photon"...see my post #7 on this thread (http://www.physicsforums.com/showthread.php?t=227254) for a discussion of the idea. But you can't really use such a limit to define the "perspective" of a photon since certain things aren't well-defined, like what speed one photon would be traveling from the "perspective" of another photon (see my post #4 from the same thread).

I had replied to this a while back, but the network disconnected me and lost the reply. Such is life. The fortunate (?) thing is that I have had a subsequent thought which I wouldn't mind addressing.

I was thinking more in terms of the diagram. If you look at the diagram, then you can't really say where the photons are. They are not somewhere on the red and green lines, they are the red and green lines.

For those who like to think about the perspective of a photon (there are some), then as you mentioned, everything happens for the photon in the same place, and at the same time. It was more of an aside than anything real. I don't think it proves or disproves predestination, in part because I don't think that information about when and where the photon's worldline ends is available until it ends. Remember that the comment was made in context of Tam Hunt (who has concerns about free will).

Anyway, I was conflating the diagram (with the worldline of the photon laid bare) and reality (where the worldline of a photon can only be extrapolated using the information to hand, not known in detail).

Right, now on to my thought. We do agree, I hope, that in 4-space individual events all have a unique location, albeit differently identified by different observers. By this I mean that while two inertial observers may have different x,y,z and t values for an event, this does not mean that the event takes place at different 4-space locations.

In light of this, I want to confirm that the diagram I posted is correct.

In A's frame, A is moving along the tA axis at a rate of one second per second while B is moving at an angle to the tA axis such that the gradient of the line is given by 1/v. I have to admit that I initially thought of B moving along what B takes to be the xB axis. I see clearly that this is wrong, since that would equate to a speed greater than c. (I was thinking of an F(x) graph, where you plot F(x) on the vertical axis and x on the horixontal axis. This diagram shows an F(t) graph, where F(t) is on the horizontal axis and the gradient is therefore t/F(t) = 1/v, rather than v as I had in mind.)

B is actually moving along the tB axis, according to A, correct?

This would make sense, since that would mean B would be moving along the tB axis according to both A and B, and therefore according to any inertial observer.

Is that right?

cheers,

neopolitan

PS Yesterday I had no luck trying to post at all.

JesseM

Apr16-09, 10:00 PM

Right, now on to my thought. We do agree, I hope, that in 4-space individual events all have a unique location, albeit differently identified by different observers. By this I mean that while two inertial observers may have different x,y,z and t values for an event, this does not mean that the event takes place at different 4-space locations.
In the same sense that dots on 2D surface like a sheet of paper or a globe have a unique location on that surface even though they can be assigned different coordinates in different coordinate systems drawn on that surface, right.
In light of this, I want to confirm that the diagram I posted is correct.

In A's frame, A is moving along the tA axis at a rate of one second per second while B is moving at an angle to the tA axis such that the gradient of the line is given by 1/v.
With the "gradient" defined as \frac{\Delta t}{\Delta x}, yes.
I have to admit that I initially thought of B moving along what B takes to be the xB axis. I see clearly that this is wrong, since that would equate to a speed greater than c. (I was thinking of an F(x) graph, where you plot F(x) on the vertical axis and x on the horixontal axis. This diagram shows an F(t) graph, where F(t) is on the horizontal axis and the gradient is therefore t/F(t) = 1/v, rather than v as I had in mind.)

B is actually moving along the tB axis, according to A, correct?
Right, the tB axis represents B's worldline.

neopolitan

Apr16-09, 11:04 PM

Ok, a new attempt. I am using a more rigorous diagram, using the figures you like, ie B is moving at 0.6c relative to A, and plotting numbers obtained from Lorentz Transformations. The green lines are the world lines of photon which pass the ends of the moving rod.

I'm just wondering, where is length contraction?

(I think I might know where it is, but drawing it clearly is difficult.)

cheers,

neopolitan

JesseM

Apr16-09, 11:11 PM

I'm just wondering, where is length contraction?
Just draw in the worldlines of either end of the orange rod--both worldlines should be parallel to the tB axis, since both ends are at rest in B's frame--and then the horizontal distance between the worldlines (along any surface of simultaneity in A's frame, which are all parallel to the xA axis) represents the length of the orange rod at a single instant in A's frame, it will be 3/gamma.

neopolitan

Apr17-09, 12:16 AM

Just draw in the worldlines of either end of the orange rod--both worldlines should be parallel to the tB axis, since both ends are at rest in B's frame--and then the horizontal distance between the worldlines (along any surface of simultaneity in A's frame, which are all parallel to the xA axis) represents the length of the orange rod at a single instant in A's frame, it will be 3/gamma.

Damn,

I had it half drawn and was called away from my desk!

Anyways, here it is.

I don't want to follow cos' example by getting all wrapped around the wheels on what is "real", but ... it seems to me that what this diagram shows is that the rod is stretched across time and space (ie 1. the ends of the rod are no longer simultaneous, think clocks at each end, simultaneous in the rest frame, are not simultaneous to A, even if the speed is taken into account; and, 2. non-simultaneously, they are further apart), but the rod appears contracted to A.

Is that right?

I think this might be approaching the issue raised elsewhere by Saw.

As for me, it seems that maybe SAFTD might be a simpler feature to champion rather than TAFLC (although there are still benefits attached to TAFLC in my way of thinking). A benefit to SAFTD manifests in the derivation of the Lorentz Transformations from the Gallilean boosts, where you remove the assumption of instantaneous transmission of information (so you are discussing an event at a distance of x where x=ct and at t=0, t'=0 and the origins of the K and K' frames are colocated).

cheers,

neopolitan

JesseM

Apr17-09, 12:35 AM

Diagram looks good...
I don't want to follow cos' example by getting all wrapped around the wheels on what is "real", but ... it seems to me that what this diagram shows is that the rod is stretched across time and space (ie 1. the ends of the rod are no longer simultaneous, think clocks at each end, simultaneous in the rest frame, are not simultaneous to A, even if the speed is taken into account;
True, if clocks on either end are synchronized in the rod's rest frame using the Einstein synchronization convention, then they will not be synchronized at a single moment in A's frame. But I think it's potentially misleading to say "the ends of the rod are no longer simultaneous", the ends are just physical objects which trace out different worldlines in spacetime, they don't have any intrinsic "opinion" about simultaneity, all claims about simultaneity are based on human definitions of coordinate systems. The rod's orange "world-sheet" can be sliced up in different ways, that's what the relativity of simultaneity is all about.
2. non-simultaneously, they are further apart), but the rod appears contracted to A.
If you say the rod "appears" to be contracted to 2.4 light-seconds in A's frame, you should also say that it "appears" to be 3 light-seconds in B's frame; alternatively you could say the rod is 2.4 l.s in A's frame and it is 3 l.s. in B's frame, the important thing is not to see one frame's definitions as "more correct" than any other's.
As for me, it seems that maybe SAFTD might be a simpler feature to champion rather than TAFLC (although there are still benefits attached to TAFLC in my way of thinking).
Yes, I would say the SAFTD is a lot simpler conceptually, and wanting to know the distance between two events in a frame where they're non-simultaneous if you know the distance in the frame where they're simultaneous is the sort of problem that might actually come up in the course of a practical SR scenario.
A benefit to SAFTD manifests in the derivation of the Lorentz Transformations from the Gallilean boosts, where you remove the assumption of instantaneous transmission of information (so you are discussing an event at a distance of x where x=ct and at t=0, t'=0 and the origins of the K and K' frames are colocated).
Can you give an example? It sounds like you might be talking about a pair of events where one is at (0,0) and the other is at (ct, t) but in that case there wouldn't be any frame where they were simultaneous, and the SAFTD applies to cases where two events are simultaneous in one of the two frames you're using.

neopolitan

Apr17-09, 01:58 AM

Diagram looks good...

True, if clocks on either end are synchronized in the rod's rest frame using the Einstein synchronization convention, then they will not be synchronized at a single moment in A's frame. But I think it's potentially misleading to say "the ends of the rod are no longer simultaneous", the ends are just physical objects which trace out different worldlines in spacetime, they don't have any intrinsic "opinion" about simultaneity, all claims about simultaneity are based on human definitions of coordinate systems. The rod's orange "world-sheet" can be sliced up in different ways, that's what the relativity of simultaneity is all about.

If you say the rod "appears" to be contracted to 2.4 light-seconds in A's frame, you should also say that it "appears" to be 3 light-seconds in B's frame; alternatively you could say the rod is 2.4 l.s in A's frame and it is 3 l.s. in B's frame, the important thing is not to see one frame's definitions as "more correct" than any other's.

Yes, I would say the SAFTD is a lot simpler conceptually, and wanting to know the distance between two events in a frame where they're non-simultaneous if you know the distance in the frame where they're simultaneous is the sort of problem that might actually come up in the course of a practical SR scenario.

Can you give an example? It sounds like you might be talking about a pair of events where one is at (0,0) and the other is at (ct, t) but in that case there wouldn't be any frame where they were simultaneous, and the SAFTD applies to cases where two events are simultaneous in one of the two frames you're using.

I've been in trouble for showing alternative derivations before.

Nevertheless, the situation I am thinking about is analogous to the diagram with a red and green photon worldline, but with one photon which could have come from anywhere (along its path) and any time. Two observers are initially colocated, one travels away at v, and the photon passes one observer then the other one.

The observers consider themselves to be at rest and the other to be moving. Both assume that the event that spawned the photon occured when they were colocated with the other observer. Each observer will come to the conclusion that the spawning event occured at x=ct (x and t in their own frame).

Using Gallilean boosts (ie ignoring relativity), both observers would calculate that x'=x-vt (where v is negative in one instance and x' is the distance in the other frame).

However, Gallilean boosts assume instantaneous transmission of information and clearly don't work in this example.

Since information is not transmitted instantaneously, both observers will work out that the other observer must be affected by some temporal and spatial skewing (this is required to reconcile a single photon apparently coming from two different locations). And it's the same temporal and spatial skewing that each concludes must affect the other, since no frame is privileged.

Using these facts, you can calculate SAFTD and TD equations, then apply them to the Gallilean boost to obtain the spatial Lorentz transformation and, given the relationship between x and t, you can also obtain the temporal Lorentz transformation.

(Actually you only need the SAFTD equation, but for completeness it helps to get both.)

If you can't work this through, I can do it for you later, but I am a little pressed for time at the moment.

cheers,

neopolitan

JesseM

Apr17-09, 03:44 AM

Nevertheless, the situation I am thinking about is analogous to the diagram with a red and green photon worldline, but with one photon which could have come from anywhere (along its path) and any time. Two observers are initially colocated, one travels away at v, and the photon passes one observer then the other one.

The observers consider themselves to be at rest and the other to be moving. Both assume that the event that spawned the photon occured when they were colocated with the other observer.
Why would they both assume that? You agree that one of them must be objectively right and the other objectively wrong if they each assume the event occurred simultaneously with their being colocated according to their own rest frame's definition of simultaneity, right?
Each observer will come to the conclusion that the spawning event occured at x=ct (x and t in their own frame).
OK.
Using Gallilean boosts (ie ignoring relativity), both observers would calculate that x'=x-vt (where v is negative in one instance and x' is the distance in the other frame).
In Galilean physics they don't disagree about simultaneity, so in this case they could both be right that the event occurred simultaneously with their being colocated. However, the meaning of your terms is a little ambiguous. For example, say the event occurs at t=0 and x=-10 light-seconds in A's frame, and they are colocated at t=0 and x=0 in A's frame, and B is moving in the +x direction at 0.6c in this frame. Suppose also that the signal from the event moves at 1c in the +x direction in A's frame (which will mean it moves at a different speed in B's frame). In this case the signal will catch up with A at t=10, x=0, and will catch up with B at t=25, x=15 (again in A's frame). In B's own frame the time will be the same but the light was only moving at 0.4c in his frame, so the initial event was at x'=-10 and t'=0, and it hit B at x'=0 and t'=25. So, if x is the distance between A and the event in A's frame, and x' is the distance between B and the event in B's frame, in which case x=x'=10, meaning your equation isn't right. On the other hand, you may have meant that x is supposed to be the distance between the original event and the event of the light hitting B as measured in A's frame, while x' is the distance between the original event and the event of the light hitting B as measured in B's frame, in which case x=25, t=25 and x'=10, so this does work because 10 = 25 - 0.6*25. So it only works if both frames are talking about the distance x and x' between the same two events.
However, Gallilean boosts assume instantaneous transmission of information and clearly don't work in this example.
I wouldn't say Galilean boosts assume instantaneous transfer of information, you could have a Galilean physics where information can travel no faster than c in both directions in one preferred frame (the aether frame, perhaps), so in a frame moving at speed v relative to the first, signals could travel at a max of c-v in one direction and c+v in another.
Since information is not transmitted instantaneously, both observers will work out that the other observer must be affected by some temporal and spatial skewing (this is required to reconcile a single photon apparently coming from two different locations).
I don't understand what you mean by that--there is no way to reconcile a single photon coming from two different locations in spacetime, that's a physical contradiction and only one of them can be correct if they both make this assumption. If the event that created the photon was simultaneous with A and B being colocated in A's frame, then it wasn't simultaneous in B's frame, and vice versa.
Using these facts, you can calculate SAFTD and TD equations, then apply them to the Gallilean boost to obtain the spatial Lorentz transformation and, given the relationship between x and t, you can also obtain the temporal Lorentz transformation.
The TD and SAFTD are meant to apply to a single well-defined pair of events, not to a case where different frames disagree about where in spacetime one of the events occurred (which is different from disagreeing about the coordinates assigned to a single well-defined event). It is true that you get equations that look like the TD and SAFTD with your assumptions though. For example, let's use the same numbers for A's frame, so in this light from the event reaches B at t=25, x=15. Because of time dilation in A's frame, B's clock will only read 0.8*25 = 20 when the light hits it, so if B assumes the event happened at t'=0 in his own frame, he must conclude it happened 20 light-seconds away at x'=-20. So A thinks the original event happened 25 light-seconds away from where B was when the light struck him, and B thinks it happened 20 light-seconds away from where B was when the light struck him. But again, conceptually this has nothing to do with TD or SAFTD because they are each making totally incompatible assumptions about the spacetime location of the original event, whereas TD and SAFTD are derived using the assumption that we are talking about two events with well-defined locations in spacetime (and also using the assumption that the events have a spacelike separation and are simultaneous in one frame in the case of TD, or a timelike separation and are colocated in one frame in the case of SAFTD).

I also don't see how you can use your scenario to get a valid derivation of either of the Lorentz transformation equations, so could you explain that?

neopolitan

Apr17-09, 10:19 PM

Why would they both assume that? You agree that one of them must be objectively right and the other objectively wrong if they each assume the event occurred simultaneously with their being colocated according to their own rest frame's definition of simultaneity, right?

I don't have time to go through it all right at this moment. All I can do is ask you to analyse the inherent assumptions in the Lorentz transformations. That's analyse, not recite because reciting will give what you know now rather than what you could know if you analysed.

To help you, think about why there is no offset (to or xo) in the equations.

Plus I reiterate:

Nevertheless, the situation I am thinking about is analogous to the diagram with a red and green photon worldline, but with one photon which could have come from anywhere (along its path) and any time. Two observers are initially colocated, one travels away at v, and the photon passes one observer then the other one.

There is no objective right or wrong here, unless you can uniquely identify individual photons. Note that I am using thought-experiment magic here, so the photon is detected by both observers without being absorbed by the first. If you prefer an expanding sphere of photons, you can have that instead so long as you don't pretend that you can see the whole sphere and work out the objective origin.

cheers,

neopolitan

JesseM

Apr17-09, 10:42 PM

I don't have time to go through it all right at this moment. All I can do is ask you to analyse the inherent assumptions in the Lorentz transformations. That's analyse, not recite because reciting will give what you know now rather than what you could know if you analysed.

To help you, think about why there is no offset (to or xo) in the equations.
There's no offset because you didn't pick a single pair of events, you picked an event on B's worldline (receiving the photon) and then gave the observers two different assumptions about where in spacetime the photon was emitted. If they both agreed on the event in spacetime where it was emitted, then they both figured out the distance and time between the event of emission and the event of B receiving the photon in their own respective rest frames, their two answers would be related by the usual Lorentz equations:

\Delta x' = \gamma * (\Delta x - v * \Delta t)
\Delta t' = \gamma * (\Delta t - v * \Delta x /c^2)

In contrast, when they both define the distance and times using the assumption that the emission event happened simultaneously with A and B being colocated according to their own frames' definitions of simultaneity, they get the equations:

\Delta x = \gamma * \Delta x'
\Delta t = \gamma * \Delta t'

(note that even besides the lack of the extra factor with the v in it, the primes and unprimes are reversed here).

I don't see how you can go from one to the other, and I'm highly dubious that you have a coherent derivation yourself, but maybe you can prove me wrong when you have more time.
Plus I reiterate:
Nevertheless, the situation I am thinking about is analogous to the diagram with a red and green photon worldline, but with one photon which could have come from anywhere (along its path) and any time. Two observers are initially colocated, one travels away at v, and the photon passes one observer then the other one.
There is no objective right or wrong here, unless you can uniquely identify individual photons. Note that I am using thought-experiment magic here, so the photon is detected by both observers without being absorbed by the first.
That assumption is fine, but what's problematic is that there has to be an objective truth about what event in spacetime corresponds to the photon emission (at least as long as we don't get into quantum uncertainty), whether or not the observers can "identify" this point; in SR problems we typically take the perspective of omniscient observers viewing spacetime from "the outside" as in spacetime diagrams, whether or not observers in the thought-experiment would actually have the technical ability to determine everything we state in the problem is not relevant. And in the form of the Lorentz transformation I posted above, we're definitely talking about the time and distance intervals between a single pair of events with well-defined locations in spacetime.

neopolitan

Apr17-09, 11:51 PM

There's no offset because you didn't pick a single pair of events, you picked an event on B's worldline (receiving the photon) and then gave the observers two different assumptions about where in spacetime the photon was emitted. If they both agreed on the event in spacetime where it was emitted, then they both figured out the distance and time between the event of emission and the event of B receiving the photon in their own respective rest frames, their two answers would be related by the usual Lorentz equations:

\Delta x' = \gamma * (\Delta x - v * \Delta t)
\Delta t' = \gamma * (\Delta t - v * \Delta x /c^2)

In contrast, when they both define the distance and times using the assumption that the emission event happened simultaneously with A and B being colocated according to their own frames' definitions of simultaneity, they get the equations:

\Delta x = \gamma * \Delta x'
\Delta t = \gamma * \Delta t'

(note that even besides the lack of the extra factor with the v in it, the primes and unprimes are reversed here).

I don't see how you can go from one to the other, and I'm highly dubious that you have a coherent derivation yourself, but maybe you can prove me wrong when you have more time.

That assumption is fine, but what's problematic is that there has to be an objective truth about what event in spacetime corresponds to the photon emission (at least as long as we don't get into quantum uncertainty), whether or not the observers can "identify" this point; in SR problems we typically take the perspective of omniscient observers viewing spacetime from "the outside" as in spacetime diagrams, whether or not observers in the thought-experiment would actually have the technical ability to determine everything we state in the problem is not relevant. And in the form of the Lorentz transformation I posted above, we're definitely talking about the time and distance intervals between a single pair of events with well-defined locations in spacetime.

You are still looking at it the wrong way.

There is a pair of events. The pair of events that the Lorentz Transformations are based on (the information that is in the transformations comes from those events).

Really there are three events:

The event which spawned the photon. I don't care where or when that event took place because the photon could have be spawned anywhere on its worldline (anywhere on the appropriate side of the observers, of course) and be indistinguishable from any other photon spawned on the same worldline (again on the correct side of the observers).
When and where the photon passes the first observer
When and where the photon passes the second observer

As I say, I don't care that much about the first event and the Lorentz Transformations only uses the second two events. Yes I know that the "where" in the Lorentz Transformation is not about events 2 and 3 but about where the photon originated to cause events 2 and 3, which requires and extrapolation and when you make that extrapolation, you are using an assumption. What is that assumption?

Does this help?

cheers,

neopolitan

PS Writing out the whole derivation will take me an hour or more with all the LaTex and typo checking, this response took close to ten minutes. I'm not being difficult, I am just using the time I do have as best I can.

JesseM

Apr18-09, 12:19 AM

You are still looking at it the wrong way.

There is a pair of events. The pair of events that the Lorentz Transformations are based on (the information that is in the transformations comes from those events).

Really there are three events:

The event which spawned the photon. I don't care where or when that event took place because the photon could have be spawned anywhere on its worldline (anywhere on the appropriate side of the observers, of course) and be indistinguishable from any other photon spawned on the same worldline (again on the correct side of the observers).
When and where the photon passes the first observer
When and where the photon passes the second observer

As I say, I don't care that much about the first event and the Lorentz Transformations only uses the second two events.
Well, the Lorentz transformation works to relate two frame's answers for the distances and times between any pair of well-defined events whatsoever. And yes, this is a helpful clarification on the two events, you didn't correct me earlier in post #169 where I interpreted t' as the time B calculated in his frame between receiving the photon and when he assumed it was emitted, and t as the time A calculated in his frame between B receiving the photon and when he assumed it was emitted. As always, it's best to be specific as possible.

So, I'll look at that numerical example again with this clarification in mind. As before, assume that in A's frame B is moving at 0.6c and they were colocated at x=0 at time t=0. And as before, assume the light hits A at x=0, t=10, and catches up with B at x=15, t=25. If delta-x and delta-t are the distance and time between this specific pair of events, then we have delta-x=15 and delta-t=15. In B's frame, B is at position x'=0 when the light hits him, and because of time dilation his clock only reads t'=20 when it hits him. At this moment A must be 0.6*20=12 light seconds away in B's frame, and the distance between A and the light increases at 1.6c in B's frame, so backtracking, the light must have crossed A's path 12/1.6 = 7.5 seconds earlier at t'=12.5, and at this moment A must have been at a distance of 0.6*12.5 = 7.5 light-seconds from B, so a position of x'=-7.5. So for B, delta-x' between these events = 7.5, and delta-t' = 7.5.

But if this is what you meant when you said "Using these facts, you can calculate SAFTD and TD equations", then I'm still confused, because this delta-t is 2 times this delta-t', which is not the gamma factor if A and B have a relative velocity of 0.6c (likewise for delta-x and delta-x'). So how are you using these two events to derive time dilation and its spatial analogue? I think we need to clarify this first, since you said that once you have already derived these equations using your scenario, you "then apply them to the Gallilean boost to obtain the spatial Lorentz transformation". Unless you think this last part can be done without worrying about the specifics of how we derived TD and SAFTD--but in that case it'll be important that you're using the normal interpretation of TD and SAFTD, where we only use TD if two events are colocated in one of the two frames, and we only use SAFTD if the two events are simultaneous in one of the two frames.

neopolitan

Apr18-09, 01:50 PM

See the three diagrams, numbered:

1 (Galilean boost),
2 (one photon passes B, then A - indistinguishable from a photon from Event E - Galilean boost not applicable) and
3 (part of the way to Lorentz Transformations).

There are six numbered equations in diagram 3.

Put (5) into (6) or (6) into (5) and you have:

G = \gamma .... (7)

Put (7) into (1) and you have "SAFTD".

Put (7) and (4) into (2) and you have the spatial Lorentz Transformation.

Because x=ct and x'=ct', then you just divide through by c to get TD and the temporal Lorentz Transformation (noting that t/c = x/c2).

You are probably going to tell me I am wrong for some reason, but it's really a heck of a lot simpler than some of the derivations I have seen.

cheers,

neopolitan

PS You can call the result of (7) into (1) SAFTD, or inverse length contraction, or length dilation, or relativistic effect on lengths, or the spatial relativistic effect. Or whatever you like. Irrespective of what you call it, can you see the physical significance of it?

neopolitan

Apr19-09, 12:13 AM

neopolitan

Apr19-09, 08:23 AM

neopolitan

Apr19-09, 11:51 PM

While awaiting a response on post #174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174), I have had a bit of time to develop the diagram from post #166 (http://www.physicsforums.com/showpost.php?p=2163783&postcount=166). That diagram only showed length contraction (LC).

I have developed three more diagrams, one for time dilation (TD), one for spatial analogue for time dilation (SAFTD) and one for temporal analogue for length contraction (TAFLC).

In each of the diagrams I have tried to be consistent with the use of colours. I hope that the sheets (light blue and light orange) are easily understood. Trying to explain briefly, depending on what is being shown:

the blue sheets have a spatial of L or a temporal width of 1 tick in the A frame
the orange sheets have a spatial width of L or a temporal width of 1 tick in the B frame.

The darker blue bar shows either a length L or the period between two consecutive ticks in the A frame. The darker orange bar shows either a length L or the period between two consecutive ticks in the B frame.

The green bars show measurements by A of a length or period in the B frame.
The purple bars show measurements by B of a length or period in the A frame.

For completeness, I will repost JesseM's diagram (which contains some of the information, but not all).

cheers,

neopolitan

It seems that I can't post diagrams at the moment. So, I have posted them all on one web page here (http://www.geocities.com/neopolitonian/index.htm).

I'm aware that only a few may understand the background to the diagrams. Therefore, I thought I might try to explain a little.

Length contraction manifests in situations such as when two rods of the same rest length pass each other. Both will observe the other to be contracted.

Time dilation manifests with in situations such as when you have two identical clocks where one is in motion relative to the other. Time dilation is not so easily observed. Part of this is because photons received from a clock moving away from you will be spaced out due to the movement of the clock away from you, even before taking into account relativity. The combined effect of standard doppler and relativity is "relativistic doppler" and in the diagrams the overall effect is to double the period between "moving" ticks compared to "stationary" ticks where v=0.6c. A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

To understand spatial analogue for time dilation (SAFTD) you really need to look at the diagram. Note that the period between ticks is stretched out for B relative to A. This is the effect of time dilation. The length of L in B's frame is stretched out in A's frame in a similar fashion.

As for the temporal analogue for length contraction (TAFLC), this is similar to the idea of two rods passing each other. Imagine there are no doppler considerations, in the time it takes the clock in A's frame to tick once, according to A, the clock in B's frame has not had enough time to tick. In other words, according to A, 1 tick in A = less than 1 tick in B. This means that the clock in A will tick more, so more time will elapse in A than in B.

Clear as mud?

JesseM

Apr21-09, 01:12 AM

It seems that I can't post diagrams at the moment. So, I have posted them all on one web page here (http://www.geocities.com/neopolitonian/index.htm).
The labels on the time dilation diagram appear to be backwards. You label the purple segment the time dilation of B looking at A, but that appears to be the time dilation that A sees when it looks at B's clock--it's the difference in A's time coordinates between two events on the worldline of B's clock that are separated by 1 second in B's frame (and the height of the purple segment is greater than 1 second in A's frame, hence the 'dilation'). Likewise, the green bar seems to show the time in B's frame between two ticks of a clock at rest in A's frame, so that should be "time dilation of B looking at A" rather than "A looking at B". Unless I'm misunderstanding what "looking at" is supposed to mean (I interpreted 'A looking at B' to mean how B's clock behaved in A's frame).

JesseM

Apr21-09, 01:27 AM

Some clarifications to the last post (done in the early hours of the morning).

G to E 01 - (Galileo to Einstein) - shows the standard Galilean boost. There is an assumption of instantaneous transfer of information (or god-like powers to see everything at once).
Why is instantaneous transfer of information relevant? Even in a Galilean universe it could be true that information has limited speed (for example, the fastest information transfer might be vibrations in the aether which travel at c in the aether frame). The coordinates an observer assigns to an event are done in retrospect, once I have already received information about an event. For example, if an event happens 12 light-seconds away from me at t=0, then if you have instantaneous transfer of information I'll learn about the event at t=0, while if information only travels at c I won't learn about it until t=12; but in the latter case I'll take into account the speed of the signal and backdate the event to t=0, so the coordinates are the same either way.
G to E 02 - shows what happens when you remove the assumption of instantaneous transfer of information. E is an "event". I stress that it could be something that causes the emission of a photon, or it just could be an event along the path of the photon. A and B see the same photon (thought experiment magic).
In this diagram, I take it t refers to the time in A's frame the light reached A, and t' refers to the time in B's frame the light reached B? If so it also seems that x refers to the position of the photon at t=0 in A's frame, while x' refers to the position of the same photon at t'=0 in B's frame (because of the relativity of simultaneity these must refer to different events on the photon's worldline). So in each frame you're calculating the distance and time between a totally different pair of events, correct?
G to E 03 - shows the reconciliation, with subscripts to show "according to ...."

In this process, it becomes explicit what sort of physical things your x, t, x', t' and, if you like, L and L' refer to.
It still isn't really clear to me...is A still measuring the distance and time between an event #1 on the photon's worldline (the event on the photon's worldline which in A's frame is simultaneous with A and B being colocated) and the event of the photon passing A, while B is still measuring the distance and time between a different event #2 on the photon's worldline (the event on the photon's worldline which in B's frame is simultaneous with A and B being colocated) and the event of the photon passing B? Or are the assumptions supposed to be different in this diagram? (or did I misunderstand the assumptions of the previous diagram?) If I'm getting the assumptions wrong, can you try to explain in clear terms what two events A is measuring the distance x and time t between, and likewise what two events B is measuring the distance x' and time t' between?

neopolitan

Apr21-09, 03:26 AM

neopolitan

Apr21-09, 04:26 AM

Why is instantaneous transfer of information relevant? Even in a Galilean universe it could be true that information has limited speed (for example, the fastest information transfer might be vibrations in the aether which travel at c in the aether frame). The coordinates an observer assigns to an event are done in retrospect, once I have already received information about an event. For example, if an event happens 12 light-seconds away from me at t=0, then if you have instantaneous transfer of information I'll learn about the event at t=0, while if information only travels at c I won't learn about it until t=12; but in the latter case I'll take into account the speed of the signal and backdate the event to t=0, so the coordinates are the same either way.

You want to reintroduce an aether?

I have never actually seen Galilean relativity done that way, but then I have never seen time brought into it at all. Galilean relativity seems to be based on a snapshot. It is certainly based on absolute space (wikipedia (http://en.wikipedia.org/wiki/Galilean_invariance)).

Really, I am just going from the Galilean boost to Lorentz Transformations though. That boost is given by x'=x-vt. Do we at least agree on that?

The Galilean assumption, in terms of my diagram, is that B is moving with an absolute velocity of v towards a location E which is a distance of x from A and, at a time t, the distance from B to E is x' = x - vt. This means that when t=0, A and B were colocated. Do we agree on that?

In Galilean relativity, at t, A has not moved, B is moving with a velocity of v and is located vt closer to E than A is. Do we agree on that?

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t) in B's frame. Do we agree on that?

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t') in B's frame, because time is absolute and t=t'. Do we agree on that?

In Galilean relativity, if B is told that E is currently x' away, and B has observed that A has been moving away at -vt, then B will calculate that A-E is currently x = x' + vt . Do we agree on that?

Do we further agree that if an event took place at (x,t) in A's frame in Special Relativity and even in a more careful analysis of Galilean relativity, that neither A nor B would know about it until a photon from the event is received?

If x = ct, in Galilean relativity, when A receives the photon at 2t, x' = x - 2vt. Do we agree that if we now talk about where a photon from the same event (x,t) hits B, this is not x' as calculated above?

I guess I could agree that Galilean relativity is based on either absolute space (ie there's an aether frame) or instantaneous transmission of information. Can you agree that it is one or the other? In a paper I put together on this, I actually had a few assumption including preferred frame and instantaneous transmission of information, I can see that I should put them as "and/or".

Can you see that if information is transmitted instantaneously and an event takes place at (x,t) in the A frame, then in the A frame that event will be detected by A at (0,t) and B at (vt,t)? And in the B frame, the event was at (x',t), B detects is at (0,t) and A at (-vt,t) where x'=x-vt? And can you see that these can all be related by the Galilean boosts?

(Because of LET, I wonder if it actually works with just an aether frame. I'd have to put more time into, and I am running out of time rapidly.)

In this diagram G to E 02, I take it t refers to the time in A's frame the light reached A, and t' refers to the time in B's frame the light reached B? If so it also seems that x refers to the position of the photon at t=0 in A's frame, while x' refers to the position of the same photon at t'=0 in B's frame (because of the relativity of simultaneity these must refer to different events on the photon's worldline). So in each frame you're calculating the distance and time between a totally different pair of events, correct?

One photon. One event spawning the photon. Two events where the photon passes B, then A. One event when A and B were colocated and t=0 and t'=0.

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

Does that help?

What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location. Correct?

It still isn't really clear to me...is A still measuring the distance and time between an event #1 on the photon's worldline (the event on the photon's worldline which in A's frame is simultaneous with A and B being colocated) and the event of the photon passing A, while B is still measuring the distance and time between a different event #2 on the photon's worldline (the event on the photon's worldline which in B's frame is simultaneous with A and B being colocated) and the event of the photon passing B? Or are the assumptions supposed to be different in this diagram? (or did I misunderstand the assumptions of the previous diagram?) If I'm getting the assumptions wrong, can you try to explain in clear terms what two events A is measuring the distance x and time t between, and likewise what two events B is measuring the distance x' and time t' between?

Hopefully the above helped. My time is up for now.

cheers,

neopolitan

JesseM

Apr21-09, 04:41 AM

The labels on the time dilation diagram appear to be backwards. You label the purple segment the time dilation of B looking at A, but that appears to be the time dilation that A sees when it looks at B's clock--it's the difference in A's time coordinates between two events on the worldline of B's clock that are separated by 1 second in B's frame (and the height of the purple segment is greater than 1 second in A's frame, hence the 'dilation'). Likewise, the green bar seems to show the time in B's frame between two ticks of a clock at rest in A's frame, so that should be "time dilation of B looking at A" rather than "A looking at B". Unless I'm misunderstanding what "looking at" is supposed to mean (I interpreted 'A looking at B' to mean how B's clock behaved in A's frame).
Time dilation continues to be a problem.

What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that tB = gamma.tA.
A value of what, though? Which two events are you supposed to be measuring the time between in both frames? The orange bar which projects through the purple line crosses the events 1 and 2 on B's worldline, so I assumed the purple bar was showing the time in A's frame between events 1 and 2 on B's worldline which have a separation of 1 second in B's frame. But if tA represents the time between this pair of events in A's frame and tB represents the between the same pair in B's frame, then tA = 1.25 and tB = 1, which means tA = gamma * tB, the reverse of what you write above.

Maybe the two events you're thinking of are not 1 and 2 on B's worldline, but just the events which actually lie at the endpoints of the purple line? But then the bar is confusing because it doesn't help you show the time between these events in B's frame, you should instead draw a slanted bar (like the light blue one) whose top and bottom cross the events at the top and bottom of the purple line, and then see where this slanted bar intersects B's time axis, which would give you tB for these two events.
Since someone said a while back:
The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame).
Right, and going by the interpretation that the purple line is supposed to show the time between 1 and 2 on B's worldline, then since these events are colocal in B's frame but not in A's frame, B should be unprimed and A should be primed.
That means that, according to A, tB = t' and tA = t
Well, not if your events are 1 and 2 on B's worldline. And why do you say "according to A"? Once you have picked the two events you want to measure the time between, according to my description of the primed vs. unprimed convention both observers will agree on which frame should be primed and which should be unprimed, the unprimed is always the frame where those particular events are colocated. But I'm not sure you're clear on the fact that we always have to have a fixed idea of what two events we're talking about in advance before we can use the time dilation equation, which is comparing the time-interval between those specific events in both frames (one of which is the frame where they are colocated, which is normally labeled as the unprimed frame).
I could learn to like time dilation more if the stretching of the time axis in B's frame is time dilation and/or I am allowed to explain something like this:
Time dilation manifests with in situations such as when you have two identical clocks where one is in motion relative to the other. Time dilation is not so easily observed. Part of this is because photons received from a clock moving away from you will be spaced out due to the movement of the clock away from you, even before taking into account relativity. The combined effect of standard doppler and relativity is "relativistic doppler" and in the diagrams the overall effect is to double the period between "moving" ticks compared to "stationary" ticks where v=0.6c. A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.
True, if "pure Doppler" is taken to mean Doppler shift when we assume no time dilation, you just multiply that by gamma to get the relativistic Doppler equation (for period rather than frequency as in your calculation).
Application of the gamma of 1.25 is time dilation and the stretching of the tB axis.
I don't follow. I'd say that application of gamma is because if the clock is emitting signals once every Temit according to its own readings, then in your frame the time interval between signal emissions is gamma*Temit and its distance from you increases by v*gamma*Temit in this time, so each successive signal has to cross this extra distance which takes an extra time of v*gamma*Temit/c in your frame, so instead of receiving the signals once every gamma*Temit (the 'actual' time between emissions in your frame), you only receive them once every gamma*Temit + v*gamma*Temit/c = gamma*Temit*(1 + v/c). So if Tobs is the time between observing successive signals from the moving clock, we get the relativistic Doppler equation Tobs = gamma*Temit*(1 + v/c), whereas the Galilean Doppler equation would just be Tobs = Temit*(1 + v/c).
This would mean that the darker orange line is time dilation for A
The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top? If so your diagram doesn't actually show anything corresponding to this vertical distance, it'd be more clear if you drew a horizontal bar whose top and bottom edge cross these two events, so you could see the times this bar's top and bottom intersect the tA axis (which would represent the dilated time in A's frame between events 2 and 3 on B's worldline that are colocated in B's frame). Alternatively you could just relocate the darker orange line so it goes between events 1 and 2 on B's worldline, since you already have a light orange horizontal bar there.
But you can see that time dilation is such a totally different thing to Length contraction, which we have covered before, and that in the SAFTD diagrams the same would apply, the darker orange line would be SAFTD for A and the dark blue line would be SAFTD for B.
I would say that the horizontal length of the orange bar in the SAFTD diagram represents the spatial distance in A's frame between events 1 and 4 on the tB axis, which are simultaneous in B's frame, and that's exactly the idea of the SAFTD...in other words, if we write SAFTD as dx' = dx * gamma, then here dx is the distance in B's frame between events 1 and 4, and dx' is the distance in A's frame between events 1 and 4 (which can also be represented by the green line). As long as that matches what you're saying above, this diagram makes sense to me.
What you can't really do though, is measure these values directly. Even to show it on the diagram is difficult
Isn't showing where the appropriate bars cross the appropriate axis "showing it on the diagram"? For example, in the SAFTD diagram, you can see the dark orange line has a length of 3 in B's coordinate system (since it goes from x=1 to x=4 on B's space axis), but you can also see that the vertical light orange bar whose edges go through the two ends of the dark orange line intersects A's x-axis at two points that are a distance of 3*1.25 = 3.75 apart (which represents the distance in A's frame between the events on the ends of the dark orange line).

neopolitan

Apr21-09, 11:37 AM

A value of what, though? Which two events are you supposed to be measuring the time between in both frames? The orange bar which projects through the purple line crosses the events 1 and 2 on B's worldline, so I assumed the purple bar was showing the time in A's frame between events 1 and 2 on B's worldline which have a separation of 1 second in B's frame. But if tA represents the time between this pair of events in A's frame and tB represents the between the same pair in B's frame, then tA = 1.25 and tB = 1, which means tA = gamma * tB, the reverse of what you write above.

The light coloured sheets are simultaneity planes, if you like:
The light orange sheet is 1 tick.gamma in the A frame, spanning 1 tick in the B frame.
The light blue sheet is 1 tick.gamma in the B frame, spanning 1 tick in the A frame.

If you must have events, the light orange sheet shows all the events which are simultaneous in the A frame with two consecutive ticks of B's clock, and all the events between. In the B frame, the two consecutive ticks are 1 tick apart and not simultaneous with all the other events which constitute the boundaries of the light orange sheet. In the A frame, the two consecutive ticks are 1 tick.gamma.

The light blue sheet shows all the events which are simultaneous in the B frame with two consecutive ticks of A's clock, and all the events between. (And so on.)

Maybe the two events you're thinking of are not 1 and 2 on B's worldline, but just the events which actually lie at the endpoints of the purple line? But then the bar is confusing because it doesn't help you show the time between these events in B's frame, you should instead draw a slanted bar (like the light blue one) whose top and bottom cross the events at the top and bottom of the purple line, and then see where this slanted bar intersects B's time axis, which would give you tB for these two events.

The purple line just shows the time between the events which are simultaneous in the A frame on clock which is at rest with A but not colocated. Any point along the sheet will do.

tB for these two events is already shown on the tB axis. That's why the light orange sheet spans two consecutive ticks on the tB axis.

But I'm not sure you're clear on the fact that we always have to have a fixed idea of what two events we're talking about in advance before we can use the time dilation equation, which is comparing the time-interval between those specific events in both frames (one of which is the frame where they are colocated, which is normally labeled as the unprimed frame).

We don't really need two specific events. We can use simultaneity sheets. It might not be immediately obvious, but you can do it.

I don't follow. I'd say that application of gamma is because if the clock is emitting signals once every Temit according to its own readings, then in your frame the time interval between signal emissions is gamma*Temit and its distance from you increases by v*gamma*Temit in this time, so each successive signal has to cross this extra distance which takes an extra time of v*gamma*Temit/c in your frame, so instead of receiving the signals once every gamma*Temit (the 'actual' time between emissions in your frame), you only receive them once every gamma*Temit + v*gamma*Temit/c = gamma*Temit*(1 + v/c). So if Tobs is the time between observing successive signals from the moving clock, we get the relativistic Doppler equation Tobs = gamma*Temit*(1 + v/c), whereas the Galilean Doppler equation would just be Tobs = Temit*(1 + v/c).

Yep, like I said. Except I said it more quickly, because I thought you would understand. In my words:

A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.

The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top? If so your diagram doesn't actually show anything corresponding to this vertical distance, it'd be more clear if you drew a horizontal bar whose top and bottom edge cross these two events, so you could see the times this bar's top and bottom intersect the tA axis (which would represent the dilated time in A's frame between events 2 and 3 on B's worldline that are colocated in B's frame). Alternatively you could just relocate the darker orange line so it goes between events 1 and 2 on B's worldline, since you already have a light orange horizontal bar there.

I would say that the horizontal length of the orange bar in the SAFTD diagram represents the spatial distance in A's frame between events 1 and 4 on the tB axis, which are simultaneous in B's frame, and that's exactly the idea of the SAFTD...in other words, if we write SAFTD as dx' = dx * gamma, then here dx is the distance in B's frame between events 1 and 4, and dx' is the distance in A's frame between events 1 and 4 (which can also be represented by the green line). As long as that matches what you're saying above, this diagram makes sense to me.

Isn't showing where the appropriate bars cross the appropriate axis "showing it on the diagram"? For example, in the SAFTD diagram, you can see the dark orange line has a length of 3 in B's coordinate system (since it goes from x=1 to x=4 on B's space axis), but you can also see that the vertical light orange bar whose edges go through the two ends of the dark orange line intersects A's x-axis at two points that are a distance of 3*1.25 = 3.75 apart (which represents the distance in A's frame between the events on the ends of the dark orange line).

You are splitting stuff up and ignoring context again.

I said it was difficult to draw, didn't I?

Imagine that the tB axis is a wiper fixed at the origin. Move it so it is parallel with the tA axis. The tB is not only rotated, but also stretched. Do you recall anyone saying that?

Keep in mind that if you draw 1-second ticks along A's time axis and 1-light second ticks along A's space axis, then also draw similar ticks along B's space and time axis, then when drawn from the perspective of A's frame, B's space and time axes are not merely rotated versions of A's, in A's frame the distances (in the diagram) between ticks on B's axes are greater than the distances between ticks on A's axes.

The extent to which the tB axis is stretched is given by time dilation.

cheers,

neopolitan

JesseM

Apr21-09, 02:00 PM

The light coloured sheets are simultaneity planes, if you like:
The light orange sheet is 1 tick.gamma in the A frame, spanning 1 tick in the B frame.
The light blue sheet is 1 tick.gamma in the B frame, spanning 1 tick in the A frame.
Right, that's what I figured.
If you must have events, the light orange sheet shows all the events which are simultaneous in the A frame with two consecutive ticks of B's clock, and all the events between.
Yes, and those two consecutive ticks are events 1 and 2 on B's worldline, as I suggested. In this case tA for these events is 1.25 and tB is 1.
The light blue sheet shows all the events which are simultaneous in the B frame with two consecutive ticks of A's clock, and all the events between. (And so on.)
Yes, here I assume the two events would be on either end of the dark blue line, the events labeled 2 and 3 on A's time axis. In A's frame tA for these events is 1, while in B's frame tB for these events is 1.25. Can I assume it's this pair of events, and the time between them in both frames, that you were referring to when you said "What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that tB = gamma.tA"? But in this case the time dilation occurs in B's frame, so I still don't really understand why you labeled the green line "time dilation A looking at B" rather than "B looking at A".
The purple line just shows the time between the events which are simultaneous in the A frame on clock which is at rest with A but not colocated.
They're not colocated with the clock at x=0 in the A frame, but the two events are colocated with one another in A's frame.
tB for these two events is already shown on the tB axis. That's why the light orange sheet spans two consecutive ticks on the tB axis.
But the light orange sheet shows the events on B's t-axis that are simultaneous with the events at either end of the purple line in A's frame, not in B's frame. The event at the bottom of the purple line does not occur at t=1 in B's frame, and the event at the top of the frame does not occur at t=2 in B's frame. Nowhere in the diagram do you show points on B's axis that are simultaneous with the events at the top and bottom of the purple line, so the diagram just doesn't show tB for those particular events.
Yep, like I said. Except I said it more quickly, because I thought you would understand. In my words:
A pure doppler effect for one observer assumed to be stationary with the other observer is in motion at v=0.6c would be: (c+0.6c)/(c+0) = 1.6. To make relativistic doppler, we apply a gamma of 1.25 to get 1.6*1.25 = 2.
Yes, and I did follow that part. What I didn't follow was the subsequent statement "Application of the gamma of 1.25 is time dilation and the stretching of the tB axis." What does it mean to consider "stretching of the tB axis" as something separate from time dilation? You can see that in my derivation of the relativistic Doppler effect I didn't consider any effect from "stretching of the tB axis", I just made use of time dilation.
This would mean that the darker orange line is time dilation for A
The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top? If so your diagram doesn't actually show anything corresponding to this vertical distance, it'd be more clear if you drew a horizontal bar whose top and bottom edge cross these two events, so you could see the times this bar's top and bottom intersect the tA axis (which would represent the dilated time in A's frame between events 2 and 3 on B's worldline that are colocated in B's frame). Alternatively you could just relocate the darker orange line so it goes between events 1 and 2 on B's worldline, since you already have a light orange horizontal bar there.
But you can see that time dilation is such a totally different thing to Length contraction, which we have covered before, and that in the SAFTD diagrams the same would apply, the darker orange line would be SAFTD for A and the dark blue line would be SAFTD for B.
I would say that the horizontal length of the orange bar in the SAFTD diagram represents the spatial distance in A's frame between events 1 and 4 on the tB axis, which are simultaneous in B's frame, and that's exactly the idea of the SAFTD...in other words, if we write SAFTD as dx' = dx * gamma, then here dx is the distance in B's frame between events 1 and 4, and dx' is the distance in A's frame between events 1 and 4 (which can also be represented by the green line). As long as that matches what you're saying above, this diagram makes sense to me.
What you can't really do though, is measure these values directly. Even to show it on the diagram is difficult
Isn't showing where the appropriate bars cross the appropriate axis "showing it on the diagram"? For example, in the SAFTD diagram, you can see the dark orange line has a length of 3 in B's coordinate system (since it goes from x=1 to x=4 on B's space axis), but you can also see that the vertical light orange bar whose edges go through the two ends of the dark orange line intersects A's x-axis at two points that are a distance of 3*1.25 = 3.75 apart (which represents the distance in A's frame between the events on the ends of the dark orange line).
You are splitting stuff up and ignoring context again.

I said it was difficult to draw, didn't I?

Imagine that the tB axis is a wiper fixed at the origin. Move it so it is parallel with the tA axis. The tB is not only rotated, but also stretched. Do you recall anyone saying that?
Yes, I understand what you mean by "stretching", but that wasn't an issue I was asking in any of the questions I asked above, if you think stretching is relevant to my questions you need to explain why. If you think I'm ignoring context, it would help if you would address the specific questions in a way that explains what context I'm missing for that particular question. I asked a bunch of questions there, so if you want we can concentrate on just one:
This would mean that the darker orange line is time dilation for A
The orange line is slanted in A's frame, so what do you mean by "time dilation for A" if this line goes in a mix of space and time directions in A's frame? Are you talking about the purely vertical distance between event 2 at the bottom of the orange line and event 3 at the top?
Answering these questions would help me understand what you mean by "the darker orange line is time dilation for A", I didn't see any context there that would help me understand this (though perhaps your last comment below shines some light on what you meant).
The extent to which the tB axis is stretched is given by time dilation.
The axis is actually stretched by more than gamma, if you're referring to the diagonal distance on the diagram between a pair of ticks on B's time axis as compared with the vertical distance between a pair of ticks on A's time axis. If you draw it so that the vertical distance between ticks 1 and 2 on A's time axis is 1 centimeter, and then you look at ticks 1 and 2 on B's slanted time axis, then the purely vertical distance between these points is gamma*1 centimeter = 1.25 centimeter, while the horizontal distance is 0.6*1.25 centimeter = 0.75 centimeter, so the diagonal distance between these points must be sqrt(1.25^2 + 0.75^2) = 1.45774 cm. So, you can see that the extent of the stretching in the diagram is not given by the gamma factor that appears in the time dilation equation.

neopolitan

Apr22-09, 09:14 PM

Yes, here I assume the two events would be on either end of the dark blue line, the events labeled 2 and 3 on A's time axis. In A's frame tA for these events is 1, while in B's frame tB for these events is 1.25. Can I assume it's this pair of events, and the time between them in both frames, that you were referring to when you said "What the diagram tried to show is a value seen from the A frame (from the A frame, ticks are all colocal) such that tB = gamma.tA"? But in this case the time dilation occurs in B's frame, so I still don't really understand why you labeled the green line "time dilation A looking at B" rather than "B looking at A".

I find time dilation really awkward. It seems I am not alone.

Can you step back, momentarily and just follow this and see if it is right?

Two events on the tA axis are consecutive ticks of a clock, so tA = 1 tick.

These two events are simultaneous in the B frame with two events on the tB axis which are separated by tB = 1 tick.gamma.

Assuming a gamma of 1.25, if tA = 1s then tB = 1.25s

Is that right?

Now the unprimed frame is that in which ticks are colocal. The primed frame in that in which ticks are not colocal.

If we take the A frame as our reference point, the A frame is the frame in which ticks are colocal and the B frame is the frame in which ticks are not colocal.

Therefore, tA = t and tB = t'.

Therefore t' = tB = tA.gamma = t.gamma, or

t' = t.gamma

Which is your time dilation equation from the A frame, considering the B frame, or "A looking at B".

They're not colocated with the clock at x=0 in the A frame, but the two events are colocated with one another in A's frame.

I thought that was staggeringly obvious, but yes.

But the light orange sheet shows the events on B's t-axis that are simultaneous with the events at either end of the purple line in A's frame, not in B's frame. The event at the bottom of the purple line does not occur at t=1 in B's frame, and the event at the top of the frame does not occur at t=2 in B's frame. Nowhere in the diagram do you show points on B's axis that are simultaneous with the events at the top and bottom of the purple line, so the diagram just doesn't show tB for those particular events.

The purple line spans two colocated events in the A frame which are simultaneous in the A frame with two consecutive ticks of the B clock. Note that if this is an issue for you with the purple line, it should be an issue for you with the green line and any issue you have with the green line should also be an issue for you with the purple line.

Did you previously understand that? Perhaps this might clarify:

The purple/green line spans two colocated events in the A/B frame which are simultaneous in the A/B frame with two consecutive ticks of the B/A clock.

Yes, and I did follow that part. What I didn't follow was the subsequent statement "Application of the gamma of 1.25 is time dilation and the stretching of the tB axis." What does it mean to consider "stretching of the tB axis" as something separate from time dilation? You can see that in my derivation of the relativistic Doppler effect I didn't consider any effect from "stretching of the tB axis", I just made use of time dilation.

Ok, this makes more sense now. I was wondering where the confusion was.

Application of the gamma of 1.25 is time dilation and the stretching of the tB axis is also representative of time dilation. Maybe that doesn't help right now, but see my response below.

Yes, I understand what you mean by "stretching", but that wasn't an issue I was asking in any of the questions I asked above, if you think stretching is relevant to my questions you need to explain why. If you think I'm ignoring context, it would help if you would address the specific questions in a way that explains what context I'm missing for that particular question. I asked a bunch of questions there, so if you want we can concentrate on just one:

Answering these questions would help me understand what you mean by "the darker orange line is time dilation for A", I didn't see any context there that would help me understand this (though perhaps your last comment below shines some light on what you meant).

The axis is actually stretched by more than gamma, if you're referring to the diagonal distance on the diagram between a pair of ticks on B's time axis as compared with the vertical distance between a pair of ticks on A's time axis. If you draw it so that the vertical distance between ticks 1 and 2 on A's time axis is 1 centimeter, and then you look at ticks 1 and 2 on B's slanted time axis, then the purely vertical distance between these points is gamma*1 centimeter = 1.25 centimeter, while the horizontal distance is 0.6*1.25 centimeter = 0.75 centimeter, so the diagonal distance between these points must be sqrt(1.25^2 + 0.75^2) = 1.45774 cm. So, you can see that the extent of the stretching in the diagram is not given by the gamma factor that appears in the time dilation equation.

I know you had a specific question, but it is sort of off track, so perhaps answering it will cause more confusion that trying to address the core issue.

In your last paragraph above, you make an incorrect assumption. Note that I didn't say that time dilation accounts for stretching of the tA that would be required to match the tB axis.

I fully admit that I wasn't being entirely clear, but I did say:

The extent to which the tB axis is stretched is given by time dilation.

I have done another couple of diagrams, showing Galilean relativity. These diagrams kill two birds with one stone, since they hopefully show that maybe I was right about instantaneous transmission of information being an issue - keeping in mind that the Galilean boost is:

x' = x - vt
(not x' = x - vt')

They are here (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg) and here (http://www.geocities.com/neopolitonian/gal_rel_neopolitan.jpg).

That aside, look at the separation between the ticks of the B clock in Galilean relativity (either diagram). If the tB axis is imagined as a wiper and swung around to line up with the tA axis then the ticks are further apart. Now go back to the original time dilation diagram and keep in mind the comment about the tB axis being stretched.

cheers,

neopolitan

PS This was written yesterday when the system wasn't letting me post because of a disc failure. I think I finished editing it, but it is possible that it is not quite the finished product I wanted it to be.

JesseM

Apr22-09, 10:31 PM

I find time dilation really awkward. It seems I am not alone.

Can you step back, momentarily and just follow this and see if it is right?

Two events on the tA axis are consecutive ticks of a clock, so tA = 1 tick.

These two events are simultaneous in the B frame with two events on the tB axis which are separated by tB = 1 tick.gamma.

Assuming a gamma of 1.25, if tA = 1s then tB = 1.25s

Is that right?
Yeah, all that is right. But if you're drawing things from the perspective of the A frame, you may find it less awkward to think about ticks on a clock which is moving in the A frame, and the time between them in the A frame (switching all the A's and B's in your explanation above). If two events on the B worldline are separated by 1 second in B's frame, then the time between these events in A's frame--which is just the vertical distance between the events in the diagram--is 1.25 seconds.
Now the unprimed frame is that in which ticks are colocal. The primed frame in that in which ticks are not colocal.

If we take the A frame as our reference point, the A frame is the frame in which ticks are colocal and the B frame is the frame in which ticks are not colocal.

Therefore, tA = t and tB = t'.

Therefore t' = tB = tA.gamma = t.gamma, or

t' = t.gamma

Which is your time dilation equation from the A frame, considering the B frame, or "A looking at B".
Your equations are right but I still just don't get the phrasing...why do you say the time dilation is "from the A frame" when the actual dilation (greater amount of time) occurs in the B frame in your example? You're taking two events on A's worldline which have a separation of 1 second in A's frame, then using that to figure out the time between these same events in the B frame, namely 1.25 seconds. So aren't you figuring out what B observing when it's "looking at" A's clock, and observing that A's ticks seem to be dilated by a factor of 1.25?

Like I said, I think if you're going to draw things in A's frame, it's much more natural to think about time dilation if you pick two events on B's worldline and then understand the dilated time to be the vertical distance between these events in the diagram of A's frame (like events 1 and 2 on B's time axis in your diagram, where the vertical distance is shown by the purple line). In this case tB is the frame where the events are colocated and tA is the frame where they're not, so you still have the equation t' = t*gamma, but now t is the time interval in B's frame and t' is the time interval in A's frame. It's this that I would call "the time dilation in A's frame", since A's time t' is dilated relative to B's time t, although the meaning of such words is ambiguous and perhaps you find it more natural to define them differently.
The purple line spans two colocated events in the A frame which are simultaneous in the A frame with two consecutive ticks of the B clock.
If that's the intended meaning of the purple line that's fine with me, although visually I think it would be more clear that the purple line is supposed to relate to those specific ticks (ticks 1 and 2) on B's worldline if the dark orange line went between those ticks rather than two other ticks. In that case the relation between the vertical purple line and the slanted dark orange line in the time dilation illustration would be exactly analogous to the relation between the horizontal green line and the slanted dark orange line in the SAFTD illustration. But strangely you label the horizontal green line "A looking at B" while you label the vertical purple line "B looking at A"--whatever "looking at" is supposed to mean, you're using it inconsistently here, because the SAFTD is supposed to be exactly analogous to time dilation but with the roles of space and time reversed.
Note that if this is an issue for you with the purple line, it should be an issue for you with the green line and any issue you have with the green line should also be an issue for you with the purple line.
But in the case of the green line, the diagram makes it clear it's supposed to relate to the same to events spanned by the dark blue line.
I know you had a specific question, but it is sort of off track, so perhaps answering it will cause more confusion that trying to address the core issue.

In your last paragraph above, you make an incorrect assumption. Note that I didn't say that time dilation accounts for stretching of the tA that would be required to match the tB axis.

I fully admit that I wasn't being entirely clear, but I did say:
The extent to which the tB axis is stretched is given by time dilation.

I done another couple of diagrams, showing Galilean relativity. These diagrams kill two birds with one stone, since they hopefully show that maybe I was right about instantaneous transmission of information being an issue - keeping in mind that the Galilean boost is:

x' = x - vt
(not x' = x - vt')

They are here (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg) and here (http://www.geocities.com/neopolitonian/gal_rel_neopolitan.jpg).
I think you need some other label for the first diagram besides Galilean relativity where t does not equal t', since in the Galilean boost (i.e. the Galilei transformation) t does equal t', and in the first diagram x' does not equal x - vt either (except in the special case of events along B's time axis of x'=0), so the first diagram would represent some other coordinate transformation that is neither Galilean nor Lorentzian. The second diagram does correctly show two coordinate systems related by the Galilei transformation though. But going back to the issue of what you meant by "The extent to which the tB axis is stretched is given by time dilation", are you saying that since the diagonal distance between ticks on B's axes already appear visually stretched in these diagrams (though of course the vertical distance between ticks on B's time axis is the same as the vertical distance between ticks on A's time axis in both diagrams, and the horizontal distance between ticks on B's space axis is the same as the horizontal distance between ticks on A's space axis in the first diagram), we can apply gamma to this preexisting stretching to get the amount of diagonal stretching seen on the diagram of the Lorentz transform?

neopolitan

Apr23-09, 01:19 AM

Yeah, all that is right. But if you're drawing things from the perspective of the A frame, you may find it less awkward to think about ticks on a clock which is moving in the A frame, and the time between them in the A frame (switching all the A's and B's in your explanation above). If two events on the B worldline are separated by 1 second in B's frame, then the time between these events in A's frame--which is just the vertical distance between the events in the diagram--is 1.25 seconds.

The B clock is moving in the A frame. So ticks on the tB axis are "ticks on a clock which is moving in the A frame" which should make you happy with this diagram (http://www.geocities.com/neopolitonian/TDv2.JPG).

Your equations are right but I still just don't get the phrasing...why do you say the time dilation is "from the A frame" when the actual dilation (greater amount of time) occurs in the B frame in your example? You're taking two events on A's worldline which have a separation of 1 second in A's frame, then using that to figure out the time between these same events in the B frame, namely 1.25 seconds. So aren't you figuring out what B observing when it's "looking at" A's clock, and observing that A's ticks seem to be dilated by a factor of 1.25?

Sort of, yes. But that is the confusing thing with time dilation. In which frame is the clock moving? Work from the A frame, in which the A clock is at rest, which means that tA = tunprimed and tB = tprimed, or more simply tA = t and tB = t'.

You want to me go from B's frame, so that means the B clock is the one at rest and the A clock is in motion, which means that tB = t and tA = t'. Then, if you want me to use the same sheet as I was using (in this diagram (http://www.geocities.com/neopolitonian/TD.JPG)), ie the light blue one which:

when it intersects the tA axis spans 1 tick, therefore t' = 1 tick
and
when it intersects the tB axis spans 1 tick.gamma, therefore t = 1 tick.gamma

therefore:

t=t'.gamma

Which is inverse time dilation and is not what I am trying to show. I don't think you want me to use the blue sheet as it is.

I think you want me to use the orange sheet or something like it as you go on to explain ...

Like I said, I think if you're going to draw things in A's frame, it's much more natural to think about time dilation if you pick two events on B's worldline and then understand the dilated time to be the vertical distance between these events in the diagram of A's frame (like events 1 and 2 on B's time axis in your diagram, where the vertical distance is shown by the purple line). In this case tB is the frame where the events are colocated and tA is the frame where they're not, so you still have the equation t' = t*gamma, but now t is the time interval in B's frame and t' is the time interval in A's frame. It's this that I would call "the time dilation in A's frame", since A's time t' is dilated relative to B's time t, although the meaning of such words is ambiguous and perhaps you find it more natural to define them differently.

But isn't that just what the light orange sheet shows?

Do you still not understand that the light orange sheet is the skewed rotation of the light blue sheet and the purple line is the skewed rotation of the green line, and that if you recast the whole thing so that the tB and xB axes were square, then you would have a (sort of) mirror image of this diagram? (Mirrored in the sense that the tA axis would have a negative slope, not mirrored in the sense that the green photon worldlines would not also be mirrored.)

I'm asking this straight out, because I just don't understand what your problem is. You seem to be saying "you are doing this completely wrong with the blue sheet, look do it this way" and then you go and do what I have done with the orange sheet, but to do that you have to change all the terminology used with the blue sheet, and lo and behold, you have the terminology that I used with the orange sheet.

Try this. Start off the way you like it. Use the orange sheet. Then, try to see how it would look if you flipped frames - and how you would draw it on the same diagram. If it is different from the way it looks on my blue sheet, then we have something to talk about. Otherwise, I really think you are at cross purposes.

If that's the intended meaning of the purple line that's fine with me, although visually I think it would be more clear that the purple line is supposed to relate to those specific ticks (ticks 1 and 2) on B's worldline if the dark orange line went between those ticks rather than two other ticks. In that case the relation between the vertical purple line and the slanted dark orange line in the time dilation illustration would be exactly analogous to the relation between the horizontal green line and the slanted dark orange line in the SAFTD illustration. But strangely you label the horizontal green line "A looking at B" while you label the vertical purple line "B looking at A"--whatever "looking at" is supposed to mean, you're using it inconsistently here, because the SAFTD is supposed to be exactly analogous to time dilation but with the roles of space and time reversed.

My error. It's fixed.

I think you need some other label for the first diagram besides Galilean relativity where t does not equal t', since in the Galilean boost (i.e. the Galilei transformation) t does equal t', and in the first diagram x' does not equal x - vt either (except in the special case of events along B's time axis of x'=0), so the first diagram would represent some other coordinate transformation that is neither Galilean nor Lorentzian. The second diagram does correctly show two coordinate systems related by the Galilei transformation though.

Jesse, it's my whole point ie "in the Galilean boost (i.e. the Galilei transformation) t does equal t'". That diagram shows Galilean relativity as you tried to tell me it would work, even if information is not transmitted instantaneously. That's why that diagram's file is called "gal_rel_JesseM.jpg".

Note that in the diagram I don't have x' = x - vt. I have x' = x - vt'. Given that, do you agree that t' does not equal t?

But going back to the issue of what you meant by "The extent to which the tB axis is stretched is given by time dilation", are you saying that since the diagonal distance between ticks on B's axes already appear visually stretched in these diagrams (though of course the vertical distance between ticks on B's time axis is the same as the vertical distance between ticks on A's time axis in both diagrams, and the horizontal distance between ticks on B's space axis is the same as the horizontal distance between ticks on A's space axis in the first diagram), we can apply gamma to this preexisting stretching to get the amount of diagonal stretching seen on the diagram of the Lorentz transform?

Yes (it is the stretching shown in diagrams TD, LC, SAFTD and TAFLC which was calculated using the Lorentz transformations).

cheers,

neopolitan

JesseM

Apr23-09, 02:28 AM

Yeah, all that is right. But if you're drawing things from the perspective of the A frame, you may find it less awkward to think about ticks on a clock which is moving in the A frame
The B clock is moving in the A frame.
But you weren't talking about ticks on the B clock when you said "Two events on the tA axis are consecutive ticks of a clock, so tA = 1 tick", you were talking about ticks on the A clock, and figuring out how far apart they were in the B frame. There's nothing wrong with doing that of course, I was just saying that if you want to think about time dilation in a diagram drawn from the A frame perspective, it's conceptually easier to think about two ticks on the B clock which is moving in this frame, and then the dilated time between these ticks in the A frame is just the vertical distance between where the ticks are drawn in the diagram.
So ticks on the tB axis are "ticks on a clock which is moving in the A frame" which should make you happy with this diagram (http://www.geocities.com/neopolitonian/TDv2.JPG).
Yes, I'm quite happy with that diagram (although to make it perfect you'd need to extend the bottom of the light orange and light blue bars to make them line up better with the bottom of the dark orange line segment and the dark blue line segment...not really important because I understand the intent though).
Your equations are right but I still just don't get the phrasing...why do you say the time dilation is "from the A frame" when the actual dilation (greater amount of time) occurs in the B frame in your example? You're taking two events on A's worldline which have a separation of 1 second in A's frame, then using that to figure out the time between these same events in the B frame, namely 1.25 seconds. So aren't you figuring out what B observing when it's "looking at" A's clock, and observing that A's ticks seem to be dilated by a factor of 1.25?
Sort of, yes. But that is the confusing thing with time dilation. In which frame is the clock moving? Work from the A frame, in which the A clock is at rest, which means that tA = tunprimed and tB = tprimed, or more simply tA = t and tB = t'.
Well, only if you're using tA to represent the time in the A frame between two events on A's own clock. I was suggesting you use tA to represent the time in the A frame between two events on B's clock, in which case tA = tprimed.
You want to me go from B's frame
No I don't, I want you to use two events on B's clock but figure out the coordinate time between these events in A's frame. This is the most natural way to think about time dilation--whatever frame you're using (A in this case), it tells you how much the time between events on a clock moving relative to that frame is dilated in that frame, if you know how much proper time the clock itself measured between them. In this case, if there are two events on B's clock which the clock itself measured as being separated by 1 second, in A's frame these same two events are separated by a dilated coordinate time of 1.25 seconds.

It's always simplest to think about time dilation in terms of a relation between a proper time on a clock and a coordinate time in a frame where the clock is moving, I think. If you try to think about two different clocks at rest in different frames, and lines of simultaneity connecting events on one clock with events on another, it gets way more confusing.
so that means the B clock is the one at rest and the A clock is in motion, which means that tB = t and tA = t'. Then, if you want me to use the same sheet as I was using (in this diagram (http://www.geocities.com/neopolitonian/TD.JPG)), ie the light blue one which:

when it intersects the tA axis spans 1 tick, therefore t' = 1 tick
and
when it intersects the tB axis spans 1 tick.gamma, therefore t = 1 tick.gamma
If you're using the events at either end of the dark blue line segment, then these events are colocated in the A frame where the time between them is 1 second, so that should be the unprimed frame, while the primed frame should be the B frame where they are not colocated and the coordinate time between them is 1*gamma (the height of the light blue sheet in the B frame).
therefore:

t=t'.gamma

Which is inverse time dilation and is not what I am trying to show. I don't think you want me to use the blue sheet as it is.
See above, you weren't sticking to the convention that the unprimed frame is the one where the events are colocated.
I think you want me to use the orange sheet or something like it as you go on to explain ...
Like I said, I think if you're going to draw things in A's frame, it's much more natural to think about time dilation if you pick two events on B's worldline and then understand the dilated time to be the vertical distance between these events in the diagram of A's frame (like events 1 and 2 on B's time axis in your diagram, where the vertical distance is shown by the purple line). In this case tB is the frame where the events are colocated and tA is the frame where they're not, so you still have the equation t' = t*gamma, but now t is the time interval in B's frame and t' is the time interval in A's frame. It's this that I would call "the time dilation in A's frame", since A's time t' is dilated relative to B's time t, although the meaning of such words is ambiguous and perhaps you find it more natural to define them differently.
But isn't that just what the light orange sheet shows?
Sure, but in the above paragraph I was responding to the verbal analysis in your previous post, which was specifically about two events on the worldline of A's clock and the time between them in A's frame and B's frame. Again, there was nothing wrong with this analysis, but I was just suggesting time dilation is easier to understand intuitively if you pick one frame (frame A in your diagram) and then consider events on a clock which is moving in that frame, the time dilation equation relating the proper time as measured by the clock to the coordinate time as measured in the frame you're using (which in this frame will just be the vertical distance between the two events).
Do you still not understand that the light orange sheet is the skewed rotation of the light blue sheet and the purple line is the skewed rotation of the green line
Of course I understand that.
I'm asking this straight out, because I just don't understand what your problem is. You seem to be saying "you are doing this completely wrong with the blue sheet, look do it this way" and then you go and do what I have done with the orange sheet, but to do that you have to change all the terminology used with the blue sheet, and lo and behold, you have the terminology that I used with the orange sheet.
Where did I say you were "completely wrong" about anything? The main issue I brought up was that I thought the words you used to describe segments, like "A looking at B", seemed backwards to me from how I would naturally interpret them (and also inconsistent with how you used the same words in the SAFTD diagram). The other minor issue I brought up was that if the light orange bar was showing the time in A's frame between events 1 and 2 on B's clock, the diagram would be more clear if the dark orange line segment went between events 1 and 2, as you have it in the new diagram here (http://www.geocities.com/neopolitonian/TDv2.JPG).
Jesse, it's my whole point ie "in the Galilean boost (i.e. the Galilei transformation) t does equal t'". That diagram shows Galilean relativity as you tried to tell me it would work, even if information is not transmitted instantaneously. That's why that diagram's file is called "gal_rel_JesseM.jpg".
You misunderstood me, my point about the speed of information transmission had nothing to do with suggesting a new type of coordinate transformation different from the standard Galilei transformation. My point was that I didn't understand why you thought the standard Galilei transformation implied instantaneous information transmission in the first place! I was saying that it would still make sense to use the standard Galilei transformation in a universe with basically Newtonian laws but where information transmission had some finite upper limit--why wouldn't it? As long as the laws of physics are such that moving clocks don't slow down and moving rulers don't shrink, then if different observers construct their coordinate systems using a lattice of inertial rulers and clocks at rest relative to themselves, and different observers use a method of synchronizing clocks that won't cause different frames to disagree about simultaneity (like bringing the clocks together to a single location and synchronizing them there, then moving them apart to their respective positions in the lattice, which unlike in relativity won't cause them to get-out-of-sync because there is no time dilation associated with movement), then the resulting coordinate systems will be related by the standard Galilei transformation. Nothing here depends on whether information is transmitted instantaneously or at finite speed, it's an irrelevant issue.
Note that in the diagram I don't have x' = x - vt. I have x' = x - vt'. Given that, do you agree that t' does not equal t?
Yeah, but in my last post I wasn't disagreeing with you that t doesn't equal t' in your first diagram (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg), I was just saying you shouldn't use the term Galilean relativity where t does not equal t' to refer to it because this coordinate transformation isn't "Galilean" at all.

neopolitan

Apr23-09, 07:17 AM

Yes, I'm quite happy with that diagram (although to make it perfect you'd need to extend the bottom of the light orange and light blue bars to make them line up better with the bottom of the dark orange line segment and the dark blue line segment...not really important because I understand the intent though).

Then you effectively end up with what I started with. I thought I would have to tell you that what is already is what I started off with, even without your suggested modification.

Take the light orange sheet from here (http://www.geocities.com/neopolitonian/TDv2.JPG), make it wider to span consecutive ticks on the tB axis.

Move the light orange sheet down so it spans tB=1 to tB=2

Then put a purple bar on the tA axis where the light orange sheet crosses it. Then to make it visually clearer (so you don't have bars lying over each other), move the purple bar to xA=6.

Then take the light blue sheet and make it wider to span consecutive ticks on the tA axis and move it down slight so it spans tA=2 to tA=3.

Then put a green bar on the tB axis where the light blue sheet crosses it. Then to make it visually clearer, move the green bar along the light blue sheet until it aligns with xB=2.

Can you see now?

If you are happy with this (http://www.geocities.com/neopolitonian/TDv2.JPG) you don't really have any reason not to be happy with this (http://www.geocities.com/neopolitonian/TD.JPG). Because they are the same.

Unless you can see this, then we will have to agree to disagree.

You misunderstood me, my point about the speed of information transmission had nothing to do with suggesting a new type of coordinate transformation different from the standard Galilei transformation. My point was that I didn't understand why you thought the standard Galilei transformation implied instantaneous information transmission in the first place! I was saying that it would still make sense to use the standard Galilei transformation in a universe with basically Newtonian laws but where information transmission had some finite upper limit--why wouldn't it? As long as the laws of physics are such that moving clocks don't slow down and moving rulers don't shrink, then if different observers construct their coordinate systems using a lattice of inertial rulers and clocks at rest relative to themselves, and different observers use a method of synchronizing clocks that won't cause different frames to disagree about simultaneity (like bringing the clocks together to a single location and synchronizing them there, then moving them apart to their respective positions in the lattice, which unlike in relativity won't cause them to get-out-of-sync because there is no time dilation associated with movement), then the resulting coordinate systems will be related by the standard Galilei transformation. Nothing here depends on whether information is transmitted instantaneously or at finite speed, it's an irrelevant issue.

Perhaps we disagree about what is relevant and irrelevant.

To see the equation x' = x - vt, in "my" Galilean relativity, which we seem to both agree is Galilean relativity rather than the other one which I ascribed to you, it requires either:

instant understanding of where A is and B and the event is at 5 ticks, rather than waiting until 8 ticks when a photon from E reaches A then working backwards to see what happened when that same photon passed B to get a similar equation,

a god like observer (but really that god like observer sees everything instantly, so you are back at square one) and the god like observer implies a preferred frame (part of the perks of being a god).

Yeah, but in my last post I wasn't disagreeing with you that t doesn't equal t' in your first diagram (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg), I was just saying you shouldn't use the term Galilean relativity where t does not equal t' to refer to it because this coordinate transformation isn't "Galilean" at all.

I don't what you were going on about a few posts ago then (post #179). Perhaps you can show a diagram in which you have an aether frame which works with Galilean relativity and you get the right equation - and you show it complete with photons and no hint of instantaneous transmission of information.

cheers,

neopolitan

JesseM

Apr23-09, 08:54 AM

Then you effectively end up with what I started with. I thought I would have to tell you that what is already is what I started off with, even without your suggested modification.

Take the light orange sheet from here (http://www.geocities.com/neopolitonian/TDv2.JPG), make it wider to span consecutive ticks on the tB axis.
Wait, so it was intentional that the light orange bar doesn't already span from one end of the dark orange line segment to the other? In that case I'm not very happy with the diagram, I thought that was just an accident of your drawing, which is why I said "(although to make it perfect you'd need to extend the bottom of the light orange and light blue bars to make them line up better with the bottom of the dark orange line segment and the dark blue line segment...not really important because I understand the intent though)". I thought the point of the light orange bar was that its vertical height showed the dilated time in the A frame between the two events on the dark orange bar which had a separation of 1 second on B's clock.
Move the light orange sheet down so it spans tB=1 to tB=2

Then put a purple bar on the tA axis where the light orange sheet crosses it. Then to make it visually clearer (so you don't have bars lying over each other), move the purple bar to xA=6.
But if you did that, can you also move the dark orange line segment down, so that it spans the same two ticks that are spanned by the light orange bar? Remember, that was my only visual criticism of your original diagram, that it was confusing that the dark orange line segment spanned two different ticks than the light orange bar, so the light orange bar wasn't clearly indicated as illustrating the dilated time between the two ends of the dark orange line segment (and hence the purple line segment wasn't clearly indicated as illustrating this either).
Then take the light blue sheet and make it wider to span consecutive ticks on the tA axis and move it down slight so it spans tA=2 to tA=3.
Again, this is what I thought your intention was, for the light blue slanted bar to span from one end of the dark blue line segment to the other. If not then again I take back what I said about being happy with the diagram, the point of each light-colored bar should be to show the dilated time in one frame between points on the worldline of a clock which is at rest in the other frame, since again it is easiest to conceptualize time dilation as relating the proper time between events on a clock to the coordinate time between the same events in the frame where the clock is moving.
Then put a green bar on the tB axis where the light blue sheet crosses it. Then to make it visually clearer, move the green bar along the light blue sheet until it aligns with xB=2.

Can you see now?
Can I see what, exactly? You talk as though there was something about the original diagram I didn't "see". Again, my main criticism of the original (http://www.geocities.com/neopolitonian/TD.JPG) was just about the "A looking at B" terminology, and the other minor criticism was that I thought the dark orange line segment should be moved down so it was spanned by the light orange bar, just as the light blue bar spanned the dark blue line segment (that way each bar is clearly showing the time interval in one frame between events on the worldline of a clock at rest in the other frame)
Perhaps we disagree about what is relevant and irrelevant.

To see the equation x' = x - vt, in "my" Galilean relativity, which we seem to both agree is Galilean relativity rather than the other one which I ascribed to you, it requires either:

instant understanding of where A is and B and the event is at 5 ticks, rather than waiting until 8 ticks when a photon from E reaches A then working backwards to see what happened when that same photon passed B to get a similar equation,

a god like observer (but really that god like observer sees everything instantly, so you are back at square one) and the god like observer implies a preferred frame (part of the perks of being a god).
I really don't understand why you think standard Galilean relativity "requires" this, since the Galilie transformation is about relating the coordinates that two observers assign to events, not about when they "know" the events occurred. The same is true of the Lorentz transform! In SR if an event E occurs 10 light-seconds away from my position and I assign it the coordinate time t=5, same as the coordinate time of my own clock reading 5 seconds, you understand that doesn't mean I actually instantaneously know about the event as soon as my clock reads 5 seconds, right? The light from this event can't actually reach me until t=15, so I won't know it occurred until then. But if I use a lattice of rulers and synchronized clocks to define my coordinate system, then when the light does reach me at t=15, I'll be able to see that the event occurred next to the 10 light-second mark on my rulers, and that the clock sitting at that mark (which is synchronized with mine) read 5 seconds as the event happened. So, in retrospect I assign this event the position coordinate x=10 and time coordinate t=5, even though I myself was totally oblivious to the existence of the event prior to t=15.

Well, it would be exactly the same in a Galilean universe with a finite speed of information transmission. Each observer would define their own coordinate system using a network of rulers and synchronized clocks at rest relative to themselves, and the coordinates of distant events would be determined in retrospect based on what ruler-marking and clock-reading the event was seen to be next to when it had happened. The difference in Galilean relativity would be that 1) one observer's rulers wouldn't be measured to be shrunk relative to another observer's if the first was moving relative to the second, 2) one observer's clock wouldn't be measured to be slowed down relative to another's if the first was moving relative to the second, and 3) different observers wouldn't disagree about simultaneity, because they could all agree to synchronize their own clocks by bringing them together to a common location and synchronizing them there before moving them to their respective positions on the lattice. As long as 1-3 are true, then naturally the coordinates that different observers assign to the same event will be related by the Galilei transform.
I don't what you were going on about a few posts ago then (post #179). Perhaps you can show a diagram in which you have an aether frame which works with Galilean relativity and you get the right equation - and you show it complete with photons and no hint of instantaneous transmission of information.
Since the coordinates of different observers would be related by the Galilei transform, different coordinate axes would be drawn just as in this diagram (http://www.geocities.com/neopolitonian/gal_rel_neopolitan.jpg). If you want to show a finite speed of information transmission in this diagram, just draw light cones emanating from events. The time that an observer learns about a given event will be represented by the time his worldline crosses into the future light cone of the event, which will be later than the time the event actually occurred--just as in SR. The only difference with SR is that if light moves at the same speed in both directions in one preferred frame, so both sides of the light cone are at the same angle from the horizontal in the diagram for this frame, then in other frames light would move at different speeds in different directions so the two sides of the light cone would have different angles if you drew things from the perspective of other frames.

neopolitan

Apr23-09, 09:54 AM

Wait, so it was intentional that the light orange bar doesn't already span from one end of the dark orange line segment to the other?

We are both bright enough to know that it is just dividing through by gamma, right? It doesn't change the equations.

We do know that don't we?

Remember, that was my only visual criticism of your original diagram, that it was confusing that the dark orange line segment spanned two different ticks than the light orange bar, so the light orange bar wasn't clearly indicated as illustrating the dilated time between the two ends of the dark orange line segment (and hence the purple line segment wasn't clearly indicated as illustrating this either).

Can I see what, exactly? You talk as though there was something about the original diagram I didn't "see". Again, my main criticism of the original (http://www.geocities.com/neopolitonian/TD.JPG) was just about the "A looking at B" terminology, and the other minor criticism was that I thought the dark orange line segment should be moved down so it was spanned by the light orange bar, just as the light blue bar spanned the dark blue line segment (that way each bar is clearly showing the time interval in one frame between events on the worldline of a clock at rest in the other frame)

So if, in the first time dilation diagram, if I moved the dark orange bar down slightly so that it spanned a specific pair of events, you'd be happier?

It wasn't overly important to me, the axis bars were just showing a tick, not any specific tick. But I see what you mean.

The odd thing is that you are happy with the same terminology as I have in the original diagram being used the second. Here (http://www.geocities.com/neopolitonian/TD.JPG) and here (http://www.geocities.com/neopolitonian/TDv2.JPG).

I need to think more about the Galilean - Lorentz thing, but I am basically talked around.

What leaves me a little stumped is ... it worked. So, I need to see what it is that makes it work.

cheers,

neopolitan

JesseM

Apr23-09, 10:03 AM

We are both bright enough to know that it is just dividing through by gamma, right? It doesn't change the equations.
When you say "it is just dividing through by gamma", what does "it" refer to? What quantity are you dividing by gamma to get what other quantity? Do you mean that if we look at the horizontal orange bar spanned between the top of the dark orange slanted line segment, then if you take the height of the orange bar and divide by gamma, you get the proper time between the ends of the dark orange segment? If so I agree of course, but if that's not what you were referring to could you clarify?
So if, in the first time dilation diagram, if I moved the dark orange bar down slightly so that it spanned a specific pair of events, you'd be happier?
By "dark orange bar" you mean the slanted line segment rather than the thick horizontal orange bar, right? And yeah, if you moved the line segment down so that its top and bottom coincided with the upper and lower edges of the horizontal orange bar, I think the visual relation between them would be clearer.
The odd thing is that you are happy with the same terminology as I have in the original diagram being used the second. Here (http://www.geocities.com/neopolitonian/TD.JPG) and here (http://www.geocities.com/neopolitonian/TDv2.JPG).
I admit I didn't actually think about the terminology in the second one, I was just saying I was happy with what was being shown visually.
I need to think more about the Galilean - Lorentz thing, but I am basically talked around.

What leaves me a little stumped is ... it worked. So, I need to see what it is that makes it work.
When you say "it worked", as before I'm not quite sure what the "it" refers to...did you do some sort of calculation to check what I was saying?

JesseM

Apr23-09, 12:54 PM

Responding also to your older post #181:
You want to reintroduce an aether?
Not in real life, but then I don't want to reintroduce Galilei-symmetric physics in real life either, I was just making the point that the Galilei transformation is not incompatible with a finite speed of information transmission. But I think we cleared this up in the most recent posts.
Really, I am just going from the Galilean boost to Lorentz Transformations though. That boost is given by x'=x-vt. Do we at least agree on that?
Yes, that's the spatial component of the boost, and of course the temporal part is t'=t.
The Galilean assumption, in terms of my diagram, is that B is moving with an absolute velocity of v towards a location E which is a distance of x from A and, at a time t, the distance from B to E is x' = x - vt. This means that when t=0, A and B were colocated. Do we agree on that?
I assume x' means the distance of B from E, so that's fine.
In Galilean relativity, at t, A has not moved, B is moving with a velocity of v and is located vt closer to E than A is. Do we agree on that?
A has not moved in the A frame, although it has moved in the B frame (there is no need to assume absolute space in Galilean relativity, although of course many classical physicists did believe in absolute space). Of course in the B frame it's E that's moving towards B, but either way, it's true that at time t (or t') B will be vt (or vt') closer to E than A is.
In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t) in B's frame. Do we agree on that?
Yes, and the Galilei transformation is telling us that if an event has coordinates x,t in A's frame, it has coordinates x'=x-vt, t'=t in B's frame.
In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t') in B's frame, because time is absolute and t=t'. Do we agree on that?
The first part just looks like a repeat of the previous paragraph, but why do you say "because time is absolute and t=t' "? It's certainly true that t=t', but I don't see how the fact that an event has coordinates x,t in A's frame and x',t' in B's frame is "because" of this.
In Galilean relativity, if B is told that E is currently x' away, and B has observed that A has been moving away at -vt, then B will calculate that A-E is currently x = x' + vt . Do we agree on that?
Yes, if an event occurs at time t' on E's worldline at position x' in B's frame, then the Galilei transformation tells us that the same event must occur at position x = x' + vt' in A's frame, and since t'=t this is also at position x = x' + vt.
Do we further agree that if an event took place at (x,t) in A's frame in Special Relativity and even in a more careful analysis of Galilean relativity, that neither A nor B would know about it until a photon from the event is received?
Yes, but in SR the time-coordinate they learned about the event would be different than tht time-coordinate they retroactively assigned to the event itself, as discussed before.
If x = ct, in Galilean relativity, when A receives the photon at 2t, x' = x - 2vt. Do we agree that if we now talk about where a photon from the same event (x,t) hits B, this is not x' as calculated above?
But x' = x - vt relates the coordinates of a single event in two frames. The event of a photon hitting B is different from the event of a photon hitting A, so we would not expect the x,t coordinates of the photon hitting A and the x',t' coordinates of the photon hitting B to be related by the Galilei transformation.
I guess I could agree that Galilean relativity is based on either absolute space (ie there's an aether frame) or instantaneous transmission of information. Can you agree that it is one or the other?
I don't see why, can you explain? If by "aether frame" you just mean there's one frame where the upper limit on the speed of information is the same in both directions, while in other frames it's different in one direction than the other, then I'd agree with that, but I'm not sure that's what you meant.
Can you see that if information is transmitted instantaneously and an event takes place at (x,t) in the A frame, then in the A frame that event will be detected by A at (0,t) and B at (vt,t)? And in the B frame, the event was at (x',t), B detects is at (0,t) and A at (-vt,t) where x'=x-vt? And can you see that these can all be related by the Galilean boosts?
Sure, but this is trivially true since the coordinates of any event in two different frames are related by the Galilei boosts if you're dealing with Galilean frames, that's the very definition of what the Galilei transformation is supposed to do! If information is transmitted at c in the -x direction of A's frame (which means it's transmitted at c+v in the -x' direction of B's frame, based on Galilean velocity addition), then we can also figure out when A and B will receive the information in both frames, and the coordinates of each event in the two frames are also related by the Galilei transformation.
In this diagram G to E 02, I take it t refers to the time in A's frame the light reached A, and t' refers to the time in B's frame the light reached B? If so it also seems that x refers to the position of the photon at t=0 in A's frame, while x' refers to the position of the same photon at t'=0 in B's frame (because of the relativity of simultaneity these must refer to different events on the photon's worldline). So in each frame you're calculating the distance and time between a totally different pair of events, correct?
One photon. One event spawning the photon. Two events where the photon passes B, then A. One event when A and B were colocated and t=0 and t'=0.

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?
But x and x' refer to the coordinates of two different events on the photon's worldline in this case, at least if we're dealing with relativity (and we must be if both observers say the photon is traveling at c). The event EA on the photon's worldline that is simultaneous in A's frame with A&B being colocated is different from the event EB on the photon's worldline that is simultaneous in B's frame with A&B being colocated, due to the relativity of simultaneity.

Do you agree that x refers to the position coordinate (in frame A) of one event EA and x' refers to the position coordinate (in frame B) of a different event EB? And do you agree that in the spatial component of the Lorentz transform, x' = gamma*(x - vt), x and x' are supposed to refer to the position coordinates of a single event in two different frames?
What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location. Correct?
Not if simultaneity is relative! And if you want to say that both observers measure the speed of photons to be c regardless of direction, it has to be (as demonstrated by Einstein's train/embankment thought-experiment).

neopolitan

Apr24-09, 12:30 AM

When you say "it is just dividing through by gamma", what does "it" refer to? What quantity are you dividing by gamma to get what other quantity? Do you mean that if we look at the horizontal orange bar spanned between the top of the dark orange slanted line segment, then if you take the height of the orange bar and divide by gamma, you get the proper time between the ends of the dark orange segment? If so I agree of course, but if that's not what you were referring to could you clarify?

The only real difference between TD.JPG and TDv2.JPG is that the light blue and light orange sheets are wider in one than the other.

The relationship between the widths of those sheets remains the same and the relationship between the width of the sheets and what they span on the t-axes remains the same.

The factor relating the sheets in TD.JPG and the sheets in TDv2.JPG is gamma, ie the TD.JPG sheets are gamma wider than those in TDv2.JPG. The dark green bar in TD.JPG is gamma longer than the dark orange bar in both. The purple bar in TD.JPG is gamma longer than the dark blue bar in both.

By "dark orange bar" you mean the slanted line segment rather than the thick horizontal orange bar, right? And yeah, if you moved the line segment down so that its top and bottom coincided with the upper and lower edges of the horizontal orange bar, I think the visual relation between them would be clearer.

Fixed. A similar issue in TAFLC is also fixed.

When you say "it worked", as before I'm not quite sure what the "it" refers to...did you do some sort of calculation to check what I was saying?

"It" refers to the derivation I showed for going from Galilean boosts to Lorentz Transformations, which are in part based on removing the assumption of instantaneous transmission of information. It was done back here (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174).

cheers,

neopolitan

neopolitan

Apr24-09, 01:15 AM

The first part just looks like a repeat of the previous paragraph, but why do you say "because time is absolute and t=t' "? It's certainly true that t=t', but I don't see how the fact that an event has coordinates x,t in A's frame and x',t' in B's frame is "because" of this.

It's not a repeat, because in the second I say (x,t) in A and (x',t') in B, whereas in the first it is (x',t) (unprimed t) in B. I could reduce " because time is absolute and t=t' " to "because t=t' ". Putting it all back in context, a common refrain:

In Galilean relativity, we could have an event at E, (x,t) in A's frame and (x',t) in B's frame.

If we had an event at E which is (x,t) in A's frame and (x',t) in B's frame, then we would have an event at E which is at (x,t) in A's frame and (x',t') in B's frame, because t=t'.

Is that better? I'm not sure what reason you have for t=t' other than time is absolute. But in any event, we agree that t=t' so the reason for it is immaterial.

I don't see why, can you explain? If by "aether frame" you just mean there's one frame where the upper limit on the speed of information is the same in both directions, while in other frames it's different in one direction than the other, then I'd agree with that, but I'm not sure that's what you meant.

That's basically what I meant. I guess if you apply Galilean invariance (http://en.wikipedia.org/wiki/Galilean_invariance) and include the speed of light as a fundamental law of physics (which by its inclusion in such things as the Planck equations, it certainly seems to be), then can't have the aether and different measured values for the speed of light, which means that for the Galilean boosts to work, you have to look for something else. I took instantaneous transmission of information.

It seems odd to have Galilean relativity without proper Galilean invariance (although omission of the speed of light as a fundamental law of physics was a forgivable lapse at the time).

One photon. One event spawning the photon. Two events where the photon passes B, then A. One event when A and B were colocated and t=0 and t'=0.

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

But x and x' refer to the coordinates of two different events on the photon's worldline in this case, at least if we're dealing with relativity (and we must be if both observers say the photon is traveling at c). The event EA on the photon's worldline that is simultaneous in A's frame with A&B being colocated is different from the event EB on the photon's worldline that is simultaneous in B's frame with A&B being colocated, due to the relativity of simultaneity.

Do you agree that x refers to the position coordinate (in frame A) of one event EA and x' refers to the position coordinate (in frame B) of a different event EB? And do you agree that in the spatial component of the Lorentz transform, x' = gamma*(x - vt), x and x' are supposed to refer to the position coordinates of a single event in two different frames?

I think you've misread.

I'll try again, and highlight what I think you have misread:

One photon (there is only one photon, which by thought experiment magic could be detected by both B and then A).

One event spawning the photon (there's only one photon, so there is only one event, so this is really tautological, but despite that you still want to have two events related to the photon, perhaps I have misread you, but you certainly are not on the same page as me on this one).

Two events where the photon passes B, then A (these are two separate events, which I agree with you on, assuming that you agree).

One event when A and B were colocated and t=0 and t'=0 (when A and B are colocated, that is one event, their colocation is the event I am talking about, I am not talking about any other event at any other time or location. To drive it in, this event is (x=0,t=0) and (x'=0, t'=0) and irrespective of their coordinate systems, this is still a unique event to each of them).

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?

What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location. Correct?Not if simultaneity is relative! And if you want to say that both observers measure the speed of photons to be c regardless of direction, it has to be (as demonstrated by Einstein's train/embankment thought-experiment).

Again, I think you have misread, or perhap misunderstood.

In any coordinate system (say A's), a photon will have a unique spacetime location when A and B are colocated. In another coordinate system (say B's), the same photon will have a unique spacetime location when A and B are colocated. Therefore, when B is colocated with A is it true that:
1. that one unique photon is in different spacetime locations or
2. A and B ascribe different coordinates to the one unique spacetime location (which, I must stress, doesn't have to be any "true" or "objectively correct" spacetime location)?

I vote for #1. You either misread or misunderstood or vote for #2.

cheers,

neopolitan

JesseM

Apr24-09, 02:28 AM

It's not a repeat, because in the second I say (x,t) in A and (x',t') in B, whereas in the first it is (x',t) (unprimed t) in B.
Ah, I missed that it was unprimed.
I'm not sure what reason you have for t=t' other than time is absolute.
Just to show the full coordinate transformation I suppose, same reason they write y'=y and z'=z in the 3D version of the Lorentz transformation.
It seems odd to have Galilean relativity without proper Galilean invariance (although omission of the speed of light as a fundamental law of physics was a forgivable lapse at the time).
True, if you have a finite speed of information transmission which is different in different frames, then if this finite speed is built into the fundamental laws of physics (rather than being a consequence of a physical substance like aether which just happens to be at rest in one particular frame, but which could in principle be moved) then the fundamental laws are not really Galilei-invariant.
But x and x' refer to the coordinates of two different events on the photon's worldline in this case, at least if we're dealing with relativity (and we must be if both observers say the photon is traveling at c). The event EA on the photon's worldline that is simultaneous in A's frame with A&B being colocated is different from the event EB on the photon's worldline that is simultaneous in B's frame with A&B being colocated, due to the relativity of simultaneity.

Do you agree that x refers to the position coordinate (in frame A) of one event EA and x' refers to the position coordinate (in frame B) of a different event EB? And do you agree that in the spatial component of the Lorentz transform, x' = gamma*(x - vt), x and x' are supposed to refer to the position coordinates of a single event in two different frames?
I think you've misread.

I'll try again, and highlight what I think you have misread:
One photon (there is only one photon, which by thought experiment magic could be detected by both B and then A).

One event spawning the photon (there's only one photon, so there is only one event, so this is really tautological, but despite that you still want to have two events related to the photon, perhaps I have misread you, but you certainly are not on the same page as me on this one).

Two events where the photon passes B, then A (these are two separate events, which I agree with you on, assuming that you agree).

One event when A and B were colocated and t=0 and t'=0 (when A and B are colocated, that is one event, their colocation is the event I am talking about, I am not talking about any other event at any other time or location. To drive it in, this event is (x=0,t=0) and (x'=0, t'=0) and irrespective of their coordinate systems, this is still a unique event to each of them).

A thinks that at colocation, the photon was located at x=ct.

B thinks that at colocation, the photon was located at x'=ct'.

What is the relationship between x' and x, and t and t'?
That doesn't really clarify the issue I had. When you said "A thinks that at colocation, the photon was located at x=ct" and "B thinks that at colocation, the photon was located at x'=ct'", are you implying that one is objectively right and one is objectively wrong? I interpreted "thinks" to just mean "A is defining x and t to be the position and time coordinates (in the A frame) of the event EA on the photon's worldline that occurs simultaneously with A&B being colocated (in the A frame)", and likewise "B is defining x' and t' to be the position and time coordinates (in the B frame) of the event EB on the photon's worldline that occurs simultaneously with A&B being colocated (in the B frame)". Since A and B must naturally disagree about simultaneity if they both measure the photon traveling at c, EA and EB would be two different events on the photon's worldline (with one or both happening after the emission event).

If that's not what you meant, could you describe in specific terms what event you are saying should be assigned coordinates x and t in the A frame, and what event you are saying should be assigned coordiantes x' and t' in the B frame? It wouldn't make sense for you to be talking about the emission event in both cases, since if the emission event happens "at colocation" in one frame, it does not happen at the moment of colocation in the other frame.
Again, I think you have misread, or perhap misunderstood.

In any coordinate system (say A's), a photon will have a unique spacetime location when A and B are colocated. In another coordinate system (say B's), the same photon will have a unique spacetime location when A and B are colocated. Therefore, when B is colocated with A is it true that:
1. that one unique photon is in different spacetime locations or
2. A and B ascribe different coordinates to the one unique spacetime location (which, I must stress, doesn't have to be any "true" or "objectively correct" spacetime location)?

I vote for #1. You either misread or misunderstood or vote for #2.
Why do you say that? My interpretation was that they were both talking about two physically separate events ('physically separate' meaning different points in spacetime) on the photon's worldline, EA and EB, so that sounds like #1 unless I'm misunderstanding what you're saying there. In your way of speaking, is saying "when A&B pass, one unique object X is in different spacetime locations" equivalent to "according to A's definition of simultaneity, when A&B pass this is simultaneous with one event on the worldline of a unique object X, but according to B's definition of simultaneity, when A&B pass this is simultaneous with a different event (at a different spacetime location) on the worldline of the same unique object X"? For example, if two distant observers A and B wanted to know my age at the moment they passed each other, and in A's frame the passing was simultaneous with my turning 20 but in B's frame the passing was simultaneous with my turning 30, would you sum this up by saying "when A&B pass there is one unique Jesse at two different spacetime locations (and thus two different ages)"?

neopolitan

Apr24-09, 03:42 AM

That doesn't really clarify the issue I had. When you said "A thinks that at colocation, the photon was located at x=ct" and "B thinks that at colocation, the photon was located at x'=ct'", are you implying that one is objectively right and one is objectively wrong? I interpreted "thinks" to just mean "A is defining x and t to be the position and time coordinates (in the A frame) of the event EA on the photon's worldline that occurs simultaneously with A&B being colocated (in the A frame)", and likewise "B is defining x' and t' to be the position and time coordinates (in the B frame) of the event EB on the photon's worldline that occurs simultaneously with A&B being colocated (in the B frame)". Since A and B must naturally disagree about simultaneity if they both measure the photon traveling at c, EA and EB would be two different events on the photon's worldline (with one or both happening after the emission event).

If that's not what you meant, could you describe in specific terms what event you are saying should be assigned coordinates x and t in the A frame, and what event you are saying should be assigned coordiantes x' and t' in the B frame? It wouldn't make sense for you to be talking about the emission event in both cases, since if the emission event happens "at colocation" in one frame, it does not happen at the moment of colocation in the other frame.

Why do you say that? My interpretation was that they were both talking about two physically separate events ('physically separate' meaning different points in spacetime) on the photon's worldline, EA and EB, so that sounds like #1 unless I'm misunderstanding what you're saying there. In your way of speaking, is saying "when A&B pass, one unique object X is in different spacetime locations" equivalent to "according to A's definition of simultaneity, when A&B pass this is simultaneous with one event on the worldline of a unique object X, but according to B's definition of simultaneity, when A&B pass this is simultaneous with a different event (at a different spacetime location) on the worldline of the same unique object X"? For example, if two distant observers A and B wanted to know my age at the moment they passed each other, and in A's frame the passing was simultaneous with my turning 20 but in B's frame the passing was simultaneous with my turning 30, would you sum this up by saying "when A&B pass there is one unique Jesse at two different spacetime locations (and thus two different ages)"?

I'm such a visual sort of fellow. Another diagram, here (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg).

The photon has a unique spacetime location both times. Do we agree on that?

It's just that for the first event (it's the first event in both frames), the photon is at (8,0) according to A and (10,-6) according to B and for the second event, the photon is at (5,3) according to A and (4,0) according to B.

But even though each event is described differently, both events describe one unique location of the photon. A and B just disagree about which event is simultaneous with their colocation with the other.

Do we agree on this?

cheers,

neopolitan

JesseM

Apr24-09, 05:24 AM

I'm such a visual sort of fellow. Another diagram, here (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg).

The photon has a unique spacetime location both times. Do we agree on that?

It's just that for the first event (it's the first event in both frames), the photon is at (8,0) according to A and (10,-6) according to B and for the second event, the photon is at (5,3) according to A and (4,0) according to B.

But even though each event is described differently, both events describe one unique location of the photon. A and B just disagree about which event is simultaneous with their colocation with the other.

Do we agree on this?
Yes, the two arrows on the diagram are pointing to exactly the events I was thinking of when I referred to EA and EB.

neopolitan

Apr24-09, 06:33 AM

Yes, the two arrows on the diagram are pointing to exactly the events I was thinking of when I referred to EA and EB.

Do you agree that they are both unique events? Irrespective of what coordinates A and B might assign to them and whether or not A and B might disagree about which is simultaneous with their colocation (unless they do some Lorentz Transformations, of course).

neopolitan

Apr24-09, 01:40 PM

I don't really want to smother the previous post even if I am pretty sure that you will agree, but I've been wondering what caused the misunderstanding which led to it.

I think it was that in #181, I wrote in haste "What we do know is that, irrespective of coordinate system, when A and B were colocated, the photon had one unique location."

I was in a real rush at the time, and it seems that although the comment made sense to me at the time I now see I should have been more careful - some words were in my head but didn't get converted to pixels.

Let me rephrase "What we do know is that, irrespective of what coordinate system you chose to use, when A and B were colocated, the photon had one unique location."

It's still not perfect, since I should add that you must chose only one coordinate system, either A or B's but not both since A and B have different ideas about what is simultaneous with the colocation event.

Anyway, it seems you did misunderstand but, given the poor wording, it was hardly your fault.

cheers,

neopolitan

JesseM

Apr24-09, 04:09 PM

Do you agree that they are both unique events?
Yes, and that was kind of my point, I was confused about how you intended to use these two different events EA and EB to derive the Lorentz transform when the Lorentz transform relates the coordinates that two different frames assign to a single event. I can't really follow what's going on in the second diagram from post #174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174), and hence I can't follow the third diagram either, could you walk me through them a bit? In the second diagram when you say "According to A, there is one event E" are you referring to the event I've called EA? In that diagram, do x and t refer to the coordinates that A assigns to EA while x' and t' refer to the coordinates that B assigns to EB? Why would A "tentatively" believe that x' = x - vt if x' refers to the position coordinate of an entirely different event EB from the event EA that x and t refer to? Is it because they don't realize the events are distinct because they're still thinking in terms of the Galilei transform where simultaneity is absolute? But if that's the case, why do they both assume the light from the event was traveling at c to reach them, given that this is impossible under the Galilei transform?

And in the third diagram, do xA and xB refer to the position coordinates in A's frame of EA and EB respectively, while x'A and x'B refer to the position coordinates of these two events in B's frame? If so why do you still show only a single yellow event E in this diagram?

neopolitan

Apr24-09, 10:15 PM

Yes, and that was kind of my point, I was confused about how you intended to use these two different events EA and EB to derive the Lorentz transform when the Lorentz transform relates the coordinates that two different frames assign to a single event. I can't really follow what's going on in the second diagram from post #174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174), and hence I can't follow the third diagram either, could you walk me through them a bit? In the second diagram when you say "According to A, there is one event E" are you referring to the event I've called EA? In that diagram, do x and t refer to the coordinates that A assigns to EA while x' and t' refer to the coordinates that B assigns to EB? Why would A "tentatively" believe that x' = x - vt if x' refers to the position coordinate of an entirely different event EB from the event EA that x and t refer to? Is it because they don't realize the events are distinct because they're still thinking in terms of the Galilei transform where simultaneity is absolute? But if that's the case, why do they both assume the light from the event was traveling at c to reach them, given that this is impossible under the Galilei transform?

And in the third diagram, do xA and xB refer to the position coordinates in A's frame of EA and EB respectively, while x'A and x'B refer to the position coordinates of these two events in B's frame? If so why do you still show only a single yellow event E in this diagram?

Because in its original form, the derivation is a staged process starting with the Galilean boost (Galilei transform) and ending up at the Lorentz transformations. In hind sight, you can see clearly that so long as event E is simultaneous with when A and B are colocated, then event E is actually two events along the photon's worldline. But in the Galilean boost, event E isn't two events.

The same goes for the question in the previous paragraph, except a little reversed, once both assume that information/light gets to them at c, then they find that the Galilean boost is invalid, except as an approximation for where v << c.

So it is step 1, show the Galilean boost. Step 2, introduce the speed of light considerations (ensuring that it is Galilean invariant). Step 3, introduce Galiliean invariance more fully. In the process of doing Steps 2 and 3, it is shown that the Galilean boost is invalid. Step 4, derive the Lorentz transformations.

Then you can do what you seem to want to do, if you like: Step 5, go back and show that event E shown in the diagrams is actually describing two events E and thereby introduce the relativity of simultaneity.

Our approaches seem fundamentally different, if we were talking about how to make a robot, you might start with instructing the class to study gyroscopes to explain how they would be used to keep the robot upright because the robot is top heavy whereas I'd be telling them to how to smelt metal. Your explanation assumes you already know how the robot is built (top heavy), whereas I am starting from close to the ground up (I didn't start right back at "climb down from the trees" or "first evolve into multicellular into lifeform"). Similarly, you seem to want me to assume relatively advanced prior knowledge on the part of the student in order to show a derivation of the Lorentz transform (ie, that the Galilean boost is invalid and that the relativity of simultaneity applies).

cheers,

neopolitan

JesseM

Apr24-09, 10:54 PM

Our approaches seem fundamentally different, if we were talking about how to make a robot, you might start with instructing the class to study gyroscopes to explain how they would be used to keep the robot upright because the robot is top heavy whereas I'd be telling them to how to smelt metal. Your explanation assumes you already know how the robot is built (top heavy), whereas I am starting from close to the ground up (I didn't start right back at "climb down from the trees" or "first evolve into multicellular into lifeform"). Similarly, you seem to want me to assume relatively advanced prior knowledge on the part of the student in order to show a derivation of the Lorentz transform (ie, that the Galilean boost is invalid and that the relativity of simultaneity applies).
Well, the normal derivation of the Lorentz transform just starts from the idea that we want a transformation where the two postulates of SR are true, and sees what conclusions can be derived from these postulates alone. That's how any "derivation" in math or physics is supposed to work, you pick some starting assumptions and then show in a step by step way what follows from the assumptions. From what you're saying it sounds like the first two diagrams are not really part of your "derivation", but are just part of a sort of pedagogical strategy of showing why we can't continue to use the Galilei transformation if we want the speed of light to be constant in all frames (which also follows in a more direct way from the Galilean velocity addition formula)--if you had made clear that these diagrams weren't part of the actual derivation I wouldn't have been confused, I have no problem with making some pedagogical points before delving into a derivation (for example, one might also point out the context of why Einstein wanted to postulate that the speed of light be the same for all observers, which would involve talking a little about Maxwell's laws and the Michelson-Morley results).

So let's skip the first two diagrams and go directly to number 3. I wrote:
And in the third diagram, do xA and xB refer to the position coordinates in A's frame of EA and EB respectively, while x'A and x'B refer to the position coordinates of these two events in B's frame? If so why do you still show only a single yellow event E in this diagram?
Your response addressed the last sentence but not the previous one, can I assume that's because my interpretation of the meaning of those symbols matched your intentions? If so I don't really understand the next step...if xA and xB both refer to the position coordinates in A's frame of the two different events, what does this have to do with A concluding that "B measures distance and/or time oddly"? Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E?

neopolitan

Apr24-09, 11:29 PM

neopolitan

Apr25-09, 12:10 AM

(for example, one might also point out the context of why Einstein wanted to postulate that the speed of light be the same for all observers, which would involve talking a little about Maxwell's laws and the Michelson-Morley results).

I was actually trying to go from Galileo to Einstein in one fell swoop. I think you can do it via Galilean invariance, but I do accept that it might be unfair since the invariance of the speed of light didn't get shown until a long time after.

I remain curious about when we had enough information to arrive at SR, I was thinking that it was around Galileo's time. There was an experiment proposed by one of Galileo's contempories as early as 1629 to measure the speed of light (http://en.wikipedia.org/wiki/Speed_of_light#Measurements_of_the_speed_of_light) .

Galileo said he had done the experiment and wikipedia has conflicting reports of the results (one article says the speed he came up with was in the order of 1000 times greater than the speed of sound - same link as above - and another said the results indicated that the speed of light was infinite (http://en.wikipedia.org/wiki/Galileo_Galilei#Physics)). When the Accademia del Cimento carried out the experiment in 1667, no delay was detected. Robert Hooke explained the negative results by saying that they don't necessarily indicate that the speed of light is infinite, but could just be extremely high.

The point being that it is not unreasonable to posit that when Galileo formulated his boosts, an assumption therein was not that there is an aether frame in which the speed of light is a constant but rather that the speed of light is infinite, and light (and thus information) is transmitted instantaneously.

The last nail in the coffin of infinite speed of light was apparently driven home by James Bradley in 1728, eighty years after Galileo's death.

So, I suppose, you could say that there was not quite enough information available in Galileo's time to arrive at SR, but there was in 1728 - 160 years before Michelson and Morely conducted their experiment and more than a century before the birth of Maxwell.

cheers,

neopolitan

JesseM

Apr25-09, 12:35 AM

Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E?
Correct.

There are four values involved (which is a fair indication to the astute student that there might be two events and therefore is already laying the ground work for introducing the relativity of simultaneity). The values which are primed are as per the Galilean boost. The subscripted values mean "according to ..."
Why do you say "as per the Galilean boost"? According to the definition you called "correct" above, primed and unprimed just refer to two different events. If we use the same subscript, like x'A and xA, then we are exclusively talking about a single frame, the rest frame of A, no boosts are involved at all.
x'A = xA - vtA
xB = x'B + vt'B

which work, even in SR. The primes relate to measurements between B and E and the unprimes relate to measurements between A and E.
I don't think it does work in SR, given the definitions above. Let's suppose in A's frame B has a velocity of 0.6c. Suppose the unprimed event is the one that's simultaneous with A&B passing each other in A's frame, so in A's frame tA = 0, and let's say this unprimed event occurs at position xA = 16 light-seconds. In this case the light will pass B at a position of 6 light-seconds and a time of 10 seconds in A's frame. Since B's clock is running slow by a factor of 0.8 in this frame, the light must hit B when B's clock reads 8 seconds, so the primed event which occurs at t'B = 0 in B's frame must occur at position x'B = 8 light-seconds in B's frame. Using the Lorentz transformation we can find the coordinates of the primed event in the A frame:

x'A = 0.8*(x'B + v*t'B) = 0.8*(8 + 0.6*0) = 6.4
t'A = 0.8*(t'B + v*x'B/c^2) = 0.8*(0 + 0.6*8) = 3.84

But in this case your first equation x'A = xA - vtA doesn't work, since 6.4 doesn't equal 16 - 0.6*0.

Maybe instead you want the primed event to be the one that's simultaneous with A&B passing each other in A's frame...in this case let's use the same numbers and say t'A = 0 and x'A = 16, and similarly tB = 0 and xB = 8. Now we can figure out the coordinates of the primed event in the B frame:

x'B = 0.8*(x'A - v*t'A) = 0.8*(16 - 0.6*0) = 12.8
t'B = 0.8*(t'A - v*x'A) = 0.8*(0 - 0.6*16) = 7.68

So in this case your second equation xB = x'B + vt'B doesn't work because 8 is not equal to 12.8 + 0.6*7.68.
Do you now agree that Galilean relativity assumes either instantaneous transmission of information (totally ignoring light) or a variant speed of light (which leads to an internal contradiction since Galilean invariance is a component of Galilean relativity)? While you may still not agree with the former, do you agree with this catch-all phrasing?
Yes, I'd agree with that. However, we can imagine a possible set of laws in which the Galilei transformation is still the most "natural" one for inertial observers to use despite the fact that the laws of physics are not all Galilei-symmetric, because the behavior of rulers and clocks is Galilei-symmetric so coordinate systems constructed out of such rulers and clocks will be related to one another by the Galilei transformation. In fact, this is exactly how 19th century physicists who believed in a preferred aether frame assumed things would work, at least prior to Michelson-Morley.

neopolitan

Apr25-09, 12:49 AM

According to A, at any time an event happens (xA,tA), how far has B travelled?

I think that distance is vtA, according to A. Therefore, using the simple Galilean boost, the distance between B and the event according to A, at that time according to A is x'A = xA - vtA.

Do you understand that? Or have I made a mistake?

cheers,

neopolitan

JesseM

Apr25-09, 01:10 AM

According to A, at any time an event happens (xA,tA), how far has B travelled?

I think that distance is vtA, according to A.
Yes.
Therefore, using the simple Galilean boost, the distance between B and the event according to A, at that time according to A is x'A = xA - vtA.
But none of your symbols were supposed to represent the distance between B and the event in A's frame. xA referred to the distance between A and the unprimed event in A's frame, xB referred to the distance between B and the unprimed event in B's frame, x'A referred to the distance between A and the primed event in A's frame, and x'B referred to the distance between B and the primed event in B's frame. Or were you mistaken when you said that my earlier statement about the notation was "correct"? If so can you please explain specifically what each of these symbols is supposed to refer to?

neopolitan

Apr25-09, 01:59 AM

Or were you mistaken when you said that my earlier statement about the notation was "correct"? Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E?

Correct.

I stand by that.

<snip> can you please explain specifically what each of these symbols is supposed to refer to?

First, before I go into lots of detail, can you confirm that you agree that this is correct:

A and B separate at v and an event or events take place closer to B than A, such that the direction that the event lies is notionally taken as the positive direction. In other words, according to A, B has a velocity of +v and according to B, A has a velocity of -v.

If according to A, an event happens at (xA,tA), then according to A B has travelled vtA, therefore according to A, the distance between B and where the event took place according to A, can at that time be given by x'A = xA - vtA.

And, if according to B, an event happens at (x'B,t'B), then according to B A has travelled -vt'B, therefore according to B, the distance between A and where the event took place according to B, can at that time be given by xB = x'B + vt'B.

If it's wrong, I'd like to know what assumptions you are making that make it wrong (which I think you will find are not in accordance with the assumptions that I have obviously not made sufficiently clear).

cheers,

neopolitan

JesseM

Apr25-09, 02:31 AM

Or was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), so that xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E?
Correct.
I stand by that.
OK, I guess I didn't state it explicitly, but I thought it was obvious that if primed and unprimed was just supposed to distinguish between the two events E and E'--since you agreed with my statement "you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"--then if you also agreed that "xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E", then naturally a corollary would be that "x'A represents the position in A's frame of the primed event E' while x'B represents the position in B's frame of the same primed event E' ". Is that not correct?
First, before I go into lots of detail, can you confirm that you agree that this is correct:

A and B separate at v and an event or events take place closer to B than A, such that the direction that the event lies is notionally taken as the positive direction. In other words, according to A, B has a velocity of +v and according to B, A has a velocity of -v.

If according to A, an event happens at (xA,tA)
So based on my statment "xA represents the position in A's frame of the unprimed event E", I assume this still refers specifically to the position and time of the unprimed event E in A's frame?
then according to A B has travelled vtA, therefore according to A, the distance between B and where the event took place according to A, can at that time be given by x'A = xA - vtA.
Yes, if you use the notation x'A to mean "the distance between B and the unprimed event E at the moment E occurred, all in A's frame." But this would seem to be inconsistent with the statement earlier that primed and unprimed was to distinguish between the two events: "you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)".
And, if according to B, an event happens at (x'B,t'B)
Which event? The unprimed event E or the primed event E'?
then according to B A has travelled -vt'B, therefore according to B, the distance between A and where the event took place according to B, can at that time be given by xB = x'B + vt'B.
Sure, if xB represents the distance between A and the event in B's frame. But this seems to contradict my statement "xB represents the position in B's frame of the same unprimed event E" which you said was "correct" earlier.

neopolitan

Apr25-09, 03:26 AM

OK, I guess I didn't state it explicitly, but I thought it was obvious that if primed and unprimed was just supposed to distinguish between the two events E and E'--since you agreed with my statement "you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"--then if you also agreed that "xA represents the position in A's frame of the unprimed event E while xB represents the position in B's frame of the same unprimed event E", then naturally a corollary would be that "x'A represents the position in A's frame of the primed event E' while x'B represents the position in B's frame of the same primed event E' ". Is that not correct?

There are four values involved (which is a fair indication to the astute student that there might be two events and therefore is already laying the ground work for introducing the relativity of simultaneity). The values which are primed are as per the Galilean boost. The subscripted values mean "according to ..." (as clarified in post #175 (http://www.physicsforums.com/showpost.php?p=2166309&postcount=175)).

I didn't take your E and E' as prescriptive but rather descriptive since I had already written twice by that time that the subscripted values mean "according to ..." and that the values which are primed are as per the Galilean boost (which I was in the process of trying to explain).

If you want to refer to the events (xA,tA) according to A and (xB,tB) according to B via a priming notation, go ahead, but it's not really sensible. Personally, I would refer to the events as EA and EB. Remember also I was using primes in the same sense as they are used in the Galilean boost, where there are no events to speak of, primed or unprimed.

Try comparing this diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg) with this one (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg). Try to restrict your response to how the events were named (we've already discussed the latter diagram in the context of this one (http://www.geocities.com/neopolitonian/gal_rel_neopolitan.jpg)).

Until I know for sure what you are calling the primed E and what you are calling the unprimed E, since you do seem to jump around a lot, I'd prefer you use either "event at (xA,tA) according to A" and "the event at (x'B,t'B) according to B" or EA and EB. Until then, I am not totally convinced that you aren't talking about completely different events.

Driving the point home even further, I specify explicitly that in this diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg), EA is "Photon when A is colocated with B, according to A" and "EB is "Photon when A is colocated with B, according to B".

Note the consistency with the "acccording to ..." notation.

cheers,

neopolitan

JesseM

Apr25-09, 04:13 AM

I didn't take your E and E' as prescriptive but rather descriptive since I had already written twice by that time that the subscripted values mean "according to ..." and that the values which are primed are as per the Galilean boost (which I was in the process of trying to explain).
I don't understand what you mean by "prescriptive rather than descriptive". Could you please just write out explicitly what each of the following refer to?

xA refers to position coordinate of ___ (or the distance between ___ and ___) in the A frame
xB refers to position coordinate of ___ (or the distance between ___ and ___) in the B frame
x'A refers to position coordinate of ___ (or the distance between ___ and ___) in the A frame
x'B refers to position coordinate of ___ (or the distance between ___ and ___) in the B frame

I put in those parentheses because it seems that at times these symbols refer to actual coordinates of individual events in SR, while at other times they refer to distance intervals between events rather than coordinates of individual events.
If you want to refer to the events (xA,tA) according to A and (xB,tB) according to B via a priming notation, go ahead, but it's not really sensible. Personally, I would refer to the events as EA and EB.
But then if I see notation like xA, there's nothing in the notation itself that tells me whether it refers to the position in A's frame of event EA or the position in A's frame of event EB? If you don't want to use the prime to tell me which event is being referred to, could you add some other indication? Like instead of calling the events EA and EB you could call them E1 and E2, then xA1 would refer to the position coordinate of event E1 in frame A, xB2 would refer to the position coordinate of event E2 in frame B, and so forth.
Remember also I was using primes in the same sense as they are used in the Galilean boost, where there are no events to speak of, primed or unprimed.
Why do you say that? All coordinate transformations are showing you the coordinates of a single event in two different frames. For example, if an event has coordinates (x,t) in the unprimed frame, then according to the Galilei transformation (which is what is meant by the 'Galilean boost') in the primed coordinate system it has coordinates (x',t') given by:

x' = x - vt
t' = t
Try comparing this diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg) with this one (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg). Try to restrict your response to how the events were named (we've already discussed the latter diagram in the context of this one (http://www.geocities.com/neopolitonian/gal_rel_neopolitan.jpg)).
But the second diagram is just a weird transformation (which I think would be horrendously complicated to write the equations for) that has nothing to do with the Galilei transformation, and was based on your misinterpretation of what I meant when I said instantaneous communication wasn't necessary in the Galilei transformation, so what's the relevance?
Until I know for sure what you are calling the primed E and what you are calling the unprimed E, since you do seem to jump around a lot
I didn't specify because I assumed you were using primed vs. unprimed to refer to the two events, since you said I was correct when I said you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), and I didn't know which was supposed to be which in your notation.
I'd prefer you use either "event at (xA,tA) according to A" and "the event at (xB,tB) according to B" or EA and EB. Until then, I am not totally convinced that you aren't talking about completely different events.
Of course I was talking about different events! We already established that you were discussing two different events, you showed them in this diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg). Are you saying the above notation which you "prefer" is not talking about two different events? If so, which of the events are you talking about with that notion, the one that was simultaneous with colocation in A's frame, or the one that was simultaneous with colocation in B's frame?
Driving the point home even further, I specify explicitly that in this diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg), EA is "Photon when A is colocated with B, according to A" and "EB is "Photon when A is colocated with B, according to B".
OK, then please tell me what notation you would use for each of the following:

1. position and time coordinates of EA in A's frame
2. position and time coordinates of EA in B's frame
1. position and time coordinates of EB in A's frame
2. position and time coordinates of EB in B's frame
Note the consistency with the "acccording to ..." notation.
Yes, I understood that subscripts referred to which frame we were using, that wasn't the issue.

JesseM

Apr25-09, 05:16 AM

But the second diagram is just a weird transformation (which I think would be horrendously complicated to write the equations for) that has nothing to do with the Galilei transformation, and was based on your misinterpretation of what I meant when I said instantaneous communication wasn't necessary in the Galilei transformation, so what's the relevance?
By the way, I spent a bit of time figuring out what the transformation equations implied by that diagram (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg) would actually be, turns out it's not so complicated, in units where c=1 I believe it'd be:

xB = [1/(1 - v^2)]*(xA - vtA)
tB = [1/(1 - v^2)]*(tA - vxA)

So, almost like the Lorentz transformation but with xB and tB multiplied by gamma. And the reverse transformation would just be:

xA = xB + vtB
tA = tB + vxB

neopolitan

Apr25-09, 05:54 AM

I don't understand what you mean by "prescriptive rather than descriptive". Could you please just write out explicitly what each of the following refer to?

Descriptive - thou could if thou wanted
Prescriptive - thou shalt

Sure you can call the events what you want. E' and E. E1 and E2, E and F. Bonny and Clyde. Frank and Ernest.

You didn't initially say not only must I use your nomenclature but I also must use your definitions.

xA refers to position coordinate of event EA (or the distance between the origin of the xA axis and Event EA) in the A frame
xB refers to position coordinate of Event EB (or the distance between the origin of the xB axis and Event EB) in the B frame
x'A refers to the distance between the location of B at tA and the location of Event EA in the A frame
x'B refers to the distance between the location of A at t'B and the location of Event EB in the B frame

I put in those parentheses because it seems that at times these symbols refer to actual coordinates of individual events in SR, while at other times they refer to distance intervals between events rather than coordinates of individual events.

I've filled it in, in red.

As you can see, yes, two of them represent both a coordinate and a distance interval between two events. Note that the origin of either axis at a specific time is "an event" since it's both a time and a location, so really any coordinate really represents an interval between two events.

But then if I see notation like xA, there's nothing in the notation itself that tells me whether it refers to the position in A's frame of event EA or the position in A's frame of event EB? If you don't want to use the prime to tell me which event is being referred to, could you add some other indication? Like instead of calling the events EA and EB you could call them E1 and E2, then xA1 would refer to the position coordinate of event E1 in frame A, xB2 would refer to the position coordinate of event E2 in frame B, and so forth.

You could prime it as well, if you like. But let's not.

EAa = (xA,tA) when viewed by A. EAb is the same event when viewed by B.
EBb = (x'B,t'B) when viewed by B. EBa is the same event when viewed by A.

Why do you say that? All coordinate transformations are showing you the coordinates of a single event in two different frames. For example, if an event has coordinates (x,t) in the unprimed frame, then according to the Galilei transformation (which is what is meant by the 'Galilean boost') in the primed coordinate system it has coordinates (x',t') given by:

x' = x - vt
t' = t

I'm going to have to take your word for that. I've only ever seen the Galilean boost talked of in terms of locations, except where SR is what is really being discussed. Events seem to be introduced in SR.

That may be just because t'=t in Galilean relativity, so it is safe to discuss just locations, but underneath, back when it was written Galileo was thinking in terms of events. (I just don't think so, I think he thought in terms of something like "vehicle moving towards shed, at t=t, what is the distance interval between vehicle and shed". Thinking in terms of events is a consequence of working backwards from SR.)

But the second diagram is just a weird transformation (which I think would be horrendously complicated to write the equations for) that has nothing to do with the Galilei transformation, and was based on your misinterpretation of what I meant when I said instantaneous communication wasn't necessary in the Galilei transformation, so what's the relevance?

You just couldn't help yourself, could you?

I repeat what I said when I directed you to that diagram "Try to restrict your response to how the events were named".

Instead, you totally ignored it. It boggles the mind.

I didn't specify because I assumed you were using primed vs. unprimed to refer to the two events, since you said I was correct when I said you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say), and I didn't know which was supposed to be which in your notation.

You were wrong. You wanted to prime one of the events. I'd specifically avoid priming one of the events because it is misleading.

Of course I was talking about different events! We already established that you were discussing two different events, you showed them in this diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg). Are you saying the above notation which you "prefer" is not talking about two different events? If so, which of the events are you talking about with that notion, the one that was simultaneous with colocation in A's frame, or the one that was simultaneous with colocation in B's frame?

"Completely" different events. Are you talking about completely different events? I was talking about two events, were you talking about two completely different events which are not the events that I was talking about?

You say "of course", but I am not so sure since you point back to the two events that I am talking about.

OK, then please tell me what notation you would use for each of the following (coordinates from my diagram here (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg)):

1. position and time coordinates of EA in A's frame - (xA , 0) (8,0)
2. position and time coordinates of EA in B's frame - (gamma.xA , gamma.(-v.c2.xA)) (10,-6)
1. position and time coordinates of EB in A's frame - (gamma.x'B , gamma.(+v.c2.x'B)) (5,3)
2. position and time coordinates of EB in B's frame - (x'B , 0) (4,0)

Getting the code right for writing this up is much more difficult than working out what I actually need to write.

cheers,

neopolitan

neopolitan

Apr25-09, 06:02 AM

By the way, I spent a bit of time figuring out what the transformation equations implied by that diagram (http://www.geocities.com/neopolitonian/gal_rel_JesseM.jpg) would actually be, turns out it's not so complicated, in units where c=1 I believe it'd be:

xB = [1/(1 - v^2)]*(xA - vtA)
tB = [1/(1 - v^2)]*(tA - vxA)

So, almost like the Lorentz transformation but with xB and tB multiplied by gamma. And the reverse transformation would just be:

xA = xB + vtB
tA = tB + vxB

Yeah, I did the same thing. But there were some c's in my answer.

JesseM

Apr25-09, 06:42 AM

Descriptive - thou could if thou wanted
Prescriptive - thou shalt

Sure you can call the events what you want. E' and E. E1 and E2, E and F. Bonny and Clyde. Frank and Ernest.

You didn't initially say not only must I use your nomenclature but I also must use your definitions.
I take it you didn't understand that my question was specifically about me trying to understand what you meant by different symbols? If you understood that, you would understand how confusing it would be to answer "correct" if what you really meant was "well, that's not what I meant by primed and unprimed, but you're free to use that notation if you like".
I've filled it in, in red.
OK, thanks. You seem to have no notation to represent the position coordinate of event EA in the B frame or of EB in the A frame, but perhaps that's not relevant to your derivation?
You could prime it as well, if you like. But let's not.

EAa = (xA,tA) when viewed by A. EAb is the same event when viewed by B.
EBb = (x'B,t'B) when viewed by B. EBa is the same event when viewed by A.
We can do it that way if you like. I was thinking that the different subscripts for E represented different physical events--that's the convention I normally see in relativity problems of this kind--but if you want them to refer to an event and its coordinates that's fine too. Again you seem to have no notation to represent the x and t coordinates associated with EAb or EBa, but perhaps you don't need it.
You just couldn't help yourself, could you?

I repeat what I said when I directed you to that diagram "Try to restrict your response to how the events were named".

Instead, you totally ignored it. It boggles the mind.
Jeez, relax, I just didn't catch the significance of that phrase, I didn't intentionally "ignore" it. There's a lot of stuff I write that you misunderstand too, I don't complain about how mind-boggling it is that you would fail to get everything I say.
You were wrong. You wanted to prime one of the events. I'd specifically avoid priming one of the events because it is misleading.
No, I asked a question about how you were using the notation: "was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"? I didn't "want" to use that notation, I was just asking if that's what you meant when you put primes on some coordinates and not others, because I was trying to follow your equations and I didn't understand what that notation meant...you answered "correct" to my question. Can you see why I was confused?
"Completely" different events. Are you talking about completely different events? I was talking about two events, were you talking about two completely different events which are not the events that I was talking about?
I think I understand what you mean now, but it wasn't obvious from the context--the word "completely" does not naturally imply "two events different from the two events I am talking about" as opposed to "two different events, as opposed to a single physical event with different coordinates in different frames". I thought you were suggesting that xA and xB were supposed to refer to the coordinates of a single event in the A frame vs. the B frame.

Anyway, now that you've given a clear definition of your spatial terms:
xA refers to position coordinate of event EA (or the distance between the origin of the xA axis and Event EA) in the A frame
xB refers to position coordinate of Event EB (or the distance between the origin of the xB axis and Event EB) in the B frame
x'A refers to the distance between the location of B at tA and the location of Event EA in the A frame
x'B refers to the distance between the location of A at t'B and the location of Event EB in the B frame
...can I ask for definitions of the temporal terms too? Comparing to the above I would initially think tA is the time coordinate of EA in the A frame, and tB is the time coordinate of EB in the B frame, but that would mean tA = 0 and tB = 0, which probably isn't what you meant. So are tA and tB just two arbitrary time-coordinates in each frame? And what's the difference between them and t'A and t'B?

neopolitan

Apr25-09, 07:07 AM

I take it you didn't understand that my question was specifically about me trying to understand what you meant by different symbols? If you understood that, you would understand how confusing it would be to answer "correct" if what you really meant was "well, that's not what I meant by primed and unprimed, but you're free to use that notation if you like".

I wasn't saying "correct" to the primes. It didn't figure highly. But I think we are past that now.

OK, thanks. You seem to have no notation to represent the position coordinate of event EA in the B frame or of EB in the A frame, but perhaps that's not relevant to your derivation?

Nope, not necessary.

We can do it that way if you like. I was thinking that the different subscripts for E represented different physical events--that's the convention I normally see in relativity problems of this kind--but if you want them to refer to an event and its coordinates that's fine too. Again you seem to have no notation to represent the x and t coordinates associated with EAb or EBa, but perhaps you don't need it.

Jeez, relax, I just didn't catch the significance of that phrase, I didn't intentionally "ignore" it. There's a lot of stuff I write that you misunderstand too, I don't complain about how mind-boggling it is that you would fail to get everything I say.

I specifically put the phrase in there because I thought that (without it) you would go off on a tangent. But you did anyway. I was actually more irritated about the "completely different" thing. Oh well, such is life.

No, I asked a question about how you were using the notation: "was I in fact mistaken about your notation, and you were actually distinguishing between the two events using primed and unprimed (calling one event E and the other E', say)"? I didn't "want" to use that notation, I was just asking if that's what you meant when you put primes on some coordinates and not others, because I was trying to follow your equations and I didn't understand what that notation meant...you answered "correct" to my question. Can you see why I was confused?

Yes. We were focussed on different things. What I was saying correct to was not what you thought I was saying correct to. As I said, I think we are past this now.

I think I understand what you mean now, but it wasn't obvious from the context--the word "completely" does not naturally imply "two events different from the two events I am talking about" as opposed to "two different events, as opposed to a single physical event with different coordinates in different frames". I thought you were suggesting that xA and xB were supposed to refer to the coordinates of a single event in the A frame vs. the B frame.

Anyway, now that you've given a clear definition of your spatial terms:

...can I ask for definitions of the temporal terms too? Comparing to the above I would initially think tA is the time coordinate of EA in the A frame, and tB is the time coordinate of EB in the B frame, but that would mean tA = 0 and tB = 0, which probably isn't what you meant. So are tA and tB just two arbitrary time-coordinates in each frame? And what's the difference between them and t'A and t'B?

You could ask for definitions of the temporal terms, but for the purposes of the derivation, you don't need them.

Since x=ct and x'=ct', you just divide through by c.

x'=gamma.(x-vt) =>
x'/c = gamma.(x/c-vt/c) =>
t' = gamma.(t - v.(x/c)/c) =>
t' = gamma.(t - vx/c2)

But if you comprehend how you get to the spatial Lorentz transformation, doing the same for time is not difficult - if you must derive it separately.

Is it possible that you could look at the post where I introduced this derivation and see if you can work it through, now you know what refers to what?

You asked a while back why the yellow E was still one event rather than two. I answered it my own sort of way along with some other questions, but perhaps you didn't think that I answered.

For each of A and B, E is one event (EA for A, and EB for B). Both A and B will think that E according to the other is another event (EBa and EAb).

cheers,

neopolitan

phyti

Apr25-09, 10:59 AM

neopolitan

Apr25-09, 12:27 PM

Neo:

Since you seem to have a handle on space-time drawings, and they do tend to simplify for you and the reader, here is an extension to the example you are working. Hope it helps.
In the 3rd case, the viewer is always the rest frame.
18619

Thanks phyti, I have pretty much always had something like that in mind.

I did wonder if E1 is misplaced on the second two. It sits on the xA axis at first, then moves to the xB which at first glance can't be right.

I think I know what you mean though, that E1 is simultaneous with the colocation of A and B, according to the viewer of the graph (for whom it is square) and at first that is someone at rest with A and then it is someone at rest with B. Which means that there are two separate E1s. Am I right?

In terms of the discussion which precedes this, then E1 is both EAa and EBb (although the assumption that EA and EB are causally linked is absent).

cheers,

neopolitan

JesseM

Apr25-09, 04:46 PM

You could ask for definitions of the temporal terms, but for the purposes of the derivation, you don't need them.

Since x=ct and x'=ct', you just divide through by c.

x'=gamma.(x-vt) =>
x'/c = gamma.(x/c-vt/c) =>
t' = gamma.(t - v.(x/c)/c) =>
t' = gamma.(t - vx/c2)

But if you comprehend how you get to the spatial Lorentz transformation, doing the same for time is not difficult - if you must derive it separately.

Is it possible that you could look at the post where I introduced this derivation and see if you can work it through, now you know what refers to what?

You asked a while back why the yellow E was still one event rather than two. I answered it my own sort of way along with some other questions, but perhaps you didn't think that I answered.

For each of A and B, E is one event (EA for A, and EB for B). Both A and B will think that E according to the other is another event (EBa and EAb).

cheers,

neopolitan
OK, going back to the third drawing in post 174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174):

Question #1: Why do both A and B conclude the same factor G will appear in the equations xA = G*xB and x'B = G*x'A? The two equations have fairly different physical meanings...the first means:

(position coordinate of EA in A's frame) = G*(position coordinate of EB in B's frame)

But the second means:

(distance between position of A at t'B and position of EB, in the B frame) = G*(distance between position of B at t'A and position of EA, in the A frame)

Can we go through a numerical example so I can try to understand what additional assumptions you might be making that lead you to think the factor relating these would be the same?

Let's suppose in A's frame B has a velocity of 0.6c. Suppose in A's frame the event EA has position xA = 16 light-seconds. In this case the light will pass B at a position of 6 light-seconds and a time of 10 seconds in A's frame. Since B's clock is running slow by a factor of 0.8 in this frame, the light must hit B when B's clock reads 8 seconds, so EB must occur at position xB = 8 light-seconds in B's frame. So here, G=2.

Now, in the A frame the distance between B and the position of EA as a function of time must be (16 - 0.6*t'A). And in the B frame the distance between A and the position of EB as a function of time must be (8 + 0.6*t'B). Obviously we can't pick an arbitrary t'A and t'B and know that these two expressions will always be related by the same G-factor of 2. So do you specifically want the relation between t'A and t'B to be such that the two expressions are related by a factor of 2, i.e.

(8 + 0.6*t'B) = 2*(16 - 0.6*t'A) ?

If so we could solve for t'B as a function of t'A

t'B = 40 - 2*t'A

For example, if we pick t'A = 3, then t'B = 34. Did you intend this sort of relation between the two primed times, or am I misunderstanding? If I am, it really would be helpful if you'd explain a little more about what the primed times are supposed to mean physically...

neopolitan

Apr25-09, 11:17 PM

OK, going back to the third drawing in post 174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174):

Question #1: Why do both A and B conclude the same factor G will appear in the equations xA = G*xB and x'B = G*x'A? The two equations have fairly different physical meanings...the first means:

(position coordinate of EA in A's frame) = G*(position coordinate of EB in B's frame)

But the second means:

(distance between position of A at t'B and position of EB, in the B frame) = G*(distance between position of B at t'A and position of EA, in the A frame)

Can we go through a numerical example so I can try to understand what additional assumptions you might be making that lead you to think the factor relating these would be the same?

Let's suppose in A's frame B has a velocity of 0.6c. Suppose in A's frame the event EA has position xA = 16 light-seconds. In this case the light will pass B at a position of 6 light-seconds and a time of 10 seconds in A's frame. Since B's clock is running slow by a factor of 0.8 in this frame, the light must hit B when B's clock reads 8 seconds, so EB must occur at position xB = 8 light-seconds in B's frame. So here, G=2.

Now, in the A frame the distance between B and the position of EA as a function of time must be (16 - 0.6*t'A). And in the B frame the distance between A and the position of EB as a function of time must be (8 + 0.6*t'B). Obviously we can't pick an arbitrary t'A and t'B and know that these two expressions will always be related by the same G-factor of 2. So do you specifically want the relation between t'A and t'B to be such that the two expressions are related by a factor of 2, i.e.

(8 + 0.6*t'B) = 2*(16 - 0.6*t'A) ?

If so we could solve for t'B as a function of t'A

t'B = 40 - 2*t'A

For example, if we pick t'A = 3, then t'B = 34. Did you intend this sort of relation between the two primed times, or am I misunderstanding? If I am, it really would be helpful if you'd explain a little more about what the primed times are supposed to mean physically...

I was all ready to get irritated when I realised that I had not put labels the third diagram (the diagram that it is taken from does have labels, which would have prevented some problems) and further, I slipped up with primes earlier (http://www.physicsforums.com/showpost.php?p=2173445&postcount=214) when defining what all the x and x' values meant. I'll start again.

Remember I said that primes are "as per Galilean boosts"? In other words,

x'=x-vt

Part of the consequence of that is that x'B is B talking about B and xA is A talking about A.

Then there is plenty of scope for confusion about what xB and x'A mean.

x'A=xA-vtA
xB=x'B+vt'B

So:

xA is the distance between A and event EA according to A ( <- position coordinate)
(value in the example we have been discussing: 8)

x'A is the distance between B and event EA at t=5 according to A
(value in the example we have been discussing: 5)

x'B is the distance between B and event EB according to B ( <- position coordinate)
(value in the example we have been discussiing: 4)

xB is the distance between A and event EB at t=-10 according to B
(value in the example we have been discussing: 10)

Therefore (after you work it all through):

xB = xA.gamma (= 4*1.25 = 5 )
and
x'A = x'B.gamma (= 8*1.25 = 10 )

I should not have cut and pasted, I should have written it all out from first principles.

I apologise for any confusion caused.

Remember, the derivation allows you go back and show that events EA and EB are not the same. But it does not initially assume it.

cheers,

neopolitan

JesseM

Apr26-09, 12:09 AM

xA is the distance between A and event EA according to A ( <- position coordinate)
(value in the example we have been discussing: 8)
OK, it was actually 16 in my example but this works fine too.
x'A is the distance between B and event EA at t=5 according to A
(value in the example we have been discussing: 5)
In this case the distance between B and EA as a function of time tA in A's frame is 8 - 0.6*tA, so that works.
x'B is the distance between B and event EB according to B ( <- position coordinate)
(value in the example we have been discussiing: 4)

xB is the distance between A and event EB at t=-10 according to B
(value in the example we have been discussing: 10)
And here the distance between A and EB as a function of time t'B in B's frame is 4 + 0.6*t'B...I guess that should be t'B=10 rather that t'B=-10, right?
Therefore (after you work it all through):

xB = xA.gamma (= 4*1.25 = 5 )
and
x'A = x'B.gamma (= 8*1.25 = 10 )
But this only works because of the two particular times you chose for tA and t'B, right? You could just as easily pick a pair of t-values such that the factor was gamma^2 or anything else, as well as a pair such that the factors in the two equations were different. Do these times have any physical significance other than that they were chosen to make these two equations include a gamma factor?

Also, in equation 3 from the third diagram you wrote x'B + vt'B = x'B + vx'B/c, so it looks like you were making the substitution t'B = x'B/c, but that doesn't work with the above numbers where t'B = 10 seconds and x'B/c = 4 seconds.

neopolitan

Apr26-09, 01:05 AM

I'm clearly going to have to go back to the beginning with a separate diagram and show you all the relationships as I build the derivation.

I need to do that because I only used a flat 1 dimensional diagram, and it doesn't easily translate to a 1+1 diagram.

You can preempt me, if you like, by looking at this (http://www.physicsforums.com/attachment.php?attachmentid=18548&d=1240077258), this (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg) and this (http://www.geocities.com/neopolitonian/gal_rel_neopolitan.jpg) and thinking about how they could be melded.

cheers,

neopolitan

neopolitan

Apr26-09, 01:57 AM

But this only works because of the two particular times you chose for tA and t'B, right? You could just as easily pick a pair of t-values such that the factor was gamma^2 or anything else, as well as a pair such that the factors in the two equations were different. Do these times have any physical significance other than that they were chosen to make these two equations include a gamma factor?

I'm obviously tiring.

xA is the distance from the origin of the xA axis to event EA.

So why the hell did I have xA=4? Fixing it:

xB = xA.gamma (= 8*1.25 = 10 )
and
x'A = x'B.gamma (= 4*1.25 = 5 )

xA = tA.c = 8
x'B = t'B.c = 4

Then we have to remember that x'=x-vt only works because t=t'

Otherwise, x'=x-vt' so, in this case (my mess up again, we really should start from the beginning):

x'A=xA - vt'A
xB=x'B + vtB

where t'A is when the photon from EA passes B (eg t=5) according to A and vtB is when the photon from EB passes A according to B (eg t=10).

Summarising:

xA=8 ..... tA=8
x'A=5 ..... t'A=5

x'B=4 ..... x'B=4
xB=10 .... tB=10

xB = xA.gamma (= 8*1.25 = 10 )
x'A = x'B.gamma (= 4*1.25 = 5 )

x'A=xA - vt'A = 8 - 0.6*5 = 5
xB=x'B + vtB = 4 + 0.6*10 = 10

which means it all works.

I very very much hope this has been written out correctly this time, I even printed it out and checked it. I will still do the diagram, because I think it might help even though the diagram is only something you can construct retrospectively.

cheers,

neopolitan

JesseM

Apr26-09, 03:27 AM

where t'A is when the photon from EA passes B (eg t=5) according to A and vtB is when the photon from EB passes A according to B (eg t=10).
Ah, OK, then that answers my question about the physical meaning of the time coordinates, thanks.

So, xB = xA * gamma can be written as:

(distance between A and EB at the time tB when the photon from EB reaches A, all in the B frame) = (distance between A and EA in the A frame) * gamma

And x'A = x'B * gamma can be written as:

(distance between B and EA at the time t'A when the photon from EA reaches B, all in the A frame) = (distance between B and EB in the B frame) * gamma

So, this does seem to work in SR (although I'm having trouble conceptualizing why it must work). I guess my next question is, if you don't assume SR from the start, what starting assumptions do you use to derive the fact that these equations should include the same factor G relating the two quantities in each equation?

Also, since the equations in diagram 3 from post 174 are xA = xB*G and x'B = x'A*G, I guess that means the G factor is actually 1/gamma? Or you could just rewrite the equations in that diagram as xB = xA*G and x'A = x'B*G if you prefer, that way G would still end up being equal to gamma.
xA = tA.c = 8
x'B = t'B.c = 4
OK, so that means tA is the time for the photon to get from EA to A in the A frame, and t'B is the time for the photon to get from EB to B in the B frame.
x'A=xA - vt'A
xB=x'B + vtB
These equations are actually different from what's written in equation 3 and 4 of the diagram, here you have t'A where the diagram has tA, and tB where the diagram has t'B. I think the revised versions make sense given what you say above, for example the first revised equation would mean:

(distance between B and EA at the time t'A when the photon from EA reaches B, all in the A frame) = (distance between A and EA in the A frame) - v*(time t'A when the photon from EA reaches B as seen in A frame)

...which does make sense, since B started at the same distance from EA as A did, and then at any later time t, B has moved closer to the position of EA by an amount vt (all as seen in A frame). But with this revised version x'A=xA - vt'A, you can't substitute xA/c in for t'A as you did in the diagram's version with tA right? So doesn't that mean the final part of equation 4 in the diagram is no longer valid? And likewise with the final part of equation 3?

I guess you would instead want to substitute x'A/c in for t'A, giving xA = x'A + vx'A/c = x'A*(1 + v/c), an altered version of equation 4. Likewise for xB=x'B + vtB you could substitute xB/c for tB, giving x'B = xB - vxB/c = xB*(1 - v/c), an altered version of equation 3. Then with the rewritten equations xB = xA*G and x'A = x'B*G (the ones I suggested earlier so G would still end up being equal to gamma) you'd be able to plug in and write xB = G*x'A*(1 + v/c) along with x'A = G*xB*(1 - v/c), which would be altered versions of your equations 6 and 5.

JesseM

Apr26-09, 05:12 AM

neopolitan

Apr26-09, 05:32 AM

So, this does seem to work in SR (although I'm having trouble conceptualizing why it must work). I guess my next question is, if you don't assume SR from the start, what starting assumptions do you use to derive the fact that these equations should include the same factor G relating the two quantities in each equation?

Galilean invariance

Also, since the equations in diagram 3 from post 174 are xA = xB*G and x'B = x'A*G, I guess that means the G factor is actually 1/gamma?

x'A=xA - vt'A
xB=x'B + vtB

and

t'A = x'A/c
tB = xB/c

so

x'A=xA - vx'A/c
xB=x'B + vxB/c

so

xA=x'A + vx'A/c = x'A * (1 + v/c )
x'B=xB - vxB/c = xB * (1 - v/c )

and

xB = xA.G
x'A = x'B.G

so

xB = xA.G = G * x'A * (1 + v/c )

and

x'A = x'B.G = G * xB * (1 - v/c )

so

xB = G * ( G * xB * (1 - v/c )
) * (1 + v/c )

so

1 = G2 (1 - v2/c2)

so

G = gamma - if you put G on the other sides of the first equations you come across it in, then you end up with G = 1/gamma (which is what you originally asked and the answer is yes), but you always end up with:

xB = xA.gamma
x'A = x'B.gamma

then taking the next step:

xB=x'B + vtB

so

x'B=xB - vtB

and

x'A = (xB - vtB).gamma

which is your spatial Lorentz transformation

then dividing through by c

t'A = (tB - vtB/c).gamma

and since tB = xB/c then

t'A = (tB - v.xB/c2).gamma

which is your temporal Lorentz transformation

These equations are actually different from what's written in equation 3 and 4 of the diagram, here you have t'A where the diagram has tA, and tB where the diagram has t'B. I think the revised versions make sense given what you say above, for example the first revised equation would mean:

(distance between B and EA at the time t'A when the photon from EA reaches B, all in the A frame) = (distance between A and EA in the A frame) - v*(time t'A when the photon from EA reaches B as seen in A frame)

...which does make sense, since B started at the same distance from EA as A did, and then at any later time t B has moved an additional vt from EA (all as seen in A frame). But with this revised version x'A=xA - vt'A, you can't substitute xA/c in for t'A as you did in the diagram's version with tA right? So doesn't that mean the final part of equation 4 in the diagram is no longer valid? And likewise with the final part of equation 3?

Sort of answered above, these supplant the diagrams and any errors therein.

cheers,

neopolitan

neopolitan

Apr26-09, 06:06 AM

So, to continue with the slightly altered version of your derivation, we can combine the equations xB = G*x'A*(1 + v/c) along with x'A = G*xB*(1 - v/c) to get xB = G*(G*xB*(1 - v/c))*(1 + v/c) = G^2*xB*(1 - v^2/c^2). So divide both sides bey xB to get 1 = G^2*(1 - v^2/c^2), or G^2 = 1/(1 - v^2/c^2), showing G is the familiar gamma factor. We can plug that back into xB = xA*G to get xB = gamma*xA. Now I think you'd want to combine that with another revised equation, x'A=xA - vt'A or equivalently xA = x'A + vt'A, and then we get xB = gamma*(x'A + vt'A). Now, you would say this is a derivation of the spatial portion of the Lorentz transformation, correct? The problem I see here is that when you actually examine the meaning of the symbols, they don't seem to be all coordinates of a specific event or pair of events as in the Lorentz transformation--xB is the difference in the B frame between the spatial coordinates of the event EB and the event of the photon from EB hitting A, but x'A is the difference in the A frame between the spatial coordinates of a different pair of events, namely the event EA and the event of the photon from EA hitting B (t'A is the difference between the time coordinates of the same pair of events as x'A). So unless I've gotten the equations wrong or the symbols mixed up, it seems that any resemblance to the spatial Lorentz transformation equation here is coincidental, this equation doesn't have the same physical meaning at all, and in fact it would only be applicable to the specific definitions of the symbols in terms of the particular physical scenario you described, rather than the Lorentz equation delta-xB = gamma*(delta-xA + v*delta-tA) which is applicable to the difference in spatial coordinates and difference in time coordinates of two arbitrary events (events that may not lie on the worldline of a photon for example), but with the understanding that delta-xB and delta-xA are the difference in spatial coordinates between the same pair of events in two frames.

What I've shown you so far is how to end up with:

A talking about the location of event EA as B sees it, modified by gamma. EA can be any event even two events which do not share the same world line which you can then find the delta between.

and

B talking about the location of event EB as A sees it, modified by gamma. EB can be any event and even two events which do not share the same world line which you can then find the delta between.

If you want to call this the Lorentz boost to distinguish it from the Lorentz Transformation, since it comes from the Galilean boost, then you are welcome to.

It's such a simple thing compared to what you have to go through to work out the matrix versions, perhaps it is better to call it a boost and note that it is a 100% accurate approximation of the Lorentz Transformation limited to two dimensions but that you can apply the same logic to all three spatial dimensions to work out what happens in y and z if v is not limited.

cheers,

neopolitan

(The derivation of the temporal Lorentz transformation does seem dodgy, I accept that, it just happens to work. I understand your concern. Note, however, that I did say at one point something along the lines of "Once you understand how to derive the spatial Lorentz transformation, it is not difficult to do the same for the temporal". Would you be more happy if you could see that I can derive the temporal Lorentz transformation so that I have find the spatial component of any event, followed by the temporal component of any event. If I was showing students, I think I would have to show the temporal derivation as well.)

JesseM

Apr26-09, 06:29 AM

What you end up with is:

A talking about event EA as B sees it, modified by gamma. EA can be any event even two events which do not share the same world line which you can then find the delta between.

and

B talking about event EB as A sees it, modified by gamma. EB can be any event and even two events which do not share the same world line which you can then find the delta between.
The problem is that when we actually look at the meaning of the symbols in your equations, they are not talking about the coordinates of the same event (or the coordinate intervals between the same pair of events) in two different frames. And there is nothing in your derivation to show it can be generalized to such a case, everything about your derivation referred to specific events on the worldline of a light ray that were chosen based on simultaneity with A&B being colocated.

For example, you say x'A = (xB - vtB).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (EA, photon hitting B), while the right side deals with the distance and time between a different pair of events (EB, photon hitting A). It so happens because of the symmetry of the problem that the distance and time between the first pair should be the same as the distance and time between the second pair in any given frame, but how do you prove that the equation would still work if we changed the meaning of the symbols so both sides were referring to the same pair of events, events which didn't necessarily have to lie on the worldline of a light ray? To derive the Lorentz transform you need an equation where the physical meaning of the symbols is the same, not just an equation that has the same form.
It's such a simple thing compared to what you have to go through to work out the matrix versions
If you want a relatively simple derivation of the 1+1 dimensional Lorentz transform that sticks to basic algebra, see my post #14 on this thread (http://www.physicsforums.com/showthread.php?t=180578).

neopolitan

Apr26-09, 06:31 AM

The problem is that when we actually look at the meaning of the symbols in your equations, they are not talking about the coordinates of the same event (or the coordinate intervals between the same pair of events) in two different frames. And there is nothing in your derivation to show it can be generalized to such a case, everything about your derivation referred to specific events on the worldline of a light ray that were chosen based on simultaneity with A&B being colocated.

For example, you say x'A = (xB - vtB).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (EA, photon hitting B), while the right side deals with the distance and time between a different pair of events (EB, photon hitting A). How do you prove that the equation would still work if we changed the meaning of the symbols so both sides were referring to the same pair of events? To derive the Lorentz transform you need an equation where the physical meaning of the symbols is the same, not just an equation that has the same form.

If you want a relatively simple derivation of the 1+1 dimensional Lorentz transform that sticks to basic algebra, see my post #14 on this thread (http://www.physicsforums.com/showthread.php?t=180578).

I modified my post. Perhaps you might want to modify your reply in light of that, then I can delete this.

JesseM

Apr26-09, 06:41 AM

What I've shown you so far is how to end up with:

A talking about the location of event EA as B sees it, modified by gamma. EA can be any event even two events which do not share the same world line which you can then find the delta between.

and

B talking about the location of event EB as A sees it, modified by gamma. EB can be any event and even two events which do not share the same world line which you can then find the delta between.
Are you still talking about the equation x'A = (xB - vtB).gamma though? If so I don't see how your description fits, x'A on the left side isn't the spatial coordinates of any individual event in A's frame, rather it's the interval (x-coordinate A assigns to event EA) - (x-coordinate A assigns to photon hitting B). Likewise xB on the right side is the interval (x-coordinate B assigns to EB) - (x-coordinate B assigns to photon hitting A), and tB does the same but for time coordinates. Also, you say "EB can be any event" but you don't justify that, your derivation of this equation made use of some facts that were specifically related to the fact that these are events on the worldline of a light ray, like the substitution tB = xB/c.

neopolitan

Apr26-09, 06:56 AM

Are you still talking about the equation x'A = (xB - vtB).gamma though? If so I don't see how your description fits, x'A on the left side isn't the spatial coordinates of any individual event in A's frame, rather it's the interval (x-coordinate A assigns to event EA) - (x-coordinate A assigns to photon hitting B). Likewise xB on the right side is the interval (x-coordinate B assigns to EB) - (x-coordinate B assigns to photon hitting A), and tB does the same but for time coordinates. Also, you say "EB can be any event" but you don't justify that, your derivation of this equation made use of some facts that were specifically related to the fact that these are events on the worldline of a light ray, like the substitution tB = xB/c.

Yes, my EA and EB are causally linked and right at the beginning, if you remember, I talked about one photon which could have been emitted anywhere so long as it subsequently passes B and then A.

This derivation is based on the fact that the event emits a photon which is subsequently detected, otherwise A and B know nothing about the event and quite possibly could never know anything about it.

cheers,

neopolitan

JesseM

Apr26-09, 07:22 AM

Yes, my EA and EB are causally linked and right at the beginning, if you remember, I talked about one photon which could have been emitted anywhere so long as it subsequently passes B and then A.

This derivation is based on the fact that the event emits a photon which is subsequently detected, otherwise A and B know nothing about the event and quite possibly could never know anything about it.
The emission event doesn't even enter into any of your equations, the photon could have been traveling along the same path for infinite time, it wouldn't make a difference. And if you agree your derivation involves a very particular choice of four events to look at, then do you agree that you aren't really deriving the Lorentz transformation? The Lorentz transform deals with the coordinates of a single arbitrary event in two frames (or the coordinate intervals between a single arbitrary pair of events in two frames), whereas your equation shows a relation between two different pairs of events that have particular properties that you defined for them (like the fact that all four events have a light-like separation from one another). It so happens that because of a symmetry in the way you defined the events, the spatial and temporal separation between the first pair is guaranteed to be the same as the spatial and temporal separation between the second pair (try drawing both the 'EA/photon hitting B' pair and the 'EB/photon hitting A' pair on the same graph and hopefully you'll see what I mean), which is why your relation looks just like the Lorentz transform equation dealing with a single pair of events, but your derivation wouldn't generalize to an arbitrarily-chosen pair of events, in particular a pair that had a time-like or space-like separation.

neopolitan

Apr26-09, 08:35 AM

The emission event doesn't even enter into any of your equations, the photon could have been traveling along the same path for infinite time, it wouldn't make a difference. And if you agree your derivation involves a very particular choice of four events to look at, then do you agree that you aren't really deriving the Lorentz transformation? The Lorentz transform deals with the coordinates of a single arbitrary event in two frames (or the coordinate intervals between a single arbitrary pair of events in two frames), whereas your equation shows a relation between two different pairs of events that have particular properties that you defined for them (like the fact that all four events have a light-like separation from one another).

You really are going to have to make up your mind.

I think the Lorentz transform is one event described in two frames. Do you agree?

Or do you want the Lorentz transform to be coordinate intervals between a pair of events described in two frames?

Or do you want it to be both?

I don't know where you are getting four events from.

It so happens that because of a symmetry in the way you defined the events, the spatial and temporal separation between the first pair is guaranteed to be the same as the spatial and temporal separation between the second pair (try drawing both the 'EA/photon hitting B' pair and the 'EB/photon hitting A' pair on the same graph and hopefully you'll see what I mean), which is why your relation looks just like the Lorentz transform equation dealing with a single pair of events, but your derivation wouldn't generalize to an arbitrarily-chosen pair of events, in particular a pair that had a time-like or space-like separation.

I think you don't understand, yes.

Take one event. Any event, E, any where any time.

Take a single observer, A, any where any time.

Now that we have a single observer and a single event (we don't stuff around by changing anything), there is a single distance interval between the spatial location of event E and the spatial location of the observer and a single time interval between the time location of event E and any specific time reading for the observer.

The difference is that x=0 makes sense, but really t=0 doesn't make much sense, except perhaps if you start when the observer is born/created/assigned the task. But if you pick any time for A and call it t=0, then there is a single time interval between the event (0,0) and the event E.

So there is a single spacetime interval between the event (0,0) and the event E (irrespective of how that spacetime interval is measured, ie what coordinate system you use to measure it - even if the coordinate system of A makes the most sense).

Do we agree on this?

I'm not going further than this. Please agree or disagree on this, then I can proceed. But if we disagree on this, or I have a fundamental misunderstanding, then we have to sort that out first.

cheers,

neopolitan

JesseM

Apr26-09, 01:08 PM

You really are going to have to make up your mind.

I think the Lorentz transform is one event described in two frames. Do you agree?

Or do you want the Lorentz transform to be coordinate intervals between a pair of events described in two frames?

Or do you want it to be both?
The Lorentz transform can be written both ways. For example, the spatial part can be written as

x' = gamma*(x - vt)

where (x,t) are the coordinates of a single event in the unprimed frame and x' is the spatial coordinate of the same event in the primed frame, or as

delta-x' = gamma*(delta-x - v*delta-t)

where delta-x and delta-t are the coordinate intervals in the unprimed frame between a pair of events, and delta-x' is the spatial coordinate interval between the same pair of events in the primed frame. It's easy to derive the second equation from the first.
I don't know where you are getting four events from.
I mentioned the four events in your equation in all three of my previous posts. For example, in post 229: you say x'A = (xB - vtB).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (EA, photon hitting B), while the right side deals with the distance and time between a different pair of events (EB, photon hitting A).
The first parentheses is one pair of events, the one x'A gives the coordinate distance between in frame A according to previous definitions, while the second parentheses is a different pair of events, the one xB and tB give the coordinate intervals between in frame B. So your equation relates one pair of events on the left side to a different pair of events on the right side.
I think you don't understand, yes.

Take one event. Any event, E, any where any time.

Take a single observer, A, any where any time.

Now that we have a single observer and a single event (we don't stuff around by changing anything), there is a single distance interval between the spatial location of event E and the spatial location of the observer and a single time interval between the time location of event E and any specific time reading for the observer.

The difference is that x=0 makes sense, but really t=0 doesn't make much sense, except perhaps if you start when the observer is born/created/assigned the task. But if you pick any time for A and call it t=0, then there is a single time interval between the event (0,0) and the event E.

So there is a single spacetime interval between the event (0,0) and the event E (irrespective of how that spacetime interval is measured, ie what coordinate system you use to measure it - even if the coordinate system of A makes the most sense).

Do we agree on this?
Sure, but the Lorentz transformation doesn't directly calculate the invariant spacetime interval x^2 - c^2*t^2. It says if you know the coordinates of E are (x,t) in one frame, then it can tell you the corresponding coordinates (x',t') of the same event E in another frame. Likewise, if you have two arbitrary events E1 and E2 and you know the coordinate distance delta-x and coordinate time delta-t between them in one frame, then it can tell you the delta-x' and delta-t' for the same specific pair E1 and E2 in another frame. Your equation does neither of these things.

neopolitan

Apr26-09, 11:09 PM

I mentioned the four events in your equation in all three of my previous posts. For example, in post 229:

you say x'A = (xB - vtB).gamma is the spatial Lorentz transform, but it just isn't, not with the stated meaning of the symbols--the left side deals with the distance between two events (EA, photon hitting B), while the right side deals with the distance and time between a different pair of events (EB, photon hitting A).

The first parentheses is one pair of events, the one x'A gives the coordinate distance between in frame A according to previous definitions, while the second parentheses is a different pair of events, the one xB and tB give the coordinate intervals between in frame B. So your equation relates one pair of events on the left side to a different pair of events on the right side.

You're not confused here Jesse, you are simply wrong. There are two relevant events in terms of the derivation, photon crosses the xA axis (at t=0 according to A and t=-6 according to B) and photon crosses the xB axis (at t=3 according to A and t=0 according to B).

Agreed that both can use another two events to work out where on their own x axis that crossing took place, but otherwise those events (photon passes B and photon passes A) are irrelevant (in this part of the derivation I really don't care when A and B work out where a photon from event EA passed their x axis).

But do you at least agree that all four events could involve the same photon? (I think you do because you were complaining about them all being on the same world line.)

Sure.

I light of that I present a semi humorous pair of drawings.

cheers,

neopolitan

JesseM

Apr27-09, 12:03 AM

You're not confused here Jesse, you are simply wrong. There are two relevant events in terms of the derivation, photon crosses the xA axis (at t=0 according to A and t=-6 according to B) and photon crosses the xB axis (at t=3 according to A and t=0 according to B).
Are you claiming that EA and EB don't also figure into the definition of x'A and xB? Here was how you defined those terms in post #221:
x'A is the distance between B and event EA at t=5 according to A
(value in the example we have been discussing: 5)

xB is the distance between A and event EB at t=-10 according to B
(value in the example we have been discussing: 10)
Note that t=5 in the A frame was the time when the light hit B according to the A frame, so according to the above x'A is the distance in the A frame between B and EA at the moment the light hits B (i.e. the distance in the A frame between the event EA and the event of light hitting B)

Likewise, t=10 (which is what I assume you mean to write in the second line rather than t=-10) in the B frame was the time when the light hit A according to the B frame (which is also how you defined tB earlier), so according to the above xB is the distance in the B frame between A and EB at the moment the light hit A (i.e. the distance in the B frame between the event EB and the event of light hitting A)

So unless you've revised your definitions of the terms x'A and xB, it appears that the first refers to the distance between two events (EA, light hits B) and the second refers to the distance between two different events (EB, light hits A). If you think I am misunderstanding something here, please explain.
But do you at least agree that all four events could involve the same photon? (I think you do because you were complaining about them all being on the same world line.)
Yes, I agree with that--where I write "photon hits A" or "photon hits B" it shouldn't be taken to imply the photon is absorbed, perhaps it would be better to write "photon passes" instead.

neopolitan

Apr27-09, 12:33 AM

Going back as far as post #212 (when I filled in forms for you), I can't see anywhere where I have spoken about photons have hit A and B and given notation for those events.

You brought those events up for the first time in post #229. I followed up #229 with #230 in which I said "I modified my post (#228). Perhaps you might want to modify your reply in light of that, then I can delete this."

You didn't modify #229 so I have subsequently ignored it thinking that #231 which referred to the modified #228 replaced #229.

After that I did talk about the photon being subsequently detected and detection was always part of the scenario, but I never formally described any event which was the photon passing either A or B. (Those events would be (0,8) and (0,4) in the frame of the observer being passed.)

I don't think you did either.

cheers,

neopolitan

JesseM

Apr27-09, 12:57 AM

Going back as far as post #212 (when I filled in forms for you), I can't see anywhere where I have spoken about photons have hit A and B and given notation for those events.
In post #224 you said:
x'A=xA - vt'A
xB=x'B + vtB
where t'A is when the photon from EA passes B (eg t=5) according to A and vtB is when the photon from EB passes A according to B (eg t=10).
xA was the coordinate position of EA in the A frame, and in this frame B is moving towards EA at velocity v starting at the origin, so naturally the distance between B and the position of EA as a function of time in this frame is xA - vt. If you define t = t'A as the time when the photon passes B as seen in the A frame, that means xA - vt'A must be the distance in this frame between B and EA at the moment the photon passes B, i.e. the distance in the A frame between the event of the photon passing B and the event EA. Do you disagree?
You brought those events up for the first time in post #229. I followed up #229 with #230 in which I said "I modified my post (#228). Perhaps you might want to modify your reply in light of that, then I can delete this."
Was the modification to post 228 supposed to modify any of the definitions of yours which I quoted above?
After that I did talk about the photon being subsequently detected and detection was always part of the scenario, but I never formally described any event which was the photon passing either A or B. (Those events would be (0,8) and (0,4) in the frame of the observer being passed.)
t'A and tB referred to the time in one observer's frame when the other observer was being passed by the photon, and they occurred at t=5 and t=10 which did figure into your calculations (remember in the second-to-last paragraph of my post #222 where I pointed out that the relations xB = xA*gamma and x'A = x'B*gamma only included the gamma factor because you chose that particular pair of times, and I asked why you had chosen them...your definitions of t'A and tB in post #224 in terms of photons passing observers was what I took as an answer to that question).

neopolitan

Apr27-09, 01:26 AM

Are you claiming that EA and EB don't also figure into the definition of x'A and xB?

Have I said that? The photon passing B and then A don't really figure as events in my derivation, otherwise I would have drawn them in as events - we know that a photon at 8ls away from A travelling towards A will pass A 8s later. But the event I am talking about is the photon being 8ls away from A, EA -the photon passing A 8s later is a consequence of that event as well as being an event in its own right.

It's just that I am focussed on the former (consequence) rather then the latter (a separate event in its own right).

Note that t=5 in the A frame was the time when the light hit B according to the A frame, so according to the above x'A is the distance in the A frame between B and EA at the moment the light hits B (i.e. the distance in the A frame between the event EA and the event of light hitting B)

Likewise, t=10 (which is what I assume you mean to write in the second line rather than t=-10) in the B frame was the time when the light hit A according to the B frame (which is also how you defined tB earlier), so according to the above xB is the distance in the B frame between A and EB at the moment the light hit A (i.e. the distance in the B frame between the event EB and the event of light hitting A)

So unless you've revised your definitions of the terms x'A and xB, it appears that the first refers to the distance between two events (EA, light hits B) and the second refers to the distance between two different events (EB, light hits A). If you think I am misunderstanding something here, please explain.

Step 1, ask if something that seems odd is me screwing up.

Step 2, wait for a response, it's not as if I am making you wait.

Step 3, work from the correction, if there is indeed one.

The value t=-10 is correct. When the photon passes EA, then according to B, t=-10. When the photon passes A, then according to B, t=10. Your incorrect correction is right, since you redefined what I was saying.

Note very very carefully, I said in #224:

vtB is when the photon from EB passes A according to B (eg t=10)

and in #221:

xB is the distance between A and event EB at t=-10 according to B

I didn't put a subscript on t=-10. It's not an error.

Yes, I agree with that--where I write "photon hits A" or "photon hits B" it shouldn't be taken to imply the photon is absorbed, perhaps it would be better to write "photon passes" instead.

I know what you mean, I have tried to use "passes", but if I say "hits" I mean "passes". I mentioned "thought experiment magic" before and I fully expect you to have the right to use the same magic in your descriptions, so a photon which hits B recovers immediately and continues on its way to A with no delay or other ill effects.

Any comments on the diagrams? They were semi-humorous in the sense that they were semi-serious.

cheers,

neopolitan

neopolitan

Apr27-09, 01:38 AM

xA was the coordinate position of EA in the A frame, and in this frame B is moving towards EA at velocity v starting at the origin, so naturally the distance between B and the position of EA as a function of time in this frame is xA - vt. If you define t = t'A as the time when the photon passes B as seen in the A frame, that means xA - vt'A must be the distance in this frame between B and EA at the moment the photon passes B, i.e. the distance in the A frame between the event of the photon passing B and the event EA. Do you disagree?

No, I agree.

Was the modification to post 228 supposed to modify any of the definitions of yours which I quoted above?

No.

t'A and tB referred to the time in one observer's frame when the other observer was being passed by the photon, and they occurred at t=5 and t=10 which did figure into your calculations (remember in the second-to-last paragraph of my post #222 where I pointed out that the relations xB = xA*gamma and x'A = x'B*gamma only included the gamma factor because you chose that particular pair of times, and I asked why you had chosen them...your definitions of t'A and tB in post #224 in terms of photons passing observers was what I took as an answer to that question).

They are consequences of the placements of the photon at events EA and EB, like I said before, I didn't consider the passing A and B by the photon to be significant events in their own right.

It's like falling down the top of the stairs (top step being JesseM's reference step) - the event "tripping" is important but really, all the little events that follow (Jesse M hits the third step down, JesseM hits the fourth and fifth step down, JesseM starts rolling and hits all steps until step 13) are consequences of the initial trip. Perhaps you could mention one other important event, trying to stop falling by reaching for the banister and failing, but once you are really falling, the end result is pretty much all scripted.

If I made step 12 a special step (it's the one I hide all my gold under, so it is my reference step) would I be obliged to refer to JesseM hits step 12 as a special event?

In the same sense, although I know that the photon eventually passes both B and A, I don't feel obliged to refer to them as special events.

cheers,

neopolitan

JesseM

Apr27-09, 01:57 AM

Have I said that? The photon passing B and then A don't really figure as events in my derivation, otherwise I would have drawn them in as events - we know that a photon at 8ls away from A travelling towards A will pass A 8s later. But the event I am talking about is the photon being 8ls away from A, EA -the photon passing A 8s later is a consequence of that event as well as being an event in its own right.

It's just that I am focussed on the former (consequence) rather then the latter (a separate event in its own right).
Regardless of whether you were "focussed" on them, what I was saying was that if we look at the physical meaning of x'A and xB as you defined them, there doesn't seem to be any way to define them that doesn't involve those events of the photons passing A and B. If you can think of a rigorous definition of x'A and xB that make no mention of these events (and don't refer to other coordinates which themselves are defined in terms of these events), then please explain.
The value t=-10 is correct. When the photon passes EA, then according to B, t=-10. When the photon passes A, then according to B, t=10. Your incorrect correction is right, since you redefined what I was saying.

Note very very carefully, I said in #224:
vtB is when the photon from EB passes A according to B (eg t=10)
And was it an error that you wrote vtB there as opposed to tB? vtB would be a distance rather than a time.
and in #221:
xB is the distance between A and event EB at t=-10 according to B
OK, but that would imply xB = 2, which doesn't fit with what you wrote elsewhere. After all, in B's frame A was moving in the -x direction at 0.6c, so at t=-10, A would have been at position x=+6, while the event EB occurred at x=+4 in B's frame.
I didn't put a subscript on t=-10. It's not an error.
When you say you "didn't put a subscript", you mean t=-10 is distinct from tB which is t=10 according to the definition above (if we remove the v), right? But then it seems you have offered two distinct definitions of xB, one of which makes use of t=-10 and one of which makes use of tB = 10:
xB is the distance between A and event EB at t=-10 according to B
xB=x'B + vtB
These two definitions would be equivalent if you had written t=10 in the first, which was part of why I assumed it was a mistake. But if it's not a mistake, the definitions appear to be incompatible--as I said, the first would seem to imply xB = 2, while the second implies xB = 4 + 0.6*10 = 10
Any comments on the diagrams? They were semi-humorous in the sense that they were semi-serious.
I couldn't quite follow the point you were making, but it seemed like you were saying my objection was that the red event might not lie on the light ray that crossed through EA and EB...if so that wasn't really my objection, I realize that you can always draw a new light ray which goes through any arbitrary event, and define a new EA and EB in terms of where this ray crosses the x-axes of A and B's frames. But even if we assume our "arbitrary event" is along this ray, my problem is that none of the coordinates you defined--xA, x'A, xB, x'B--have anything to do with that event specifically as opposed to any of an infinite number of other possible events along the same ray, all the events on this ray would yield the same values for those coordinates that you defined. So, the relation between these coordinates doesn't really demonstrate anything about how the coordinates of the event itself in each frame will be related to one another, it only tells us about relations between coordinates of events involved in the definitions of xA and your other coordinates.

neopolitan

Apr27-09, 04:01 AM

Regardless of whether you were "focussed" on them, what I was saying was that if we look at the physical meaning of x'A and xB as you defined them, there doesn't seem to be any way to define them that doesn't involve those events of the photons passing A and B. If you can think of a rigorous definition of x'A and xB that make no mention of these events (and don't refer to other coordinates which themselves are defined in terms of these events), then please explain.

Do you disagree that the photon passing A at a specific time (in either coordinate frame) is a consequence of that photon being at EA at another specific time and having a specific direction? I agree that the photon being at that spacetime location leads to the photon passing B and then A. I've never come close to denying that.

Just as much as event EA has a specific spacetime location, so too does the photon pass B and A (constituting two new events in your parlance). But which comes first?

I'm giving priority to the first consequential event in each frame (either EA or EB, not the detection of the event(s) (a detection which is in itself a new event in your parlance).

It gets more complicated if we use the B frame, because if the photon is spawned by EB then it never passed the xA axis and so there was no location of the photon simultaneous with the colocation of A and B, according to A. That's why I started with the idea of a photon which just passes the x axes, and call those passings events EA and EB.

And was it an error that you wrote vtB there as opposed to tB? vtB would be a distance rather than a time.

Yeah, I've been having all sorts of problems with cutting and pasting code. Delete the v.

OK, but that would imply xB = 2, which doesn't fit with what you wrote elsewhere. After all, in B's frame A was moving in the -x direction at 0.6c, so at t=-10, A would have been at position x=+6, while the event EB occurred at x=+4 in B's frame.

I can only refer you back to posts #227 and #224.

xA is the distance between the origin of the xA axis and EA, according to A, which is 8.

x'B is the distance between the origin of the xB axis and EB, according to B, which is 4.

tA is the time it takes a photon to travel from event EA to the origin of the xA axis, according to A, which is 8.

t'B is the time it takes a photon to travel from event EB to the origin of the xB axis, according to B, which is 4.

t'A is the time it takes a photon to travel from event EA and pass the tB axis (and hence B), according to A, which is 5.

tB is the time it takes a photon to travel from event EB and pass the tA axis (and hence B), according to B, which is 10.

x'A is the distance beween B and event EA when the photon passes B (which is, I stress, just a consequence of the spacetime location of event EA), according to A, which is 5.

xB is the distance beween A and event EB when the photon passes A (which is, I stress, just a consequence of the spacetime location of event EB), according to B, which is 10.

When you say you "didn't put a subscript", you mean t=-10 is distinct from tB which is t=10 according to the definition above (if we remove the v), right?

Yes, t=-10 is not shown anywhere on the diagram, but if you took the tB axis and relocated it so it crossed EB and then followed it down until it crossed the tA axis, then the that crossing would be t = -10 on the tB axis.

I couldn't quite follow the point you were making, but it seemed like you were saying my objection was that the red event might not lie on the light ray that crossed through EA and EB...if so that wasn't really my objection, I realize that you can always draw a new light ray which goes through any arbitrary event, and define a new EA and EB in terms of where this ray crosses the x-axes of A and B's frames. But even if we assume our "arbitrary event" is along this ray, my problem is that none of the coordinates you defined--xA, x'A, xB, x'B--have anything to do with that event specifically as opposed to any of an infinite number of other possible events along the same ray, all the events on this ray would yield the same values for those coordinates that you defined. So, the relation between these coordinates doesn't really demonstrate anything about how the coordinates of the event itself in each frame will be related to one another, it only tells us about relations between coordinates of events involved in the definitions of xA and your other coordinates.

Pick any event, relocate (conceptually) your axes, and you can work out xB in terms of xA by seeing where a photon from the event which now lies on the xA axis crosses the xB axis.

Do a similar thing with the t axes and you can work out tB in terms of tA.

Perhaps I have confused you. I talked about an event that happens anywhere on the world line defined by EA and EB. Really, I only want to talk about one "real" event which I purposely shift my axes to align up so that the event is on the xA axis for the purposes of deriving the spatial Lorentz transformation.

To do the same with the temporal Lorentz transformation you can shift the axes so that the event is on the tA axis. When does (or did) a photon which crosses the tA axis at tA cross the tB axis?

The relationship will actually be the same (just shifted) as the relationship between the two events you want to add to the mix, photon passing B and photon passing A.

t'A = (tB - v.xB/c2).gamma

5 = ( 10 - 0.6*10 ) * 1.25 = 4 * 1.25 = 5

or (noting v in the other direction)

tB = (t'B + v.x'B/c2).gamma

10 = ( 5 + 0.6*5 ) * 1.25 = 8 * 1.25 = 10

If this doesn't help then, without some animation, I really wonder if there is any way to get this through to you.

I don't really have the facilities here to do animation. If there is anyone following this thread who understands what I am trying to explain and can do animation, then perhaps you could help by showing the temporal relocation of the xA and xB axes and the spatial relocation of the tB axis to align with any event that JesseM would like to choose.

For example, the event in my diagram (http://www.geocities.com/neopolitonian/uniquespacetimelocation.jpg) is nominally:

(xA,tA)=(8,0).

Say we chose an event:

(xA,tA)=(5,-4)

the relocation would make this event be:

(xA,tA+4)=(5,0)

The animation I am thinking of is the three axes in question sliding down four to align with the new event. Is that possible?

cheers,

neopolitan

(There may be some cut and paste, or failure to subscript errors in here. I am really getting tired, physically and intellectually tired, of explaining something that seems quite obvious to me, but clearly isn't obvious to everybody, or perhaps anybody else. And the more I write, the more chances there are that something I write is not perfect.)

JesseM

Apr27-09, 05:03 AM

Do you disagree that the photon passing A at a specific time (in either coordinate frame) is a consequence of that photon being at EA at another specific time and having a specific direction? I agree that the photon being at that spacetime location leads to the photon passing B and then A. I've never come close to denying that.
I don't disagree with you in causal terms, but I'm not talking about causality, I'm just talking about the definition of terms. Since EB happens at a later time than EA you could equally well say that the photon passing through the point EB is a "consequence" of it having been at EA, but there's no need to refer to two events in the definition of x'B which just refers to the coordinate position of the photon at time 0 in the B frame (and this event is of course EB). In contrast, your definitions of x'A and xB was in terms of the distance between two events, and you don't have any other way to define the meaning of these terms. So, your equation x'A=gamma*(xB - vtB) is not physically equivalent to the Lorentz transform, despite the fact that it looks the same if you forget about the definitions of the terms.
and in #221:
xB is the distance between A and event EB at t=-10 according to B
OK, but that would imply xB = 2, which doesn't fit with what you wrote elsewhere. After all, in B's frame A was moving in the -x direction at 0.6c, so at t=-10, A would have been at position x=+6, while the event EB occurred at x=+4 in B's frame.
I can only refer you back to posts #227 and #224.

xA is the distance between the origin of the xA axis and EA, according to A, which is 8.

x'B is the distance between the origin of the xB axis and EB, according to B, which is 4.

tA is the time it takes a photon to travel from event EA to the origin of the xA axis, according to A, which is 8.

t'B is the time it takes a photon to travel from event EB to the origin of the xB axis, according to B, which is 4.

t'A is the time it takes a photon to travel from event EA and pass the tB axis (and hence B), according to A, which is 5.

tB is the time it takes a photon to travel from event EB and pass the tA axis (and hence B), according to B, which is 10.

x'A is the distance beween B and event EA when the photon passes B (which is, I stress, just a consequence of the spacetime location of event EA), according to A, which is 5.

xB is the distance beween A and event EB when the photon passes A (which is, I stress, just a consequence of the spacetime location of event EB), according to B, which is 10.
How does referring me back to these definitions (which I don't object to) answer my question about your comment in post #221, where you defined xB in a different way, not in terms of the distance between A and EB at tB=10 as above, but rather in terms of the distance between A and EB at t=-10?
Yes, t=-10 is not shown anywhere on the diagram, but if you took the tB axis and relocated it so it crossed EB and then followed it down until it crossed the tA axis, then the that crossing would be t = -10 on the tB axis.
"Relocated it"? Your definition in post #221 didn't say anything about such a relocation. Also, are you talking about shifting the point in spacetime that you label the crossing point of A&B (in which case you'd have to change which point you call EB and EA), or are you talking about keeping that point the same but having B's time axis no longer pass through it, so it's as if A and B are actual physical observers who cross at some point, but B is using a coordinate system where he's at rest but not located at x=0?
I couldn't quite follow the point you were making, but it seemed like you were saying my objection was that the red event might not lie on the light ray that crossed through EA and EB...if so that wasn't really my objection, I realize that you can always draw a new light ray which goes through any arbitrary event, and define a new EA and EB in terms of where this ray crosses the x-axes of A and B's frames. But even if we assume our "arbitrary event" is along this ray, my problem is that none of the coordinates you defined--xA, x'A, xB, x'B--have anything to do with that event specifically as opposed to any of an infinite number of other possible events along the same ray, all the events on this ray would yield the same values for those coordinates that you defined. So, the relation between these coordinates doesn't really demonstrate anything about how the coordinates of the event itself in each frame will be related to one another, it only tells us about relations between coordinates of events involved in the definitions of xA and your other coordinates.
Pick any event, relocate (conceptually) your axes, and you can work out xB in terms of xA by seeing where a photon from the event which now lies on the xA axis crosses the xB axis.

Do a similar thing with the t axes and you can work out tB in terms of tA.

Perhaps I have confused you. I talked about an event that happens anywhere on the world line defined by EA and EB. Really, I only want to talk about one "real" event which I purposely shift my axes to align up so that the event is on the xA axis for the purposes of deriving the spatial Lorentz transformation.
OK, I don't think you said before that you wanted to relocate the axes so to ensure that the event lies on the xA axis (in which case the event would be the new EA). But then you haven't really proved the general Lorentz transformation which says that events at arbitrary coordinates (x,t) in one frame and (x',t') in the other are related by x'=gamma*(x - vt), you've only shown that if you pick a pair of coordinate systems such that the event lies on the x-axis of one of the frames, then something like this relation holds. And I say "something like" because your equation does not actually relate the coordinates of the individual event EA in the A frame with the coordinates of the same individual event in the B frame, rather it relates the interval (EA, photon passing B) in the A frame to the interval (EB, photon passing A) in the B frame.

neopolitan

Apr27-09, 06:08 AM

I don't disagree with you in causal terms, but I'm not talking about causality, I'm just talking about the definition of terms. Since EB happens at a later time than EA you could equally well say that the photon passing through the point EB is a "consequence" of it having been at EA, but there's no need to refer to two events in the definition of x'B which just refers to the coordinate position of the photon at time 0 in the B frame (and this event is of course EB). In contrast, your definitions of x'A and xB was in terms of the distance between two events, and you don't have any other way to define the meaning of these terms. So, your equation x'A=gamma*(xB - vtB) is not physically equivalent to the Lorentz transform, despite the fact that it looks the same if you forget about the definitions of the terms.

I'm glad that you point out that EB is causally linked to EA. I thought you had grasped that (perhaps not consciously) the whole time.

Think like this, if you can. According B, B is stationary, so the distance between B and the location of EA never changes, correct? So the distance between B and the location of EA at any time, according to B, is invariant (Lorentz invariant but B doesn't need to say that).

When A and B are colocated, tB = 0 and 4 time units later B passes a photon, so B "knows" that the separation between B and the photon when A and B were colocated was 4 space units. Correct?

If the photon was spawned by EA it will pass EB, so a photon spawned by EA is indistinguishable from a photon spawned by B. Correct?

My equations reflect this. How I can word that so that it makes you happy, I don't know.

What I do know is that somehow I have single handedly come up with a way to derive equations which are indistinguishable from the Lorentz transformations. Not sure what I should call them though, since if I tell people I have derived these new equations, they will tell me "No, that is just a recasting of the Lorentz transformations". I'm pretty damn sure that if I started off like that, saying I had new equations which just look like Lorentz transformations, you would be telling me that they are not new, they actually are the Lorentz transformations recast. But that's ok, I've come up with new equations. I'm happy with that.

How does referring me back to these definitions (which I don't object to) answer my question about your comment in post #221, where you defined xB in a different way, not in terms of the distance between A and EB at tB=10 as above, but rather in terms of the distance between A and EB at t=-10?

Because in post #224 I said what I had written (in post #224) supersedes what had come earlier. I thought if I did it all again, you could work from that, rather than going to something that is superseded.

"Relocated it"? Your definition in post #221 didn't say anything about such a relocation. Also, are you talking about shifting the point in spacetime that you label the crossing point of A&B (in which case you'd have to change which point you call EB and EA), or are you talking about keeping that point the same but having B's time axis no longer pass through it, so it's as if A and B are actual physical observers who cross at some point, but B is using a coordinate system where he's at rest but not located at x=0?

In post #221 I had no inkling of the lengths I would have to go to try to get you to understand. But anyway, #221 is superseded.

OK, I don't think you said before that you wanted to relocate the axes so to ensure that the event lies on the xA axis (in which case the event would be the new EA). But then you haven't really proved the general Lorentz transformation which says that events at arbitrary coordinates (x,t) in one frame and (x',t') in the other are related by x'=gamma*(x - vt), you've only shown that if you pick a pair of coordinate systems such that the event lies on the x-axis of one of the frames, then something like this relation holds. And I say "something like" because your equation does not actually relate the coordinates of the individual event EA in the A frame with the coordinates of the same individual event in the B frame, rather it relates the interval (EA, photon passing B) in the A frame to the interval (EB, photon passing A) in the B frame.

Again, I never thought I would have to go to such lengths.

I'm going to hope you get an idea of what I am getting at, and later I will try to do a be all and end all diagram (but only from one perspective) that will help you understand.

cheers,

neopolitan

JesseM

Apr27-09, 09:41 AM

I'm glad that you point out that EB is causally linked to EA. I thought you had grasped that (perhaps not consciously) the whole time.
Yes, this has always been obvious to me, but I don't see the relevance.
Think like this, if you can. According B, B is stationary, so the distance between B and the location of EA never changes, correct? So the distance between B and the location of EA at any time, according to B, is invariant (Lorentz invariant but B doesn't need to say that).
It's certainly invariant in B's frame, but if you think it's "Lorentz invariant" you misunderstand the meaning of the term. "Lorentz invariant" means "invariant under the Lorentz transform", i.e. something which is the same in all inertial frames, like the invariant interval ds^2 = dx^2 - c^2*dt^2. The distance between B and the location of EA is not something that's the same in every frame (in fact in most frames it's not constant with time), so it's not Lorentz invariant.
When A and B are colocated, tB = 0 and 4 time units later B passes a photon, so B "knows" that the separation between B and the photon when A and B were colocated was 4 space units. Correct?
Sure.
If the photon was spawned by EA it will pass EB, so a photon spawned by EA is indistinguishable from a photon spawned by B. Correct?

My equations reflect this. How I can word that so that it makes you happy, I don't know.
I'm not sure how you think this is relevant. Yes, obviously EA, EB, the event of the photon passing B, and the event of the photon passing A all lie on the worldline of a single photon. This doesn't change the fact that x'A is defined as the difference in position between the first and third event in the A frame, while xB is defined as the difference in position between the second and fourth event in the B frame, and that your derivation assumes all four events have a light-like separation from one another. So your equation x'A = gamma*(xB - vtB) does not have the same physical meaning as the Lorentz equation x' = gamma*(x - vt) despite the fact that it looks similar, because in the Lorentz equation x' and x either represent the coordinates of a single event in the primed and unprimed frame (which can be at any arbitrary position, not necessarily on one of the frame's spatial axes), or else x' and x can represent the coordinate intervals in two frames between a single pair of events (which can also be located at arbitrary positions, and which need not have a light-like separation from one another).
What I do know is that somehow I have single handedly come up with a way to derive equations which are indistinguishable from the Lorentz transformations.
But they're not indistinguishable, not when you keep in mind the physical meaning of the terms in your equations vs. the physical meaning of the terms in the Lorentz transformation.
Not sure what I should call them though, since if I tell people I have derived these new equations, they will tell me "No, that is just a recasting of the Lorentz transformations". I'm pretty damn sure that if I started off like that, saying I had new equations which just look like Lorentz transformations, you would be telling me that they are not new, they actually are the Lorentz transformations recast.
Not if you explained the physical meaning of the terms. I'm going to try to draw some diagrams of my own to show the difference in the meaning of the terms visually.
But that's ok, I've come up with new equations. I'm happy with that.
Yes, new equations which are only applicable to the very specific definitions of the events you've given (all lying on the path of a single light ray, all lying on either the space or the time axis of one of the two frames), as opposed to the Lorentz transformation which can apply to any arbitrary event or pair of events.

JesseM

Apr27-09, 10:25 AM

OK, here are the four diagrams I drew up, apologies for the messiness. The first one shows my understanding of what the terms in your equation mean in a spacetime diagram. The second shows what the terms mean in the spatial part of the Lorentz transform equation delta-x = gamma*(delta-x' + v*delta-t'). The third shows a particular symmetry in the scenario that you use to define your terms, and in the fourth diagram this symmetry shows how the meaning of your terms xB and tB can be changed so that now the modified version of your equation does deal with only a single pair of events, making it a special case of the Lorentz transformation equation, which explains why both equations look the same. But notice that your derivation only works in the case where the pair of events have a light-like separation, and where one event is on the space axis of one frame and the other event is on the time axis of the second frame, whereas the Lorentz transformation deals works for arbitrary pairs of events that don't need to have these properties.

By the way, to make your equation more consistent with the Lorentz transformation equation I made a slight tweak to your definitions--where you defined x'A and xB in terms of the "distance" between a pair of events, and so made these positive, I picked the rule that they were defined in terms of (position coordinate of later event) - (position coordinate of earlier event), which means x'A = -5 rather than 5, and xB = -10 rather than 10. So the tweaked version of your equation relating them ends up being x'A = gamma*(xB + v*tB).

neopolitan

Apr27-09, 10:28 AM

It's certainly invariant in B's frame, but if you think it's "Lorentz invariant" you misunderstand the meaning of the term. "Lorentz invariant" means "invariant under the Lorentz transform", i.e. something which is the same in all inertial frames, like the invariant interval ds^2 = dx^2 - c^2*dt^2. The distance between B and the location of EA is not something that's the same in every frame (in fact in most frames it's not constant with time), so it's not Lorentz invariant.

Fair cop, I was thinking originally about the separation between B and the event (either of them) when A and B are colocated, which is invariant and Lorentz invariant, and B takes that separation in the B frame to be invariant (but that's not Lorentz invariant).

I'm not sure how you think this is relevant. Yes, obviously EA, EB, the event of the photon passing B, and the event of the photon passing A all lie on the worldline of a single photon. This doesn't change the fact that x'A is defined as the difference in position between the first and third event in the A frame, while xB is defined as the difference in position between the second and fourth event in the B frame, and that your derivation assumes all four events have a light-like separation from one another. So your equation x'A = gamma*(xB - vtB) does not have the same physical meaning as the Lorentz equation x' = gamma*(x - vt) despite the fact that it looks similar, because in the Lorentz equation x' and x either represent the coordinates of a single event in the primed and unprimed frame (which can be at any arbitrary position, not necessarily on one of the frame's spatial axes), or else x' and x can represent the coordinate intervals in two frames between a single pair of events (which can also be located at arbitrary positions, and which need not have a light-like separation from one another).

What is an axis to you?

The x-axis is normally a line with a constant value of t, in my diagram it happens to be 0. Does it have to be 0?

Since in this instance I only want to know what the x value is in both frames, t can be whatever. Can't it?

So, if relative to A, B is travelling at v, I can use an xA-axis with any tA value, and an xB-axis with any tB value and a tB-axis with any xB value. And if I chose the right values, I can use my diagram (and my derivation) to work out that if an event lies at xA from A in the A frame, then it lies at x'B from B in the B frame ... irrespective of what the tA value of the event is.

Similarly, I can use a tA-axis with any xA value, and an xB-axis with any tB value and a tB-axis with any xB value. And if I chose the right values, I can use my diagram (and my derivation) to work out that if an event happens at tA in the A frame (which is relative to an event common to A and B, usually colocation, but not necessarily), then happens at t'B in the B frame (again relative to an event common to A and B) ... irrespective of what the xA value of the event is.

But I stress yet again, these diagrams all retrospective. My derivation doesn't call for them. I'm only using the diagrams to try to explain to JesseM what the physical meaning of the values in the original derivation are (and to be honest, I was not originally as curious about that).

But they're not indistinguishable, not when you keep in mind the physical meaning of the terms in your equations vs. the physical meaning of the terms in the Lorentz transformation.

Write it on a piece of paper, compare them.

My end equations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c2)

Lorentz Transformations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c2)

I can't see the difference, can you see the difference? (If there is a difference then cutting and pasting isn't what it used to be.)

Not if you explained the physical meaning of the terms. I'm going to try to draw some diagrams of my own to show the difference in the meaning of the terms visually.

Ok, I look forward to that. I will look forward to you showing me that even if I moved the axes (ie if I did not feel obliged to use an (x,0) axis and a (0,t) axis) that I couldn't line up my diagram to match yours.

Yes, new equations which are only applicable to the very specific definitions of the events you've given (all lying on the path of a single light ray, all lying on either the space or the time axis of one of the two frames), as opposed to the Lorentz transformation which can apply to any arbitrary event or pair of events.

I disagree. The little red dot that doesn't lie on the world line defined by EA and EB disagrees too.

cheers,

neopolitan

(I may shortly have to take a break from this. Other things are demanding my attention.)

JesseM

Apr27-09, 11:56 AM

What is an axis to you?

The x-axis is normally a line with a constant value of t, in my diagram it happens to be 0. Does it have to be 0?

Since in this instance I only want to know what the x value is in both frames, t can be whatever. Can't it?
The x-value of what? Neither x'A nor xB refer to the x-coordinate of any individual event in either frame. Rather, they both refer to the difference in x-coordinates between a pair of events.
Write it on a piece of paper, compare them.

My end equations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c2)

Lorentz Transformations: x'=gamma.(x-vt) and t'=gamma.(t-vx/c2)

I can't see the difference, can you see the difference? (If there is a difference then cutting and pasting isn't what it used to be.)
See, this is the basic problem that comes up in a lot of our discussions, you don't seem to understand that equations in physics are not just defined by how they look algebraically, but also by the actual physical meaning of the terms involved. This is why I often disagreed whenever you would say that any equation of the form t1 = t2 / gamma was the "temporal analogue of length contraction" even though the physical meaning of t1 and t2 was different as far as I could tell.

Here's a word problem: "my age is gamma times what your age was v times as many years in the past as your little brother's current age." Well, if we define t=your brother's current age, x as your current age, and my age as x', then algebraically this would be represented as x' = gamma*(x - vt). But would it be accurate to refer to this equation, with the terms defined in this way, as "the spatial component of the Lorentz transformation equation"?
Ok, I look forward to that. I will look forward to you showing me that even if I moved the axes (ie if I did not feel obliged to use an (x,0) axis and a (0,t) axis) that I couldn't line up my diagram to match yours.
Well, see the previous post. Your equation conceptually relates two different pairs of events, and even though we can exploit a symmetry in that setup to show it's equivalent to relating a single pair, your derivation only shows that the equation holds for a pair of events with a light-like separation, whereas the second of my 4 diagrams shows the Lorentz transformation equation works for events with arbitrary separation (time-like in that diagram). Also, your derivation as presented does assume that all the events lie on axes that go through the origins of your two coordinate systems...if you wanted to avoid that condition, I suppose you could prove a lemma that shows that the distance and time intervals between a pair of events in a given coordinate system will be unchanged in a second coordinate system with the origin at a different location but which is at rest relative to the first (i.e. a simple coordinate transformation of the form x' = x + X0 and t' = t + T0 where X0 and T0 are constants).
I disagree. The little red dot that doesn't lie on the world line defined by EA and EB disagrees too.
But isn't the idea that you draw a new light ray through that red dot, and shift the position of the coordinate axes so the red dot now lies on A's spatial xA axis, and redefine the meaning of EA and EB in terms of these changes? If not, maybe you could give a non-joking explanation of those diagrams from post 236. But if you are shifting the positions of the axes, then without a lemma of the type I mentioned above, you haven't proved anything about how the coordinates of the red dot in the original two coordinate systems were related (I should add that now that I think about such a lemma, which I hadn't prior to this post, it occurs to me that it would be very trivial to prove).

neopolitan

Apr28-09, 12:39 AM

I'm not going to try to explain in your terms, since your diagrams show that you have gone off on some tangent.

I've tried explaining in your terms and that doesn't seem to work.

Can you try to understand in my terms?

Here is a very busy little diagram and a less busy diagram. I've gone all the way back to the beginning so anything I have said in between to try to explain in your terms is defunct, so please try to start from here.

The diagrams may well contain all you need to understand. If they don't, then we can discuss the diagrams in my terms. Once we are both happy that you understand what I am actually talking about in my terms, then we can try to convert things into your terms. Does that sound fair?

(Note that the temporal component is not there, let's get the spatial component sorted out before we complicate things.)

cheers,

neopolitan

PS I noticed that the definitions were difficult to read on the very busy little diagram, so I have attached them separately in a more readable format.

JesseM

Apr28-09, 12:56 AM

I'm not going to try to explain in your terms, since your diagrams show that you have gone off on some tangent.
That's rather dismissive of you. Just focus on my first diagram, since my next three diagrams were merely intended to show why your equation looks just like the Lorentz transformation equation, and can actually be interpreted as a special case of it. Do you see any significant differences between my first diagram and your first diagram? They look the same to me, except that I included visual depictions of the meaning of symbols like x'A and xB whereas you didn't include them in your diagram. I also don't understand what you mean by "your terms", since except for an unimportant tweak about the signs of the distances (to make your equation consistent with the Lorentz transformation equation, which I thought is what you wanted), I've used the same terms that you used, following your definitions from post 243.
I've tried explaining in your terms and that doesn't seem to work.

Can you try to understand in my terms?
How have I not been? Again, please explain where you see any significant difference between my first diagram and your first diagram.

neopolitan

Apr28-09, 01:31 AM

That's rather dismissive of you. Just focus on my first diagram, since my next three diagrams were merely intended to show why your equation looks just like the Lorentz transformation equation, and can actually be interpreted as a special case of it. Do you see any significant differences between my first diagram and your first diagram? They look the same to me, except that I included visual depictions of the meaning of symbols like x'A and xB whereas you didn't include them in your diagram. I also don't understand what you mean by "your terms", since except for an unimportant tweak about the signs of the distances (to make your equation consistent with the Lorentz transformation equation, which I thought is what you wanted), I've used the same terms that you used, following your definitions from post 243.

How have I not been? Again, please explain where you see any significant difference between my first diagram and your first diagram.

My first diagram posted on this thread was yours. Otherwise there was one at #156 (http://www.physicsforums.com/showpost.php?p=2160139&postcount=156). I don't think that is the one you mean though.

You could mean the first one posted here (http://www.geocities.com/neopolitonian/index.htm).

Or maybe you mean the first of my most recent drawings (a couple of posts ago).

In any event, I sort of see what you are getting at but your first drawing (and in fact the rest) implies that I am focussed on something that I am not focussed on. Since we don't agree about what I am talking about, it is better than I start again, rather than trying to talk to a drawing which isn't about what I am talking about. Sorry if that sounds dismissive.

I put quite a bit of time into the most recent diagrams. Did they help at all? Hopefully you can now better understand what I was getting at when I last mentioned Lorentz invariance.

cheers,

neopolitan

PS About the unimportant tweak, move your xB so it ends in Event EA, rather than beginning at photon hits A, and you will see that it crosses the tB axis at t = -6. Then move your tB so it spans t = -6 and the event which is the colocation of the photon and B. (<- this was an edit)

That's more like what I had in mind.

PPS Diagram added which shows what I mean.

JesseM

Apr28-09, 02:10 AM

My first diagram posted on this thread was yours. Otherwise there was one at #156 (http://www.physicsforums.com/showpost.php?p=2160139&postcount=156). I don't think that is the one you mean though.

You could mean the first one posted here (http://www.geocities.com/neopolitonian/index.htm).

Or maybe you mean the first of my most recent drawings (a couple of posts ago).
Sorry, lot of diagrams posted on this thread, I meant to compare the first of my diagrams from the most recent post where I posted diagrams (post 247) with the first of your diagrams from the most recent post where you posted diagrams (post 250).
In any event, I sort of see what you are getting at but your first drawing (and in fact the rest) implies that I am focussed on something that I am not focussed on. Since we don't agree about what I am talking about, it is better than I start again, rather than trying to talk to a drawing which isn't about what I am talking about. Sorry if that sounds dismissive.
I was focused on the meaning of the individual terms in your equation which looked similar to the spatial Lorentz transformation equations. Correct me if I'm wrong, but I thought that what we're arguing about here is whether you've really derived the Lorentz transformation, or whether (as I claim) a close look at the meaning of the terms in the equation you derived shows you did not actually derive an equation which applies to the coordinates of arbitrary events or coordinate intervals between arbitrary pairs of events as with the Lorentz transformation, but only an equation that applies to events which have some more specific properties that were part of your original derivation (like the fact that the events have a light-like separation between them). I don't see how we can settle this without actually focusing on the physical meaning of individual terms like x'A and xB, which was what I was trying to depict in that first diagram.
I put quite a bit of time into the most recent diagrams. Did they help at all? Hopefully you can now better understand what I was getting at when I last mentioned Lorentz invariance.
What comment about Lorentz invariance do you mean, and which part of the diagram is supposed to relate to it specifically? I looked at the two diagrams, but as I said I don't really see how they contain any information that I didn't already understand and hadn't included in my own diagram.
PS About the unimportant tweak, move your xB so it ends in Event EA, rather than beginning at photon hits A, and you will see that it crosses the tB axis at t = -10. Then move your tB so it span t = -10 and the event which is the colocation of A and B.
What I called my "tweak" wasn't about changing the actual events spanned by the intervals (I did show how you could do that in diagram 4 using the symmetry argument from diagram 3, but in the other diagrams I kept the events the same), it was just about being consistent with the order of the events so that if tB referred to (time in B frame of light passing A) - (time in B frame of EB), then xB should also take the events in that order, i.e. (position in B frame of light passing A) - (position in B frame of EB) which would make xB negative, as opposed to reversing the order and defining xB as (position in B frame of EB) - (position in B frame of light passing A). The reason for this tweak is just that this is how it's done in the Lorentz transformation equation dealing with intervals between a pair of events, so making your equation have a consistent convention makes it easier to see how your equation can be interpreted as a special case of the Lorentz transformation equation.

But OK, as something unrelated to my own tweak, if you take a spatial interval in the B frame which has length 10 (as xB did) and you place one end at EA, then since EA has coordinates x=10 and t=-6 in the B frame, the other end of this interval will be at position x=0 and t=-6, so it seems to me it crosses the t axis of the B frame at -6 rather than -10. Unless I've gotten the algebra wrong, which is quite possible (if you think it's wrong, is it because you disagree about the coordinates of EA in the B frame?)

neopolitan

Apr28-09, 04:34 AM

Sorry, lot of diagrams posted on this thread, I meant to compare the first of my diagrams from the most recent post where I posted diagrams (post 247) with the first of your diagrams from the most recent post where you posted diagrams (post 250).

Ok, thanks.

I was focused on the meaning of the individual terms in your equation which looked similar to the spatial Lorentz transformation equations. Correct me if I'm wrong, but I thought that what we're arguing about here is whether you've really derived the Lorentz transformation, or whether (as I claim) a close look at the meaning of the terms in the equation you derived shows you did not actually derive an equation which applies to the coordinates of arbitrary events or coordinate intervals between arbitrary pairs of events as with the Lorentz transformation, but only an equation that applies to events which have some more specific properties that were part of your original derivation (like the fact that the events have a light-like separation between them). I don't see how we can settle this without actually focusing on the physical meaning of individual terms like x'A and xB, which was what I was trying to depict in that first diagram.

See my previous post, a new drawing!

What comment about Lorentz invariance do you mean, and which part of the diagram is supposed to relate to it specifically? I looked at the two diagrams, but as I said I don't really see how they contain any information that I didn't already understand and hadn't included in my own diagram.

Post #245 (http://www.physicsforums.com/showpost.php?p=2175785&postcount=245), where I mentioned Lorentz invariance but incorrectly (what was in my head did not end up in pixels).

Did the fact that there is only one Lorentz invariant interval, which is clearly identified at least in the second diagram, not make anything clearer?

What I called my "tweak" wasn't about changing the actual events spanned by the intervals (I did show how you could do that in diagram 4 using the symmetry argument from diagram 3, but in the other diagrams I kept the events the same), it was just about being consistent with the order of the events so that if tB referred to (time in B frame of light passing A) - (time in B frame of EB), then xB should also take the events in that order, i.e. (position in B frame of light passing A) - (position in B frame of EB) which would make xB negative, as opposed to reversing the order and defining xB as (position in B frame of EB) - (position in B frame of light passing A). The reason for this tweak is just that this is how it's done in the Lorentz transformation equation dealing with intervals between a pair of events, so making your equation have a consistent convention makes it easier to see how your equation can be interpreted as a special case of the Lorentz transformation equation.

Hopefully the diagram in the previous post clarifies things.

But OK, as something unrelated to my own tweak, if you take a spatial interval in the B frame which has length 10 (as xB did) and you place one end at EA, then since EA has coordinates x=10 and t=-6 in the B frame, the other end of this interval will be at position x=0 and t=-6, so it seems to me it crosses the t axis of the B frame at -6 rather than -10. Unless I've gotten the algebra wrong, which is quite possible (if you think it's wrong, is it because you disagree about the coordinates of EA in the B frame?)

I made a correction, after doing the diagram and clearly also after you posted this. -6 is right.

cheers,

neopolitan

JesseM

Apr28-09, 07:32 AM

I guess I was paying too much attention to your first diagram in post 250 and not enough to your second, because now that I look at it more carefully, I understand the top part but I'm having trouble understanding the bottom part...
I made a correction, after doing the diagram and clearly also after you posted this. -6 is right.
So I take it in bottom "In the B frame" part of the diagram, the caption "Location of A at -10 before the photon spawned" should instead by "Location of A at -6 before the photon spawned"?

The top "In the A frame" part of the diagram seems straightforward enough, on the right when you say "Location of event: photon spawned" you mean EA, correct? So the shorter line in the A-frame diagram is the distance between EA and the position where the photon passed B (x'=5) while the longer line A-frame diagram is the distance between EA and the event of the position where the photon passed A (x=8).

But I'm confused by the bottom "in the B frame" part of the diagram. When you say "Location of event: photon spawned" in the bottom part, which event are you referring to, EA or EB? In the B frame the distance between EB and the photon passing B is 4, so that would seem to be what the shorter line in the B-frame diagram refers to. But what should the longer line in the B-frame diagram refer to? At t=-6 A is at position x=3.6 in B's frame, so the distance between A at that moment and EB's position is only 0.4, while the distance between A at that moment and EA's position is 6.4. In the first case the bottom line should actually be shorter than the top line that goes from EB to the photon passing B, not longer. But in the second case the B-frame diagram would be using a different event for "location of event: photon spawned" for the bottom line than it uses for the top line, which would be confusing.

Also, when you say the "this is the only interval which is Lorentz invariant", in the A frame diagram you seem to be pointing to the interval between the events EA and the photon passing B (events which have a spatial separation of 5 in the A frame), while in the B frame diagram you seem to be pointing to the interval between the events EB and the photon passing B (events which have a spatial separation of 4 in the B frame). Am I misunderstanding? Also, when you say the "interval" is Lorentz invariant, are you referring to the interval of coordinate distance between some pair of events, or to the spacetime interval dx^2 - c^2dt^2 between some pair of events?

Finally, I do understand what you're talking about here:
PS About the unimportant tweak, move your xB so it ends in Event EA, rather than beginning at photon hits A, and you will see that it crosses the tB axis at t = -6. Then move your tB so it spans t = -6 and the event which is the colocation of the photon and B. (<- this was an edit)
It's true that in the B frame, the spatial distance between the event at x=0, t=-6 and EA is 10, and the time between this event and the photon passing B is 10 (since the photon passes B at t=4 in this frame). So, this is the same as xB and tB when they were defined in terms of EA and the photon passing A, and I can see why this works based on the symmetry of the diagram, similar to my own diagram #3 in post 247 but with the second isosceles triangle flipped over. However, I don't see how this relates to what I was referring to when I talked about the "tweak", which again didn't involve changing the events that xB and tB were defined in terms of. And if this is supposed to be related to your second "less busy diagram" in post 250, I'll have to ask you to elaborate because I don't see that either.

neopolitan

Apr28-09, 08:47 AM

I guess I was paying too much attention to your first diagram in post 250 and not enough to your second, because now that I look at it more carefully, I understand the top part but I'm having trouble understanding the bottom part...

So I take it in bottom "In the B frame" part of the diagram, the caption "Location of A at -10 before the photon spawned" should instead by "Location of A at -6 before the photon spawned"?

Frustratingly enough, I saw that the -10 was in the diagram after I got home, and the program I drew it in is at work.

The top "In the A frame" part of the diagram seems straightforward enough, on the right when you say "Location of event: photon spawned" you mean EA, correct? So the shorter line in the A-frame diagram is the distance between EA and the position where the photon passed B (x'=5) while the longer line A-frame diagram is the distance between EA and the event of the position where the photon passed A (x=8).

But I'm confused by the bottom "in the B frame" part of the diagram. When you say "Location of event: photon spawned" in the bottom part, which event are you referring to, EA or EB? In the B frame the distance between EB and the photon passing B is 4, so that would seem to be what the shorter line in the B-frame diagram refers to. But what should the longer line in the B-frame diagram refer to? At t=-6 A is at position x=3.6 in B's frame, so the distance between A at that moment and EB's position is only 0.4, while the distance between A at that moment and EA's position is 6.4. In the first case the bottom line should actually be shorter than the top line that goes from EB to the photon passing B, not longer. But in the second case the B-frame diagram would be using a different event for "location of event: photon spawned" for the bottom line than it uses for the top line, which would be confusing.

Also, when you say the "this is the only interval which is Lorentz invariant", in the A frame diagram you seem to be pointing to the interval between the events EA and the photon passing B (events which have a spatial separation of 5 in the A frame), while in the B frame diagram you seem to be pointing to the interval between the events EB and the photon passing B (events which have a spatial separation of 4 in the B frame). Am I misunderstanding? Also, when you say the "interval" is Lorentz invariant, are you referring to the interval of coordinate distance between some pair of events, or to the spacetime interval dx^2 - c^2dt^2 between some pair of events?

There is no event EB. There is an event which spawns a photon (the event formerly known as EA) and there is the event when that photon passes B. Remember I said I was going back to the beginning, so I am trying another tack.

The event EA, if you still want to call it that, and the event when the photon passes B are both unique events, and there is a unique spacetime interval between them which is Lorentz invariant. The magnitude of the spatial component of this spacetime interval in the A frame and the B frame are in both diagrams (in the two dimensional one, x'B is displaced.

Attached are modified diagrams, highlighting something. They are messy because I don't have all the tools I need, but you should see that I have cut a bit out of mine and moved it up. If you can do it with t (your diagram) you can do it x (my diagram).

Finally, I do understand what you're talking about here:

It's true that in the B frame, the spatial distance between the event at x=0, t=-6 and EA is 10, and the time between this event and the photon passing B is 10 (since the photon passes B at t=4 in this frame). So, this is the same as xB and tB when they were defined in terms of EA and the photon passing A, and I can see why this works based on the symmetry of the diagram, similar to my own diagram #3 in post 247 but with the second isosceles triangle flipped over. However, I don't see how this relates to what I was referring to when I talked about the "tweak", which again didn't involve changing the events that xB and tB were defined in terms of. And if this is supposed to be related to your second "less busy diagram" in post 250, I'll have to ask you to elaborate because I don't see that either.

This will have to wait, I am currently busier than my drawing.

cheers,

neopolitan

JesseM

Apr28-09, 09:17 AM

There is no event EB. There is an event which spawns a photon (the event formerly known as EA) and there is the event when that photon passes B. Remember I said I was going back to the beginning, so I am trying another tack.
OK, I didn't realize that by going back to the beginning you meant starting the proof again without referring to EB. So in the second diagram from post 250, the shorter line in the "In the B frame" part of the diagram is supposed to go from the photon-spawning event (formerly known as EA) to the event of the photon passing B? But if the spawning event occurs at x=8,t=0 in the A frame, then in the B frame it must occur at position x=1.25*(8 - 0.6*0) = 10, while of course B is always at position x=0...so shouldn't that line say x'=10 rather than x'=4? (or x=-10 if you want to define it as 'position of photon passing B' - 'position of spawning event' as in my 'tweak') And since A is at position x=3.6 at t=-6, should the bottom line representing the distance between A and the spawning event at that time be x = x' + vt = 10 + 0.6*-6 = 6.4? (or x = x' - vt = -10 - 0.6*-6 = -6.4 in the tweaked version, since A's position is further in the -x direction than the spawning event).
Attached are modified diagrams, highlighting something. They are messy because I don't have all the tools I need, but you should see that I have cut a bit out of mine and moved it up. If you can do it with t (your diagram) you can do it x (my diagram).
Is the circled line segment intended to represent the distance between the spawning event and the event of the photon passing B, as measured in the B frame? If so it needs to be longer, because you don't want the ends of the segment to lie on vertical lines of constant x extending from each event in the A frame as you seem to have drawn it, rather you want the two ends of the segment to lie on two slanted lines of constant x in the B frame (lines parallel to B's time axis) which extend from the two events. If this isn't clear I can draw my own sketch to illustrate.
This will have to wait, I am currently busier than my drawing.
No problem, take your time.

neopolitan

Apr28-09, 10:53 AM

Recall in post #191, I said:

What leaves me a little stumped is ... it worked. So, I need to see what it is that makes it work.

I'm still doing that. Which means I am still trying to work this out. I don't know if you still have this in mind.

OK, I didn't realize that by going back to the beginning you meant starting the proof again without referring to EB. So in the second diagram from post 250, the shorter line in the "In the B frame" part of the diagram is supposed to go from the photon-spawning event (formerly known as EA) to the event of the photon passing B? But if the spawning event occurs at x=8,t=0 in the A frame, then in the B frame it must occur at position x=1.25*(8 - 0.6*0) = 10, while of course B is always at position x=0...so shouldn't that line say x'=10 rather than x'=4? (or x=-10 if you want to define it as 'position of photon passing B' - 'position of spawning event' as in my 'tweak') And since A is at position x=3.6 at t=-6, should the bottom line representing the distance between A and the spawning event at that time be x = x' + vt = 10 + 0.6*-6 = 6.4? (or x = x' - vt = -10 - 0.6*-6 = -6.4 in the tweaked version, since A's position is further in the -x direction than the spawning event).

(1) The distance between A and the event when it happens at 0 is 8, according to A. (xA = 8)

(2) A period of 5 later, at 5, according to A, the photon passes B. (t'A = 5)

(3) A period of 8 later, at 8, according to A, the photon passes A. (tA = 8)

(4) According to A, at that time, B has moved 3 towards the event's location, so the separation between B and where the photon was when A and B were colocated is 5. (x'A = 5)

(5) According to B, at that time, B has not moved and the separation between B and where the photon was when A and B were colocated is 4. (x'B = 4)

(6) The distance between B and the event when it happens at -6 is 10, according to B (xB = 10)

(7) A period of 10 later, at 4, according to B, the photon passes B. (tB = 10)

(8) According to B, at that time, A has moved 6 away from the event's location, so the separation between A where the photon was when A and B were colocated is 10. (xB = 10)

(9) According to A, at that time, A has not moved and the separation between A and where the photon was when A and B were colocated is 8. (xA = 8)

Events

(E1) A and B are colocated

(E2) Photon is emitted

(E3) Photon passes B

(E4) Photon passes A

After a walk and some further thought, I am beginning to wonder if the spatial intervals being measured are not:

From location where photon passes B to where photon was when A and B were colocated.

According to A, B has moved 3 closer to what was 8 away = 5.

According to B, B has not moved, but meets the photon at 4 so at 0, the photon was at 4.

I can't spend more time on this, but it may shed some light.

Is the circled line segment intended to represent the distance between the spawning event and the event of the photon passing B, as measured in the B frame? If so it needs to be longer, because you don't want the ends of the segment to lie on vertical lines of constant x extending from each event in the A frame as you seem to have drawn it, rather you want the two ends of the segment to lie on two slanted lines of constant x in the B frame (lines parallel to B's time axis) which extend from the two events. If this isn't clear I can draw my own sketch to illustrate.

I think it depends on which direction you are going (B to A) or (A to B). I've tried to show both. Now, I am just trying to show one.

I think to show what you want to see, I could take a section of the xA axis and cross the xB axis (reflecting events A-B colocation and photon-crosses xB axis).

I'm going the other way.

Must go,

cheers,

neopolitan

JesseM

Apr28-09, 05:01 PM

What leaves me a little stumped is ... it worked. So, I need to see what it is that makes it work.
I'm still doing that. Which means I am still trying to work this out. I don't know if you still have this in mind.
By "works" do you just mean the fact that the equation you get ends up looking just like the Lorentz transformation? My diagrams from post 247 were intended to show why this was the case, showing how your equation could be interpreted as a special case of the Lorentz transform when dealing with two events that have a light-like separation.
(1) The distance between A and the event when it happens at 0 is 8, according to A. (xA = 8)

(2) A period of 5 later, at 5, according to A, the photon passes B. (t'A = 5)

(3) A period of 8 later, at 8, according to A, the photon passes A. (tA = 8)

(4) According to A, at that time, B has moved 3 towards the event's location, so the separation between B and where the photon was when A and B were colocated is 5. (x'A = 5)
When you say "at that time", you're referring to the time in (2) rather than (3) I take it. Also, when you refer to "where the photon was when A and B were colocated" in A's frame, that was the earlier definition of EA.
(5) According to B, at that time, B has not moved and the separation between B and where the photon was when A and B were colocated is 4. (x'B = 4)
But in B's frame, "where the photon was when A and B were colocated" is how the event EB was defined earlier, so you're still including this event in your definition of x'B.
(6) The distance between B and the event when it happens at -6 is 10, according to B (xB = 10)
I take it by "the event when it happens" you still mean the event formerly known as EA. So, was it a mistake in the second diagram from post 250 when in the B frame diagram you had the distance of 10 be the distance between the photon-spawning and "Location of A" at the time that should be -6? The distance between B and EA in the B frame is 10 (this is time-invariant in the B frame, so the time of -6 is irrelevant here), but the distance between A and EA at -6 is 6.4.
(7) A period of 10 later, at 4, according to B, the photon passes B. (tB = 10)

(8) According to B, at that time, A has moved 6 away from the event's location, so the separation between A where the photon was when A and B were colocated is 10. (xB = 10)
Now when you refer to "where the photon was when A and B were colocated" you seem to mean in the B frame, but that would be the event we defined as EB, so you still seem to be including this event in your definition of xB. Also, when you say "at that time", are you referring to the time of 4 in the B frame from (7)? At that time A is at position -0.6*4 = -2.4 on B's x-axis, so the distance between A and EB at that time is not 10, it's 6.4 just like the distance between A and EA at a time of -6 in B's frame. Your original definition of xB was the distance from EB and A at the time the photon passes A in B's frame (t=10 in B's frame), and in that case the distance is 10. So either your above verbal definition is mistaken, or you got the value of xB wrong with that definition.
(9) According to A, at that time, A has not moved and the separation between A and where the photon was when A and B were colocated is 8. (xA = 8)
Yes, the separation between A and EA is always 8 in the A frame.
Events

(E1) A and B are colocated

(E2) Photon is emitted

(E3) Photon passes B

(E4) Photon passes A

After a walk and some further thought, I am beginning to wonder if the spatial intervals being measured are not:

From location where photon passes B to where photon was when A and B were colocated.
But "where photon was when A and B were colocated" depends on which frame's definition of simultaneity you're using, so again you seem to be talking about both EA and EB.
According to A, B has moved 3 closer to what was 8 away = 5.
In A's frame, the distance between B and EA at the moment the photon passes B is 5, yes. This was your definition of x'A, both in older posts and above.
According to B, B has not moved, but meets the photon at 4 so at 0, the photon was at 4.
The photon was at position x=4 on B's space axis at time t=0 in B's frame, yes. This was the position in B's frame of the event EB, which is how you defined x'B in older posts, and also above although you didn't use the term EB any more (if you're going to keep talking about the event on the photon's worldline that happens at t=0 in B's frame when A and B were colocated, then can we bring back the notation of EA and EB?)
Is the circled line segment intended to represent the distance between the spawning event and the event of the photon passing B, as measured in the B frame? If so it needs to be longer, because you don't want the ends of the segment to lie on vertical lines of constant x extending from each event in the A frame as you seem to have drawn it, rather you want the two ends of the segment to lie on two slanted lines of constant x in the B frame (lines parallel to B's time axis) which extend from the two events. If this isn't clear I can draw my own sketch to illustrate.
I think it depends on which direction you are going (B to A) or (A to B). I've tried to show both. Now, I am just trying to show one.
I don't understand what you mean by "direction" and "B to A" vs. "A to B". Are you talking about converting something from one frame to another? If so what, specifically?
I think to show what you want to see, I could take a section of the xA axis and cross the xB axis (reflecting events A-B colocation and photon-crosses xB axis).
The photon crosses the xB axis at event EB, are you just talking about a horizontal line in the A frame between EB and A's time axis (x=0)? If so, that is definitely not what I "wanted to see" above, I was talking about "the distance between the spawning event (EA) and the event of the photon passing B, as measured in the B frame". As I said, the way to represent this would be to draw in two lines parallel to B's time axis which go through these two events (EA and the photon passing B), then draw a segment parallel to B's space axis with each end touching one of the parallel lines. On the other hand, your circled diagram in post 256 seemed to be based on imagining two vertical lines parallel to A's time axis, one line going through the event EA and the other line going through the event of the photon passing B, and then drawing a line segment parallel to B's space axis with each end touching one of the parallel lines. This would not be the distance between EA and the event of the photon passing B in either frame. Am I misunderstanding what you were trying to represent in that diagram?

phyti

Apr28-09, 07:15 PM

Neo;
How close does this drawing match yours, and is this what you are trying to show?

18676

neopolitan

Apr28-09, 11:22 PM

By "works" do you just mean the fact that the equation you get ends up looking just like the Lorentz transformation? My diagrams from post 247 were intended to show why this was the case, showing how your equation could be interpreted as a special case of the Lorentz transform when dealing with two events that have a light-like separation.

Originally it wasn't a special case. The only way I could give you numbers (which is your preferred approach, nothing wrong with it) was to present a special case.

When you say "at that time", you're referring to the time in (2) rather than (3) I take it. Also, when you refer to "where the photon was when A and B were colocated" in A's frame, that was the earlier definition of EA.

Cut and paste error, (3) should have been one up (the numbering came later), so you are right, the "at that time" in (4) refers to (2).

Yes, there is the event formally known as "EA".

But in B's frame, "where the photon was when A and B were colocated" is how the event EB was defined earlier, so you're still including this event in your definition of x'B.

I know. I was just laying out all the intervals, noting that some intervals either appear in different places, or I have just reworded the description of the exact same interval. Note what I said further down in my post.

I take it by "the event when it happens" you still mean the event formerly known as EA. So, was it a mistake in the second diagram from post 250 when in the B frame diagram you had the distance of 10 be the distance between the photon-spawning and "Location of A" at the time that should be -6? The distance between B and EA in the B frame is 10 (this is time-invariant in the B frame, so the time of -6 is irrelevant here), but the distance between A and EA at -6 is 6.4.

Yes, it is B and the event at t=-6 so that time 10 later at 4, the photon hits B after having travelled 10. This accords with (6) meaning xB, right? (Noting that x unprimed is the location of the event formerly known as EA, the subscript means according to B.)

I have an evening walk inspired idea for showing the relationships visually, which I will address shortly.

Now when you refer to "where the photon was when A and B were colocated" you seem to mean in the B frame, but that would be the event we defined as EB, so you still seem to be including this event in your definition of xB. Also, when you say "at that time", are you referring to the time of 4 in the B frame from (7)? At that time A is at position -0.6*4 = -2.4 on B's x-axis, so the distance between A and EB at that time is not 10, it's 6.4 just like the distance between A and EA at a time of -6 in B's frame. Your original definition of xB was the distance from EB and A at the time the photon passes A in B's frame (t=10 in B's frame), and in that case the distance is 10. So either your above verbal definition is mistaken, or you got the value of xB wrong with that definition.

That value appears twice as (6) and (8). I know that. Clearly if we are tying ourselves to unique physical definitions for each term (and I am not necessarily saying that we shouldn't), then one of these is the wrong definition of xB, if not both.

But "where photon was when A and B were colocated" depends on which frame's definition of simultaneity you're using, so again you seem to be talking about both EA and EB.

Yes, and no, but then again yes. But sort of no. Hopefully the diagram will make this clearer (and I know it can't make things less clear.)

The photon was at position x=4 on B's space axis at time t=0 in B's frame, yes. This was the position in B's frame of the event EB, which is how you defined x'B in older posts, and also above although you didn't use the term EB any more (if you're going to keep talking about the event on the photon's worldline that happens at t=0 in B's frame when A and B were colocated, then can we bring back the notation of EA and EB?)

I certainly don't want a separate photon spawning event. I return to the diagram that I need to draw again, in which the event formally known as EB sort of makes a reappearance. This should make more sense, once I finish responding, find time to actually draw the diagram and post it.

I don't understand what you mean by "direction" and "B to A" vs. "A to B". Are you talking about converting something from one frame to another? If so what, specifically?

Yes, converting the spatial component of a spacetime interval from one frame to another.

My diagram shows converting the spatial component of a spacetime interval in the A frame (example: x'A ... a horizontal line, length 5) to the spatial component of a spacetime interval in the B frame (example: x'B ... a line parallel to the xB axis line, length 4).

You seemed to talking about converting the spatial component of a spacetime interval in the B frame (example: xB ... a line parallel to the xB axis line, length 10) to the spatial component of a spacetime interval in the A frame (example: xA ... a horizontal line, length 8).

See what I mean?

The photon crosses the xB axis at event EB, are you just talking about a horizontal line in the A frame between EB and A's time axis (x=0)? If so, that is definitely not what I "wanted to see" above, I was talking about "the distance between the spawning event (EA) and the event of the photon passing B, as measured in the B frame". As I said, the way to represent this would be to draw in two lines parallel to B's time axis which go through these two events (EA and the photon passing B), then draw a segment parallel to B's space axis with each end touching one of the parallel lines. On the other hand, your circled diagram in post 256 seemed to be based on imagining two vertical lines parallel to A's time axis, one line going through the event EA and the other line going through the event of the photon passing B, and then drawing a line segment parallel to B's space axis with each end touching one of the parallel lines. This would not be the distance between EA and the event of the photon passing B in either frame. Am I misunderstanding what you were trying to represent in that diagram?

I think I explained that just above. If not, let me know.

Diagram to follow, as other priorities permit.

cheers,

neopolitan

neopolitan

Apr28-09, 11:38 PM

Neo;
How close does this drawing match yours, and is this what you are trying to show?

There's only one photon emission, so I would prefer any diagram to have only one location. The diagram isn't what I would have drawn, but I am not saying it is wrong. I'd like to get the diagram I do want to draw done without having to work out another depiction.

I do know that some of the figures you have noted do not appear on my diagrams (6.4 and 2.4) which seem to relate to a distinctly different event.

cheers,

neopolitan

JesseM

Apr28-09, 11:58 PM

Originally it wasn't a special case. The only way I could give you numbers (which is your preferred approach, nothing wrong with it) was to present a special case.
When I say "special case" I'm not talking about this specific numerical example though, I'm talking about the fact that all the intervals are between events that lie on the same light ray and therefore have a light-like separation, and your derivation wouldn't be applicable to events with a time-like or space-like separation. The Lorentz transform deals with intervals between arbitrary events which may not have a light-like separation, like in the second diagram from my post 247.
(6) The distance between B and the event when it happens at -6 is 10, according to B (xB = 10)
I take it by "the event when it happens" you still mean the event formerly known as EA. So, was it a mistake in the second diagram from post 250 when in the B frame diagram you had the distance of 10 be the distance between the photon-spawning and "Location of A" at the time that should be -6? The distance between B and EA in the B frame is 10 (this is time-invariant in the B frame, so the time of -6 is irrelevant here), but the distance between A and EA at -6 is 6.4.
Yes, it is B and the event at t=-6 so that time 10 later at 4, the photon hits B after having travelled 10. This accords with (6) meaning xB, right?
Yeah, the distance between B and EA is 10 in the B frame (regardless of time).
(7) A period of 10 later, at 4, according to B, the photon passes B. (tB = 10)

(8) According to B, at that time, A has moved 6 away from the event's location, so the separation between A where the photon was when A and B were colocated is 10. (xB = 10)
Now when you refer to "where the photon was when A and B were colocated" you seem to mean in the B frame, but that would be the event we defined as EB, so you still seem to be including this event in your definition of xB. Also, when you say "at that time", are you referring to the time of 4 in the B frame from (7)? At that time A is at position -0.6*4 = -2.4 on B's x-axis, so the distance between A and EB at that time is not 10, it's 6.4 just like the distance between A and EA at a time of -6 in B's frame. Your original definition of xB was the distance from EB and A at the time the photon passes A in B's frame (t=10 in B's frame), and in that case the distance is 10. So either your above verbal definition is mistaken, or you got the value of xB wrong with that definition.
That value appears twice as (6) and (8). I know that. Clearly if we are tying ourselves to unique physical definitions for each term (and I am not necessarily saying that we shouldn't), then one of these is the wrong definition of xB, if not both.
Well, I think (8) has to be wrong if my numbers above are right (the distance between A and EB being 6.4 at time t=4 in the B frame).
The photon was at position x=4 on B's space axis at time t=0 in B's frame, yes. This was the position in B's frame of the event EB, which is how you defined x'B in older posts, and also above although you didn't use the term EB any more (if you're going to keep talking about the event on the photon's worldline that happens at t=0 in B's frame when A and B were colocated, then can we bring back the notation of EA and EB?)
I certainly don't want a separate photon spawning event.
I wasn't suggesting a separate photon spawning event. Are we using the same definition of "event"? Normally in SR an event just refers to a particular geometric point in spacetime (such that all frames agree on the spacetime interval between it and other events), there doesn't need to be anything of interest actually happening at that point. So if we define EB as "the point on the photon's worldline that's simultaneous with A&B in B's frame", that's enough to define a unique "event" even if nothing special is happening to the photon at that point on its worldline.
I don't understand what you mean by "direction" and "B to A" vs. "A to B". Are you talking about converting something from one frame to another? If so what, specifically?
Yes, converting the spatial component of a spacetime interval from one frame to another.

My diagram shows converting the spatial component of a spacetime interval in the A frame (example: x'A ... a horizontal line, length 5) to the spatial component of a spacetime interval in the B frame (example: x'B ... a line parallel to the xB axis line, length 4).

You seemed to talking about converting the spatial component of a spacetime interval in the B frame (example: xB ... a line parallel to the xB axis line, length 10) to the spatial component of a spacetime interval in the A frame (example: xA ... a horizontal line, length 8).
No, I wasn't talking about conversion at all, I was just talking about drawing a line segment to represent the spatial distance in the B frame between two specific events. If you want to draw the distance in the B frame between the event EA and the event of the photon passing B, you draw one line parallel to B's time axis that goes through EA (representing the set of events which have the same position coordinate as EA in B's frame), another line parallel to B's time axis that goes through the event of the photon passing B (representing the set of events which have the same position coordinate as the photon passing B in B's frame), and then a line segment parallel to B's space axis whose endpoints lie on these parallel lines (representing the distance in B's frame between the position coordinate of the first event in B's frame and the position coordinate of the second event in B's frame). None of this involves any conversions to the A frame, although the lines will be skewed if you draw this in the context of an A frame diagram. It's a lot easier to visualize if you imagine doing this in the context of a B frame drawing, where you'd just draw two vertical lines going through the events, and then the horizontal distance between these lines would be the same as the distance between the events in the B frame.

I don't really understand what you mean by "the spatial component of the spacetime interval"--what spacetime interval, specifically? Do you agree a spacetime interval is always defined in terms of a pair of events? If so, are you talking about the same events I was, the event EA and the event of the photon passing B? And is the line segment in your drawing from post 256, which is parallel to B's space axis, supposed to represent the spatial distance in B's frame between these two events?

neopolitan

Apr29-09, 05:01 AM

Here is the diagram I have mentioned.

According to B, B does not move. According to A, B does move, so the photon which eventually passes B is, at the event photon passes the xB axis (formerly known as event EB) a spacetime interval away from the photon spawning event (formerly known as EA) as shown by the green line.

I've been far too busy today to describe the green lines. Hopefully you'll work it out.

cheers,

neopolitan

Just quickly, the diagram should be viewed together with the second drawing from post #250. What I am focussing on is x' (both of them, ie x'A (5) and x'B (4)). With a little effort you can find vt'A (3), vtB (6), xA (8) and xB (10). I think these make more physical sense.

Something that remains to be seen is if this is a special case or not. I don't think so, but how to convince anyone else is a task for another day (or week even).

JesseM

Apr29-09, 07:25 AM

neopolitan

Apr29-09, 10:13 AM

Ah, I see what you mean. Yes, that relationship works. I suppose we could name the event where the bottom red and green lights meet as EC, which could be defined as "the event that is at the same position in the A frame as the light passing B, and is simultaneous in the B frame with EA." Then it would be true that in the B frame, the distance from B to EB is equal to the distance from EC to EA.

But aren't we drawing on our prior knowledge of how spacetime diagrams in SR work (which are based on already knowing the full Lorentz transform) to conclude that this relationship holds? Are you saying you could derive this relationship from first principles without first knowing (or first deriving) the full Lorentz transform? If not, how does this relate to your attempt to derive the Lorentz transform?

Yes, I draw on my existing knowledge to draw this and how does it relate? By giving physical meanings to the terms in the equation.

I suspect that what I might be doing is akin to what we do when we take two events with a spacelike separation and define the line joining them as the x-axis so the spatial interval between them is x. To the extent that that is a special case, I agree that what I am doing is a special case.

I think that the two absolutely key events are 1) the event formerly known as EA and 2) the event where we say that A and B are colocated: this is the spacetime interval of interest. This might sound like a special case, but really A and B don't really ever need to be colocated, we can rearrange axes and label A and B appropriately and get equations that work. (It's sort of like working out how far a boat is off an island when we are sitting on the shore of the mainland. We can work out how far the island is from us, how far the boat is from us and then how far the boat is from the island, and we can make the line joining the island and the boat the x axis and make the island the origin of the x axis, even though we may never actually visit the island and after making our measurements we take off vertically in a balloon. I don't know about you, but I always have a tendancy to think of where I am headed as my own personal x axis - even though I could say that is the axis along which things seem to approach me :smile:)

cheers,

neopolitan

JesseM

Apr29-09, 12:31 PM

neopolitan

Apr29-09, 10:09 PM

Well, in your original derivation, the terms in the equation x'A = gamma*(xB - vt[sub]B) definitely referred to coordinate distances and times between pairs of events with a lightlike separation...if you're saying that you think you could derive a similar-looking equation but where the terms had a different physical meaning, and derive it from first principles without relying on preexisting knowledge of how spacetime diagrams in SR look, then to convince me of that I think you'd really have to go back and go through the steps of the altered derivation from the beginning. I don't know if that's worth the effort at this point though, it's up to you. Also, are you saying you think you could derive it for events at arbitrary pairs of coordinates, or only in the case where we've oriented the x-axis so both events are simultaneous (or colocated) in one of the frames? If the latter that's still not as general as the Lorentz transformation, which can be applied to events that need not be simultaneous or colocated in either of the two frames.

My original derivation at post #174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174).

In that post I said to put (7) and (4) into (2), where:

(2) x'B = G.x'A

(4) x'A = xA - v.tA

(7) G = \gamma

giving

x'B = G.( xA - v.tA )

so that x'A actually disappears. I'm not saying that x'A has no meaning at all, but I do wonder if it should (or at least could) perhaps be used as an interim value. The value xA in this equation is the spatial separation between the origin of the xA axis and the event formerly known as EA. These two events don't have a lightlike separation. (And here I can point out that I am totally aware that events don't have to have anything happen at them. I was mistakenly under the impression that you wanted to tie "happenings" to events as part of your desire to have physical meanings to all the various values. The notional colocation of A and B event is part of this, even though in reality A and B don't ever have to be colocated - they can start separated and head off in opposite directions in a 1+1 universe, or just have a nearest point of approach in a 3+1 universe.)

There is no altered derivation from post #174. It stands. Each of the values may well have a better physical definition, but those better definitions don't change the derivation.

Everything between #174 and here has been to get those better definitions and while it has been quite a journey, I don't think that it has been in vain. At first I was confident that my derivation works, but that confidence was based on a rather nebulous mental picture which was clearly difficult to express in words. Now I am confident that my derivation works, and my confidence is based on a much clearer mental picture which I believe I can present in a spacetime diagram.

A clarification, do you realise that the image given in post #264 (http://www.physicsforums.com/showpost.php?p=2178288&postcount=264) only refers to the spatial component of a spacetime interval?

cheers,

neopolitan

JesseM

Apr29-09, 11:03 PM

My original derivation at post #174 (http://www.physicsforums.com/showpost.php?p=2165684&postcount=174).

In that post I said to put (7) and (4) into (2), where:

(2) x'B = G.x'A

(4) x'A = xA - v.tA

(7) G = \gamma

giving

x'B = G.( xA - v.tA )

so that x'A actually disappears.
OK, I was thinking of the equation you derived in post 227, x'A = (xB - vtB).gamma, and that was also the Lorentz-like equation you talked about in later posts. I think your derivation in post 227 was using different definitions than the earlier one you quote above from post 174, because in later posts you had x'B = 4 and x'A = 5, but if G=gamma that would mean x'A = G*x'B (in fact you wrote this equation in post 227), which is the reverse of what you have above.
I'm not saying that x'A has no meaning at all, but I do wonder if it should (or at least could) perhaps be used as an interim value. The value xA in this equation is the spatial separation between the origin of the xA axis and the event formerly known as EA. These two events don't have a lightlike separation.
I don't think the equation from post 174 could actually be derived using your later definitions that you were using in post 227 and later posts; in post 227 you wrote:
xB = xA.gamma
x'A = x'B.gamma

then taking the next step:

xB=x'B + vtB

so

x'B=xB - vtB
Substituting, the equations you could get from this would be either x'A = (xB - vtB)*gamma, which is what you derived, or x'B + vtB = xA*gamma, which doesn't really look like a Lorentz transformation equation at all.
Everything between #174 and here has been to get those better definitions and while it has been quite a journey, I don't think that it has been in vain. At first I was confident that my derivation works, but that confidence was based on a rather nebulous mental picture which was clearly difficult to express in words. Now I am confident that my derivation works, and my confidence is based on a much clearer mental picture which I believe I can present in a spacetime diagram.
As I said I don't think you can actually derive x'B = G.( xA - v.tA ) using the definitions from post 227 and subsequently. Even if you could, this would not really be much like the Lorentz transformation equation, because it doesn't relate the coordinates of a single event or single pair of events in two different frames--x'B represents the position of EB in the B frame (or equivalently the distance between EB and the event of light passing B in the B frame), while xA represents the position of EA in the A frame (or equivalently the distance between EA and the event of light passing A in the A frame) and tA represents the time of the light passing A in the the A frame (or equivalently the time between EA and the event of light passing A in the A frame, the same pair of events you might use to define xA, except that if you want to define both xA and tA in terms of this pair of events then one of them must be negative if you're consistent about the order you take the events).
A clarification, do you realise that the image given in post #264 (http://www.physicsforums.com/showpost.php?p=2178288&postcount=264) only refers to the spatial component of a spacetime interval?
Yes, I understood that.

neopolitan

Apr30-09, 01:59 AM

OK, I was thinking of the equation you derived in post 227, x'A = (xB - vtB).gamma, and that was also the Lorentz-like equation you talked about in later posts. I think your derivation in post 227 was using different definitions than the earlier one you quote above from post 174, because in later posts you had x'B = 4 and x'A = 5, but if G=gamma that would mean x'A = G*x'B (in fact you wrote this equation in post 227), which is the reverse of what you have above.

I did think about that shortly after I wrote #227. You can eliminate x'B and xA or xB and x'A depending on what you are after. When I wrote #227, I was really just showing that G could be gamma or 1/gamma, depending on where you initially put G but what you ended up with (a Lorentz transformation - or "Lorentz-like") didn't change. I should have either not gone any further than that or, if I did, I should have eliminated xB and x'A to remain consistent with #174.

I don't think the equation from post 174 could actually be derived using your later definitions that you were using in post 227 and later posts; in post 227 you wrote:

Substituting, the equations you could get from this would be either x'A = (xB - vtB)*gamma, which is what you derived, or x'B + vtB = xA*gamma, which doesn't really look like a Lorentz transformation equation at all.

As I said I don't think you can actually derive x'B = G.( xA - v.tA ) using the definitions from post 227 and subsequently. Even if you could, this would not really be much like the Lorentz transformation equation, because it doesn't relate the coordinates of a single event or single pair of events in two different frames--x'B represents the position of EB in the B frame (or equivalently the distance between EB and the event of light passing B in the B frame), while xA represents the position of EA in the A frame (or equivalently the distance between EA and the event of light passing A in the A frame) and tA represents the time of the light passing A in the the A frame (or equivalently the time between EA and the event of light passing A in the A frame, the same pair of events you might use to define xA, except that if you want to define both xA and tA in terms of this pair of events then one of them must be negative if you're consistent about the order you take the events).

With the understanding of what x'B is, and how it relates to x'A and xA and xB in the diagram at #264, do you agree that the equations at #174 work?

I repeat yet again that I have been trying to work out why this thing works. And I think it does.

Part of the process of trying to work out why works was a discussion which we have conducted which has involved what I can see are some false starts and some dead ends from which I have had to retreat and start off again. Add to that the problems I have had with coding up replies (misplaced primes, misplaced subscripts) and it's a mess.

How about we take a pause for a bit while I fix what is wrong in #174 (as far as I can tell), then I make a temporal version of what is explained in the diagram in #264, I can repose the question from three paragraphs up and we can go from there? When I redo #174, I will put what is in the post into a diagram, because I find it so much better to use a WYSIWYG interface than the Latex reference interface - especially when I am strapped for time.

cheers,

neopolitan

neopolitan

Apr30-09, 04:45 AM

Some diagrams posted here (http://www.geocities.com/neopolitonian/gen.htm). Some more to come, when I get to the machine the originals were created on.

If there are comments at this stage, please keep them to where I have stuffed up (there's bound to be something).

cheers,

neopolitonian

JesseM

Apr30-09, 09:27 AM

I did think about that shortly after I wrote #227. You can eliminate x'B and xA or xB and x'A depending on what you are after. When I wrote #227, I was really just showing that G could be gamma or 1/gamma, depending on where you initially put G but what you ended up with (a Lorentz transformation - or "Lorentz-like") didn't change. I should have either not gone any further than that or, if I did, I should have eliminated xB and x'A to remain consistent with #174.
But how do you actually derive x'B = gamma*(xA - vtA)? Are you still defining x'A and xB like this?

x'A=xA - vt'A
xB=x'B + vtB

Then if you want to write the G equations as in #174 (which will make G = 1/gamma) that'd be:

xA = xB*G
x'B = x'A*G

(by the way, I should add that although I never got into it, I wasn't really happy with your answer in post 227 to why you assumed the G factor would be the same in both equations--you cited 'Galilean invariance', but how can you use Galilean invariance as a starting assumption in a derivation that's supposed to give SR equations, when Galilean invariance explicitly contradicts SR? Maybe you just meant 'invariance of the laws of physics in different inertial frames', i.e. the 'principle of relativity' that works in both Galilean relativity and SR? Even then I think more work would be needed to justify this step, because the physical situations aren't totally symmetric, in A's frame B is moving towards the photon while in B's frame A is moving away from the photon...)

Then I guess the next step would be the substitution t'A = x'A/c and tB = xB/c...I guess the justification here is that these terms can be defined using events on the path of a photon, for example t'A is (time in the A frame between EA and photon reaching B) while x'A basically means (distance in A frame between EA and photon reaching B).

Anyway, with that substitution you'd have:

x'A=xA - vx'A/c --> xA = x'A*(1 + v/c)

and

xB=x'B + vxB/c --> x'B = xB*(1 - v/c)

Then you can take xA = x'A*(1 + v/c) and x'B = xB*(1 - v/c) and combine with the G equations xA = xB*G and x'B = x'A*G giving:

x'A*(1 + v/c) = xB*G
xB*(1 - v/c) = x'A*G

which combine to give [x'A*(1 + v/c)/G] * (1 - v/c) = x'[/sub]A[/sub]*G which implies G^2 = (1 + v/c)*(1 - v/c), so G = 1/gamma.

So, x'B = x'A*G becomes x'A = gamma*x'B, and xA = xB*G becomes xB = gamma*xA.

But then what's next? Is there a way to combine those equations with the original definitions of x'A and xB, namely x'A=xA - vt'A and xB=x'B + vtB, to yield the desired equation x'B = gamma*(xA - vtA)? If there is I'm not seeing how.
With the understanding of what x'B is, and how it relates to x'A and xA and xB in the diagram at #264, do you agree that the equations at #174 work?
Just in terms of the numbers, sure, if x'B = 4 and xA = 8 and tA = 8, then x'B = gamma*(xA - vtA) works out. But I don't see how to actually derive that equation from your starting definitions. Also, as I said, the physical meaning of the terms, in particular why xA and tA are both positive, is ambiguous to me:
Even if you could, this would not really be much like the Lorentz transformation equation, because it doesn't relate the coordinates of a single event or single pair of events in two different frames--x'B represents the position of EB in the B frame (or equivalently the distance between EB and the event of light passing B in the B frame), while xA represents the position of EA in the A frame (or equivalently the distance between EA and the event of light passing A in the A frame) and tA represents the time of the light passing A in the the A frame (or equivalently the time between EA and the event of light passing A in the A frame, the same pair of events you might use to define xA, except that if you want to define both xA and tA in terms of this pair of events then one of them must be negative if you're consistent about the order you take the events).
So, which of these definitions is the reason that xA and tA are both positive?

1) they are the coordinates of two different individual events--xA is the position coordinate of EA in the A frame while tA is the time coordinate of the event of light passing A in the A frame

2) they represent the coordinate distance and time between a single pair of events, but taken in different order--xA = (position of event EA) - (position of event of light passing A), while tA = (time of event of light passing A) - (time of event EA)

3) Similar to 3, but they are defined as the absolute value of (coordinate of one event) - (coordinate of other event), so the order doesn't matter

4) Something else?

I assume whichever definition you adopt, the definitions for x'B and t'B would be analogous. Also, if you want to change the definitions a little so xA involves taking the events in the same order as tA, meaning that xA will actually be negative (my 'tweak'), that would make things a lot simpler so feel free to take that option (in this case the rest of the proof should work if you do some other tweaks as well, like defining x'A as xA + vt'A so that it refers to [position of light passing B] - [position of EA]).
How about we take a pause for a bit while I fix what is wrong in #174 (as far as I can tell), then I make a temporal version of what is explained in the diagram in #264, I can repose the question from three paragraphs up and we can go from there? When I redo #174, I will put what is in the post into a diagram, because I find it so much better to use a WYSIWYG interface than the Latex reference interface - especially when I am strapped for time.
Sounds good, all of that would be helpful. If you want to go back and redo derivations I do recommend using the tweak above where coordinate distances and times between pairs of events are always consistent about the order so that they can sometimes be negative...it's up to you though.

neopolitan

Apr30-09, 11:11 AM

JesseM

Apr30-09, 12:26 PM

The diagrams are here (http://www.geocities.com/neopolitonian/g2ev2.htm) and here (http://www.geocities.com/neopolitonian/gen.htm).

cheers,

neopolitan

PS Just read your previous post. I am defining Galilean invariance as saying that "fundamental physical laws are invariant across all inertial frames". I didn't go into that in the diagram, but since there are physical laws which involve the speed of light as a constant (permittivity of free space comes to mind) , then I take invariant speed of light to be a fundamental physical law.
The idea that the laws of physics are invariant across inertial frames is known as the "principle of relativity" (which can mean either Galilean relativity or SR relativity depending on the context), while Galilean invariance is defined as the principle that the laws of physics are invariant under the Galilei transform (x' = x - vt and t' = t). Using these definitions, while it makes sense to assume the principle of relativity in a derivation of the Lorentz transformation, it wouldn't make sense to assume Galilean invariance since this assumption is logically incompatible with the idea that light moves at c in all frames, so I'd suggest changing your terminology here to bring it in line with the way physicists would understand these terms. In any case, even if you're talking about the principle of relativity I don't think that's sufficient to justify the claim that the same G factor should appear in xA = xB*G and x'B = x'A*G because these symbols represent coordinate distances and times for particular events in a particular physical scenario, these equations don't represent general laws of physics.

Starting on your diagrams, I'm a little confused about something on the page here (http://www.geocities.com/neopolitonian/g2ev2.htm) where you say "According to the Galilean boost, t'=t so the time that photon hits B is the same for A and B". But you can't assume the coordinates of the two observers are related by the Galilean boost if you're deriving the Lorentz transform (and in fact we know in our numerical example that the time coordinate the photon hits B is different in the A frame than it is in the B frame)--is this just another pedagogical remark about what would be true in Galilean physics that is not actually part of the derivation? Hopefully when you write x' = x - vt at the top of that page you'd agree that this equation cannot actually be interpreted as relating A's frame to B's frame if we're assuming they both use frames where the speed of light is c as in SR...x' and x are both just different distances in A's frame, right? (If we're talking the distance between B and EA as a function of time, x would be the position coordinate of EA in A's frame while x'(t) would be the distance between B and EA in A's frame, for example.) So in this context x' = x - vt is not a Galilean boost because it's not relating the coordinates of two different frames.

Also, on that page you appear to have changed the meaning of tA and t'B from what they were previously. For example, in prior posts tA represented the time in the A frame that the photon passed A (giving tA = 8, as you wrote in post 243), but based on equation (1) from that page it seems you're now using tA to represent the time in the A frame that the photon passes B (giving tA = 5...in previous posts you had t'A = 5). Do you want to change the definitions from here on out, or do you want to just treat that as an error and keep things consistent with the notation used earlier on this thread? In any case, with tA as the time the photon passes B (tA = 5), and xA as the position of EA (xA = 8), it would no longer make sense to do the substitution tA = xA/c as you did in the line between (1) and (3).

neopolitan

Apr30-09, 10:08 PM

The idea that the laws of physics are invariant across inertial frames is known as the "principle of relativity" (which can mean either Galilean relativity or SR relativity depending on the context), while Galilean invariance is defined as the principle that the laws of physics are invariant under the Galilei transform (x' = x - vt and t' = t). Using these definitions, while it makes sense to assume the principle of relativity in a derivation of the Lorentz transformation, it wouldn't make sense to assume Galilean invariance since this assumption is logically incompatible with the idea that light moves at c in all frames, so I'd suggest changing your terminology here to bring it in line with the way physicists would understand these terms. In any case, even if you're talking about the principle of relativity I don't think that's sufficient to justify the claim that the same G factor should appear in xA = xB*G and x'B = x'A*G because these symbols represent coordinate distances and times for particular events in a particular physical scenario, these equations don't represent general laws of physics.

I was using "Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames." - wikipedia (http://en.wikipedia.org/wiki/Galilean_invariance)

I do note that subsequently in that article it gives the axioms of Newtonian relativity which are the absoluteness of space and the universality of time.

Remember I originally said something along the lines of "according to each,the other measures space oddly". I've tired to avoid that terminology. Instead I showed that there is some difference between how each measures space and how the other measures space, and kept an element of invariance (or relativity) - that neither frame is privileged.

That would mean that according to A, a spatial interval measured in A's frame (between two events) would be measured differently in B's frame and the relationship between those spatial measurements would be identical to when, according to B, a spatial interval measured in B's frame (between the same two events) would be measured differently in A' frame.

In otherwords, according to A, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is xA. According to B, that distance (which is not when A and B are colocated), is xB. The interval is a pure distance in the A frame, so:

xB = (a factor times or fuction of).xA

According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'B. According to A, that distance (which is not when A and B are colocated), is x'A. The interval is a pure distance in the B frame, so:

x'A = (a factor times or fuction of).x'B

I've taken the step of saying it is not a function, but a factor. I'm using prior knowledge here, but if I were to be very very particular, I could say it might be a function, but let's try a factor first then once I've found that a factor works, I can say we don't need a function.

Therefore:

xB = (a factor times).xA
x'A = (a factor times).x'B

Now I need to give (a factor times) a useful notation.

I originally chose a Thai letter, but to make it easier, instead I decided to use a Roman letter. Initially, I made (a factor times) = G. But you can quickly work out that that makes G=1/gamma. So, to make it easier - I thought - I expressed everything so that (a factor times) = 1/gamma = 1/G.

Do I need to spell that out in the derivation?

Starting on your diagrams, I'm a little confused about something on the page here (http://www.geocities.com/neopolitonian/g2ev2.htm) where you say "According to the Galilean boost, t'=t so the time that photon hits B is the same for A and B". But you can't assume the coordinates of the two observers are related by the Galilean boost if you're deriving the Lorentz transform (and in fact we know in our numerical example that the time coordinate the photon hits B is different in the A frame than it is in the B frame)--is this just another pedagogical remark about what would be true in Galilean physics that is not actually part of the derivation?

Pedagogical remark

Hopefully when you write x' = x - vt at the top of that page you'd agree that this equation cannot actually be interpreted as relating A's frame to B's frame if we're assuming they both use frames where the speed of light is c as in SR...x' and x are both just different distances in A's frame, right? (If we're talking the distance between B and EA as a function of time, x would be the position coordinate of EA in A's frame while x'(t) would be the distance between B and EA in A's frame, for example.) So in this context x' = x - vt is not a Galilean boost because it's not relating the coordinates of two different frames.

Look at drawings 4 and 5 in the sequence. In those you can see x', x and vt mapped according to A (so according to A, you would know that these can be subscripted with A).

We are starting with the Galilean boost (perhaps just the equation) to go through a process to obtain the spatial Lorentz transformation (perhaps just the equation) and during that process there will points where what we have is not quite either.

x'A=xA - vtA works in both GalRel and SR, correct?

Perhaps we need to define what we mean by "boost", I just mean the equation, I am not using it to compare two frames. I am using it to tell me the answer to: "with an initial (t=0)separation of x between a body and a distant location, if that body moves towards that location with a speed of v, then what is the separation between that body and the distance location at a time t?" I am effectively comparing two frames, because I can continue doing that, for all values of t and build up a description of the frame according the body and implied frame in which the body is in motion. But that just means I can use the equation as a tool later on, if I feel like it.

Also, on that page you appear to have changed the meaning of tA and t'B from what they were previously. For example, in prior posts tA represented the time in the A frame that the photon passed A (giving tA = 8, as you wrote in post 243), but based on equation (1) from that page it seems you're now using tA to represent the time in the A frame that the photon passes B (giving tA = 5...in previous posts you had t'A = 5). Do you want to change the definitions from here on out, or do you want to just treat that as an error and keep things consistent with the notation used earlier on this thread?

I did say I wanted to start again. I think I have said that a few times, but I don't want to trawl through old posts to show you that I have. So I will just repeat, I wish to start again.

I do notice that on that page, I have errors.

Firstly, time for a photon to get from YDE to A (according to A) is xA/c and the time at which a photon from YDE gets to B (according to A) is x'A/c. (Note the wording.)

The time for a photon to get from YDE to A (according to B) is xB/c and the time at which a photon from YDE gets to B (according to B) is x'B/c. (Note the wording.)

I will have to fix this because it flows on further through the document.

The end result will be the same, but it will be using better defined values. (edit - I know this, because I just jotted it down on paper and it works. It works the way I thought it did right back before I started this thread, it's just that I have a better grasp on what each of the values are.) I thought about posting the document (on geocities) and looking at it again in the cold light of day, but it's often more difficult to see mistakes in your own work.

I'll have to go back to the original document I did on this derivation (not posted in this thread) and see whether I have similar errors. That particular document was put together with much less pressure :smile:

cheers,

neopolitan

JesseM

May1-09, 12:06 AM

I was using "Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames." - wikipedia (http://en.wikipedia.org/wiki/Galilean_invariance)

I do note that subsequently in that article it gives the axioms of Newtonian relativity which are the absoluteness of space and the universality of time.
Yes, the wikipedia page was implicitly referring to Newtonian inertial frames.
Remember I originally said something along the lines of "according to each,the other measures space oddly". I've tired to avoid that terminology. Instead I showed that there is some difference between how each measures space and how the other measures space, and kept an element of invariance (or relativity) - that neither frame is privileged.

That would mean that according to A, a spatial interval measured in A's frame (between two events) would be measured differently in B's frame and the relationship between those spatial measurements would be identical to when, according to B, a spatial interval measured in B's frame (between the same two events) would be measured differently in A' frame.
But in your equations with G you weren't talking about measuring the distance between a single pair of events in two frames.
In otherwords, according to A, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is xA. According to B, that distance (which is not when A and B are colocated), is xB.
Previously xB referred to the distance between EB and the event of the photon passing A. Are you changing the definition completely now? When you say "that distance", it's unclear whether you mean the distance between B and the YDE at the moment it occurs, or the distance between A and the YDE at the moment it occurs (either way it'd be the distance in the B frame, and the moment it occurs in the B frame, presumably). Whichever way you choose, the equation xB = G*xA is not talking about the distance between a single pair of events in two different frames.
xB = (a factor times or fuction of).xA

According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'B.
But in B's frame the YDE doesn't occur when A and B are colocated. Do you mean the distance between B and the position the yellow dot occurred in the past? But in B's frame B isn't moving, so the distance between B and the position where the YDE was in the past will be the same as the distance between B and the YDE at the moment it occurred, which might already be the definition of xB, unless xB referred to the distance between A and the YDE at the moment it occurred in the B frame (see my question above).
According to A, that distance (which is not when A and B are colocated), is x'A.
Distance between what two events? Is one of them the YDE? But the YDE did occur when they were colocated in A's frame, did it not? Your way of defining these terms is extremely confusing, you really need to be much more specific. Illustrating the definitions in terms of a numerical example would be helpful, then you could say things like "the YDE occurred at x=8 and t=0 in A's frame" and "I want the distance between the YDE event, which occurred at t=-6 in B's frame, and the event on B's worldline which also occurred at t=-6 in B's frame", stuff like that.
The interval is a pure distance in the B frame, so:

x'A = (a factor times or fuction of).x'B

I've taken the step of saying it is not a function, but a factor. I'm using prior knowledge here, but if I were to be very very particular, I could say it might be a function, but let's try a factor first then once I've found that a factor works, I can say we don't need a function.

Therefore:

xB = (a factor times).xA
x'A = (a factor times).x'B
I don't understand what you mean by "a factor works" ('works' in what sense? Do you just mean it gives results consistent with your prior knowledge of the Lorentz transform?), and I also don't see where you justified the idea that it would be the same numerical factor in both equations.
Look at drawings 4 and 5 in the sequence. In those you can see x', x and vt mapped according to A (so according to A, you would know that these can be subscripted with A).

We are starting with the Galilean boost (perhaps just the equation) to go through a process to obtain the spatial Lorentz transformation (perhaps just the equation) and during that process there will points where what we have is not quite either.

x'A=xA - vtA works in both GalRel and SR, correct?
That depends on what the terms mean. It would work if xA referred to the distance between B and the YDE at t=0 in the A frame (assuming the YDE occurs at that time in the A frame, so it's just a renamed EA), and x'A referred to the distance between B and the position of the YDE at some later time tA (what event defines the term tA?)
Perhaps we need to define what we mean by "boost", I just mean the equation, I am not using it to compare two frames. I am using it to tell me the answer to: "with an initial (t=0)separation of x between a body and a distant location, if that body moves towards that location with a speed of v, then what is the separation between that body and the distance location at a time t?" I am effectively comparing two frames, because I can continue doing that, for all values of t and build up a description of the frame according the body and implied frame in which the body is in motion.
I don't really see how you're comparing two frames--aren't all distances and times here defined in terms of frame A?
I did say I wanted to start again. I think I have said that a few times, but I don't want to trawl through old posts to show you that I have. So I will just repeat, I wish to start again.
If you're going to redefine all kinds of terms though, you really need to provide detailed definitions of what they mean.
Firstly, time for a photon to get from YDE to A (according to A) is xA/c and the time at which a photon from YDE gets to B (according to A) is x'A/c. (Note the wording.)
So x'A is indeed the distance between B and the position of the YDE at the moment the photon passes B, all as measured in the A frame? If the YDE occurs at x=8, t=0 in the A frame then x'A = 5?
The time for a photon to get from YDE to A (according to B) is xB/c and the time at which a photon from YDE gets to B (according to B) is x'B/c. (Note the wording.)
OK, so using the above numbers, the YDE occurs at t=-6, x=10 in the B frame, therefore it will reach B at time t=4. So that means x'B = 4 in this example? But earlier you said "According to B, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is x'B." According to that definition, if A and B were colocated at x=0 in the B frame and the YDE's position was x=10, shouldn't x'B = 10?

neopolitan

May1-09, 02:05 AM

neopolitan

May1-09, 03:03 AM

Generality 6, all values (http://www.geocities.com/neopolitonian/generality6_all_values.jpg) has been added to here (http://www.geocities.com/neopolitonian/gen.htm).

All the values are there. Do I have to put them into words, or does the diagram suffice?

cheers,

neopolitan

neopolitan

May1-09, 08:14 AM

The g2ev2 (http://www.geocities.com/neopolitonian/g2ev2.htm) page has been revised.

cheers,

neopolitan

JesseM

May1-09, 08:32 AM

Previously xB referred to the distance between EB and the event of the photon passing A. Are you changing the definition completely now?
I don't recall ever meaning that, or writing it. I can understand how you might think it is the distance between EA and the event of the photon passing A, since this is what xA is, in the A frame. I'm not going to trawl back through old posts to look for what I said in order to defend what might well have been a typo.
From post 243, here were your old definitions:
I can only refer you back to posts #227 and #224.

xA is the distance between the origin of the xA axis and EA, according to A, which is 8.

x'B is the distance between the origin of the xB axis and EB, according to B, which is 4.

tA is the time it takes a photon to travel from event EA to the origin of the xA axis, according to A, which is 8.

t'B is the time it takes a photon to travel from event EB to the origin of the xB axis, according to B, which is 4.

t'A is the time it takes a photon to travel from event EA and pass the tB axis (and hence B), according to A, which is 5.

tB is the time it takes a photon to travel from event EB and pass the tA axis (and hence B), according to B, which is 10.

x'A is the distance beween B and event EA when the photon passes B (which is, I stress, just a consequence of the spacetime location of event EA), according to A, which is 5.

xB is the distance beween A and event EB when the photon passes A (which is, I stress, just a consequence of the spacetime location of event EB), according to B, which is 10.
Also see post 224 when you wrote xB=x'B + vtB where "tB is when the photon from EB passes A according to B (eg t=10)"; since the distance between A and EB is increasing over time, if x'B was the position of EB in the B frame (which is also what you said in the quote above), then that equation also fits perfectly with the notion that xB is the distance between A and EB at the time tB when the photon passes A. Clearly you were using these definitions at one point, it wasn't a "typo".
According to B, that distance (which is not when A and B are colocated), is xB.
In context, I thought this made sense. The distance between B and the YDE, when? Well, B is an observer, or a body, or a frame, while the YDE is an event so "at the time of the YDE" has to be "when". I even state that this is not when A and B is colocated.
OK, thanks. I realized you were talking about some distance at the time of YDE in B's frame, but the reason this was ambiguous was because your previous definition was "according to A, when A and B are colocated, the distance between A and B and the YDE (yellow dot event) is xA", so when you referred to "that distance" in the next sentence it was unclear if you meant the distance between A and YDE or the distance between B and YDE at the moment YDE occurred in the B frame (when A and B weren't colocated).
Are the spacetime diagrams of no help at all? They have numbers all over them.
That was forgetful on my part, I was trying to go back through the derivation on that page in order, so when I got confused about the meaning of the terms I didn't think to skip to the end to check the diagram. So OK, I think based on the diagram I see what the definitions are (you don't show xB in the diagram, but you explained that above), but please check to see if these are right:

xA is the distance between YDE and A (in the A frame). In the example this would be 8.

xB is the distance between the YDE and B (in the B frame). That distance is 10.

x'A is the distance between YDE and the event of the light passing B (in the A frame). In this example it would be 5.

x'B is the distance between B and the event on the worldline of the light from the YDE that's simultaneous with A&B being colocated in the B frame (this is the event that was formerly known as EB--unless you have a way of defining x'B without referring to this event, could we give it some label? We could stick with EB or use some other label since you're no longer referring to the YDE as EA). In this example it would be 4.

(Based on the diagram, x'B could be defined in terms of either of the identical red lines, so I chose the top one since it was easier to state in words...if you wanted to use the bottom one, we could define another event EC which was at the meeting point of the bottom green and red lines, it would be the event which is colocated in the A frame with the photon passing B and simultaneous in the B frame with the YDE, and then x'B would be defined as the distance between EC and the YDE.)

Incidentally, if these definitions and numbers are correct then x'A = xA - vtA would imply tA is the time in the A frame that the light passes B (so tA = 5), is that right? And in xB = x'B + vt'B implies that t'B = 10...what is the physical definition of t'B, or of the equation x(t) = x'B + vt' in general? The equation x'(t) = xA - vt in the A frame had an obvious physical interpretation, x'(t) referred to the distance between B and the YDE as a function of time in the A frame, since at t=0 B was at a distance of xA from the YDE (just as A was at that moment, since they were colocated), and B was moving towards the position of the YDE with velocity v. I suppose in this case x(t) = x'B + vt' can be taken to give the distance between A and the event EB as a function of time, since x'B is the distance between EB and A&B at t'=0 in the B frame, and A is moving away from that position at velocity v.

The only problem with this definition is that when we write xB = x'B + vt'B, xB was not originally defined to mean the distance between A and EB at some time t'B. But if we choose the time t'B when the light passes A, we find that in the B frame this event occurs at position -6 and time t'B = 10, so the distance between A and EB at this moment is 10, just like the distance between B and the YDE, so I guess we can say that xB can be defined as either of these. But here I was relying on my prior knowledge of the Lorentz transform to show that the distance in the B frame (YDE to B's position) is identical to the distance in the B frame (EB to light passing A), so I think that means if you want to use the equation xB = x'B + vt'B in your derivation without assuming what you're trying to prove, you really need to define xB as the distance between EB and the light passing A...exactly the same definition you denied when I quoted it at the beginning of this post! If instead you define xB as the distance between B and the YDE at the moment it occurs, how can you justify the equation xB = x'B + vt'B ? Why should we expect that relationship to hold if we don't already know the Lorentz transformation?

Note that if we do define xB in terms of the distance between EB and the event of the light passing A, and we also return to the term EA for the YDE, then the symmetry in the definitions is much more readily apparent:

xA is the distance between EA and A (in the A frame). It would be 8.

xB is the distance between EB and the light passing A (in the B frame). It would be 10.

x'A is the distance between EA and the light passing B (in the A frame). It would be 5.

x'B is the distance between EB and B (in the B frame). It would be 4.
A factor gives you a relationship which is symmetric. It could have (in another universe) demanded a function to have a symmetric relationship. But in our universe "a factor works".
Symmetry would also demand that the same numerical factor be in both equations. Remove symmetry and you have a privileged frame (not necessarily either of the frames in question, but a privileged frame somewhere that is more closely aligned to one of these two frames than to the other).
Why do you think "symmetry" demands that the same factor/function appear in both equations though? You haven't really justified this. The meaning of the terms in the two equations doesn't appear very symmetrical if we use your definitions--in the equation xB = (a factor times).xA, xA refers to the position of the YDE in the A frame while xB refers to the position of the YDE in the B frame, but then in the equation x'A = (a factor times).x'B, x'A is the distance in the A frame between YDE and the light passing B, while x'B is the distance in the B frame between B and EB. If we use my equivalent-but-stated-differently definitions above, then there is more of an apparent symmetry, xB = (a factor times).xA becomes:

the distance between EB and the light passing A (in the B frame) = (a factor times) the distance between EA and A (in the A frame)

And x'A = (a factor times).x'B becomes:

the distance between EA and the light passing B (in the A frame) = (a factor times) the distance between EB and B (in the B frame)

When stated this way, you can see that the second is just the first with all the A's and B's reversed. So here it is at least intuitive that it would turn out the same factor appears in both equations, although I still don't think it's really justified since the actual physical situation does not look the same in both frames (in A's frame, B is moving towards the position of EA, while in B's frame, A is moving away from the position of EB at the same speed).

neopolitan

May1-09, 10:27 AM

Post #275

I did say I wanted to start again. I think I have said that a few times, but I don't want to trawl through old posts to show you that I have. So I will just repeat, I wish to start again.
Post #270

How about we take a pause for a bit while I fix what is wrong in #174 (as far as I can tell), then I make a temporal version of what is explained in the diagram in #264, I can repose the question from three paragraphs up and we can go from there?
Post #250
I've gone all the way back to the beginning so anything I have said in between to try to explain in your terms is defunct, so please try to start from here.

JesseM, Post #280
From post 243, here were your old definitions:

I don't what I can do more to make it plain. I am not trying to defend anything after #174 and prior to #250.

cheers,

neopolitan

JesseM

May1-09, 10:42 AM

If you look at the context you'll see I only posted that quote from post 243 as a response to your comment "I don't recall ever meaning that, or writing it" and your comment that it "might well have been a typo" (The tone of this comment seemed dismissive, like you were saying I was the one who was confused when I pointed out that your new definition of xB was different from the old one. You have repeatedly made comments on this thread suggesting that that the problem in our communication is all about me not trying hard enough to follow what you're saying, rather than acknowledging that your presentation might be confusing or inconsistent, so maybe you can understand why I'd be a little defensive). I was just pointing out that you had been using that definition previously, and it clearly wasn't a typo. Notice that the rest of my post was concerned with trying to understand your new definitions, so yes, I understood that you wanted to start from the beginning.

neopolitan

May1-09, 11:57 AM

Sticking with the graphic approach, does this (http://www.geocities.com/neopolitonian/g2ev2_2.jpg) help? I moved the equations all to the next diagram in the sequence.

cheers,

neopolitan

Purely as an aside:

I did not mean to be dismissive, I actually held back from repeating that I didn't want to go over old things again, and in part that is because I have been plagued by typing issues. Each post just gets longer and more difficult to scope because it drags a lot of baggage, some intended and right (I think it is right), some intended at the time but now I can see is wrong, and quite a few things that were written in haste or in the middle of the night (like this section) and were just pure mistakes.

I apologise anyway, can we put it aside?

JesseM

May1-09, 12:31 PM

Sticking with the graphic approach, does this (http://www.geocities.com/neopolitonian/g2ev2_2.jpg) help? I moved the equations all to the next diagram in the sequence.
A question about that: you say "the time it takes a photon to get from the YDE to B, according to B, is x'B/c". But on the last diagram here (http://www.geocities.com/neopolitonian/g2ev2.htm) you have x'B = 4, are you saying it takes 4 seconds for the photon to get from the YDE to B? That wouldn't be right, because in B's frame the YDE occurs at x=10 (assuming the YDE still occurs at t=0 and x=8 in A's frame). I think you meant the time it takes a photon to get from EB to B, where EB is the meeting point of the top red line and the top green line in that last diagram (the point on the photon's worldline that's simultaneous with A and B being colocated in B's frame).

Likewise, you say "the time it takes a photon to get from the YDE to A, according to B is xB/c." But in post 277 you said xB was:
According to B, that distance (which is not when A and B are colocated), is xB.
In context, I thought this made sense. The distance between B and the YDE, when? Well, B is an observer, or a body, or a frame, while the YDE is an event so "at the time of the YDE" has to be "when". I even state that this is not when A and B is colocated.
So, here you were saying xB was the distance from B to the YDE at the time of the YDE in B's frame, which would be 10, so combining that with the above you're saying the time it takes a photon to get from the YDE to A, according to B is 10 seconds. But again that doesn't match with previous numbers, since the YDE occurs at t=-6 in B's frame, and the light reaches A at t=10 and x=-6 in B's frame, so the actual time for the photon to get from the YDE to A is 16 seconds. Again I think you may have put in the YDE when you really should have put in EB, since the time for the light to get from EB to A, according to B, is in fact 10 seconds.
Purely as an aside:

I did not mean to be dismissive, I actually held back from repeating that I didn't want to go over old things again, and in part that is because I have been plagued by typing issues. Each post just gets longer and more difficult to scope because it drags a lot of baggage, some intended and right (I think it is right), some intended at the time but now I can see is wrong, and quite a few things that were written in haste or in the middle of the night (like this section) and were just pure mistakes.

I apologise anyway, can we put it aside?
OK, sorry if I was being oversensitive, and yes I realize it easy for typos and other mistakes to happen in these long posts, so let's put it aside.

neopolitan

May1-09, 08:42 PM

A question about that: you say "the time it takes a photon to get from the YDE to B, according to B, is x'B/c". But on the last diagram here (http://www.geocities.com/neopolitonian/g2ev2.htm) you have x'B = 4, are you saying it takes 4 seconds for the photon to get from the YDE to B? That wouldn't be right, because in B's frame the YDE occurs at x=10 (assuming the YDE still occurs at t=0 and x=8 in A's frame). I think you meant the time it takes a photon to get from EB to B, where EB is the meeting point of the top red line and the top green line in that last diagram (the point on the photon's worldline that's simultaneous with A and B being colocated in B's frame).

Likewise, you say "the time it takes a photon to get from the YDE to A, according to B is xB/c." But in post 277 you said xB was:

So, here you were saying xB was the distance from B to the YDE at the time of the YDE in B's frame, which would be 10, so combining that with the above you're saying the time it takes a photon to get from the YDE to A, according to B is 10 seconds. But again that doesn't match with previous numbers, since the YDE occurs at t=-6 in B's frame, and the light reaches A at t=10 and x=-6 in B's frame, so the actual time for the photon to get from the YDE to A is 16 seconds. Again I think you may have put in the YDE when you really should have put in EB, since the time for the light to get from EB to A, according to B, is in fact 10 seconds.

OK, sorry if I was being oversensitive, and yes I realize it easy for typos and other mistakes to happen in these long posts, so let's put it aside.

Diagrams generality6_all_values.jpg (http://www.geocities.com/neopolitonian/generality6_all_values.jpg) and g2ev2_2.jpg (http://www.geocities.com/neopolitonian/g2ev2_2.jpg) apply here.

My wording is poor.

If you look at the all values diagram, you can see that x'B is a reflection of the time it takes the photon from YDE to get to B with the measurement started at colocation. Better said, perhaps, is "x'B corresponds to the time between colocation and the photon from YDE hitting B". From that time, B can work out how distant the photon from YDE was at colocation - which is the event formerly known as Eb.

If you look at the all values diagram, you can see that xB is the location of YDE at the time that YDE occurred, according to B.

That makes x'B = 4 and xB = 10.

I think you already understand this, but I'll make it explicit, the derivation does not call on the spacetime diagrams such as generality6_all_values.jpg (http://www.geocities.com/neopolitonian/generality6_all_values.jpg), we are using them so that we pin down what we are referring to and confirming that the values used in the derivations have appropriate physical meaning. But once the derivation was completed, the spacetime diagrams could be constructed.

cheers,

neopolitan

neopolitan

May1-09, 11:32 PM

x'B is the distance between B and the event on the worldline of the light from the YDE that's simultaneous with A&B being colocated in the B frame (this is the event that was formerly known as EB--unless you have a way of defining x'B without referring to this event, could we give it some label? We could stick with EB or use some other label since you're no longer referring to the YDE as EA). In this example it would be 4.

I'm a little reluctant to do this, because in my visualisation of this (an internal visualisation), A still considers that A is not in motion. Therefore while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is - if B considers that B is stationary - because the photon from it hits B a period of 4 later.

So A's conception of what B thinks is that the photon at colocation is at the other corner of the photon worldline-x'B parallogram.

So, I think that when A is trying to work out x'B, A will have the lower length in mind, rather than the upper length. (A will also have the xA which is parallel with the xA axis in mind.)

I think a similar thing happens when B is trying to work out xA.

Do this words make any sense, or will I have to make a notation on an existing diagram?

cheers,

neopolitan

JesseM

May2-09, 12:25 AM

I'm a little reluctant to do this, because in my visualisation of this (an internal visualisation), A still considers that A is not in motion. Therefore while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is - if B considers that B is stationary - because the photon from it hits B a period of 4 later.
But if A considers this, A is simply wrong about how things look in B's frame--in B's frame the YDE occurred farther from the position of colocation, and the reason this is compatible with B's clock only reading 4 when the photon hits it has to do with the relativity of simultaneity (B thinking YDE occurred much earlier than the time of colocation). It wouldn't make sense to use a wrong assumption of a derivation of a valid conclusion, so is this just another pedagogical comment, somehow? If so I think it's one that's likely to make things more confusing to people trying to follow your derivation, not less so.
So A's conception of what B thinks is that the photon at colocation is at the other corner of the photon worldline-x'B parallogram.
By "other corner" you mean the one where the YDE occurs in the diagram (http://www.geocities.com/neopolitonian/generality6_all_values.jpg), right? The corner that's opposite to the corner that marks the point where A and B pass next to one another?
So, I think that when A is trying to work out x'B, A will have the lower length in mind, rather than the upper length.
I don't really understand the "So" here. What's the connection between A assuming (incorrectly) that B thinks the YDE occurred at the moment they were colocated, and A having the lower length in mind? If there were no disagreements about simultaneity then it wouldn't even look like a parallelogram, since the bottom corner of the parallelogram is defined as a point in spacetime that's colocated with the position of the photon when it passes B in A's frame, but simultaneous with the YDE in B's frame. So if A assumes B agrees with him about simultaneity A will draw this event as occurring at the same moment as colocation, at a position between the colocation event and the YDE, so the bottom red line would just be a horizontal line extending from the YDE to that point. But perhaps I am totally misunderstanding what you meant when you said "while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is", and you were not implying here that A was making the incorrect assumption that B agrees with A about simultaneity--if so please clarify.

neopolitan

May2-09, 01:58 AM

But if A considers this, A is simply wrong about how things look in B's frame--in B's frame the YDE occurred farther from the position of colocation, and the reason this is compatible with B's clock only reading 4 when the photon hits it has to do with the relativity of simultaneity (B thinking YDE occurred much earlier than the time of colocation). It wouldn't make sense to use a wrong assumption of a derivation of a valid conclusion, so is this just another pedagogical comment, somehow? If so I think it's one that's likely to make things more confusing to people trying to follow your derivation, not less so.

It's really the same thing. Let's not get wrapped around the axles on it. Put it this way, I have a reason for not wanting to identify a separate event and label it EB. I do understand that you do.

By "other corner" you mean the one where the YDE occurs in the diagram (http://www.geocities.com/neopolitonian/generality6_all_values.jpg), right? The corner that's opposite to the corner that marks the point where A and B pass next to one another?

There are two black dots. One dot you want to label, I was talking about the one that is opposite to that.

I don't really understand the "So" here. What's the connection between A assuming (incorrectly) that B thinks the YDE occurred at the moment they were colocated, and A having the lower length in mind? If there were no disagreements about simultaneity then it wouldn't even look like a parallelogram, since the bottom corner of the parallelogram is defined as a point in spacetime that's colocated with the position of the photon when it passes B in A's frame, but simultaneous with the YDE in B's frame. So if A assumes B agrees with him about simultaneity A will draw this event as occurring at the same moment as colocation, at a position between the colocation event and the YDE, so the bottom red line would just be a horizontal line extending from the YDE to that point. But perhaps I am totally misunderstanding what you meant when you said "while A considers that when A and B are colocated, A considers that B must think that YDE is closer than it is", and you were not implying here that A was making the incorrect assumption that B agrees with A about simultaneity--if so please clarify.

It's probably better to use your event labelling. I agree.

I wanted to keep the lower x'B as something significant because it is on the line joining the location of A simultaneous with YDE according to B (xB) but I can see that perhaps, I should move the purple line there up to run along the xB axis which would make it more consistent with xA which is not offset from the xA axis.

In that case, what I was worried about disappears entirely.

What about the post that was prior to the last you responded to?

cheers,

neopolitan

JesseM

May2-09, 04:50 AM

It's really the same thing. Let's not get wrapped around the axles on it. Put it this way, I have a reason for not wanting to identify a separate event and label it EB. I do understand that you do.
I just think it's good to label any event used in the definitions. You can define all the terms without referring to that event, but then you still need to use the black dot at the bottom of the parallelogram in your definitions, the one I have labeled EC.
There are two black dots. One dot you want to label, I was talking about the one that is opposite to that.
Yes, that's the one I was calling EC, defined as the event that is colocated with the event of the light hitting B in the A frame, and which is simultaneous with the YDE in the B frame.

Did you have any comments on my post 280? You only responded to the fact that I mentioned some of your old definitions at the beginning, but the rest of the post was an attempt to deal with your new definitions. As I said there, it seems like you currently want to define the terms this way:
So OK, I think based on the diagram I see what the definitions are (you don't show xB in the diagram, but you explained that above), but please check to see if these are right:

xA is the distance between YDE and A (in the A frame). In the example this would be 8.

xB is the distance between the YDE and B (in the B frame). That distance is 10.

x'A is the distance between YDE and the event of the light passing B (in the A frame). In this example it would be 5.

x'B is the distance between B and the event on the worldline of the light from the YDE that's simultaneous with A&B being colocated in the B frame (this is the event that was formerly known as EB--unless you have a way of defining x'B without referring to this event, could we give it some label? We could stick with EB or use some other label since you're no longer referring to the YDE as EA). In this example it would be 4.

(Based on the diagram, x'B could be defined in terms of either of the identical red lines, so I chose the top one since it was easier to state in words...if you wanted to use the bottom one, we could define another event EC which was at the meeting point of the bottom green and red lines, it would be the event which is colocated in the A frame with the photon passing B and simultaneous in the B frame with the YDE, and then x'B would be defined as the distance between EC and the YDE.)
(based on your current comments, should I assume you would actually rather use the second paranthetical definition of x'B above, where it's defined in terms of the bottom black dot on the parallelogram EC and the YDE?)

Is this correct? If so, I pointed out there were some problems with justifying other equations if you defined the terms this way...the equation xB = x'B + vt'B was easy to justify using your old definitions (which I realized were actually equivalent to your new ones in SR, but you need to already know the Lorentz transformation to show this equivalence), but I don't really see how you can justify it using the new definitions above. Likewise, the similarity between the equations xB = (a factor times).xA and x'A = (a factor times).x'B was much more apparent under the old definitions, with the new definitions there's no obvious way to show there is anything analogous about the quantities in the two equations.
What about the post that was prior to the last you responded to?
Sure:
Diagrams generality6_all_values.jpg (http://www.geocities.com/neopolitonian/generality6_all_values.jpg) and g2ev2_2.jpg (http://www.geocities.com/neopolitonian/g2ev2_2.jpg) apply here.

My wording is poor.

If you look at the all values diagram, you can see that x'B is a reflection of the time it takes the photon from YDE to get to B with the measurement started at colocation. Better said, perhaps, is "x'B corresponds to the time between colocation and the photon from YDE hitting B". From that time, B can work out how distant the photon from YDE was at colocation - which is the event formerly known as Eb.
For clarity I think it's good to define all coordinate intervals in terms of a pair of events...so x'B can either be defined as the distance in the B frame between EC (which I defined earlier) and the YDE, which would be the bottom red line in the parallelogram, or it can be defined as the distance between the event of A&B being colocated and the event EB, which would be the top red line. Based on SR we can see these definitions are equivalent, but for the sake of a derivation we can't assume that, so I think it does make a difference which one we choose to use. Which of these two do you want to use as the definition of x'B?
If you look at the all values diagram, you can see that xB is the location of YDE at the time that YDE occurred, according to B.
Yes, and again based on SR we can see this is actually equivalent to the distance in the B frame between EB and the event of the light passing A, which was your older definition of xB (I can draw a diagram if it isn't clear why this should be true in general, but note that in our numerical example both would give the same value of 10). And again, without assuming SR to begin with I don't think there's any way to prove that these are equivalent, so it matters which one you choose as the definition. As I was saying in post 280, it's easy to see why the equation xB = x'B + vt'B should be expected to hold using the old definitions, but I don't know if there's any way to justify this equation under the new ones.
I think you already understand this, but I'll make it explicit, the derivation does not call on the spacetime diagrams such as generality6_all_values.jpg (http://www.geocities.com/neopolitonian/generality6_all_values.jpg), we are using them so that we pin down what we are referring to and confirming that the values used in the derivations have appropriate physical meaning.
Yes, I understand, and this is exactly why I'm skeptical that some of the steps in the derivation are justifiable under the new definitions.

neopolitan

May2-09, 05:28 AM

I just think it's good to label any event used in the definitions. You can define all the terms without referring to that event, but then you still need to use the black dot at the bottom of the parallelogram in your definitions, the one I have labeled EC.

Yes, that's the one I was calling EC, defined as the event that is colocated with the event of the light hitting B in the A frame, and which is simultaneous with the YDE in the B frame.

Noted.

Did you have any comments on my post 280? You only responded to the fact that I mentioned some of your old definitions at the beginning, but the rest of the post was an attempt to deal with your new definitions. As I said there, it seems like you currently want to define the terms this way:

(based on your current comments, should I assume you would actually rather use the second paranthetical definition of x'B above, where it's defined in terms of the bottom black dot on the parallelogram EC and the YDE?)

Is this correct? If so, I pointed out there were some problems with justifying other equations if you defined the terms this way...the equation xB = x'B + vt'B was easy to justify using your old definitions (which I realized were actually equivalent to your new ones in SR, but you need to already know the Lorentz transformation to show this equivalence), but I don't really see how you can justify it using the new definitions above. Likewise, the similarity between the equations xB = (a factor times).xA and x'A = (a factor times).x'B was much more apparent under the old definitions, with the new definitions there's no obvious way to show there is anything analogous about the quantities in the two equations.

I did read them, but was taking them as "contaminated" since you used "xB = x'B + vt'B", which was in sequence I had recently said I had to rework. I've done that, here (http://www.geocities.com/neopolitonian/g2ev2.htm).

Sorry about not responding to it before, but I thought it was pointless under the circumstances.

Sure:

For clarity I think it's good to define all coordinate intervals in terms of a pair of events...so x'B can either be defined as the distance in the B frame between EC (which I defined earlier) and the YDE, which would be the bottom red line in the parallelogram, or it can be defined as the distance between the event of A&B being colocated and the event EB, which would be the top red line. Based on SR we can see these definitions are equivalent, but for the sake of a derivation we can't assume that, so I think it does make a difference which one we choose to use. Which of these two do you want to use as the definition of x'B?

On reflection, I think I would have to go with colocation to EB.

Yes, and again based on SR we can see this is actually equivalent to the distance in the B frame between EB and the event of the light passing A, which was your older definition of xB (I can draw a diagram if it isn't clear why this should be true in general, but note that in our numerical example both would give the same value of 10). And again, without assuming SR to begin with I don't think there's any way to prove that these are equivalent, so it matters which one you choose as the definition. As I was saying in post 280, it's easy to see why the equation xB = x'B + vt'B should be expected to hold using the old definitions, but I don't know if there's any way to justify this equation under the new ones.

Looking specifically at the bottom of this (http://www.geocities.com/neopolitonian/g2ev2_2.jpg), can you see why the equations above (1) and (2) here (http://www.geocities.com/neopolitonian/g2ev2_3.jpg) hold?

You can see it together if you view this (http://www.geocities.com/neopolitonian/g2ev2.htm).

cheers,

neopolitan

JesseM

May2-09, 06:08 AM

I did read them, but was taking them as "contaminated" since you used "xB = x'B + vt'B", which was in sequence I had recently said I had to rework. I've done that, here (http://www.geocities.com/neopolitonian/g2ev2.htm).

Sorry about not responding to it before, but I thought it was pointless under the circumstances.
OK, didn't notice the change to that equation. Looking at the new equation x'B = xB - vtB, I think the same basic argument holds. If we use the old definition of xB where it refers to the distance between the event EB and the event of the light passing A in the B frame (and also define tB as the time of the light passing A), then it's fairly easy to understand why this equation works--if you rearrange it as xB = x'B + vtB, then it's just a special case of the general equation x(t) = x'B + vt, where x(t) represents the distance between A and the position of EB as a function of time t, and where x'B was the distance between A and EB at time t=0. It's easy to see why this equation should hold since A is moving away from the position of EB at speed v. And based on this general equation, the distance xB at the time tB when the light passes A would have to be x'B + vtB.

On the other hand, if we start out defining xB as the distance between the YDE and B in B's frame, I don't see what argument you would use to justify the relation x'B = xB - vtB if you don't already know from the Lorentz transformation that this definition of xB is equivalent to the older one. If you think there is a justification for the equation which uses this definition of xB rather than the older one, can you explain it?
Looking specifically at the bottom of this (http://www.geocities.com/neopolitonian/g2ev2_2.jpg), can you see why the equations above (1) and (2) here (http://www.geocities.com/neopolitonian/g2ev2_3.jpg) hold?
I'm unclear what the different dots represent in that first diagram. You have x'B as the distance from the orange dot to the yellow dot in the right-hand drawing from B's perspective, and we know that x'B can be defined either as the distance from the event of A&B being colocated to the EB, or it can be defined as the distance from EC (bottom dot on parallelogram) to the YDE. So does the orange dot represent EC, or have you redefined the meaning of the yellow dot to mean EB in this picture? If the former I can't figure out what the purple dot would be (what event to we know to be a distance of vtB from EC?), but if the latter then I suppose it's the position of A at the time tB when the light reaches it. But in this case, I don't see why you label the distance from the purple dot to the yellow dot (which would really be EB rather than the YDE) as xB unless you are reverting to the old definition of xB (distance between EB and event of light passing A) as opposed to the newer one (distance between B and YDE).

neopolitan

May2-09, 06:25 AM

OK, didn't notice the change to that equation. Looking at the new equation x'B = xB - vtB, I think the same basic argument holds. If we use the old definition of xB where it refers to the distance between the event EB and the event of the light passing A in the B frame (and also define tB as the time of the light passing A), then it's fairly easy to understand why this equation works--if you rearrange it as xB = x'B + vtB, then it's just a special case of the general equation x(t) = x'B + vt, where x(t) represents the distance between A and the position of EB as a function of time t, and where x'B was the distance between A and EB at time t=0. It's easy to see why this equation should hold since A is moving away from the position of EB at speed v. And based on this general equation, the distance xB at the time tB when the light passes A would have to be x'B + vtB.

On the other hand, if we start out defining xB as the distance between the YDE and B in B's frame, I don't see what argument you would use to justify the relation x'B = xB - vtB if you don't already know from the Lorentz transformation that this definition of xB is equivalent to the older one. If you think there is a justification for the equation which uses this definition of xB rather than the older one, can you explain it?

Right, remember, I said that the spacetime diagram is only (repeat only) to check the meanings of the various values.

As soon as you can agree that there are meanings, we can ditch the spacetime diagrams totally (at least figuratively) and work only from the Galilean boost like situation.

I think we both agree that it is unfair to use the spacetime diagrams in the derivation.

I'm unclear what the different dots represent in that first diagram. You have x'B as the distance from the orange dot to the yellow dot in the right-hand drawing from B's perspective, and we know that x'B can be defined either as the distance from the event of A&B being colocated to the EB, or it can be defined as the distance from EC (bottom dot on parallelogram) to the YDE. So does the orange dot represent EC, or have you redefined the meaning of the yellow dot to mean EB in this picture? If the former I can't figure out what the purple dot would be (what event to we know to be a distance of vtB from EC?), but if the latter then I suppose it's the position of A at the time tB when the light reaches it. But in this case, I don't see why you label the distance from the purple dot to the yellow dot (which would really be EB rather than the YDE) as xB unless you are reverting to the old definition of xB (distance between EB and event of light passing A) as opposed to the newer one (distance between B and YDE).

It seems I am taking two positions, and I think I know why. Remember I said we were working from Galilean to Lorentz, so in the middle things are a little murky.

I think one definition of the distance makes more sense from one side (Galilean) and then, once you can draw the spacetime diagrams, another definition suddenly makes more sense.

Can you at least see that might be the case? Pinning down one definition certainly seems to be extremely difficult and this would explain why.

So, can we agree that there are a number of possible definitions for the values, and hypothesize that from one perspective one set of definitions makes sense, and from another perspective, another set of definitions makes sense?

The reason I say this is because I can understand arguments for both definitions of (for example) xB, and my general approach is that "if two arguments are sound then possibly both are right from different viewpoints".

Can we agree to at least explore this avenue?

cheers,

neopolitan

Rather than having two posts, I add this:

On this diagram here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg), I show all the values x' and x with subscripts A and B, plus vt'A and -vtB. This is starting from the Galilean perspective so:

xA = distance from A to YDE at the time of YDE (simultaneous with colocation of A and B), according to A ( = 8)
xB = distance from A to YDE at the time of YDE (not simultaneous with colocation of A and B), according to B ( = 10)
x'A = distance from A to B when the photon from YDE passes B, according to A ( = 5)
x'B = distance (at the time of colocation of A and B) from B to the photon which subsequently passes B ( = 4)
t'A = time at which the photon passes B, according to A (measured from the time of colocation of A and B) ( = 5)
tB = time at which the photon passes B, according to B (measured from the time of colocation of A and B) ( = 10)

Not in the diagram per se, but implied:

tB = time between YDE and time that the photon from YDE passes A, according to A ( = 8)
t'B = time between the colocation of A and B and the time that the photon from YDE passes B, according to B ( = 4)

Going from the the spacetime diagram here (http://www.geocities.com/neopolitonian/generality6_all_values.jpg):

xA = separation between A and event EA (YDE), according to A ( = 8)
xB = separation between A and event EA (YDE) at the time of YDE, according to B ( = 10)
x'A = separation between B and the location of event EA (YDE) when the photon from YDE passes B, according to A ( = 5)
x'B = separation between B and the location of event EB (the photon from YDE passes the xB axis) when the photon from YDE passes B, according to B ( = 4)
t'A = the time between colocation of A and B and the photon passing B, according to A ( = 5)
tB = the time between when YDE occurred and when the photon passes A, according to B ( = 10)
tA = the time between colocation of A and B and the photon passing A, according to A ( = 8)
t'B = the time between colocation of A and B and the photon passing B, according to B ( = 4)

Not sure if you understand how much of a struggle it is to keep both viewpoints straight. I think I have it right, but there may be typos (late at night again).

The point is, it seems that I have may have been inconsistent, and that has bothered me but I think I have worked out why. If we talk about the g2ev2.htm (http://www.geocities.com/neopolitonian/g2ev2.htm) page, it is initially "viewpoint 1". If we talk about what is shown on the gen.htm (http://www.geocities.com/neopolitonian/gen.htm) page, it is "viewpoint 2" all the way.

In the next couple of days, I will head off on travel, so I will go quiet. I am not ignoring you, I will just be doing something other than bouncing the numbers 4, 5, 8 and 10 around in my head along with pretty coloured vectors. I will try to respond to any responsed to this, if I have enough time before I go.

cheers,

neopolitan

JesseM

May2-09, 04:52 PM

It seems I am taking two positions, and I think I know why. Remember I said we were working from Galilean to Lorentz, so in the middle things are a little murky.
But you also said the Galilean stuff was just pedagogical, it's not part of the real substance of the derivation. Maybe by "Galilean" you mean the equation xB = x'B + vtB, but as long as all these coordinates refer to the B frame, there's nothing specifically Galilean about this equation--in any coordinate system, if an object is moving away from some point it must be true that (distance at time t) = (distance at time 0) + (rate distance is changing)*t.
I think one definition of the distance makes more sense from one side (Galilean) and then, once you can draw the spacetime diagrams, another definition suddenly makes more sense.
Which one is which? Are you agreeing with me that defining xB as the distance in the B frame between A and EB at the moment the light reaches A is needed in order to justify xB = x'B + vtB? I agree that if we later decide to plot things on a spacetime diagram based on the Lorentz transform, we can see that this is equivalent to the definition of xB as the distance in the B frame from B to the YDE, although I don't really think either definition "makes more sense" than the other in a spacetime diagram, they both are reasonably simple to illustrate. But from the perspective of a step-by-step derivation where every new equation has to be justified only in terms of what came before, I do think it's necessary to use the first definition.
So, can we agree that there are a number of possible definitions for the values, and hypothesize that from one perspective one set of definitions makes sense, and from another perspective, another set of definitions makes sense?
Well, what criteria are you using to say whether a definition makes sense or not? I don't really understand why there would be a context where the second definition is clearly superior to the first definition, and in the context of a derivation it seems the first definition is the only one that makes sense if you want to justify xB = x'B + vtB.
The reason I say this is because I can understand arguments for both definitions of (for example) xB, and my general approach is that "if two arguments are sound then possibly both are right from different viewpoints".

Can we agree to at least explore this avenue?
I'm fine with using either definition if we're just talking about the consistency of the different values or analyzing their relationships in a spacetime diagram. As always, though, I'm not convinced it makes sense to use the second definition in an actual derivation.
On this diagram here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg), I show all the values x' and x with subscripts A and B, plus vt'A and -vtB.
Could you define what events the different colored dots represent?
This is starting from the Galilean perspective so:

xA = distance from A to YDE at the time of YDE (simultaneous with colocation of A and B), according to A ( = 8)
xB = distance from A to YDE at the time of YDE (not simultaneous with colocation of A and B), according to B ( = 10)
That should be "distance from B to YDE at the time of YDE", I assume. Unless somehow "starting from a Galilean perspective" means B is assuming A agrees with him about simultaneity? I should tell you that I've never really understood your comments about starting from a Galilean perspective, you said it was just pedagogical so I thought it wasn't important to understand and moved on, but if you want to keep discussing it maybe you should try to explain in more detail what the concept is here, because I just find the whole thing totally confusing.
x'A = distance from A to B when the photon from YDE passes B, according to A ( = 5)
I think that should be the distance from the YDE to B when the photon from YDE passes B, according to A.
x'B = distance (at the time of colocation of A and B) from B to the photon which subsequently passes B ( = 4)
t'A = time at which the photon passes B, according to A (measured from the time of colocation of A and B) ( = 5)
tB = time at which the photon passes B, according to B (measured from the time of colocation of A and B) ( = 10)
That last one should be the time at which the photon passes A, according to B, right? In B's frame the photon passes B at t=4.
Not in the diagram per se, but implied:

tB = time between YDE and time that the photon from YDE passes A, according to A ( = 8)
Since you defined tB one line earlier, that should be tA, right?
t'B = time between the colocation of A and B and the time that the photon from YDE passes B, according to B ( = 4)

Going from the the spacetime diagram here (http://www.geocities.com/neopolitonian/generality6_all_values.jpg):

xA = separation between A and event EA (YDE), according to A ( = 8)
xB = separation between A and event EA (YDE) at the time of YDE, according to B ( = 10)
That should be separation between B and YDE at time of YDE.
x'A = separation between B and the location of event EA (YDE) when the photon from YDE passes B, according to A ( = 5)
x'B = separation between B and the location of event EB (the photon from YDE passes the xB axis) when the photon from YDE passes B, according to B ( = 4)
t'A = the time between colocation of A and B and the photon passing B, according to A ( = 5)
tB = the time between when YDE occurred and when the photon passes A, according to B ( = 10)
Should be time between when YDE occurred (t=-6) and when the photon passed B (t=4), according to B. You have it drawn correctly in the diagram. The alternate definition would be the time between EB and the event of the light passing A, which could also be depicted on the diagram if you wanted. (You'd just take that purple line and place the top end at the event of the light passing A, then the bottom end will naturally lie on B's x-axis which represents the set of all events at t=0 in the B frame, which is when EB occurred. Likewise you could represent the alternate definition of xB by moving the purple xB line so the left end was on the event of the light passing A, and then if you drew an axis of constant x in the B frame which was parallel to B's time axis and which passed through EB, the other end of the purple line would lie on this axis.)
tA = the time between colocation of A and B and the photon passing A, according to A ( = 8)
t'B = the time between colocation of A and B and the photon passing B, according to B ( = 4)

Not sure if you understand how much of a struggle it is to keep both viewpoints straight. I think I have it right, but there may be typos (late at night again).
No problem, let me know if you disagree with any of my suggested corrections. I'm confused by what you mean when you say "both viewpoints" though, since (assuming you agree with my corrections, maybe you won't) I don't actually see any differences in the two sets of definitions above. Maybe I'm missing something though.
The point is, it seems that I have may have been inconsistent, and that has bothered me but I think I have worked out why. If we talk about the g2ev2.htm (http://www.geocities.com/neopolitonian/g2ev2.htm) page, it is initially "viewpoint 1". If we talk about what is shown on the gen.htm (http://www.geocities.com/neopolitonian/gen.htm) page, it is "viewpoint 2" all the way.
On that first page, I take it that when you say "YDE" you are actually talking about different events depending on the context? You say x'A/c is the time it takes the photon to get from the YDE event to B in A's frame (implying the YDE is EA), but then you say x'B/c is the time it takes the photon to get from the YDE to B in B's frame (implying YDE is EB). Or are you in some sense having the two of them make the erroneous Galilean assumption that they agree about simultaneity?

neopolitan

May2-09, 10:06 PM

But you also said the Galilean stuff was just pedagogical, it's not part of the real substance of the derivation. Maybe by "Galilean" you mean the equation xB = x'B + vtB, but as long as all these coordinates refer to the B frame, there's nothing specifically Galilean about this equation--in any coordinate system, if an object is moving away from some point it must be true that (distance at time t) = (distance at time 0) + (rate distance is changing)*t.

Which one is which? Are you agreeing with me that defining xB as the distance in the B frame between A and EB at the moment the light reaches A is needed in order to justify xB = x'B + vtB? I agree that if we later decide to plot things on a spacetime diagram based on the Lorentz transform, we can see that this is equivalent to the definition of xB as the distance in the B frame from B to the YDE, although I don't really think either definition "makes more sense" than the other in a spacetime diagram, they both are reasonably simple to illustrate. But from the perspective of a step-by-step derivation where every new equation has to be justified only in terms of what came before, I do think it's necessary to use the first definition.

In answer to "which is which?" - I tried to detail all the values lower in my post. However, if there is an inconsistency between my words and the diagram, the diagram is the one to use.

Well, what criteria are you using to say whether a definition makes sense or not? I don't really understand why there would be a context where the second definition is clearly superior to the first definition, and in the context of a derivation it seems the first definition is the only one that makes sense if you want to justify xB = x'B + vtB.

I'm fine with using either definition if we're just talking about the consistency of the different values or analyzing their relationships in a spacetime diagram. As always, though, I'm not convinced it makes sense to use the second definition in an actual derivation.

In the derivation only the first definitions would be used.

Could you define what events the different colored dots represent?

From left to right: photon passes A, A and B colocated, photon passes B and photon spawned.

That should be "distance from B to YDE at the time of YDE", I assume. Unless somehow "starting from a Galilean perspective" means B is assuming A agrees with him about simultaneity?

The diagram takes primacy over my words. Your words are right.

I should tell you that I've never really understood your comments about starting from a Galilean perspective, you said it was just pedagogical so I thought it wasn't important to understand and moved on, but if you want to keep discussing it maybe you should try to explain in more detail what the concept is here, because I just find the whole thing totally confusing.

It started off quite simple for me. Perhaps we should have tried suspending judgment on the definitions of the terms, done the derivation, then analysed the terms afterwards. That's probably totally alien to your mindset though.

Explaining in more detail will have to happen another day.

I think that should be the distance from the YDE to B when the photon from YDE passes B, according to A.

Yes. Again, the diagram takes primacy. This applies to all your similar comments. I've said before that I am very visual, the structure of this coding system doesn't help me at all.

Should be time between when YDE occurred (t=-6) and when the photon passed B (t=4), according to B. You have it drawn correctly in the diagram. The alternate definition would be the time between EB and the event of the light passing A, which could also be depicted on the diagram if you wanted. (You'd just take that purple line and place the top end at the event of the light passing A, then the bottom end will naturally lie on B's x-axis which represents the set of all events at t=0 in the B frame, which is when EB occurred. Likewise you could represent the alternate definition of xB by moving the purple xB line so the left end was on the event of the light passing A, and then if you drew an axis of constant x in the B frame which was parallel to B's time axis and which passed through EB, the other end of the purple line would lie on this axis.)

No problem, let me know if you disagree with any of my suggested corrections. I'm confused by what you mean when you say "both viewpoints" though, since (assuming you agree with my corrections, maybe you won't) I don't actually see any differences in the two sets of definitions above. Maybe I'm missing something though.

Again, another day.

On that first page, I take it that when you say "YDE" you are actually talking about different events depending on the context? You say x'A/c is the time it takes the photon to get from the YDE event to B in A's frame (implying the YDE is EA), but then you say x'B/c is the time it takes the photon to get from the YDE to B in B's frame (implying YDE is EB). Or are you in some sense having the two of them make the erroneous Galilean assumption that they agree about simultaneity?

Dots defined above.

YDE is fixed, it spawns the photon. Photon passes B is fixed. A and B colocated is fixed. Photon passes A is fixed. By fixed I mean "fast, invariable", but not "corrected" and not "the same in all coordinate systems". What is the same in all coordinate systems

The lengths that are compared between frames are:

"colocation-photon passes B" (which is x'B and x'A), and

"colocation-photon passes A" (which is xA and xB).

Time's up for now.

cheers,

neopolitan

neopolitan

May18-09, 01:10 AM

That should be "distance from B to YDE at the time of YDE", I assume. Unless somehow "starting from a Galilean perspective" means B is assuming A agrees with him about simultaneity? I should tell you that I've never really understood your comments about starting from a Galilean perspective, you said it was just pedagogical so I thought it wasn't important to understand and moved on, but if you want to keep discussing it maybe you should try to explain in more detail what the concept is here, because I just find the whole thing totally confusing.

It started off quite simple for me. Perhaps we should have tried suspending judgment on the definitions of the terms, done the derivation, then analysed the terms afterwards. That's probably totally alien to your mindset though.

Explaining in more detail will have to happen another day.

Try to put your brain into "pre-SR mode" which means you have to knowledge of a high school student who has paid enough attention to know that:

x' = x - vt

but you don't know anything more than that. That would give you the sort of knowledge that a person we are introducing to SR would have. Such a person won't have all the simultaneity issues you have, because they don't know enough to realise that there are simultaneity issues.

Then introduce the concept that if a photon is released from a distance of x away then it takes a period of t to reach you (remember you still don't know enough to realise there are simultaneity issues). Therefore:

x = ct and x' = x - vt

Now we know that the value t is not necessarily the same in both equations - but say we specifically want to know where the photon release location is in relation to an object moving at v away from us towards where the photon was released ... when the photon reaches us.

Then, we want to know how things look in the rest frame of that object. That is, how far from the photon release point are we when the photon reaches the object which is moving towards the photon release point (relative to us).

x' = ct' and x' = x - vt' or x = x' + vt'

We still know nothing about the relativity of simultaneity nor have any idea that the photon release location is not universally agreed. So we can try to make sense of what we have so far.

x' = x - vt and x = x' + vt'

so x' = (x' + vt') - vt

so vt' = vt which means t = t' which we know can't be right.

This is the very first step in the process. We have shown the student that just by thinking about a photon travelling past two observers in relative motion to each other, we prove that we need to have a better explanation than that given to us by Galileo and Newton.

I personally think that this is a very useful step, it engages the student's interest (at least if the student has a problem solving type of mindset) and shows that Einstein's relativity is necessary.

At this point it would probably be useful to discuss with the student the fact that whenever we measure the speed of light in an inertial frame, it is c - but note in the equations immediately above, c didn't come into it. (The x = c.t and x' = c.t' equations come into play in following steps.)

Can you understand the pedagogical process thus far?

No problem, let me know if you disagree with any of my suggested corrections. I'm confused by what you mean when you say "both viewpoints" though, since (assuming you agree with my corrections, maybe you won't) I don't actually see any differences in the two sets of definitions above. Maybe I'm missing something though.
Again, another day.

By "both viewpoints" I mean a viewpoint in which A considers A to be at rest and is considering how the universe looks from B's perspective:

x'B = gamma.(xA - v.tA)
t'B = gamma.(tA - v.xA/c2)

and a viewpoint in which B considers B to be at rest and is considering how the universe looks from A's perspective:

xA = gamma.(x'B - v.t'B)
tA = gamma.(t'B - v.x'B/c2)

Trying to keep it all straight in drawings, equations and words has been a bit of struggle even without the introduction of typos.

I'm hoping that these two responses go some way to giving you the answer to other questions you have posed (and perhaps negating some of the questions which are based on a misunderstanding or other uncertainty).

cheers,

neopolitan

JesseM

May19-09, 08:24 PM

Just to let you know, right now I'm the one who's away on a trip, I'll get back to this sometime after I get home in about a week.

neopolitan

May19-09, 10:22 PM

Just to let you know, right now I'm the one who's away on a trip, I'll get back to this sometime after I get home in about a week.

Enjoy your trip then :smile:

cheers,

neopolitan

neopolitan

May27-09, 09:42 PM

JesseM

Jun3-09, 11:42 PM

Hi again, I'm back now so I'll return to our discussion:
Try to put your brain into "pre-SR mode" which means you have to knowledge of a high school student who has paid enough attention to know that:

x' = x - vt

but you don't know anything more than that. That would give you the sort of knowledge that a person we are introducing to SR would have. Such a person won't have all the simultaneity issues you have, because they don't know enough to realise that there are simultaneity issues.

Then introduce the concept that if a photon is released from a distance of x away then it takes a period of t to reach you (remember you still don't know enough to realise there are simultaneity issues). Therefore:

x = ct and x' = x - vt
But are you assuming both that x'=x-vt and that the light moves at c in both frames? As I'm sure you'd agree, these two assumptions aren't compatible, so is your pedagogical point just to show that they aren't compatible? If so, wouldn't it be a little easier to start from the Newtonian velocity addition equation w = v + u (where u is the speed of an object in the rest frame of observer A, and observer A is moving at speed v in the same direction in the frame of observer B, and we want to know the speed w of the original object in the frame of observer B)? This follows in a pretty direct way from x' = x - vt and it should in any case be familiar to anyone who's familiar with the most basic ideas of Newtonian frames.
Now we know that the value t is not necessarily the same in both equations
Why not? If the student knows x' = x - vt he should also know that this equation relates the coordinates of a single event x,t in one frame to the coordinates x',t' of the same event in the other frame, or else it relates the coordinate intervals between a single pair of events in one frame to the coordinate intervals between the same pair of events in the other frame...in either case t' = t. If you're talking about doing something different, like having x be the distance between where the photon was released and where it hit the unprimed observer as measured in the unprimed frame, while x' is the distance between where the photon was released and where it hit the primed observer in the primed frame, then the equation x' = x - vt should not be used.
but say we specifically want to know where the photon release location is in relation to an object moving at v away from us towards where the photon was released ... when the photon reaches us.

Then, we want to know how things look in the rest frame of that object. That is, how far from the photon release point are we when the photon reaches the object which is moving towards the photon release point (relative to us).

x' = ct' and x' = x - vt' or x = x' + vt'

We still know nothing about the relativity of simultaneity nor have any idea that the photon release location is not universally agreed.
Do x,t and x',t' represent the coordinates of the single event of the photon being released in each frame? If so, what do you mean by "nor have any idea that the photon release location is not universally agreed"? Even in basic Newtonian mechanics the same event can have different position coordinates in two frames, that's the whole point of x' = x - vt.
So we can try to make sense of what we have so far.

x' = x - vt and x = x' + vt'

so x' = (x' + vt') - vt

so vt' = vt which means t = t' which we know can't be right.
Again, what do t and t' represent so that the student knows t = t' can't be right?By "both viewpoints" I mean a viewpoint in which A considers A to be at rest and is considering how the universe looks from B's perspective:

x'B = gamma.(xA - v.tA)
t'B = gamma.(tA - v.xA/c2)

and a viewpoint in which B considers B to be at rest and is considering how the universe looks from A's perspective:

xA = gamma.(x'B - v.t'B)
tA = gamma.(t'B - v.x'B/c2)
But how does the difference between these two perspectives related to the difference between this:
On this diagram here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg), I show all the values x' and x with subscripts A and B, plus vt'A and -vtB. This is starting from the Galilean perspective so:

xA = distance from A to YDE at the time of YDE (simultaneous with colocation of A and B), according to A ( = 8)
xB = distance from A to YDE at the time of YDE (not simultaneous with colocation of A and B), according to B ( = 10)
x'A = distance from A to B when the photon from YDE passes B, according to A ( = 5)
x'B = distance (at the time of colocation of A and B) from B to the photon which subsequently passes B ( = 4)
t'A = time at which the photon passes B, according to A (measured from the time of colocation of A and B) ( = 5)
tB = time at which the photon passes B, according to B (measured from the time of colocation of A and B) ( = 10)

Not in the diagram per se, but implied:

tB = time between YDE and time that the photon from YDE passes A, according to A ( = 8)
t'B = time between the colocation of A and B and the time that the photon from YDE passes B, according to B ( = 4)
...and this?
Going from the the spacetime diagram here (http://www.geocities.com/neopolitonian/generality6_all_values.jpg):

xA = separation between A and event EA (YDE), according to A ( = 8)
xB = separation between A and event EA (YDE) at the time of YDE, according to B ( = 10)
x'A = separation between B and the location of event EA (YDE) when the photon from YDE passes B, according to A ( = 5)
x'B = separation between B and the location of event EB (the photon from YDE passes the xB axis) when the photon from YDE passes B, according to B ( = 4)
t'A = the time between colocation of A and B and the photon passing B, according to A ( = 5)
tB = the time between when YDE occurred and when the photon passes A, according to B ( = 10)
tA = the time between colocation of A and B and the photon passing A, according to A ( = 8)
t'B = the time between colocation of A and B and the photon passing B, according to B ( = 4)

Not sure if you understand how much of a struggle it is to keep both viewpoints straight. I think I have it right, but there may be typos (late at night again).
Or was I misunderstanding, and these two different ways of defining things aren't meant to map to the two viewpoints you were talking about?

JesseM

Jun4-09, 12:08 AM

From another thread:
Is it perhaps worthwhile to make sure everyone knows what proper time, proper length, coordinate time and coordinate length are?

http://en.wikipedia.org/wiki/Proper_time - In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock.

Proper length (distance) - In relativistic physics, proper length is an invariant quantity which is the rod distance between spacelike-separated events in a frame of reference in which the events are simultaneous.

Coordinate time - In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. An event is specified by one time coordinate and three spatial coordinates. The time measured by the time coordinate is referred to as coordinate time to distinguish it from proper time.

Coordinate distance is not described on wikipedia but we can extrapolate thus - In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. An event is specified by one time coordinate and three spatial coordinates. The distance measured by the spatial coordinates can be referred to as coordinate distance to distinguish it from proper distance.

Interpreting all of those is the (relatively) difficult part.

Proper time - say you have an inertial clock, elapsed time on that clock is proper time. (t')

Proper length (distance) - say you have an inertial rod, the rest length of the rod (ie where the ends of the rod are simultaneous) is proper length (or proper distance between the ends of the rod). (L)

Coordinate time - we have an implied observer, the time on the observer's clock when events take place is coordinate time. (t)

Coordinate distance - we have an implied observer, the distance between the observer and an event is coordinate distance. A coordinate length would be the delta between two events, for a rod that would mean the ends of that rod. (L')

The relationship between coordinate time and proper time is given by:

t'=\frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}

The relationship between coordinate length (or distance) and proper length (or distance) is given by:

L'=L\sqrt{1-\frac{v^{2}}{c^{2}}}
Do you mean t' to be the proper time between a pair of events on the clock's own worldline as measured by that clock, while t is the coordinate time between those same events? If so you have the equation backwards, it should be:

t = \frac{t'}{\sqrt{1 - v^2/c^2}}

Of course the usual convention is to have unprimed t be proper time between events on the clock's worldline and primed t' be coordinate time between these same events in the frame where the clock is moving, so with that convention your equation above would be right.

Also, I'm confused by your definition of "coordinate distance"--are you talking about the delta-x' in the primed frame between a single pair of events which are simultaneous in the unprimed frame (so you're looking at the coordinate distance in the primed frame between events that are not simultaneous in the primed frame), or are you talking about the length in the primed frame of the same rod whose proper length you measured in the unprimed frame (with the understanding that 'length in the primed frame' means the coordinate distance between ends of the rod at a single instant in the primed frame)? Your length equation above only works under the second interpretation.
I want to say "the odd thing is that the speed in question is given by proper distance over coordinate time" but I hesitate for two reasons. Firstly, perhaps it is not so odd after all and secondly, while I can get my head around "proper distance over coordinate time" it might not be completely kosher.
v is defined in terms of the difference in position coordinate interval over difference in time coordinate interval in the unprimed frame for a pair of events on the worldline of an object at rest in the primed frame (like two events which occur at the origin of the primed coordinate system at different times).
Just in case it is not a standard thing, I would see proper distance as the distance between the ends of a rod traversed by an observed body where the rod is at rest with respect to the observer.
But of course this is the same as the coordinate distance between the event of the body passing one end and the event of the body passing the other end in this frame, since in the frame where the rod is at rest its proper length is equal to the coordinate distance between one end and the other end. Also, the term "proper distance" has a slightly different meaning than "proper length", since "proper length" refers to the length of some physical object like a rod in its own rest frame, while "proper distance" refers to taking a specific pair of spacelike-separated events and looking at the distance between them in the frame where they are simultaneous (which means if each event takes place on the end of a rod which is at rest in the frame where they are simultaneous, the proper distance between events is the same as the proper length of the rod).

neopolitan

Jun4-09, 01:29 AM

It seems I overextended and messed up in the process. Now looking at it, I can't remember what I was thinking at the time, but the end result is certainly wrong when it comes to time.

The point remains that it would be helpful to clarify what proper time and coordinate time mean. It would just be good to do a better job of it than I did.

cheers,

neopolitan

neopolitan

Jun4-09, 02:07 AM

But are you assuming both that x'=x-vt and that the light moves at c in both frames? As I'm sure you'd agree, these two assumptions aren't compatible, so is your pedagogical point just to show that they aren't compatible? If so, wouldn't it be a little easier to start from the Newtonian velocity addition equation w = v + u (where u is the speed of an object in the rest frame of observer A, and observer A is moving at speed v in the same direction in the frame of observer B, and we want to know the speed w of the original object in the frame of observer B)? This follows in a pretty direct way from x' = x - vt and it should in any case be familiar to anyone who's familiar with the most basic ideas of Newtonian frames.

Sure x'=x-vt and speed of light = c are compatible. You can only say they are incompatible if you have defined t to be something which makes them incompatible, which makes your question below a bit odd, because you are telling me you don't know what t is.

After a period of t (in the unprimed frame) the unprimed observer receives a photon. At t=0, that photon would have been a distance of x=ct away from the unprimed observer. At t=t, the primed observer, moving at v relative to the unprimed observer, is a distance of vt from the unprimed observer (assuming that at t=0 they were colocated, otherwise we just say the extra separation since t=0 is given by vt). According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt.

Happy?

I said the student paid enough attention to know the galilean equation x'=x-vt. You can't introduce a new equation from newton and say start there. You are welcome to do that as your own proposal, if you like.

Why not? If the student knows x' = x - vt he should also know that this equation relates the coordinates of a single event x,t in one frame to the coordinates x',t' of the same event in the other frame, or else it relates the coordinate intervals between a single pair of events in one frame to the coordinate intervals between the same pair of events in the other frame...in either case t' = t. If you're talking about doing something different, like having x be the distance between where the photon was released and where it hit the unprimed observer as measured in the unprimed frame, while x' is the distance between where the photon was released and where it hit the primed observer in the primed frame, then the equation x' = x - vt should not be used..

Actually, if a student knows about x' = x - vt, he or she should know about all the other conditions (for example, at t=0 the observers are colocated).

I explained x' above, but if you define it the way you have proposed - a way which doesn't really make sense - you are right, x' = x - vt should not be used.

Do x,t and x',t' represent the coordinates of the single event of the photon being released in each frame? If so, what do you mean by "nor have any idea that the photon release location is not universally agreed"? Even in basic Newtonian mechanics the same event can have different position coordinates in two frames, that's the whole point of x' = x - vt.

I just mean that we do not know about the rotation that happens with relativity, so as far as the student knows, if x' = x - vt then x = x' + vt'. (And in the very next paragraph of the original post, I show this can't be the case, so don't rush into knocking that straw man over.)

Again, what do t and t' represent so that the student knows t = t' can't be right?.

I had just explained in the original post that x'=ct' so you know that t' = x/c.

In the paragraphs before that, I explained what t is, specifically, x=ct so t=x/c.

I'm struggling to see what is confusing about this. Are you trying to jump forward, despite my request to try putting your brain into "pre-SR mode"?

But how does the difference between these two perspectives related to the difference between this:

...and this?

Or was I misunderstanding, and these two different ways of defining things aren't meant to map to the two viewpoints you were talking about?

There is at least one typo in what you quoted. I think you are confused anyway.

There are two perspectives in the equations I showed in the most recent post, there are two frames, one perspective is that one frame is at rest, the other perspective is that the other frame is at rest. I even wrote that. I am really not sure what your confusion is here.

Please try to go from the question posed by your statement "I'm confused by what you mean when you say "both viewpoints"", to the answer I provided where I specifically described the two viewpoints. Do you understand what I meant originally by "both viewpoints"? Does it help if I tell you that I meant viewpoint to have the same meaning as "perspective" in this context?

cheers,

neopolitan

JesseM

Jun4-09, 02:21 PM

Sure x'=x-vt and speed of light = c are compatible.
It's not compatible with the speed of light being c in both frames--do you disagree?
You can only say they are incompatible if you have defined t to be something which makes them incompatible, which makes your question below a bit odd, because you are telling me you don't know what t is.
Not if you mean x'=x-vt to be part of the Galilei transformation where you are assigning coordinates to particular events and you also know that for any given event, t'=t. If you are suggesting that somehow the student wasn't even paying enough attention to understand the physical meaning of x'=x-vt, and just knows that the equation exists without knowing how it is actually used in the context of the Galilei transformation, then it seems like a rather bizarre pedagogical approach to cater to this one particular student with a very idiosyncratic misunderstanding as opposed to the typical student who can at least be expected to know the physical meaning of any equation he wants to use.
After a period of t (in the unprimed frame) the unprimed observer receives a photon.
A period of t between the photon emission event and the event of the photon passing the unprimed observer, presumably? In this case, if the student knows the Galilei transformation he'll also know that the period in the primed frame between these same two events is t'=t. Using this along with x'=x-vt, he'll find that the photon's distance/time in the primed frame was not c but rather c+v.
At t=0, that photon would have been a distance of x=ct away from the unprimed observer. At t=t, the primed observer, moving at v relative to the unprimed observer, is a distance of vt from the unprimed observer (assuming that at t=0 they were colocated, otherwise we just say the extra separation since t=0 is given by vt). According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt.
I don't understand that last sentence, why would you say that the separation between where the primed observer is now and where the photon was at t=0 should be x'=x-vt? The equation x'=x-vt tells you the x' coordinate of an event with coordinates x,t in the unprimed frame, or the distance interval x' between a pair of events which have a distance and time interval of x and t in the unprimed frame, but you don't seem to be dealing with either type of question here. It's true of course that if you pick some fixed position x in the unprimed frame (like the position of the photon at t=0), and want to find the separation between the unprimed observer and that position at time t, then the answer will be x-vt, but it doesn't really make any sense to me to see this as an application of the Galilei transformation since you aren't even considering the primed frame here, and for that reason I also don't understand what it would mean to set this equal to x' if you're just calculating a separation in the unprimed frame. Note that it is also true that in SR if you had an observer moving at speed v (and located at the origin at t=0), and wanted to know the separation between this observer and some fixed position x at time t, then the answer would still be x-vt, in spite of the fact that the coordinate transformation equation x'=x-vt is wrong in SR. I think maybe you're getting confused by the superficial similarity between the Galilean coordinate transformation equation relating two different frames, namely x'=x-vt, and the equation for calculating the separation between an object moving at v and a fixed position x in the context of a single inertial frame, namely x-vt. The second does look like the right-hand side of the first but the physical meaning of what the equations are supposed to calculate is different.

Also, nowhere in the above paragraph do you calculate the distance/time for the light in the primed frame--again, when you said one of your assumptions was "speed of light = c" were you not talking about the assumption that the speed of light should be equal c in all inertial frames?
Actually, if a student knows about x' = x - vt, he or she should know about all the other conditions (for example, at t=0 the observers are colocated).
And one of the other conditions is that t'=t, yes?
I explained x' above, but if you define it the way you have proposed - a way which doesn't really make sense - you are right, x' = x - vt should not be used.
Why do you say "a way which doesn't really make sense"? Do you disagree that the standard interpretation of the Galilei transformation is that it either relates the coordinates of a single event in two different frames, or that it relates the distance and time intervals between a single pair of events in two different frames? If you're not addressing one of these questions then you shouldn't label whatever equations you use the "Galilei transformation".
Do x,t and x',t' represent the coordinates of the single event of the photon being released in each frame? If so, what do you mean by "nor have any idea that the photon release location is not universally agreed"? Even in basic Newtonian mechanics the same event can have different position coordinates in two frames, that's the whole point of x' = x - vt.
I just mean that we do not know about the rotation that happens with relativity, so as far as the student knows, if x' = x - vt then x = x' + vt'. (And in the very next paragraph of the original post, I show this can't be the case, so don't rush into knocking that straw man over.)
You didn't really answer my question at all, nor was I intending to debate the idea that x = x' + vt', I don't know how you got that from that question. My question was about the physical meaning of the symbols x, t, x', and t'. If you are using the Galilei transformation, these should either represent coordinates of a single event, or coordinate intervals between a single pair of events; if one of those is the case, please specify the event or events in question. If you are not using the symbols this way, then whatever you are doing cannot be seen as an application of the Galilei transformation, even if the equations you happen to use might look superficially similar like x-vt for the separation between an object moving at v and a fixed position x.
I had just explained in the original post that x'=ct' so you know that t' = x/c.

In the paragraphs before that, I explained what t is, specifically, x=ct so t=x/c.
Again, you're just giving equations without telling me their physical meaning in terms of specific events. When you write x=ct, do x and t represent the distance and time intervals in the unprimed frame's coordinates between the event of the photon being emitted and the event of the photon passing the unprimed observer? If so then we know by the Galilei transform that the distance between this same pair of events in the primed frame is x'=x-vt (and substituting x=ct back into this gives x'=ct-vt), and the time between this same pair of events in the primed frame is t'=t. But then you write x'=ct' which is incompatible with this.
I'm struggling to see what is confusing about this. Are you trying to jump forward, despite my request to try putting your brain into "pre-SR mode"?
No, I'm not talking about SR at all, just about the physical meaning of the Galilei transformation equations.
There is at least one typo in what you quoted. I think you are confused anyway.

There are two perspectives in the equations I showed in the most recent post, there are two frames, one perspective is that one frame is at rest, the other perspective is that the other frame is at rest. I even wrote that. I am really not sure what your confusion is here.
I don't understand how the two different blocks of equations I quoted correspond in any way to the two different rest frames of A and B. Each block of equations internally seems to contain both perspectives (for example, the first block defines t'A as a time 'according to A' and tB as a time 'according to B'), it's not like the first block shows only values calculated from the perspective of A's frame and the second shows only values calculated from the perspective of B's frame. So how does the difference between the two blocks relate to the difference between the two perspectives (A's rest frame and B's rest frame)? I don't see any connection at all.

neopolitan

Jun4-09, 09:07 PM

I'm breaking this up, because it may get (even more) confusing.

The least important bit first - from my perspective.

I don't understand how the two different blocks of equations I quoted correspond in any way to the two different rest frames of A and B. Each block of equations internally seems to contain both perspectives (for example, the first block defines t'A as a time 'according to A' and tB as a time 'according to B'), it's not like the first block shows only values calculated from the perspective of A's frame and the second shows only values calculated from the perspective of B's frame. So how does the difference between the two blocks relate to the difference between the two perspectives (A's rest frame and B's rest frame)? I don't see any connection at all.

Each block of equations internally contains both perspectives, yes. Why each block should represent different perspectives as a whole, I have no idea. I can't really answer your question because it wasn't a position I was taking.

What I can say is that my original comment followed references to two diagrams (here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg) and here (http://www.geocities.com/neopolitonian/generality6_all_values.jpg)) which are two ways of looking at the same situation. I only introduced the second as part of a long discussion with you to show the physical meaning of the terms in a way that you would understand, and possibly accept. The first is what I had originally, and I would like to stick with that until (and if) we ever get to the point where we can progress further.

But the point is that both diagrams show the same thing, exactly the same thing, in a different way - different "viewpoints" (a pair of galilean-like diagrams and one spacetime diagram) on the same scenario which incorporates both perspectives (primed and unprimed).

The issue I was having at the time, was making sure that when I transitioned from one way at looking at the scenario to another, and back again, I didn't mess up with subscripts and prime notations.

cheers,

neopolitan

JesseM

Jun4-09, 09:37 PM

Each block of equations internally contains both perspectives, yes. Why each block should represent different perspectives as a whole, I have no idea. I can't really answer your question because it wasn't a position I was taking.
OK, thanks. I wasn't assuming that the two blocks mapped to the two "perspectives" you talked about, I just wondered if that was the case (and I presented it as a question, saying 'But how does the difference between these two perspectives related to the difference between this [first block] ...and this? [second block] Or was I misunderstanding, and these two different ways of defining things aren't meant to map to the two viewpoints you were talking about?') You had presented the two blocks and then immediately said "Not sure if you understand how much of a struggle it is to keep both viewpoints straight", so it seemed natural to think that "both viewpoints" might refer to the difference between the two blocks.
What I can say is that my original comment followed references to two diagrams (here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg) and here (http://www.geocities.com/neopolitonian/generality6_all_values.jpg)) which are two ways of looking at the same situation. I only introduced the second as part of a long discussion with you to show the physical meaning of the terms in a way that you would understand, and possibly accept. The first is what I had originally, and I would like to stick with that until (and if) we ever get to the point where we can progress further.
If you'd like to stick to the first diagram, which I find more confusing, could you lay out specifically what each of the colored dots is supposed to represent? I asked about this earlier when I said:
I'm unclear what the different dots represent in that first diagram. You have x'B as the distance from the orange dot to the yellow dot in the right-hand drawing from B's perspective, and we know that x'B can be defined either as the distance from the event of A&B being colocated to the EB, or it can be defined as the distance from EC (bottom dot on parallelogram) to the YDE. So does the orange dot represent EC, or have you redefined the meaning of the yellow dot to mean EB in this picture? If the former I can't figure out what the purple dot would be (what event to we know to be a distance of vtB from EC?), but if the latter then I suppose it's the position of A at the time tB when the light reaches it. But in this case, I don't see why you label the distance from the purple dot to the yellow dot (which would really be EB rather than the YDE) as xB unless you are reverting to the old definition of xB (distance between EB and event of light passing A) as opposed to the newer one (distance between B and YDE).
No need to address this right away if you want to deal with other issues first, just whenever you want to return the discussion to that first diagram.

neopolitan

Jun4-09, 11:19 PM

If you'd like to stick to the first diagram, which I find more confusing, could you lay out specifically what each of the colored dots is supposed to represent?

The diagram in question is here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg).

There are four dots:

Yellow Dot - location of an event associated with a photon, it could be the emission of a photon, or a colocated with a photon which just happened to be passing - we'd not be able to tell the difference. In the text above the diagram, I call this "the Yellow Dot Event" or YDE

Orange Dot - location of B and photon when B and photon are colocated

Green Dot - location of A when B and photon are colocated (according to A) - the logic is this: in A's rest frame, A is at rest and B is in motion towards the YDE with speed of v. The time at which B and the photon are colocated is a period of t'a after colocation (nominally t=t'=0). Therefore, in a period of t'a, B must have moved a distance of vt'a towards the YDE and the photon has travelled a distance of ct'a towards B. Since the photon and B are colocated at this time, simple additon gives xa = ct'a + vt'a = x'a + vt'a.

Purple Dot - location of A when A and photon are colocated (according to B) - the logic is similar to abovein B's rest frame, B is at rest and A is in motion away from the YDE with speed of v. The time at which A and the photon are colocated is a period of tb after colocation (nominally t=t'=0). Therefore, in a period of tb, A must have moved a distance of vtb away from the YDE and the photon has travelled a distance of ctb towards A. Since the photon and A are colocated at this time, simple additon gives x'b = ctb - vtb = xb + vtb.

If I've done it right, "towards" is exchanged with "away from", A is exchanged with B and vice versa, a is exchanged with b and vice versa and primed is exchanged with unprimed and vice versa.

Note that in A's rest frame, the distance between the location of YDE and A does not change - therefore xa does not change with time, but x'a does (because x'a is the distance between the location of YDE and B, according to A).

Similarly, note that in B's rest frame, the distance between the location of YDE and B does not change - therefore x'b does not change with time, but xb does (because xb is the distance between the location of YDE and A, according to B).

Can you reply first with whether or not you understand this explanation. If you don't please explain specifically what it is that you don't understand. If you do understand and think it is wrong, then by all means explain where it is wrong, but first state that you understand. Thanks.

neopolitan

PS Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no.

We can tie it up with the spacetime diagram and my earlier attempts to explain at a later date, if we get that far.

JesseM

Jun5-09, 01:10 AM

The diagram in question is here (http://www.geocities.com/neopolitonian/g2ev2_2.jpg).

There are four dots:

Yellow Dot - location of an event associated with a photon, it could be the emission of a photon, or a colocated with a photon which just happened to be passing - we'd not be able to tell the difference. In the text above the diagram, I call this "the Yellow Dot Event" or YDE
But not just any event associated with the photon, right? Aren't you still assuming the YDE is the specific event on the photon's worldline that occurs at the same time as A&B being colocated in A's frame?
Orange Dot - location of B and photon when B and photon are colocated

Green Dot - location of A when B and photon are colocated (according to A)
OK, so this occurs at time t'A in A's frame, which in the numerical example is equal to 5.
Purple Dot - location of A when A and photon are colocated (according to B) - the logic is similar to above
But is the meaning of the yellow dot unchanged in this right-hand diagram, or does the yellow dot now refer to the event on the photon's worldline that occurs at the same time as A&B being colocated in B's frame? Of course if we used Galilean frames this would be the same event as the one that occurred simultaneously with A&B being colocated in A's frame, but the Galilei transformation would be inconsistent with the actual numbers you gave for some of these quantities, and as I said it would also be inconsistent with the idea that the photon moves at c in both frames.
Note that in A's rest frame, the distance between the location of YDE and A does not change - therefore xa does not change with time, but x'a does (because x'a is the distance between the location of YDE and B, according to A).
Isn't x'A the distance between the YDE and a specific event on B's worldline indicated by the orange dot, namely the event of the photon passing B? If so it wouldn't change with time.

neopolitan

Jun5-09, 03:09 AM

But not just any event associated with the photon, right? Aren't you still assuming the YDE is the specific event on the photon's worldline that occurs at the same time as A&B being colocated in A's frame?

Not really. I am saying that A and B colocated is one event and the YDE is another event. A certain time later a photon from that event passes B and then A, which both constitute events (photon colocated with B, photon colocated with A). Then I work with those.

Consider, A works out that, if ta after colocation of A and a photon passes A, then at colocation of A and B that photon was c.ta distant. This may have been the spawning of that photon - simultaneous with colocation of A and B in A's frame - (it doesn't have to be) and if it was, then it was a unique event (and even if it wasn't spawning of the photon, the photon's location is still a unique event) and the spacetime interval between A and B colocated and the photon's location simultaneous with colocation of A and B in A's frame is invariant. The same applies for B (ie thinking about the location of the photon simultaneous with the colocation of A and B, which could be the spawning of that photon but doesn't have to be).

Using a different coordinate system does not move the event.

But is the meaning of the yellow dot unchanged in this right-hand diagram, or does the yellow dot now refer to the event on the photon's worldline that occurs at the same time as A&B being colocated in B's frame? Of course if we used Galilean frames this would be the same event as the one that occurred simultaneously with A&B being colocated in A's frame, but the Galilei transformation would be inconsistent with the actual numbers you gave for some of these quantities, and as I said it would also be inconsistent with the idea that the photon moves at c in both frames.

I didn't give numbers so I don't know what you are talking about.

PS Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no.

I repeat, YDE is an event which could be the spawning of a photon, A and B colocated is an event, photon colocated with B is an event, photon colocated with A is an event.

Try not to focus on simultaneity (it might be tough, since that seems to be your preferred avenue into relativity).

Think: YDE happens and colocation of A and B happens (not necessarily in that order, and not necessarily together), colocation of photon and B happens, colocation of photon and A happens.

There are two events colocated with A, ie colocation with B and colocation with the photon. That gives A a time, ta and a distance (to photon when A and B were colocated) xa. Using x'a=ct'a, A can also work out where and when B and the photon were colocated (in A's frame, if A were inclined to think in such terms) ... x'a = xa - vt'a. In A's rest frame, the distance between where the photon was when A and B were colocated and A does not change. Note that it is this apparently unchanging distance that is the subject of one of my final equations (on a later drawing - here (http://www.geocities.com/neopolitonian/g2ev2_3.jpg)). This is silvered out because it is an aside. Just note it, I am not trying to prove it at this time.

The same goes for B (being careful with primes): B has a time t'b and a distance (to photon when A and B were colocated) x'b. Using xb=ctb, B can also work out where and when A and the photon will be colocated (in B's frame, if B were inclined to think in such terms) ... xB = x'b + vtb. In B's rest frame, the distance between where the photon was when A and B were colocated and B does not change. Note that it is this apparently unchanging distance that is the subject of one of my final equations (on a later drawing).

Isn't x'A the distance between the YDE and a specific event on B's worldline indicated by the orange dot, namely the event of the photon passing B? If so it wouldn't change with time.

I was unclear, I mean that as far as A is concerned, the separation between the location of the YDE and the location of A is fixed but the separation between the location of the YDE and the location of B is not fixed. My thinking, at that precise moment, was that x'a = the separation (hence the x) between B and the YDE (hence the prime) according to A (hence the a), which varies with time.

The selection of two specific times (photon passes B and photon passes A) is handy, but not essential. In the diagram the values apply for the moment depicted so x'a is really x'a(t'a) for a very specific t'a (when B and the photon are colocated according to A). There's nothing stopping you from looking at another time when B and the photon are not colocated and x'a would still be the separation (x) between B and the location of YDE (') according to A (a).

Does that make sense to you?

cheers,

neopolitan

JesseM

Jun5-09, 09:44 AM

But not just any event associated with the photon, right? Aren't you still assuming the YDE is the specific event on the photon's worldline that occurs at the same time as A&B being colocated in A's frame?
Not really. I am saying that A and B colocated is one event and the YDE is another event. A certain time later a photon from that event passes B and then A, which both constitute events (photon colocated with B, photon colocated with A). Then I work with those.

Consider, A works out that, if ta after colocation of A and a photon passes A, then at colocation of A and B that photon was c.ta distant. This may have been the spawning of that photon - simultaneous with colocation of A and B in A's frame - (it doesn't have to be) and if it was, then it was a unique event (and even if it wasn't spawning of the photon, the photon's location is still a unique event)
I didn't say anything about the "spawning" of the photon, I just asked whether it was part of the definition of the YDE in the left-hand diagram that it is the specific event on the photon's worldline that is simultaneous with A&B being colocated as defined in A's frame. Can I take it from this reply that the answer is "yes"? Likewise, can I assume that it's part of the definition of the YDE in the right-hand diagram that it is the specific event on the photon's worldline that is simultaneous with A&B being colocated as defined in B's frame?
Using a different coordinate system does not move the event.
It does if the different coordinate systems disagree about simultaneity. If you use Galilean coordinate systems then they won't disagree about simultaneity so it'll be the same event, but then it is easy to show using the Galilei transform that the photon did not move at c in both frames. For example, say in A's frame the YDE occurred at x=cta, t=0, and the orange dot event occured at x=vt'a, t=t'a, then if we assume the light was moving at c in this frame the relation between ta and t'a must be c = (cta - vt'a)/t'a, so multiplying both sides by t'a gives ct'a = cta - vt'a which tells us that ta = (ct'a + vt'a)/c. So, in A's frame the coordinates of the YDE are:

x = (ct'a + vt'a), t=0

and the coordinates of the orange dot are:

x = vt'a, t = t'a

Then if we want to find the coordinates of these same events in B's frame under the assumption that A and B's coordinates are related by the Galilei transformation, this would give the coordinates of the yellow dot event as:

x' = (ct'a + vt'a), t'=0

And the coordinates of the orange dot as:

x' = vt'a - vt'a = 0, t = t'a

So in this case the light has traveled a distance of (ct'a + vt'a) in a time of t'a, meaning its speed was (c + v) in this frame.
I didn't give numbers so I don't know what you are talking about.
Probably you do know what I'm talking about and are just trying to tell me that you want me to forget the numerical example that you had previously used to give values to symbols like x'a and ta (as in the blocks of text I quoted earlier). If so that's fine, but even without numbers, as long as we can assign abstract coordinates to the yellow dot and the orange dot as I did above, then it should be possible to show that the speed of light cannot be c in both frames if their coordinates are related by the Galilei transform.
Try not to focus on simultaneity (it might be tough, since that seems to be your preferred avenue into relativity).

Think: YDE happens and colocation of A and B happens (not necessarily in that order, and not necessarily together), colocation of photon and B happens, colocation of photon and A happens.
But you just said "Consider, A works out that, if ta after colocation of A and a photon passes A, then at colocation of A and B that photon was c.ta distant." Are you not assuming the YDE event is the same as the event of the photon being at a distance of c*ta from A? If the YDE occurred "at colocation of A and B", presumably you mean at the same time that A and B were colocated, i.e. simultaneously with their being colocated.
There are two events colocated with A, ie colocation with B and colocation with the photon. That gives A a time, ta
Gives A a time between what and what? I thought ta represented the time between the YDE and the event of the photon being colocated with A (which is only the same as the time between the two events you mention if we assume the YDE is simultaneous with A and B being colocated), because I thought the YDE was supposed to occur at a distance of c*ta from A. Is this incorrect?
and a distance (to photon when A and B were colocated) xa.
But by talking about where the photon was "when A and B were colocated" you are using the concept of simultaneity. We don't have to get into the *relativity* of simultaneity of course, if we want to use the Galilei transform then simultaneity is non-relative. But in this case it's impossible that the light could move at c in both frames, as I keep saying.
I was unclear, I mean that as far as A is concerned, the separation between the location of the YDE and the location of A is fixed but the separation between the location of the YDE and the location of B is not fixed. My thinking, at that precise moment, was that x'a = the separation (hence the x) between B and the YDE (hence the prime) according to A (hence the a), which varies with time. The selection of two specific times (photon passes B and photon passes A) is handy, but not essential. In the diagram the values apply for the moment depicted so x'a is really x'a(t'a) for a very specific t'a (when B and the photon are colocated according to A).
OK, makes sense. You might consider changing the label next to the blue arrow from just x'a to x'a(t'a), in order to make it consistent with the brown arrow whose label refers to the distance between A and B at the specific time t'a.

neopolitan

Jun6-09, 12:24 AM

JesseM

Jun6-09, 02:52 AM

I have been trying to get specific confirmation from you whether you understand the overall explanation and are nitpicking (or understand and feel that the explanation is wrong) or whether you still don't understand the explanation and are trying to get aspects explained so that you can understand (even if you may feel that the explanation is wrong). Is there any chance that you could address that in terms of the post where I specifically said:
Please also answer only in terms of the diagram and this explanation, do you understand this explanation of the diagram I linked? Yes or no
Didn't I do so in the purely symbolic notation above? Or do you disagree that in A's frame, the yellow dot event has coordinates x=cta and t=0, while the orange dot event has coordinates x=vt'a and t=t'a? (assuming we set the origin so the event of A and B being colocated has coordinates x=0 and t=0). If you think this is wrong, then I guess my answer would have to be "no", since in that case I don't really understand what ta and t'a are supposed to represent physically (I thought they were the time coordinates in A's frame of the photon passing A and B respectively).
Additionally, in the post that I gave the explanation of the diagram in question, #295, I got down to
so vt' = vt which means t = t' which we know can't be right
And as I said in my responses to that post, the meaning you were assigning these symbols was unclear to me...when you wrote "x = ct and x' = x - vt", what physically do x, t, and x' represent? This is especially confusing because x' = x - vt looks like the spatial component of the Galilei transform, a general equation that holds for arbitrary events that have coordinates x,t in one frame and x',t' in the other, whereas x = ct is clearly not a general relation that is supposed to apply to arbitrary events. If you want to refer to the positions and times of specific events as opposed to relationships that are supposed to hold between arbitrary sets of coordinates, it's really helpful if you put some subscripts to indicate this, and actually name in English the specific events they are the coordinates of (or the pairs of events that they represent distance and time intervals between). For example, if look at the following two events:

1. The event on the photon's worldline that occurs at t=0 in the frame of the unprimed observer, when he is colocated with the primed observer
2. The event of the photon being colocated with the unprimed observer

...then if xa is used to denote the spatial coordinate of event #1 and ta is used to denote the temporal coordinate of event #2, it would indeed be true that xa = cta. However, since these are the coordinates of two different events, we cannot plug xa and ta in for x and t in the equation x' = x - vt to get the space coordinate of either event in the primed frame, since if you want the left side of this equation to be the primed space coordinate of a particular event, you have to plug in the x and t coordinates of that same single event in the right side.

On the other hand, the Galilei transform also gives us the equation dx' = dx - v*dt for the intervals between a pair of events, so if event #1 has coordinates (x=xa, t=0) and event #2 has coordinates (x=0, t=ta), then subtracting the coordinates of the first from the second gives dx = -xa and dt = ta. Then it would be valid to plug this value for dx and dt into the equation dx' = dx - v*dt to find the spatial interval in the primed frame between event #1 and event #2 above.

Either way, as I said in post #303, it only makes sense to use the Galilean equation x' = x - vt when you have a specific event that you know the x and t coordinates of in the unprimed frame, or a specific pair of events that you know the distance and time intervals between in the unprimed frame. The Galilei transformation equation x' = x - vt has no meaning outside of one of those specific contexts. You never addressed my post #303 so I don't know if you agree or disagree with this, if you disagree please say so.
You are right if you mean that "the speed of light is c in all frames and x' = x - vt and x = x' + vt' and t' = t" is invalid. 100% But I never claim that.

I claim that "x' = x - vt and the speed of light is c is in all frames" is compatible, it's only a problem if you assert that "t' = t and x' in one frame = x' in another frame and x in one frame and x in another frame" is also valid. I introduce subscripts specifically because I know that this is not valid.
Again, it would help if you would address post #303. What does the equation x' = x - vt mean if it isn't being written in the context of the full Galilei transformation? As I said in that post, it didn't really make sense to me when you wrote "According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt", because the x' seemed superfluous here...you were just calculating the separation between the primed observer's position at time t and and the position "where the photon was at t=0", a calculation expressed entirely in terms of the unprimed frame, so the answer should just be x-vt, an equation that has nothing specifically to do with the Galilei transformation because it doesn't deal with multiple frames (the answer would still be x-vt in SR after all, something I also pointed out in post 303). Unless of course you were totally redefining the meaning of x' here, so that it no longer had jack squat to do with the coordinates of anything in the primed observer's own rest frame, but just was being used as a variable x'(t) to refer to the distance in the same unprimed frame between the primed observer and the position where the photon had been at t=0. But in this case it would be very strange to introduce the equation x'=x-vt without mentioning that the physical meaning of x' is totally different from what it means in the Galilei transformation which is the only context this equation would appear in physics books.
The YDE is obviously key to you. I'm not totally fussed about where or when it is. There's a specific reason for this lack of concern.
It's not the YDE specifically that's key to me, I just want to know the space and time coordinates of all three events (expressed in abstract rather than numerical terms is fine), otherwise the diagram and the terms don't seem very well-defined to me. In the left-hand diagram, do you agree or disagree that if the event of A and B being colocated is assigned coordinates x=0 and t=0, then xa represents the position of the photon at t=0, ta represents the time the photon passes A at x=0, and t'a represents the time the photon passes B at x=vt'a? That's all I want to know about the left-hand diagram.
First, the uncertainty about when and where the event is located comes later, once you get into the relativity of simultaneity.
I'm not talking about the relativity of simultaneity here, which involves multiple frames, just about whether the YDE is simultaneous with the event of A and B being colocated in any individual frame; as above, if they are colocated at t=0 in this frame, then does the YDE represent the event on the photon's worldline which also occurs at t=0? Or are you saying it would make no difference to you if we defined the YDE to be an event on the photon's worldline which occurred at some totally different time in this frame, say at t=-1000*ta?
Secondly, there are two sets of Lorentz transformations that you can arrive at, one pair from the perspective of A looking at B, and one pair from the perspective of B looking at A. We really only have to arrive at one pair.
I never understand what your "looking at" terminology means, but presumably you refer to difference between a set of equations that takes as inputs the coordinates of an event in the A frame and gives as outputs the coordinates of the same event in the B frame, vs. a set of equations that takes B-coordinates as inputs and gives A-coordinates as outputs. Which of these sets corresponds in your terminology to A looking at B vs. B looking at A I'm not sure.
I'm not worried that one pair might speaks about a YDE that is simultaneous in A's frame with A and B being colocated while the other speaks about a YDE that is simultaneous in B's frame with A and B being colocated. All I care about is whether the photon involved in each pair is the same photon, spawned by the same event.
Huh? These equations wouldn't "speak" about any event in particular, they relate the coordinates of any arbitrary event in one frame to the coordinates of the same event in the other frame, but either way it is necessary that you have a specific physical event in mind. And if you're going to define terms like ta or xa in terms of relationships between specific events, you have to clearly specify what the events are, or else your terms aren't well-defined.
Think about it. xa is the separation between A and where the photon was at colocation of A and B in A's frame.
OK, then what you've just said is that xa is defined as the position coordinate of the event on the photon's worldline that is simultaneous with the colocation of A and B in A's frame.
x'b is the separation between B and where the photon was at colocation of A and B in B's frame.
And here you've said that x'b is defined as the position coordinate of the event on the photon's worldline that is simultaneous with the colocation of A and B in B's frame. These are perfectly good ways of defining xa and x'b in terms of coordinates of specific events, and that's all I was asking for. We don't have to worry (yet) about whether the event on the photon's worldline used in the first definition is identical to or different from the event o the photon's worldline used in the second definition.
Both are at rest in their own rest frames, so both consider that the other has a separation from that distant location that changes with time (x'a and xb respectively).
Sure (although again, if certain symbols are going to be variables as opposed to be constants, it would be helpful if you'd indicate them as such using notation like x'a(t) and xb(t)...or maybe it'd be xb(t'), I dunno, this is another confusing aspect of your notation since you don't seem to follow the convention that unprimed terms always refer to coordinates in the first frame and primed terms always refer to coordinates in the second frame)
Taking just A's side of the story, A doesn't move, B does. The photon from YDE reaches A at a time ta. That same photon passed B, and on B's clock at that time it said t'b. According to A, that photon was at xa when A and B were colocated. But according to B, that photon was at x'b when A and B were colocated. The photon is the same. It's the photon that passes B and reaches A.
Sure, the photon is the same, but the event on the photon's worldline that occurred at a position of x=xa in A's frame may or may not be the same event as the event on the photon's worldline that occurred at a position of x'=x'b in B's frame. Terms like xa must have well-defined physical definitions if we want to use them in a physics context.
If you like, this not worring about the specific spacetime location of YDE is a little like a lie to children (http://en.wikipedia.org/wiki/Lie-to-children). If you want to focus heavily on the YDE and fix it in space and time, then I have to give you an overt "lie to children" and tell you it's the same event.
You can't have a valid derivation that starts from a wrong premise, unless you're doing a proof by contradiction. In any case, the lie seems totally superfluous here. Why not have a yellow dot event in the left diagram that occurs at t=0 in the A frame, and a pink dot event in the right diagram that occurs at t'=0 in the B frame, and just not say anything one way or another about whether these two events are identical or different? What exactly would be lost?
The thing is, the average student being introduced to relativity would not be like you and want to know the precise spacetime location of the YDE. Can you understand that?
Anyone who's familiar with the use of coordinate systems at all (Galilean or otherwise) will want to know the coordinates of any event that's introduced, even if they are presented in abstract rather than numerical notation. As an example, how do you expect the student would understand that xa (the space coordinate of the YDE in the A frame) should equal cta (where ta is the time coordinate the the photon passes A) if they don't assume the photon was traveling at c and the YDE occurred at t=0?
Addressing the second first, I'm not really using simultaneity. I'm using extrapolation.

If A and B are colocated and a photon passes A a period of ta later, then A can extrapolate that the photon must have been at xa=c.ta when A and B were colocated.
How does this contradict the idea that you're using simultaneity to define the YDE? If you define the YDE as where the photon was "when A and B were colocated", that's exactly equivalent to defining the YDE event as the point on the photon's worldline that's simultaneous with A and B being colocated--to say two events are simultaneous is just another way of saying one event happened when the other event did. The fact that you can then use this definition (along with the fact that the photon passed A at ta, and the assumption that the light was traveling at c) to extrapolate the position of the YDE doesn't somehow invalidate the fact that simultaneity with A & B's colocation was key to the original definition.
No explicit relativity of simultaneity.
I just said that the YDE was defined in terms of simultaneity with A&B's colocation in each frame, I didn't say anything about the relativity of simultaneity. Again, simultaneity just means "at the same time coordinate", it's perfectly OK to use the word simultaneity in a discussion of Galilean frames when there is no relativity of simultaneity because all frames agree whether or not two events are simultaneous. I made this point in my last post too:
But by talking about where the photon was "when A and B were colocated" you are using the concept of simultaneity. We don't have to get into the *relativity* of simultaneity of course, if we want to use the Galilei transform then simultaneity is non-relative. But in this case it's impossible that the light could move at c in both frames, as I keep saying.
And hopefully I have explained here why I am not fussed about the when and where of the YDE and what ta is (between what and what).
No, nothing you have said helps me to make any sense of what it could mean to have a well-defined problem where you introduce events without any notion of their coordinates, or introduce terms without knowing their physical meaning. If you're "not fussed" about the coordinates of the YDE or what ta means, will it make no difference to your derivation if I secretly choose to assume the YDE occured at t=-1000*ta (still assuming that A and B were colocated at t=0), or that ta refers to the time coordinate of A marking the 30th anniversary of the photon having passed him?

neopolitan

Jun6-09, 03:18 AM

A common refrain, but your dividing strategy makes replying difficult. I did partially reply to #303. I may not have replied to something specific in #303 because I am not going to always give an individual reply to each of your paragraphs.

I did intend to reply to the totality of #303 in separate posts because there were two strands in there, which would lead to confusion (once you split everything into individual paragraphs, there is no longer a clear separation between those strands). I didn't get around to it but I do think I addressed most of what was in the first part of #303 in later posts.

Is there still a specific question in #303 which you need a specific answer to which I have not addressed since in response to a later question?

I have labelled this #303 strand.

cheers,

neopolitan

neopolitan

Jun6-09, 03:33 AM

Or do you disagree that in A's frame, the yellow dot event has coordinates x=cta and t=0, while the orange dot event has coordinates x=vt'a and t=t'a? (assuming we set the origin so the event of A and B being colocated has coordinates x=0 and t=0).

I can live with that, yes. I don't really want to bring in a pink dot. There must be a mutually satisfactory way to avoid that.

These equations wouldn't "speak" about any event in particular, they relate the coordinates of any arbitrary event in one frame to the coordinates of the same event in the other frame, but either way it is necessary that you have a specific physical event in mind.

I want to focus in on this for the moment, if I may. Can we try to focus on one thing at a time?

For me, this is precisely what a good coordinate transformation equation would do. It would relate the coordinates of an arbritrary event in one frame to the coordinates of the same arbitrary event in another frame. I don't think that is a fault, it's an important feature of the coordinate transformation equation.

Do you agree that the event that results in (0,ta) is arbitrary, if we chose a handy event (xa=cta,0) as the initiating event and we haven't actually pinned down what numerical value ta has?

We can certainly make it more general by selecting an initiating event which was not simultaneous with (0,0), but do you agree that that would be awkward?

Again, I really would like to focus on this, because it may be key. Can we do that?

cheers,

neopolitan

JesseM

Jun6-09, 03:59 AM

A common refrain, but your dividing strategy makes replying difficult. I did reply to #303. I may not have replied to something specific in #303 because I am not going to always give an individual reply to each of your paragraphs.
You only responded to one incidental question in 303 about the relation between the two blocks of text and the "two perspectives" (which was just a continuation of a discussion about an incidental question I had asked in the second-to-last paragraph of 293), you didn't respond to the main body of the post which was part of the discussion about the equations you wrote out in 295 (I had responded to 295 in 299, then you responded to 299 in 302, and my 303 was in response to that).
I did intend to reply to #303 in separate posts because there were two strands in there, which would lead to confusion (once you split everything into individual paragraphs, there is no longer a clear separation between those strands). I didn't get around to it but I do think I addressed most of what was in the first part of #303 in later posts.

Is there still a specific question in #303 which you need a specific answer to which I have not addressed since in response to a later question?
Yes, the basic question I was focused on in 303 was whether you understood that the equation defining the separation between B and the position of the photon at t=0 as a function of time t in the A frame, namely x-vt, has only a superficial resemblance to the Galilei transformation equation, x'=x-vt, but their physical meanings are really quite different because the first is just calculating the distance between two things exclusively in A's frame with no reference to B's frame, while the fundamental purpose of the second is to relate the x,t coordinates of an event in the A frame to the x' coordinate of the same event in the B frame. So given this, I didn't understand why you had written "According to the unprimed observer, the separation between where the primed observer is now, and where the photon was at t=0, is x'=x-vt"...if you were just calculating the separation in the unprimed frame, that would just be x-vt, the x' doesn't make any sense there unless you have redefined x' to mean the separation in the unprimed frame, which would be very confusing and totally different from the meaning of the equation x'=x-vt in the Galilei transformation equation (and you said the student knew the equation x'=x-vt based on the fact that he 'paid enough attention' in class, so it would be pretty weird if what you really meant was that he knew the equation but totally misunderstood its physical meaning).

I had some other questions in post 303 about the meaning of the equations in your post 295, but I think I restated these questions a little more clearly in post 311 so maybe you could just respond to this section:
And as I said in my responses to that post, the meaning you were assigning these symbols was unclear to me...when you wrote "x = ct and x' = x - vt", what physically do x, t, and x' represent? This is especially confusing because x' = x - vt looks like the spatial component of the Galilei transform, a general equation that holds for arbitrary events that have coordinates x,t in one frame and x',t' in the other, whereas x = ct is clearly not a general relation that is supposed to apply to arbitrary events. If you want to refer to the positions and times of specific events as opposed to relationships that are supposed to hold between arbitrary sets of coordinates, it's really helpful if you put some subscripts to indicate this, and actually name in English the specific events they are the coordinates of (or the pairs of events that they represent distance and time intervals between). For example, if look at the following two events:

1. The event on the photon's worldline that occurs at t=0 in the frame of the unprimed observer, when he is colocated with the primed observer
2. The event of the photon being colocated with the unprimed observer

...then if xa is used to denote the spatial coordinate of event #1 and ta is used to denote the temporal coordinate of event #2, it would indeed be true that xa = cta. However, since these are the coordinates of two different events, we cannot plug xa and ta in for x and t in the equation x' = x - vt to get the space coordinate of either event in the primed frame, since if you want the left side of this equation to be the primed space coordinate of a particular event, you have to plug in the x and t coordinates of that same single event in the right side.

On the other hand, the Galilei transform also gives us the equation dx' = dx - v*dt for the intervals between a pair of events, so if event #1 has coordinates (x=xa, t=0) and event #2 has coordinates (x=0, t=ta), then subtracting the coordinates of the first from the second gives dx = -xa and dt = ta. Then it would be valid to plug this value for dx and dt into the equation dx' = dx - v*dt to find the spatial interval in the primed frame between event #1 and event #2 above.
...or am I totally on the wrong track about the meaning of x'=x-vt, and is it not supposed to have the same meaning as in the Galilei transformation at all? As I suggested above, perhaps you are just redefining x' to mean the separation between B and the position of the YDE in the unprimed frame, so that every part of x'=x-vt deals with the unprimed frame and despite appearances it is not meant to be a coordinate transformation equation at all?

JesseM

Jun6-09, 04:09 AM

I want to focus in on this for the moment, if I may. Can we try to focus on one thing at a time?

For me, this is precisely what a good coordinate transformation equation would do. It would relate the coordinates of an arbritrary event in one frame to the coordinates of the same arbitrary event in another frame. I don't think that is a fault, it's an important feature of the coordinate transformation equation.
Sure, that's just the standard meaning of what coordinate transformation equations are meant to do (although a coordinate transformation can also transform the intervals between an arbitrary pair of events in one frame to the intervals between the same pair of events in another frame).
Do you agree that the event that results in (0,ta) is arbitrary, if we chose a handy event (xa=cta,0) as the initiating event and we haven't actually pinned down what numerical value ta has?
Yes, as I said I'm fine with defining events in abstract notation rather than numerical values.

neopolitan

Jun6-09, 04:41 AM

Sure, that's just the standard meaning of what coordinate transformation equations are meant to do (although a coordinate transformation can also transform the intervals between an arbitrary pair of events in one frame to the intervals between the same pair of events in another frame).

Yes, as I said I'm fine with defining events in abstract notation rather than numerical values.

Then I can't understand why you said this:

These equations wouldn't "speak" about any event in particular, they relate the coordinates of any arbitrary event in one frame to the coordinates of the same event in the other frame, but either way it is necessary that you have a specific physical event in mind.

I took that to be criticism, although I couldn't understand what alternative you were presenting.

By the way, you talk about coordinates and intervals. To avoid having to do that, can we agree that coordinates are intervals (just a specific one where one of the implied pair is the origin of the axis)? I have no problem with the idea of rearranging my axes to transform any interval into a single coordinate, or the reverse.

On the image, we could call the red dot a coordinate, but it is also an interval from the orange dot to the red dot. Between the red dot and the green dot is an interval, but if I redefined my axes so that (0,0) was the red dot, then the green dot would be a coordinate from the red dot. I don't see a huge difference between coordinate and interval for this very reason.

cheers,

neopolitan

JesseM

Jun6-09, 04:52 AM

I took that to be criticism, although I couldn't understand what alternative you were presenting.
It was less of a criticism and more of a puzzlement about your statement here:
I'm not worried that one pair might speaks about a YDE that is simultaneous in A's frame with A and B being colocated while the other speaks about a YDE that is simultaneous in B's frame with A and B being colocated. All I care about is whether the photon involved in each pair is the same photon, spawned by the same event.
You seemed to be saying that one pair would speak specifically speak about "a YDE that is simultaneous in A's frame with A and B being colocated", while the other pair would speak specifically about a YDE that was simultaneous with colocation in B's frame. But there's no specificity here, since each pair of equations is totally general, either pair could deal with either type of YDE. For example, one pair of equations for the Lorentz transform would be x'=gamma*(x-vt) and t'=gamma*(t-vx/c^2); if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the unprimed A frame I could plug those in and get the same event's x',t' coordinates, and likewise if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the primed B frame I could plug those in and get the same event's x',t' coordinates. Of course that's assuming I started out knowing the x,t coordinates of each type of event, I suppose in the case of a YDE that was simultaneous with colocation in the primed frame it's more plausible I would start out knowing the event's x',t' coordinates, so maybe that's what you meant (but then again events aren't 'native' to any particular coordinate system, it's possible I would focus on this particular event while working in unprimed coordinates for some other reason without knowing in advance it had the property of being simultaneous with colocation in the primed frame).
By the way, you talk about coordinates and intervals. To avoid having to do that, can we agree that coordinates are intervals (just a specific one where one of the implied pair is the origin of the axis)? I have no problem with the idea of rearranging my axes to transform any interval into a single coordinate, or the reverse.
Sure, that's a good way of thinking about it.

neopolitan

Jun6-09, 04:53 AM

neopolitan

Jun6-09, 05:10 AM

You seemed to be saying that one pair would speak specifically speak about "a YDE that is simultaneous in A's frame with A and B being colocated", while the other pair would speak specifically about a YDE that was simultaneous with colocation in B's frame. But there's no specificity here, since each pair of equations is totally general, either pair could deal with either type of YDE. For example, one pair of equations for the Lorentz transform would be x'=gamma*(x-vt) and t'=gamma*(t-vx/c^2); if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the unprimed A frame I could plug those in and get the same event's x',t' coordinates, and likewise if I knew the x,t coordinates of a YDE that was simultaneous with colocation in the primed B frame I could plug those in and get the same event's x',t' coordinates.

x' = \gamma. (x - vt)
t' = \gamma. (t - vx/c^2)

x = \gamma. (x' + vt')
t = \gamma. (t' + vx'/c^2)

gamma = 1.25
v = 0.6c
event (x,t) = (10,0)

x' = 12.5
t' = - 7.5

This event was not simultaneous with t' = 0

But we can plug these right back into the second pair of equations and get the original x and t back out.

Alternatively, we could start with

event (x',t') = (10,0)

which is a totally different event, just one which is simultaneous (in the primed frame) with t' = 0 - and we can do the same sorts of substitutions.

Notionally, we could start with an event which is not simultaneous with either.

I'm not that fussed. It's just much simpler to start with an event that is simultaneous with the colocation of the origins of the axes and work from there.

What I do know is that no matter what event you start from, that event can be associated with a photon which could result in two more events (such as: photon colocated with A and photon colocated with B).

Am I making any sort of headway?

cheers,

neopolitan

JesseM

Jun6-09, 05:27 AM

I'm going to try to address the x' = x - vt thing.

Rather than go over each of your paragraphs, I will try to just explain it. I hope this will satisfy you.

The implication with x' = x - vt is that you have an unprimed observer who is considering what things would be like for someone else who is moving with a speed of v towards a location, or event, at a distance of x away.

At a time t, x will be unchanged for our unprimed observer in the unprimed frame. However, the separation between the someone else and that location, or the location where the event took (or will take or takes) place, will have changed.

Our primed observer can then work out that at t this equation applies:

x' = x - vt

or to be more specific, since we are describing a function of time,

x'(t) = x - vt
So was I right in guessing that you are using x'(t) to refer to the distance in the unprimed frame between the primed observer and the location, despite the fact that the symbol is primed? If so, why do you say "our primed observer" works out this equation, if all of the terms are defined in terms of the unprimed frame? And if this is what you mean, can you see why this notation might be extremely confusing, especially since you said that the high school student was aware of this equation because he paid attention in class, and yet in any textbook which used standard notation conventions the meaning of this equation would always be in the context of the Galilei transformation relating one frame to another? If you're going to take standard textbook equations and change the meaning of the terms this is really something you need to explain in advance to avoid confusion.
This equation can be used to obtain the spatial interval between the someone else and any event that happens at any time at any location under Galilean relativity.
But it has nothing specifically to do with Galilean relativity, since if x'(t) just means the distance in the unprimed frame between B and the location at x, then the equation x'(t)=x-vt isn't even relating multiple frames, it would be equally valid in SR.
In Galilean relativity, that interval is also the spatial coordinate for the event in the someone else frame.
Yes.
Can you see that while you are 100% right about x'(t) = x - vt applying to spatial coordinate, that that is not 100% of the story?
Not really sure what you mean by "applying to spatial coordinate". Do you just mean that x'=x-vt normally is understood to relate one Galilean frame's coordinates to another's, but that we can in principle redefine the meaning of x' so that it refers to the spatial separation between a moving object and a fixed location x in a single frame? If so I agree (and I already asked you if you were making such a redefinition in several earlier posts). But when you change the physical definition of the terms you're dealing with a different physical equation even if the symbols are the same, so it's not like there are two different ways of looking at the same physical equation. What's more, writing a new physical equation using notation that has a different preexisting established meaning is kind of perverse from a pedagogical point of view (sort of like if I wrote the equation E=mc^2 and said that E stands for force and c stands for the square root of acceleration), there's already an accepted convention about the physical interpretation of primed vs. unprimed coordinates and about the equation x'=x-vt, if you want to write down an equation giving a spatial separation in the unprimed frame then it would be much better to use some different notation which wouldn't be so likely to confuse people who were trying to understand you, like s(t) = xa - vt.

JesseM

Jun6-09, 05:34 AM

x' = \gamma. (x - vt)
t' = \gamma. (t - vx/c^2)

x = \gamma. (x' + vt')
t = \gamma. (t' + vx'/c^2)

gamma = 1.25
v = 0.6c
event (x,t) = (10,0)

x' = 12.5
t' = - 7.5

This event was not simultaneous with t' = 0

But we can plug these right back into the second pair of equations and get the original x and t back out.
Right, that was exactly my point. Neither pair of equations deals specifically with this event as your earlier comment sounded like it was saying, we could either start with the event's x and t coordinates and use the first pair of equations to get its x' and t' coordinates, or we could start with the event's x' and t' coordinates and use the second pair of equations to get its x and t coordinates. It may be a little more "natural" to start with the coordinate system where the time coordinate is zero, so that's probably what you meant in that earlier comment. In any case I don't really think we have any remaining disagreement here.

neopolitan

Jun6-09, 07:46 AM

So was I right in guessing that you are using x'(t) to refer to the distance in the unprimed frame between the primed observer and the location, despite the fact that the symbol is primed? If so, why do you say "our primed observer" works out this equation, if all of the terms are defined in terms of the unprimed frame?

Thankfully I only ever mentioned one observer, so it should be obvious that this was a typo. Note I mentioned "observer" and "someone else". Wherever observer was written it should have been prefixed with unprimed. Sorry.

With that fixed (I edited the post) can you revisit your questions and see if they still apply.

thanks,

neopolitan

Oh, and by the way, I never primed a frame. That assumption on your part might be confusing you. Remember my description of x'[sub]a[\sub] from an earlier post?

JesseM

Jun6-09, 12:49 PM

Thankfully I only ever mentioned one observer, so it should be obvious that this was a typo. Note I mentioned "observer" and "someone else". Wherever observer was written it should have been prefixed with unprimed. Sorry.

With that fixed (I edited the post) can you revisit your questions and see if they still apply.
Yes, aside from that one sentence about the primed observer all my other points still apply; I still think it's extremely perverse from a pedagogical point of view to take an equation with an established meaning like x'=x-vt and redefine the meaning of the terms so it the equation's physical meaning becomes completely different (without stating explicitly that this is what you're doing), and to say that a high school student is aware of this equation from his classes without mentioning that his interpretation of the symbols has nothing to do with what he actually would have been taught in class. Do you not see how ridiculously confusing this is for readers? Again, if you want to write an equation expressing only the displacement in the unprimed frame, write something like d(t) = xa - vt.
Oh, and by the way, I never primed a frame.
It doesn't matter if you did, what matters is that the equation x'=x-vt, whenever it appears in any physics textbook, always appears in the context of the Galilei transform where it relates coordinates in an unprimed frame to coordinates in a primed frame. And you said that the high school student was aware of this equation from his classes (presumably not taught by a teacher who was making up his own idiosyncratic notation), not to mention you even said that you were starting your derivation from a "Galilean" perspective! Like I said, this is akin to writing the equation E=mc^2 without mentioning that you've mentally redefined E to mean force and c^2 to mean acceleration.

neopolitan

Jun7-09, 12:22 AM

So, what you are saying is that I am using an equation which looks a lot like the Galilean boost, but isn't, using a process which you may not understand, and arriving at a equations which look a lot like the Lorentz Transforms but are not?

This is a rhetorical question, and I am taking the answer to be yes.

So, I've found a new set of equations. I am so proud. I hereby name them "the Neopolitonian boost" and "the Neopolitonian Transforms".

The Neopolitonian boost

This equation is used in a scenario where there are two observers (A and B) and an Event.

According to A that Event is at (xa,0). A photon from that event reaches A at event (0,ta)

According to A, B has a velocity of v towards the Event and the boost is used to find the interval between B and that Event (according to A) at ta

x'a = xa - v.ta

(In our numerical example, this is 5 = 8 - 0.6 * 5 )

The Neopolitonian Transform

This pair of equations is used to transform coordinates between two inertial observers (or two inertial frames). For example, in the scenario above, all the values given are according to A. The Neopolitonian Transform gives a value of x' and t' according to B (x'b and t'b) - which make B's frame the primed frame. This can also be called "A looking at B" since the values of x'b and t'b are given in terms of xa and ta, ie "what do B's values look like in terms of A's values?"

x'b = \gamma.(xa - v.ta)

t'b = \gamma.(ta - v.xa/c2)

Ok. Are you happy to talk about the process used to derive the Neopolitonian Transform from the Neopolitonian boost?

Plus, are you absolutely certain that no-one is going to accuse me of just rebadging the Galilean boost and the Lorentz Transform, because it is sooooo obvious that they are not the same as the Neopolitonian boost and the Neopolitonian Transform?

cheers,

neopolitan

PS I want make explicit the fact that I still remember that we can convert coordinates to intervals and back again. I also know that it is applicable in the above.

JesseM

Jun7-09, 01:29 AM

So, what you are saying is that I am using an equation which looks a lot like the Galilean boost, but isn't, using a process which you may not understand, and arriving at a equations which look a lot like the Lorentz Transforms but are not?
I was mainly expressing frustration that you don't seem to consider how your idiosyncratic way of explaining yourself will lead naturally to some obvious misunderstandings which have to be laboriously worked out over a long series of subsequent posts, which could have been easily avoided if you explained the difference between your equations and the "standard" ones to begin with, or made your notation different from that of standard equations so such misunderstandings would be less likely to occur. Now I understand what you meant by x'=x-vt in post 295, but a lot of wasted time could have been avoided if you had been more explicit about the way you were redefining things at the start. If you got confused by someone's explanation of a series of equations which included the equation E=mc^2, and only after a long discussion did he make clear that he was defining E to mean force and c to mean the square root of acceleration, wouldn't you find this a little frustrating too?
So, I've found a new set of equations. I am so proud. I hereby name them "the Neopolitonian boost" and "the Neopolitonian Transforms".

The Neopolitonian boost

This equation is used in a scenario where there are two observers (A and B) and an Event.

According to A that Event is at (xa,0). A photon from that event reaches A at event (0,ta)

According to A, B has a velocity of v towards the Event and the boost is used to find the interval between B and that Event (according to A) at ta

x'a = xa - v.ta

(In our numerical example, this is 5 = 8 - 0.6 * 5 )
It isn't really proper to use the term "boost" here, as the word "boost" in physics refers to a transformation between two frames (for example see the last paragraph in this section (http://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_form) of wikipedia's Lorentz transformation (http://en.wikipedia.org/wiki/Lorentz_transformation) article). Your equation is just a simple kinematical equation (http://en.wikipedia.org/wiki/Kinematics) for the separation between an object moving at constant coordinate speed and another object at fixed coordinate position, it would hold in absolutely any coordinate system whatsoever (like an inertial SR frame or even a non-inertial frame) and therefore has no specific relation to anything "Galilean".
The Neopolitonian Transform

This pair of equations is used to transform coordinates between two inertial observers (or two inertial frames). For example, in the scenario above, all the values given are according to A. The Neopolitonian Transform gives a value of x' and t' according to B (x'b and t'b) - which make B's frame the primed frame. This can also be called "A looking at B" since the values of x'b and t'b are given in terms of xa and ta, ie "what do B's values look like in terms of A's values?"

x'b = \gamma.(xa - v.ta)

t'b = \gamma.(ta - v.xa/c2)
Why do you call this the "Neopolitan transform"? If it really holds for events at arbitrary coordinates in the A frame, and gives you the corresponding coordinates of the same event in the B frame, then it obviously has the same physical meaning as the Lorentz transformation, unlike your equation x'a = xa - v.ta which did not have the same physical meaning as the Galilei transformation equation x' = x - vt since you were not relating the coordinates of two different frames. I'm trying to get you to communicate your ideas in a non-confusing way which avoids hours of back-and-forth posts clarifying your meaning which could have been easily avoided if you had stated things more clearly at that outset--in order to do this, you need to think carefully about the physical meaning of the equations you write in relation to the physical meaning of standard textbook equations, and if you realize your equation has a different physical meaning in spite of superficial similarities to an existing equation then don't write it in exactly the same form without pointing out the difference, but likewise if your equations do have the same physical meaning as existing equations then you should be clear on this and not talk as though the equation is original to you just because you've altered the notation slightly. This issue of your not thinking through the physical meaning of equations is one that seems to come up again and again in our discussions--it arose in our discussions of the difference between the time dilation equation and the "temporal analogue for length contraction" too--so if you don't really follow the point I'm making (as suggested by your indiscriminately renaming equations above regardless of whether they do or don't have the same physical meaning as existing equations) then I'd like to try to focus on this until you understand.

But as a sort of aside to this, although the words you use to describe the equations above indicate they have the same physical meaning as the Lorentz transformation, I am in fact skeptical that your actual derivation would in fact prove something as general as the words suggest, unless you have totally changed the proof from what you offered before. I expressed my view of the physical meaning of the equations you had derived back in post 247 (http://www.physicsforums.com/showpost.php?p=2175921&postcount=247)--as far as I could tell, your derivation only proved that x'a would equal gamma*(xb + vtb) in the specific case where x'a was defined as the interval between Ea and the light passing B, while xb and tb were the intervals between a different pair of events, namely the event of Eb and the event of the light passing A. As I explained in those diagrams, by exploiting the symmetry of your scenario we can see that all the intervals between the first pair of events are identical to the intervals between the second pair, explaining why your equation comes out looking just like the Lorentz transformation even though it's not dealing with a single pair of events like the Lorentz transformation, but it seemed to me we only know about this symmetry because we already know how the two frames are related by the Lorentz transformation. In any case, even if we "allow" this symmetry to be exploited in the proof, the scope of the proof would still be limited to showing that the separation between events on the path of a light beam obey an equation like x'a = gamma*(xb + vtb), the proof you presented simply wouldn't tell cover the case of pairs of events with a timelike or spacelike separation (though we know from other more general proofs that the equation would be the same). Finally, there was a step midway through the proof that didn't seem justified to me, where you said that we could assume the factor in x'A = (a factor times).x'B was the same as the factor in xB = (a factor times).xA--I explained my objection to this at the very bottom of post 280 (http://www.physicsforums.com/showpost.php?p=2180963&postcount=280), for example. Basically, unless you have made really large changes to the proof you already presented, it's unlikely that these criticisms will change.
Plus, are you absolutely certain that no-one is going to accuse me of just rebadging the Galilean boost and the Lorentz Transform, because it is sooooo obvious that they are not the same as the Neopolitonian boost and the Neopolitonian Transform?
I am certain that if you stated the physical meanings of the equations explicitly in the way you did above, anyone knowledgeable about physics could be convinced that the first equation does not have the same physical meaning as the Galilean boost (because it's an intrinsic part of the definition of the 'Galilean boost' x' = x - vt that it relates one frame to another, whereas you defined your equation to only involve the coordinates of a single frame), while the second set of equations does have the same physical meaning as the Lorentz transformation (as you defined the meaning of the second set of equations in words above, not saying anything about whether your derivation would actually prove those words). If there is any doubt in your mind about either of these points I suggest we focus on the issue of the physical meaning of the Galilean boost x'=x-vt and the Lorentz transformation equations and how it relates to the equations you wrote, and leave other issues for later.

neopolitan

Jun7-09, 05:00 AM

I disagree with your analogy, but I understand your frustration.

Can we agree that it would be silly for me to claim that I have found a new equation which looks pretty much like the Lorentz Transform, and claim it as my own?

That leaves the method for getting there and the boost.

Did you see my PS in the previous post?

It was there specifically because I wanted you to remember that fact and know that I remember it.

In Galilean relativity, the (spatial) interval from B to an event is the coordinate of the event in B's rest frame.

Do we disagree about that?

If we can agree on that, then I think I could call x' = x - vt a (spatial) boost, and I think I would not get away with calling x'a = x - vta the Neopolitonian (not neopolitan) boost. It would swiftly be recognised as a slight rewording of the Galilean (spatial) boost.

The thing that would distinguish it, possibly is the implication that if it is just a slight rewording of the Galilean boost ie

x'a = x - vta

then

t'a = ta - NOTE, I am not saying this, this is the second part of an if-then statement

It is something that I go on to disprove. But until this second part is disproved, I would consider it to be the Galilean boost.

cheers,

neopolitan

JesseM

Jun7-09, 01:27 PM

In Galilean relativity, the (spatial) interval from B to an event is the coordinate of the event in B's rest frame.

Do we disagree about that?
They happen to be equal, but an equation telling you about the spatial interval has a different physical meaning than an equation telling you about the coordinate in B's rest frame. In a given physics equation every symbol must have a single well-defined meaning.
If we can agree on that, then I think I could call x' = x - vt a (spatial) boost
Not if you have defined x' to mean the spatial interval in A's frame. You may have outside knowledge that the spatial interval in A's frame is equal to the coordinate in B's frame, but the equation itself, with the symbols defined in this way, doesn't tell you anything about B's frame.
and I think I would not get away with calling x'a = x - vta the Neopolitonian (not neopolitan) boost. It would swiftly be recognised as a slight rewording of the Galilean (spatial) boost.
No, it wouldn't be a "boost" at all because it only deals with one frame. If we define x'a as merely the coordinate separation between an object at position x and an object B moving towards x at coordinate speed v (which started at x=0 at t=0), all in A's coordinate system, do you agree that this is a purely kinematical equation which is independent of the laws of physics, or of what type of coordinate system you're using (inertial or non-inertial)?

Think of actually writing out the physical meaning of all the terms in any equations in words. One equation is:

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

This equation, again, would work in absolutely any type of coordinate system whatsoever. The second equation is different in that it only works when using Galilean inertial frames, and it can be written as:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

Of course, we also happen to know that in Galilean relativity the following is true:

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position in B's frame of object at position x in A's frame at time t in A's frame)

But this is knowledge external to the first two equations themselves. Think of physics equations as very stupid things that can only give you an answer to one specific type of question, and they have no knowledge of any larger context. The only way two equations can be considered "the same" is if they are answering exactly the same physical question, and only the notation is different.

It may also help to point out that someone could easily have "discovered" the first equation before the discovery of the Galilei transformation, since the first equation doesn't involve multiple coordinate systems. Do you think it would be fair for this person to demand that the Galilei transformation be renamed after themselves, since they had already discovered the equation even though the physical meaning of what was being calculated was different and they hadn't even been thinking about the relation between multiple frames?

neopolitan

Jun7-09, 08:48 PM

sylas

Jun7-09, 09:02 PM

While I applaud your dedication to rigour, I think you take it too far.

With respect, I don't think so. I think you need to be more rigorous and careful in your definitions. This is not an excessive hurdle, and it's getting better; but I think you need to keep being as rigourous as you can manage. The comparison with Einstein is invalid. Being rigourous does not mean "mentioning frames". It means being unambiguous and precise in whatever terminology or diagrams you use. It's not that you need full derivations and proofs of everything; just less ambiguity.

No offense intended... but I've been watching with some interest and I think the major problem is a lack of precision and rigour, and it should not be that hard to fix.

Cheers -- sylas

neopolitan

Jun7-09, 09:14 PM

With respect, I don't think so. I think you need to be more rigorous and careful in your definitions. This is not an excessive hurdle, and it's getting better; but I think you need to keep being as rigourous as you can manage. The comparison with Einstein is invalid. Being rigourous does not mean "mentioning frames". It means being unambiguous and precise in whatever terminology or diagrams you use. It's not that you need full derivations and proofs of everything; just less ambiguity.

No offense intended... but I've been watching with some interest and I think the major problem is a lack of precision and rigour, and it should not be that hard to fix.

Cheers -- sylas

Thanks, it helps to get another perspective.

JesseM

Jun7-09, 09:40 PM

While I applaud your dedication to rigour, I think you take it too far.

Just out of curiosity, I look here (http://www.fourmilab.ch/etexts/einstein/specrel/www/) and searched for the word "frame". It appears exactly once, in the introduction, in the phrase "frames of reference" in the context of describing the first postulate.
The entire paper is about what we now call frames, Einstein just doesn't use that term. When he introduces a "a system of co-ordinates in which the equations of Newtonian mechanics hold good" at the beginning of section 1, what do you think that is if not an inertial frame? And he talks about different systems of coordinates throughout the paper, sometimes just using the word "system" (it's clear he means coordinate system and not some other type of physical system from the context)--for example, part 3, where he actually derives the Lorentz transformation, is titled "Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former".
While trying not to go so far as committing an "appeal to authority", I do want to know why I am being held to such high standards of rigour (specifying that the Galilean boost addresses a single question about frames) when the genius who came up with Special Relativity in his own way didn't really mention frames at all?
Again, the whole paper is about frames. The precise word is irrelevant as long as the concept is understood; I'd be equally happy with saying the Galilean boost is about relating the coordinates of an event in one "system of coordinates" to the coordinates of the same event in another "system". Whatever wording you use, this is conceptually quite different from just telling you how the coordinate separation between two objects is changing in a single coordinate system.
I'm not saying you are wrong that the Galilean boost is about frames, and the Lorentz Transformation is about frames, I am just wondering if your demands are truly warranted.
Just the fact that I was confused for so long by the meaning of the equation x' = x - vt in post 295 shows that they are warranted; I'd rather not get into more lengthy discussions over such trivial stuff in the future. Even if you incorrectly described the equation as the Galilean boost, the problem could have been avoided if you had spelled out in words what each symbol meant physically; if you had said at the outset that x' was supposed to represent a separation in the same frame that x and t referred to, then I might have offered a quick correction about terminology but there wouldn't have been all the confusion about what you were trying to demonstrate with your equations. But the combination of not giving physical definitions of your symbols at the outset, using the term "Galilean boost", and writing your equation using exactly the same notation as is usually used for the Galilean boost naturally led me to draw the wrong conclusions about the physical meaning of the equation. Hopefully you agree that, spelled out in words, this:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

...is telling us something physically from this?

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

Would you be happy if there was an extra step added in which I address Galilean frames, say the equation is x' = x - vt and that there is also a kinematic equation x' = x - vt and while they talk about different things, the relationship x' = x - vt holds equally for whatever values of x and t you enter into it? (Since the conditions under which the equation holds are the same for x' = x - vt and for x' = x - vt).
As long as you define the physical meaning of whatever equations you use I'll be OK, although from a pedagogical point of view I don't really like the approach of using identical notation for two physically different equations. In any case, is it necessary to discuss Galilean relativity at all in your derivation? Isn't the kinematical equation the only one you actually make use of?
It may also help to point out that someone could easily have "discovered" the first equation before the discovery of the Galilei transformation, since the first equation doesn't involve multiple coordinate systems. Do you think it would be fair for this person to demand that the Galilei transformation be renamed after themselves, since they had already discovered the equation even though the physical meaning of what was being calculated was different and they hadn't even been thinking about the relation between multiple frames?
No, and my suggestion to rename equations was entirely facetious.
I understood it was meant to be facetious...but my point in the above comment was, if you agree this hypothetical pre-Galileo guy shouldn't get credit for the Galilei transformation despite writing down an equation like x' = x - vt, doesn't that mean you should also agree we shouldn't use the same terminology for his kinematical equation that we do for the spatial component of the Galilei transformation, even if they look the same symbolically?
I do wonder if you have the visual ability to see that what I am doing is not really invalid.
It has nothing to do with visual abilities, I get visually why it works out that the separation in A's frame between B and the object at position x is always going to be equal to the position coordinate assigned to that object in B's frame. The point is that the equations are telling you different things physically, and that since I naturally thought you were introducing x'=x-vt in post 295 to transform into B's frame, I was confused since under the Galilei transformation the light could not be moving at c in B's frame.
In short, are you happy with:

Introduce Galilean frames (hopefully already done by the education system)
Introduce a kinematic equation in the form x' = x - vt (partially done)
As I said I don't like using the same notation for two equations with different physical meanings, and I think from a pedagogical point of view it's more confusing than helpful.
Point out that both equations operate on the same conditions
By "operate on the same conditions", I take it you mean if we pick a given x,t in A's frame, we get the same value for the answer? That's fine as long as you point out the physical meaning of the "answer" is different.
PS Am I going to have to draw another diagram? I've already been thinking of the best way to try to show you that what I am doing is not as whacky as you seem to think it is. Part of the problem might be that I am an engineer, manipulating equations is partly what I do. As someone with more of a physics bent, you don't seem to like the actual use of equations (or what you might term "abuse of equations" :smile:)
Now that I understand the physical meaning of your symbols I don't object to "what you are doing" in the derivation so far, only to how you are explaining it. And I have no dislike of equations (a scurrilous charge for a student of physics, my good sir! :wink:), I just need to be clear on the physical meaning of any variables/constants that appear in them.

neopolitan

Jun7-09, 10:36 PM

I am happy to use the kinematic equation from the start. Historically, how long ago was that equation (or at least that relationship) first identified? (Part of my interest is to see how long ago we could have got to SR. If the kinematic equation is all that is required, rather than Galilean relativity, that might actually push the possible date back to before the 1100's given an early Islamic mathematician's work. "In dynamics and kinematics, Biruni was the first to realize that acceleration is connected with non-uniform motion, which is part of Newton's second law of motion." - wikipedia (http://en.wikipedia.org/wiki/Al-Biruni) Sadly, the coverage of this fellow's work is less visible to me than that for da Vinci, Galileo and Newton, but it seems to me that if Al-Biruni got so far as to consider non-uniform motion then uniform motion was probably understood. Al-Biruni's contribution to optics as claimed in the same article is interesting as well, apparently being among the first to consider the speed of light to be finite (but faster than sound) - it makes one wonder why someone like this has been pretty much invisible. There is what seems to be an inconsistency in that article, did people prior to Al-Biruni think that the speed of sound was infinite along with the speed of light? {Since Al-Biruni is credited with not only being among to consider the speed of light to be finite, but also the first to find that the speed of light is much faster than the speed of sound. I would have thought that infinitely fast is much faster than the speed of sound.} But this is merely an aside.)

Amusingly, I did try something like your:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

but with subscripts. I found it too unwieldy and distracting. So, I do see the value in it.

Anyway, since with the riders, you don't object with my derivation so far, perhaps I should try to sum up where we are right now, before we go further. If I do that, being as rigorous as I can, are you happy to work from that point onwards?

cheers,

neopolitan

JesseM

Jun7-09, 11:06 PM

I am happy to use the kinematic equation from the start. Historically, how long ago was that equation (or at least that relationship) first identified?
Not sure, I guess as soon as people came up with the notion of measuring how distances between things change with time (which would require at least somewhat fine-grained clocks), along with the concept of speed as distance traveled/time elapsed, they could have realized that the distance between a thing moving at constant speed v and a stationary thing would be shrinking at v times the time elapsed. Maybe this would come up in seafaring or something, even if it wasn't written as an algebraic equation. On the other hand, to write an equation like x - vt for a distance between a moving object and a stationary one, you need some notion of assigning objects position coordinates on a coordinate grid (or at least a coordinate line), and of choosing your origin so the moving object starts at position x=0 at time t=0, don't know if people would have thought in those terms until Descartes invented Cartesian geometry (incidentally, Galileo was about thirty years older than Descartes so I'd guess he never actually wrote the 'Galilei transformation' in algebraic terms, even if it's implicit in his work that he was saying the laws of physics would be invariant under this transformation).
Amusingly, I did try something like your:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

but with subscripts. I found it too unwieldy and distracting. So, I do see the value in it.
Yeah, it would be unwieldy to include the full description in the symbol itself, but it would be helpful if each time a new symbol is introduced, you could say something like "define x' as the position in B's frame of the event that had coordinates x,t in A's frame", something along those lines.
Anyway, since with the riders, you don't object with my derivation so far, perhaps I should try to sum up where we are right now, before we go further. If I do that, being as rigorous as I can, are you happy to work from that point onwards?
Sure, as long as being rigorous means defining the physical meaning of any new terms you introduce.

neopolitan

Jun8-09, 12:02 AM

Can we use this notation?

x'a = the separation (hence the x) between B and an Event (hence the prime) according to A (hence the a)

This perhaps should be expanded a little to specify clearly that unprimed means "between A and an Event". And capitalisation of Event is used to clarify that I am referring to a specific event, not just any event.

So you have two frames (A -> Event and B -> Event), as many perspectives as you like (a, b, c, d - but we will only use two) and we will have two dimensions (x and t).

I might also need to clarify something that I have firmly in mind about my t values.

When we draw a spacetime diagram, we can draw an interval between, say, A and an Event in A's future. But in the real world (another engineering trait coming out perhaps), A will not know about that Event until enough time has elapsed for the Event to take place (in the A frame) and for a photon to travel from the Event to A.

For that reason, I see utility in moving the origin of the ta axis to be simultaneous with the Event in the A frame. If I do that then, as a consequence, xa = c.ta.

Notionally, A and B are colocated at (0,0) - now this does not have to physically take place because we calculate intervals so any initial offset will cancel out.

Are you happy with this? Clarifying what I am asking:

1. Are you happy with the notation regime suggested?
2. Are you happy with the concept that A and B are notionally colocated at (0,0), but they don't actually have to be colocated at that time or place?

cheers,

neopolitan

JesseM

Jun8-09, 12:54 AM

Can we use this notation?
If x'a is meant to be a variable, as opposed to the separation between B and the Event at some specific time, can we write it as x'a(t)?
I might also need to clarify something that I have firmly in mind about my t values.

When we draw a spacetime diagram, we can draw an interval between, say, A and an Event in A's future. But in the real world (another engineering trait coming out perhaps), A will not know about that Event until enough time has elapsed for the Event to take place (in the A frame) and for a photon to travel from the Event to A.

For that reason, I see utility in moving the origin of the ta axis to be simultaneous with the Event in the A frame. If I do that then, as a consequence, xa = c.ta.
In other recent posts (such as 308) you used ta to refer to a specific time interval (between A and B being colocated and the photon passing A) in A's frame, so it's potentially confusing to refer to "the ta axis"--I assume you just mean the t-axis in A's frame? Whereas in the equation xa = c.ta, does ta still refer to that time interval I mentioned? And when you talk about moving the origin to be "simultaneous with the Event in the A frame", do you just mean the origin has the same time coordinate as the Event (i.e. the t-coordinate of the Event in A's frame is 0), not that the Event is actually at the origin? So it's still true that the event has coordinates x=xa, t=0 in A's frame, and it's still true that the photon reaches A at x=0, t=ta? If so I don't really understand why you talk about "moving the origin", since this is exactly how things were before.
Notionally, A and B are colocated at (0,0) - now this does not have to physically take place because we calculate intervals so any initial offset will cancel out.
Do the origins of their coordinate systems still coincide at a time coordinate of 0 in both systems? If so, of course it is not necessary for observers at rest in these coordinate systems to be located at x=0 in each system, they can be at any position coordinate we like. But in this case I'd like a redefinition of ta--does it refer to the time in A's frame that the photon passes x=0, or the time it passes A, or something else?
1. Are you happy with the notation regime suggested?
2. Are you happy with the concept that A and B are notionally colocated at (0,0), but they don't actually have to be colocated at that time or place?
See above, I'm not sure I understand what you're saying here.

neopolitan

Jun8-09, 01:31 AM

I use ta as a time interval in the A frame between an Event, and the reception of the photon coincident with that Event (or spawned by that Event).

I started writing the summary and lost it all. Very annoying.

What I mean about moving the origin of the axes is that the Event can take place whenever. But despite that, we shift the origin of the axes so that the Event is simultaneous in the A frame with (0,0).

In other words, we can work backwards. We get a photon today and discover from other reliable sources that the photon was released a distance of 10 light years away (in our frame), so we shift the origin of our t axis to back when the photon was released (in our frame) making today t = 10 years. Equally, we can be told that in three years from now, a photon will be released from the same location. We can shift the origin of our t axis forward 3 years from today, making today t = minus 3 years (knowing that the photon won't reach us until t = 10 years.

So I can shift the origin of the t-axis backwards or forwards as I like, which means I can consider any event, at any time.

If either of A and B were to not be located at the origin of their frame of reference, I would make it B.

I'm a bit perplexed by the idea that x'a wouldn't be variable with different values of ta.

To the same extent that x' in the equation x' = x - vt is variable with t, so to is x'a in the equation x'a = xa - vta variable with ta.

Also, I do believe that in the standard Lorentz Transformation x' is variable as you vary t (and indeed x).

I would agree that I would have to write x'a(ta) = xa - vta, if I routinely saw the Lorentz Transformation written as:

x'(x,t)= \gamma.(x - vt)
t'(x,t)= \gamma.(t - vx/c^2)

But I don't.

I feel as if you are demanding more than is justified. I can do as requested if I must though.

Can you confirm that I absolutely must specify that x'a varies as ta varies?

cheers,

neopolitan

neopolitan

Jun8-09, 02:06 AM

Summarising where I think we are at (including corrections in an attempt to be more rigorous).

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if a photon is released from a distance of xa away from A and it takes a period of ta to reach A, then:

xa = cta

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'a = xa - vta

or

x'a(ta) = xa - vta

which is, in words:

(separation between B and the location of the Event, in A's frame, when a photon from the Event reaches A, in A's frame) = (location of the Event, in A's frame) - (velocity of B in A's frame)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Then, we want to know how things look in B's rest frame.

x'b = ct'b and
xb = x'b + vt'b {or xb(t'b)}

which are, in words:

(location of Event, in B's frame) = (speed of light)*(time interval between the Event and reception of photon from the Event, in B's frame) and

(separation between B and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between the Event and when a photon from the Event reaches B, in B's frame)

Since

ta = time interval between the Event and when a photon from the Event reaches A, in A's frame

and

t'b = time interval between the Event and when a photon from the Event reaches B, in B's frame

we have no expectation that ta = t'b

Happy with that?

cheers,

neopolitan

JesseM

Jun8-09, 02:29 AM

I use ta as a time interval in the A frame between an Event, and the reception of the photon coincident with that Event (or spawned by that Event).
Again, are you assuming A is at position x=0 in A's own rest frame? If not, when you say "reception of the photon" do you mean when the photon crosses A's worldline, or when the photon crosses the x=0 axis?
What I mean about moving the origin of the axes is that the Event can take place whenever. But despite that, we shift the origin of the axes so that the Event is simultaneous in the A frame with (0,0).
"Take place whenever" relative to what coordinate system? If you're using a coordinate system where it takes place at t=0, then it doesn't take place whenever, and if that's the starting point of your proof then everything else in the proof follows from that assumption and whatever conclusions you reach cannot simply be assumed to still hold if the Event is located somewhere else besides t=0 (if that's what you're getting at, I'm not sure). If it helps, suppose you end up proving that if in A's frame the spatial and temporal intervals between the Event at t=0 and some second event are x and t, then the spatial interval between these same pair of events in B's frame is gamma*(x - vt) and the temporal interval is gamma*(t - vx/c^2). In that case, even though your proof started from the assumption that the Event occurred at t=0 in A's frame, it would be easy to prove a lemma of the type I talked about back in post 249 (http://www.physicsforums.com/showpost.php?p=2175999&postcount=249):
I suppose you could prove a lemma that shows that the distance and time intervals between a pair of events in a given coordinate system will be unchanged in a second coordinate system with the origin at a different location but which is at rest relative to the first (i.e. a simple coordinate transformation of the form x' = x + X0 and t' = t + T0 where X0 and T0 are constants).
So with this lemma added, you could then show that the relation between the intervals in A's frame and the intervals in B's frame will be the same even if you move the origins so that the Event is at some totally arbitrary set of coordinates. This lemma could be added to the very end of the proof. However, this will still not necessarily mean your proof is fully general; if in your proof you assume that the first Event and the second event (which together define the intervals you're dealing with in each frame) both lie along the path of a light ray, then you haven't proved that the same relation would hold for a pair of events with a timelike or spacelike separation.
In other words, we can work backwards. We get a photon today and discover from other reliable sources that the photon was released a distance of 10 light years away (in our frame), so we shift the origin of our t axis to back when the photon was released (in our frame) making today t = 10 years.
Why assume the origin was somewhere else to begin with? You don't even have to pick the placement of your axes until you've already received the photon, and at that point it's easy to position them so that the Event 10 light years away occurred at t=0, if that's all you're worried about. When dealing with SR problems you don't really have to concern yourself with these sorts of practical issues, just assume either an omniscient perspective on spacetime, or assume all coordinates are assigned indefinitely far into the future when all the events in the region of spacetime you're interested in are already known.
If either of A and B were to not be located at the origin of their frame of reference, I would make it B.
If B is not at the origin of its own frame, then does that mean B is not necessarily colocated with A at t=0 in A's frame? If it's not, then don't you have to modify the equation
I'm a bit perplexed by the idea that x'a wouldn't be variable with different values of ta.
ta is a constant in any given physical scenario, is it not? It's the time coordinate of when the photon passes A, right? So if x'a represents the distance between B and the position xa as a function of time, this distance is varying with the time coordinate t in A's frame, not varying with ta (unless you are using ta to represent A's time variable as well as the specific time the photon passes A, something I requested you not do in my last post because it'd be confusing). On the other hand, if you just want to define x'a as the distance between B and xa at the specific time ta when the photon passes A (or alternatively, at the specific time t'a when the photon passes B), that's fine with me, in this case x'a would be a constant rather than a variable. But you seemed to want x'a to represent a distance that could vary with time rather than a distance at a specific time in post 306 when you said:
Note that in A's rest frame, the distance between the location of YDE and A does not change - therefore xa does not change with time, but x'a does (because x'a is the distance between the location of YDE and B, according to A).
To the same extent that x' in the equation x' = x - vt is variable with t, so to is x'a in the equation x'a = xa - vta variable with ta.
Again, in any specific physical setup isn't ta a constant? Of course you can vary the physical setup itself, but that's not what I meant when I said I thought you were making x'a a variable--I thought that even given a particular setup (a particular choice of position xa for the Event on the photon's worldline), x'a represented the changing distance between B and xa as a function of time in A's coordinate system, not the distance between B and xa at some specific time like ta (I based this on your comment from post 306 above).
Also, I do believe that in the standard Lorentz Transformation x' is variable as you vary t (and indeed x).
If you can vary x and t, sure, but if you pick some specific physical event then x', x, and t for that choice of event are all constants, echoing my comment about ta and xa being constants for a particular choice of physical setup in your scenario.
Can you confirm that I absolutely must specify that x'a varies as ta varies?
See above for a clarification of my meaning. If you want x'a to be the distance between B and xa at a specific time corresponding to some event in your setup like the photon passing A at ta or the photon passing B at t'a (and remember that you had actually defined x'a in the latter way in our earlier discussions, not the former), then there's no need to call it a variable. But your comment in post 306 seemed to insist that x'a is defined in such a way that it changes with time rather than being the distance between B and xa at some specific time.

neopolitan

Jun8-09, 02:59 AM

Again, are you assuming A is at position x=0 in A's own rest frame?

Yes

As for the rest, I'm a bit confused why something that seems so obvious to me is confusing for you.

I'll try again.

In A's rest frame, A is at rest.

In A's rest frame, B is not at rest.

In A's rest frame, the separation between where the Event takes, took or will take place and A is constant.

In A's rest frame, the separation between where the Event takes, took or will take place and B is not constant.

xa is the separation between where the Event takes, took or will take place and A in A's rest frame.

x'a is the separation between where the Event takes, took or will take place and B in A's rest frame.

Therefore, xa is constant and x'a is not constant.

It seems so simple to me, I can't quite grasp why it warrants such a long post about it.

For a specific time in A's frame, x'a is defined, not varying, but not quite a constant either (because to me a constant is only a constant if you can vary something and once you pick a specific time, you don't have anything to vary in A's frame anymore, so long as you continue to talk about the same Event).

cheers,

neopolitan

JesseM

Jun8-09, 03:00 AM

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if a photon is released from a distance of xa away from A and it takes a period of ta to reach A, then:

xa = cta

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'a = xa - vta
You should also add the assumption that the Event occurs at time coordinate t=0 in A's frame, and that A is located at x=0, since otherwise this substitution wouldn't work.
(separation between B and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between the Event and when a photon from the Event reaches B, in B's frame)
I assume you meant to write separation between A and the location of the Event in B's frame, right? After all, the separation between B and any given location in B's frame will be constant. Assuming that's what you meant, then it seems to me the equation would only hold if we assume that A and B are colocated at the same time as the Event occurs in B's frame. But that's obviously problematic, because we already assumed the Event was simultaneous with A and B being colocated in A's frame, and as we know they can't both be true in relativity.

JesseM

Jun8-09, 03:48 AM

I'll try again.

In A's rest frame, A is at rest.

In A's rest frame, B is not at rest.

In A's rest frame, the separation between where the Event takes, took or will take place and A is constant.

In A's rest frame, the separation between where the Event takes, took or will take place and B is not constant.

xa is the separation between where the Event takes, took or will take place and A in A's rest frame.

x'a is the separation between where the Event takes, took or will take place and B in A's rest frame.

Therefore, xa is constant and x'a is not constant.
Yes, that's exactly what I originally thought you meant, until you made the confusing comment that x'a varies with ta, rather than saying it varies with t, which is something I was asking about (I know you don't like my habit of responding to your posts line-by-line, but your habit of responding to my posts in a 'gestalt' manner often means you don't answer the specific questions I ask about, and instead just repeat things I already understand without answering the questions I specifically asked for the purposes of clarifying). Again, I thought ta was a constant (given a particular physical setup) just like xa--ta represents the time the photon passes A, and for a particular physical setup there's only one unique time that this happens. After all, you wrote xa = c*ta--if you call xa a constant, then based on this equation you must call ta a constant too. It's possible you are using the symbol ta to represent both the abstract time variable in A's frame and the specific time coordinate when the photon passes A, but I already speculated in two previous posts that you might be doing this and asked you to please not use that sort of ambiguous notation if that's what's going on.

Then you also made the point that you shouldn't have to spell out that x'a varies with time when x' is not written as x'(x,t) in the Lorentz transformation. I pointed out that given a particular choice of physical event, x and t are not variables, analogous to how given a particular choice of physical setup (distance between the Event on the path of the light ray and A at time t=0 in A's frame), xa and ta are not variables in your equations. Do you understand this point, and if so do you drop this particular argument for why it's unreasonable for me to ask you to write x'a(t) in your equations?
For a specific time in A's frame, x'a is defined, not varying
Sure, for any variable with a function of t, if you pick a specific time t it doesn't vary at that time! This is totally trivial, but it hardly proves that a quantity that varies with t is not a "variable". The point is, in a given physics problem with a well-defined physical setup, I'd call a symbol a "variable" if its value can change depending on the value of some other symbol, and a "constant" its value depends only on the setup. Is it my request that you differentiate between the two in your notation really so onerous? Just to help add some context, one of the reasons I make this request is that I anticipate that later in the proof you're probably going to want to talk about the value of x'a(t) at a different time other than ta, perhaps at the time t'a in A's frame when the photon passes A; at that point it really could become genuinely confusing if you use exactly the same symbol, a problem that will be avoided if you write x'a(ta) for the first and x'a(t'a) for the second. And if I'm wrong, and your proof will never make use of x'a at any time other than ta, in that case making a big deal out of the fact that x'a is a variable seems pointless, it would be much simpler just to define x'a as the distance between B and the position of the event at the specific time ta when the photon passes A.

neopolitan

Jun8-09, 04:29 AM

Fixing (corrections and clarifications are highlighted - I accept that wherever I need to make a clarification this is as bad as being wrong, I don't intend to defend being wrong - or being so unclear as to necessitate a clarification);

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if at t=0 a photon is released from a distance of xa away from A and it takes a period of ta to reach A and A is located at x=0, then:

xa = cta

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'a = xa - vta

or

x'a(ta) = xa - vta

which is, in words:

(separation between B and the location of the Event, in A's frame, when a photon from the Event reaches A, in A's frame) = (location of the Event, in A's frame) - (velocity of B in A's frame)*(time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame)

Then, we want to know how things look in B's rest frame. If at t'=0 the photon released as described before is at a distance of x'b away from B and it takes a period of t'b to reach B and B is located at x'=0, then:

x'b = ct'b and
xb = x'b + vt'b {or xb(t'b)}

which are, in words:

(location of Event, in B's frame) = (speed of light)*(time interval between colocation of A and B and reception of photon from the Event, in B's frame) and

(separation between A and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between colocation of A and B and when a photon from the Event reaches B, in B's frame)

Since

ta = time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame

and

t'b = time interval between colocation of A and B and when a photon from the Event reaches B, in B's frame

we have no expectation that ta = t'b

I'm aware that the event simultaneous with t=0 and the event simultaneous with t'=0 are not simultaneous with each other. However, each event used is colocated with the relevant observer (notionally, if A and B are timing events, the only events they can give accurate time values to are "colocation of self with other observer" and "colocation of self with photon").

The point I have to make clear again is that we only know what happened at an event once information (or photon from the event) reach us. Then we work backwards.

If A receives a photon at ta from an event at t=0, then when did that same photon pass B? What is t'b in terms of ta and xa?

What is x'b in terms of ta and xa?

Can we work it out?

I think we can.

neopolitan

Jun8-09, 04:56 AM

Yes, that's exactly what I originally thought you meant, until you made the confusing comment that x'a varies with ta, rather than saying it varies with t, which is something I was asking about (I know you don't like my habit of responding to your posts line-by-line, but your habit of responding to my posts in a 'gestalt' manner often means you don't answer the specific questions I ask about, and instead just repeat things I already understand without answering the questions I specifically asked for the purposes of clarifying). Again, I thought ta was a constant (given a particular physical setup) just like xa--ta represents the time the photon passes A, and for a particular physical setup there's only one unique time that this happens. After all, you wrote xa = c*ta--if you call xa a constant, then based on this equation you must call ta a constant too. It's possible you are using the symbol ta to represent both the abstract time variable in A's frame and the specific time coordinate when the photon passes A, but I already speculated in two previous posts that you might be doing this and asked you to please not use that sort of ambiguous notation if that's what's going on.

Then you also made the point that you shouldn't have to spell out that x'a varies with time when x' is not written as x'(x,t) in the Lorentz transformation. I pointed out that given a particular choice of physical event, x and t are not variables, analogous to how given a particular choice of physical setup (distance between the Event on the path of the light ray and A at time t=0 in A's frame), xa and ta are not variables in your equations. Do you understand this point, and if so do you drop this particular argument for why it's unreasonable for me to ask you to write x'a(t) in your equations?

Sure, for any variable with a function of t, if you pick a specific time t it doesn't vary at that time! This is totally trivial, but it hardly proves that a quantity that varies with t is not a "variable". The point is, in a given physics problem with a well-defined physical setup, I'd call a symbol a "variable" if its value can change depending on the value of some other symbol, and a "constant" its value depends only on the setup. Is it my request that you differentiate between the two in your notation really so onerous? Just to help add some context, one of the reasons I make this request is that I anticipate that later in the proof you're probably going to want to talk about the value of x'a(t) at a different time other than ta, perhaps at the time t'a in A's frame when the photon passes A; at that point it really could become genuinely confusing if you use exactly the same symbol, a problem that will be avoided if you write x'a(ta) for the first and x'a(t'a) for the second. And if I'm wrong, and your proof will never make use of x'a at any time other than ta, in that case making a big deal out of the fact that x'a is a variable seems pointless, it would be much simpler just to define x'a as the distance between B and the position of the event at the specific time ta when the photon passes A.

I don't intend to use a value of x'a that is different from its value at ta.

I don't intend to use a value of xb that is different from its value at t'b.

I don't intend use a value of xa other than such that xa = c.ta

I don't intend use a value of x'b other than such that x'b = c.t'b.

Does that make things easier?

If I do find myself using anything other than these, I will try to mark them accordingly (but I really don't think that I will).

cheers,

neopolitan

sylas

Jun8-09, 05:19 AM

I don't intend to use a value of x'a that is different from its value at ta.

What does this even mean?

Is x'a a constant, or a variable? If you mean a constant, why the heck are you speaking of its value at certain time? If you mean a variable, why the heck would you only use a single value?

Isn't (x'a, t'a) just the co-ordinates of the event "photon passes A" in the frame of reference of B?

neopolitan

Jun8-09, 05:39 AM

Isn't (x'a, t'a) just the co-ordinates of the event "photon passes A" in the frame of reference of B?

The post you quoted was in response to a specific concern from JesseM that I would go changing the meaning of x'a. I don't intend to.

He got to that because I noted at one point that (in general) while in the A frame xa does not change, x'a does. But my (specific scenario driven) derivation centres around events which lock in values of xa, ta and hence x'a. So in the scenario I describe, xa, ta and x'a all have one value. So fixed are they that in earlier posts we assigned them numbers.

In answer to the question you posed here, no.

x'a is the separation between B and where the photon was at t = 0 in the A frame (x = separation, ' = between B and the Event, a = in the A frame or according to A).

cheers,

neopolitan

It may be worth mentioning that I am keeping the prime from the kinematic equation x' = x - vt

While I understand that this may cause concern because while I focus on observer A, and that that would make a few people consider that primes refer to B's rest frame, those people might be happier to know that the equations I end up with are:

x'b = gamma.(xa - vta)
t'b = gamma.(ta - vxa/c2)

which is a confluence of the primed is the B frame, unprimed is the A frame and b is the B frame, a is the A frame.

sylas

Jun8-09, 07:19 AM

The post you quoted was in response to a specific concern from JesseM that I would go changing the meaning of x'a. I don't intend to.

He got to that because I noted at one point that (in general) while in the A frame xa does not change, x'a does. But my (specific scenario driven) derivation centres around events which lock in values of xa, ta and hence x'a. So in the scenario I describe, xa, ta and x'a all have one value. So fixed are they that in earlier posts we assigned them numbers.

In answer to the question you posed here, no.

x'a is the separation between B and where the photon was at t = 0 in the A frame (x = separation, ' = between B and the Event, a = in the A frame or according to A).

cheers,

neopolitan

Thanks... but I am still finding this incredibly hard to follow.

You've said "x'a is the separation between B and where the photon was at t = 0 in the A frame". Distance WHEN? At t=0 also? t according to whom? You say it has one value. But then you've also said that x'a can "change" in the A frame? How can that possibly be?

I have trouble following along when you speak of a "location". What is a fixed location in one frame is not a fixed location in another. I think it would be clearer if you stick to "events", so that you can sensibly speak of one event in several different frames.

You scenario is this, isn't it? It involves three particles: A, B and photon. A and B are moving at constant velocity v relative to each other. The events of interest occur in this order.

A passes by B (co-located).
Photon passes by B.
Photon passes by A.

Is that right? You've also added another event of photon being "emitted".

The distance between events A and B (photon passing by A and photon passing by B) as observed by the particles A and B are related by the Doppler shift factor, are they not? Multiply, or divide the distance by
\sqrt{\frac{c-v}{c+v}}
to get the distance for the other observer. The distance is greater for the particle that the photon passes by first.

Cheers -- sylas

neopolitan

Jun8-09, 07:37 AM

sylas,

I do appreciate your interest, but you might notice that a lot has come before this. If I reply to you, Jesse will reply to my replies to you (it's happened before) and we will end up going over old ground again which is something I am trying to avoid.

In general, x'a(t) is variable with t. Specifically, x'a(t) is fixed with a fixed value of t=ta.

The scenario is framed such that xa(0) = c.ta, in other words a time interval of ta after t=0 (in the A frame), a photon passes A since it was initially the right distance away to cover that distance in that time.

At the time at which the photon passes A, B has travelled a distance towards where the event took place and in A's frame that is:

x'a(ta) = xa - v.ta

(the separation between B and where the event took place at the time at which the photon from the event passes A, in the A frame) = (the separation between A and where the event took place) - (the distance that B has moved towards where the event took place in the time it took for a photon to travel from the event to A)

Note, I am not specifically writing this to explain to you, I am writing it in a format that JesseM has said is necessary for it to be explained.

Now, I suspect that the method for explaining to JesseM just possibly won't be as suitable for explaining to you. If that is the case, can I suggest that you go back into the earlier posts, find an explanation which seems to suit you more, and I can try to address your questions in a separate thread?

cheers,

neopolitan

JesseM

Jun8-09, 09:54 AM

I don't intend to use a value of x'a that is different from its value at ta.

I don't intend to use a value of xb that is different from its value at t'b.

I don't intend use a value of xa other than such that xa = c.ta

I don't intend use a value of x'b other than such that x'b = c.t'b.

Does that make things easier?
In that case, why introduce the complication of saying x'a is a variable but xa is a constant? You could easily make x'a a constant just by specifically defining it as the separation at time ta when the photon passes A, not the separation at an arbitrary time t in A's frame.

Just as a reminder though, back when we were going through the numerical example where you put numbers to these values, you did define the symbol x'a in terms of t'a, the time the photon passed B (both x'a and t'a had the value 5).

sylas

Jun8-09, 10:17 AM

Now, I suspect that the method for explaining to JesseM just possibly won't be as suitable for explaining to you. If that is the case, can I suggest that you go back into the earlier posts, find an explanation which seems to suit you more, and I can try to address your questions in a separate thread?

S'okay. I am perfectly comfortable with relativity and don't need it explained to me. I can see I am not helping here, and withdraw. Sorry for the distraction!

My main aim was to suggest, gently, that you are not doing a very good job of giving clear and unambiguous definitions of what you mean. I'm glad you didn't take offense at that; I wanted to say it without coming across as being too negative. But I still find it really hard to follow what you mean with notation or use of language, and I don't think this is just me, or because it is non-standard. The problem is that it is almost always ambiguous. You evidently have a clear idea what you mean. I don't.

It should be possible to express whatever it is you mean with less words and repetition. All you need is to avoid any potential ambiguity for what notation refers to; and then pretty much any of the regulars who have struggled to follow these threads will get it, IMO. It's not a problem of finding the "right" explanation for different people.

When explanations refer to the distance to a "location", rather than an event, there's a potential ambiguity as to what the location means in different frames and times. A "location" without an associated particle usually means a fixed distance co-ordinate, or worldline with zero velocity; but that depends on the observer and I often don't know what observer is intended. Referring to a specific event, however, is nearly always crystal clear.

The main answer here is that observers A and B measure different distances from the event "photon passes B" to the event "photon passes A". If the photon passes by B and then A, and if v is their relative velocity (+ve for moving apart), then the distance between these two events dA for observer A is related to distance dB for observer B by
d_A = d_B \sqrt{\frac{c-v}{c+v}}
You can show this with the Lorentz transformations; and there may be other ways to get the right answer.

JesseM

Jun8-09, 10:28 AM

Fixing (corrections and clarifications are highlighted - I accept that wherever I need to make a clarification this is as bad as being wrong, I don't intend to defend being wrong - or being so unclear as to necessitate a clarification);
OK, the clarified version looks clear to me.
I'm aware that the event simultaneous with t=0 and the event simultaneous with t'=0 are not simultaneous with each other. However, each event used is colocated with the relevant observer
Each event you use to define the time intervals, yes. The "Event(s)" on the photon's worldline used to define xa and x'b aren't colocated with the observers, of course.
If A receives a photon at ta from an event at t=0, then when did that same photon pass B?
You mean, what time in A's frame did the photon pass B? This is a time you haven't defined a symbol for yet, although in the earlier discussion you defined this as t'a.
What is t'b in terms of ta and xa?
Note that t'b and ta/xa don't refer to intervals between the same pair of events, so if you show a certain relation between these values it won't necessarily prove anything about the intervals in different frames between a single pair of events as in the the Lorentz transformation. Also note that in the Lorentz transformation equation it's not only assumed you're talking about intervals between a single pair of events, but it's also assumed that when calculating the intervals you're being consistent about the order in which you're taking the events. For example, xa and ta could both be understood as space and time intervals between the same pair of events (photon passing A) and (Event on photon's worldline that's simultaneous with A&B being colocated) even if you didn't choose to define them in terms of this pair, so if ta was to be defined as (time coordinate of photon passing A) - (time coordinate of Event on photon's worldline that's simultaneous with A&B being colocated) in order to make it a positive number, then that means in order to be consistent we would have to define xa as (position coordinate of photon passing A) - (position coordinate of Event on photon's worldline that's simultaneous with A&B being colocated), so if you assume this Event has a positive position position coordinate that would make the interval xa negative according to the above definition. I think that in your notation you are just defining xa as the absolute value of the distance between A and the Event, so it would be positive rather than negative; in this case the physical meaning of the equation you derive relating these quantities will be quite different from the physical meaning of the Lorentz transformation relating intervals between a single pair of events calculated using a consistent order for the events.

Just to check where you're going with this, do you intend to derive an equation that has the same physical meaning as the Lorentz transformation, or do you just intend to derive an equation which looks superficially similar but whose physical meaning is different?

neopolitan

Jun8-09, 11:46 AM

OK, the clarified version looks clear to me.

Then shortly I will move onto the next stage. I will try to hold back (which means: write a reply, leave it for a while, then check it for typos then post)

I've started replying to other parts of your post but run out of time. Will try to address them later.

cheers,

neopolitan

JesseM

Jun8-09, 02:59 PM

Then shortly I will move onto the next stage. I will try to hold back (which means: write a reply, leave it for a while, then check it for typos then post)

I've started replying to other parts of your post but run out of time. Will try to address them later.
OK, but in order to avoid getting into an extended discussion of the steps of your proof only to find that the ending conclusion isn't actually equivalent to the Lorentz transformation, can you state in advance the final equation(s) you intend to derive, including the physical definitions of any symbols appearing in the final equation(s) in terms of the setup you've outlined (space and time intervals involving a photon traveling towards A and B which passes them at particular points along with some Event or Events on its worldline, presumably)?

neopolitan

Jun8-09, 08:21 PM

OK, but in order to avoid getting into an extended discussion of the steps of your proof only to find that the ending conclusion isn't actually equivalent to the Lorentz transformation, can you state in advance the final equation(s) you intend to derive, including the physical definitions of any symbols appearing in the final equation(s) in terms of the setup you've outlined (space and time intervals involving a photon traveling towards A and B which passes them at particular points along with some Event or Events on its worldline, presumably)?

Alright, I'll put some thought into that as well.

cheers,

neopolitan

Rasalhague

Jun9-09, 12:56 AM

I haven't caught up with all the most recent posts yet, but I hope no one minds if I butt in with some general observations.

I just googled the expression "moving clocks run slow" (3190 hits), then tried "moving clocks run fast" (128 hits), several of the latter apparently referring to unorthodox theories and rejections of relativity, although not all of them. Presumably authors who present the slogan "moving clocks run slow" as a verbal equivalent to the time dilation equation are taking the word dilation to mean a bigger number when it refers to time, but a smaller number when it refers to space! On the other hand, Taylor and Wheeler in Spacetime Physics explicitly define dilation as a bigger number, and I think this is the more standard idea.

So if some (most?) of the authors who use the expression "moving clocks run slow" are, in fact, referring to the temporal analogue of length contraction, it seems strange that they avoid the obvious term "time contraction" and its formula. (Unfortunately Google results for "time contraction" are mixed up with pages about timing the contractions of labour, so I can't make a fair comparison.) The only book I've yet found to mention "time contraction" is What Does a Martian Look Like by Jack Cohen and Ian Stewart, about the possible forms that extraterrestrial life might take, rather than physics as such.

This preference for the expression "moving clocks run slow" is presumably as much a matter of convention as the preference for the expression "time dilation" and the pairing of the time dilation formula with that of length contraction. Its popularily suggests that it may be no less natural a way of conceptualising the relationship. All of this strongly inclines me to agree with Neopolitan's comments in the very first post of this thread: that the pairing of time dilation with length contraction is a source of needless confusion. Thanks to Jesse's diagram and explanation of some reasons why contraction is a more natural way to view what happens to the length of a physical object when viewed as moving as opposed to standing still, it seems to me that it would indeed be more logical to pair time contraction and length contraction (i.e. use the same terminology for time as we use for space), for the sake of comparing like with like, and of avoiding the false impression of asymmetry which the traditional pairing creates.

That said, there are genuine asymmetries between time and space, or between the ways we relate to them, which might lead us to view length contraction as more natural while either transformation (dilation or contraction) seems equally natural for time. Here is a list I came up with. It could be that some of the items are essentially the same as others, stated in different words. The fourth point about determinism is based on what Jesse said in an earlier post.

1. Persistence. Clocks and rulers are both objects with sharply defined spatial limits; they both persist in time. To make a thought experiment more symmetrical, we could imagine everlasting clocks, each static at the origin of their respective rest frames, and infinite rulers (each existing for one moment only, as defined in its own rest frame). While this may be convenient for the thought experiment, its unphysicality could point to a real difference in how we relate to time and space.

2. Degrees of freedom. Objects are free to move in either direction along a line through space, but their motion in time is unidirectional.

3. Speed. Speed is defined as length divided by time, regardless of which component we're calculating a change in.

4. Determinism. Physical laws predict events in limited space over unlimited time (with certain limits on accuracy); they have less power to predict events in limited time over unlimited space. We're used to ideas like “what goes up must come down”, but it's harder to imagine a universe where what goes up here would be a reliable guide to what goes up simultaneously somewhere else.

Rasalhague

Jun9-09, 01:09 AM

Some thoughts on terminology and notation.

In my notes, I've taken to using the term input frame (or source frame) for the frame for which we know the coordinate values, and output frame (or target frame) for the frame for which we want to calculate the coordinate values. I've been using the terms left frame and right frame, respectively, for the frame moving left (i.e. in the negative x direction) past the other, and the frame moving right (i.e. in the positive x direction) past the other. The initials L and R stand for left and right. For example, "Clock L is at rest in frame L, the left frame, so frame L is clock L's rest frame. Clock L is moving in frame R, so frame R is a moving frame for clock L."

The terms input and output depend on the question asked, and may change their referents (change the frames they refer to) if a different question is asked. The terms left and right depend on the definition of the frames, and keep the same referents so long as the same frames are being used. The terms rest and moving are defined relative to a particular object; they may change their referents if a different object is discussed. Contraction and dilation questions can be asked of both time and space with any of these terms.

Some authors use primed and unprimed for what I'm calling input and output, but others use primed and unprimed for left and right. Others again use primed to refer to whichever frame is defined as moving in a particular example. These definitions don't necessarily coincide!

Some authors do as Wolfram Alpha does and use a subscript zero for the coordinates of the input frame if your question involves time dilation or length contraction, and for the output frame if your question involves time contraction or length dilation. Such coordinates change which frames they're referring to whenever you go from asking a dilation to a contraction question of the same coordinate, or from asking a contraction to a dilation question of the same coordinate. Wolfram Alpha calls dilated time "moving time", and contracted length "moving length".

http://www34.wolframalpha.com/input/?i=time+dilation
http://www34.wolframalpha.com/input/?i=length+contraction

To me this feels like an arbitrary switch in terminology purely for the sake of maintaining this artificial, traditional association of dilation exclusively with time, and contraction exclusively with length.

JesseM

Jun9-09, 01:25 AM

I just googled the expression "moving clocks run slow" (3190 hits), then tried "moving clocks run fast" (128 hits), several of the latter apparently referring to unorthodox theories and rejections of relativity, although not all of them. Presumably authors who present the slogan "moving clocks run slow" as a verbal equivalent to the time dilation equation are taking the word dilation to mean a bigger number when it refers to time, but a smaller number when it refers to space! On the other hand, Taylor and Wheeler in Spacetime Physics explicitly define dilation as a bigger number, and I think this is the more standard idea.
There is no contradiction between the phrase "time dilation" and the phrase "moving clocks run slow". The "dilation" in question is not of the clock's rate of ticking, but of the period between a given pair of readings. For example, if a clock ticks forward by 10 seconds between two events on its worldline, but the time interval between these two events is 30 seconds in my frame, then that period of 10 seconds between the events as measured by the clock itself has been "dilated" by a factor of 3 from my perspective. But at the same time, if it takes 30 seconds of my time for the clock to tick forward by 10 seconds, obviously I can also say this clock is "running slow" from my perspective.
So if some (most?) of the authors who use the expression "moving clocks run slow" are, in fact, referring to the temporal analogue of length contraction
No, I'm sure that not a single one of them is referring to this, "the temporal analogue of length contraction" is a fairly arcane idea I brought up for the sake of my discussion with neopolitan that would probably never be used in practice. The idea (illustrated in the diagram I drew that neopolitan posted in post #5) is that if you have two events on a clock's worldline that are separated by a time t (say 10 seconds again) according to the clock's own readings, and then you draw surfaces of simultaneity (surfaces of constant t) in the clock's own rest frame that pass through these two events, and then consider how those surfaces would look in the frame of an observer who sees the clock in motion (where the surfaces will be 'slanted'), and work out the time between these surfaces along the vertical time axis of this observer's frame, it will be less than the time of 10 seconds, even though the time in this frame between those two events on the first clock's worldline (which is what the regular time dilation equation gives you) is greater than 10 seconds. This is analogous to length contraction where you look at two lines of constant x in an object's rest frame that represent the worldlines of the object's endpoints, then switch to a different frame where the object is moving so these same lines are slanted, and consider the distance between these slanted lines along the horizontal space axis of this frame, which is the "length" of the object in this frame.

If this is hard to follow even after looking at the diagram, it's not really worth worrying about, since like I said the "temporal analogue of length contraction" is just an artificial concept I came up with for the purposes of showing that you could imagine something analogous to length contraction, it's defined in such a weird way that it's not a concept that anyone would actually be likely to find useful for any other purpose besides illustrating that such an analogous notion is possible.

Rasalhague

Jun9-09, 01:48 AM

Some notes I made to get my head around the symmetry which the traditional pairing of time dilation and length contraction disguises. All criticism welcome!

Assume a (-1, 1)-dimensional Minkowski spacetime described by two reference frames moving relative to one another with speed u.

Time. Let clocks be synchronised at the intersection of the origins of the two frames so that the coordinates of this coincidence are t_{L} = t_{R} = 0 and x_{L} = x_{R} = 0. The clocks last for all time. Clock L is confined in space to the location (line of collocality/syntopy) x_{L} = 0, clock R to x_{R} = 0.

Length. Let two rulers have their zero ends lined up at the intersection of the origins of the two frames so that the coordinates of this coincidence are t_{L} = t_{R} = 0 and x_{L} = x_{R} = 0. The rulers extend through all space. Ruler L exists only at the instant (line of contemporality/synchrony) t_{L} = 0, ruler R at t_{R} = 0.

We can ask contraction questions of time or of length. We can ask dilation questions of time or of length. The contraction questions we ask of time are formally the same as those we ask of length (except for the difference in coordinate). The dilation questions we ask of time are formally the same as those we ask of length (except for the difference in coordinate).

1. CONTRACTION

\frac{1}{\gamma} = \sqrt[]{1-\left(\frac{u}{c}\right)^{2}} = \frac{1}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}

1.1. Time contraction.

1.1.1. At a moment defined in frame R, the frame where clock L is moving, clock L shows this fraction of the time shown by clock R.

1.1.2. At a moment defined in frame L, the frame where clock L is still, clock R shows this fraction of the time shown by clock L.

1.2. Length contraction.

1.2.1. At a location defined in frame R, the frame where ruler L is moving, ruler L shows this fraction of the length shown by ruler R.

1.2.2. At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.

2. DILATION

\gamma = \frac{1}{\sqrt[]{1-\left(\frac{u}{c}\right)^{2}}} = cosh\left(artanh\left(\frac{u}{c} \right) \right)

2.1. Time dilation.

2.1.1. At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.

2.1.2. At a moment defined in frame L, the frame where clock L is still, clock L shows this multiple of the time shown by clock R.

2.2. Length dilation.

2.2.1. At a location defined in frame R, the frame where ruler L is moving, ruler R shows this multiple of the length shown ruler L.

2.2.2. At a location defined in frame L, the frame where ruler L is still, ruler L shows this multiple of the length shown by ruler R.

JesseM

Jun9-09, 02:05 AM

1. CONTRACTION

\frac{1}{\gamma} = \sqrt[]{1-\left(\frac{u}{c}\right)^{2}} = \frac{1}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}

1.1. Time contraction.

1.1.1. At a moment defined in frame R, the frame where clock L is moving, clock L shows this fraction of the time shown by clock R.
But that's just what is usually called "time dilation"--if some quantity is greater in the frame of the observer who sees the instrument moving than it is when measured by the instrument itself, that's caused dilation; if some quantity is smaller in the observer's frame, that's called contraction. In this case, the quantity is the time between two events on the clock's worldline; the time between these events will be greater in the observer's frame than as measured by the clock itself, so this is time dilation. It seems unnecessarily confusing to change this convention and to say that it's "contraction" if the quantity measured by the instrument is smaller than the corresponding quantity measured in the observer's frame.

Rasalhague

Jun9-09, 02:56 AM

But that's just what is usually called "time dilation"--if some quantity is greater in the frame of the observer who sees the instrument moving than it is when measured by the instrument itself, that's caused [called] dilation;

Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction! Wouldn't it be less confusing to call the same thing contraction for both dimensions?

if some quantity is smaller in the observer's frame, that's called contraction. In this case, the quantity is the time between two events on the clock's worldline; the time between these events will be greater in the observer's frame than as measured by the clock itself, so this is time dilation. It seems unnecessarily confusing to change this convention and to say that it's "contraction" if the quantity measured by the instrument is smaller than the corresponding quantity measured in the observer's frame.

But when we talk about rulers, the terminology is traditionally reversed. There too the distance between the corresponding pair of events is greater in the observer's frame, so why do we not call that dilation?

JesseM

Jun9-09, 03:20 AM

Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction!
How do you figure it's the "same phenomenon?" In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.
But when we talk about rulers, the terminology is traditionally reversed. There too the distance between the corresponding pair of events is greater in the observer's frame, so why do we not call that dilation?
In the case of length you aren't measuring distance between a single pair of events, you're measuring the distance between the endpoints of the ruler at a single moment in time of whatever frame you're using. The distance between the endpoints of the ruler at a single moment of time in the observer's frame is smaller than the distance between the endpoints of the ruler at a single moment of time in the ruler's own rest frame.

If you want to talk about the distance between a single pair of events in two frames, you're right that the distance is larger in the observer's frame where they're non-simultaneous than it is in the frame where the events were simultaneous (this is what I called the 'spatial analogue for time dilation'). But this is not the same thing as measuring the length of an object in two different frames, since "length" always means the distance between the endpoints at a single moment in time.

Rasalhague

Jun9-09, 04:04 AM

There is no contradiction between the phrase "time dilation" and the phrase "moving clocks run slow". The "dilation" in question is not of the clock's rate of ticking, but of the period between a given pair of readings.

Yes, this is the way Taylor and Wheeler define it: a dilation (lengthening) of the period. And yet, I'm not sure everyone understands it this way. We also find statements like this on Wikipedia: "Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock. This is often interpreted as time "slowing down" for the other clock, [...]" But from any perspective in which a clock is ticking slower, it will show a shorter (contracted) period as having elapsed. So presumably the writer of this article, rightly or wrongly, took dilation to refer to something other than the period. Perhaps they conceptualised it as a dilation of the units. Either way, it seems arbitrary to switch terminology when talking about space.

For example, if a clock ticks forward by 10 seconds between two events on its worldline, but the time interval between these two events is 30 seconds in my frame, then that period of 10 seconds between the events as measured by the clock itself has been "dilated" by a factor of 3 from my perspective. But at the same time, if it takes 30 seconds of my time for the clock to tick forward by 10 seconds, obviously I can also say this clock is "running slow" from my perspective.

Equally, you could say of the clock that's running slow: this clock has measured a shorter period; the period has been contracted, behold "time contraction". What else would you call it if a lengthening of the period is a dilation. Since the relationship between the events involved in this measurement is exactly analogous to the relationship between the events involved in calculating "length contraction", why not present these equations as parallel, showing the same symmetry between time and space that the full Lorentz transformation does?

If this is hard to follow even after looking at the diagram, it's not really worth worrying about, since like I said the "temporal analogue of length contraction" is just an artificial concept I came up with for the purposes of showing that you could imagine something analogous to length contraction, it's defined in such a weird way that it's not a concept that anyone would actually be likely to find useful for any other purpose besides illustrating that such an analogous notion is possible.

At this stage, it seems to me no more articial or arcane than "time dilation". It's just the reverse calculation. Wolfram Alpha calls it a transformation from "moving time" to "stationary time" ( http://www34.wolframalpha.com/input/?i=time+dilation ). What seems artificial and potentially confusing to me is their use of different definitions of "moving" and "stationary" when calculating distance as opposed to time.

Rasalhague

Jun9-09, 05:28 AM

How do you figure it's the "same phenomenon?" In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.

Here I think is where the asymmetry creeps in. In the case of time dilation, you define the moment (the end of the period to be measured) in the output frame.

(1) At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.

In the case of length contraction, you define the location (the end of the distance to be measured) in the input frame.

(2) At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.

So you're asking different questions and getting different answers. But this is only a matter convention. We could just as well ask the latter question of time:

(3) At a moment defined in frame L, the frame where clock L is still, clock R shows this fraction of the time shown by clock L.

Pairing question (3) with question (2) shows what happens when we ask the same kind of question of time as we ask of space. We get the same kind of answer. The spatial interval is less, the temporal interval is less. This seems to me a more intuitive way to compare time and space.

In the case of length you aren't measuring distance between a single pair of events, you're measuring the distance between the endpoints of the ruler at a single moment in time of whatever frame you're using. The distance between the endpoints of the ruler at a single moment of time in the observer's frame is smaller than the distance between the endpoints of the ruler at a single moment of time in the ruler's own rest frame.

Exactly, and this is the source of the "inconsistency" (if such it be), the fact that "in the case of length (contraction)" a different operation is being carried out with respect to length from the operation being carried out in time dilation with respect to time. Or rather, the inconsistency is to present these different operations are somehow equivalent or parallel to each other.

We can look at the time calculation and the space calculation as each involving three events.

In the case of the clocks, supposing them to be arbitrarily longlasting and arbitrarily small, thus each describing a line through spacetime: (A) a coincidence at the origin (of the two clocks meeting and being set to zero), (B) the event of the clock collocal with the origin in the input frame showing a certain value, (C) the event of the clock collocal in the output frame with the origin in the output frame showing a certain value.

In the case of the rulers, supposing them to be arbitrarily long and arbitrarily shortlived, thus each describing a line through spacetime: (A) a coincidence at the origin (of the zero end of the two rulers meeting), (B) the event of the ruler contemporaneous with the origin in the input frame showing a certain value, (C) the event of the ruler contemporaneous with the origin in the output frame showing a certain value.

Which events we choose to label (B) and (C) in each case depends on what information we have and what we want to calculate. Obviously it's possible to ask whatever questions we need to get any of the available answers. As far as I can see, calling the reverse questions from the traditional ones "time contraction" and "length dilation" would be less ambiguous than, for example, Wolfram Alpha's "stationary time" and "stationary length". After all, to compare times, there have to be two notional clocks (one stationary and one moving in each frame). And when we compare lengths, we're comparing two rulers (one stationary and one moving in each frame). In the case of length, because real objects persist in time, we think naturally of an object having a certain length when stationary being contracted when seen as moving. So why not use the same convention for time, and many authors do verbally, and use the appropriate formula to match the common statement that "a moving clock ticks slow" (and therefore shows a shorter time period), just as we think of a moving object as having a shorter length?

JesseM

Jun9-09, 11:07 AM

Yes, this is the way Taylor and Wheeler define it: a dilation (lengthening) of the period. And yet, I'm not sure everyone understands it this way. We also find statements like this on Wikipedia: "Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock. This is often interpreted as time "slowing down" for the other clock, [...]" But from any perspective in which a clock is ticking slower, it will show a shorter (contracted) period as having elapsed.
Huh? No it won't, if a clock is ticking at a slower rate it shows a longer period. For example, if a clock is slowed down by a factor of 3 in my frame, it will take a period of 30 seconds of time in my frame to tick forward by 10 seconds.
Equally, you could say of the clock that's running slow: this clock has measured a shorter period; the period has been contracted, behold "time contraction".
But then you'd be adopting the convention that dilation/contraction refers to whether a quantity measured in the instrument's own frame is greater or smaller than the corresponding quantity measured in the observer's frame. This is simply not the convention that has been adopted, dilation/contraction always refers to whether the quantity in the observer's frame is greater or smaller.
Since the relationship between the events involved in this measurement is exactly analogous to the relationship between the events involved in calculating "length contraction"
No, it isn't. Length contraction follows the normal convention that you're talking about the value in the observer's frame.
At this stage, it seems to me no more articial or arcane than "time dilation". It's just the reverse calculation.
No, it certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:

(time in observer's frame) = (time in clock's frame)/gamma

(note that this equation only works if 'time in observer's frame' refers to something other than the time between two events on the clock's worldline, such as the time in the observer's frame between planes of simultaneity from the clock's frame which I talked about earlier)

Whereas the normal time dilation equation is:

(time in observer's frame) = (time in clock's frame)*gamma

You can take the normal time dilation equation and divide both sides by gamma, but this doesn't give you the TAFLC above, instead it gives you what I called the "inverse time dilation equation":

(time in clock's frame) = (time in observer's frame)/gamma

See the difference?
What seems artificial and potentially confusing to me is their use of different definitions of "moving" and "stationary" when calculating distance as opposed to time.
It's potentially confusing if you don't make clear whether you're talking about the distance between a set pair of events in two frames or about the length of a physical object in two frames. The latter is what length contraction is dealing with, and in this case "moving" and "stationary" has an obvious meaning, it just refers to the object whose length is being measured in two frames.

JesseM

Jun9-09, 11:46 AM

Here I think is where the asymmetry creeps in. In the case of time dilation, you define the moment (the end of the period to be measured) in the output frame.

(1) At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.
Time dilation deals with the time interval between a pair of events on the clock's worldline, not just the reading at a single moment. So you should say:

(1) For a given pair of events on the clock L's worldline, frame R measures this multiple of the time interval measured between the events by clock L.
In the case of length contraction, you define the location (the end of the distance to be measured) in the input frame.

(2) At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.
Just as you weren't comparing readings at a single time in time dilation, "length" does not refer to position coordinates at a single location, it refers to the distance between the endpoints of the thing being measured at a single moment of time in whatever frame you use. Clocks "naturally" measure the time between events that take place on their own worldline, and rulers "naturally" measure the distance between points along the ruler, like the distance between their own endpoints. So in both cases we are starting from something that is naturally measured in the instrument's own frame, and figuring out whether the corresponding quantity in the observer's frame is smaller or larger, and if it's smaller we say "contraction" and if it's larger we say "dilation".
We can look at the time calculation and the space calculation as each involving three events.
Time dilation involves only two events which occur on the clock's worldline. Length contraction can be thought of two involve three events if you like; just pick one event A on the worldline of the ruler's left end, then the distance in the ruler's rest frame between this event and the event B on the worldline of the right side that is simultaneous with event A in the ruler's rest frame[i] can be understood as the "rest length" of the ruler, while the distance in the observer's frame between event A and the event C on the worldline of the right side [i]that is simultaneous with A in the observer's frame can be understood as the "length" of the ruler in the observer's frame.
In the case of the clocks, supposing them to be arbitrarily longlasting and arbitrarily small, thus each describing a line through spacetime: (A) a coincidence at the origin (of the two clocks meeting and being set to zero)
The time dilation equation is not restricted to dealing with cases where one of the events is at the origin.
(B) the event of the clock collocal with the origin in the input frame showing a certain value, (C) the event of the clock collocal in the output frame with the origin in the output frame showing a certain value.
If the first event occurs at the origin, why would you be interested in an event colocal with the origin in the observer's frame? You're only interested in the difference in time coordinates between two events on the clock's worldline, so naturally since the clock is moving in the observer's frame, if the first event occurred at the origin then the second event did not.
In the case of the rulers, supposing them to be arbitrarily long and arbitrarily shortlived, thus each describing a line through spacetime: (A) a coincidence at the origin (of the zero end of the two rulers meeting), (B) the event of the ruler contemporaneous with the origin in the input frame showing a certain value, (C) the event of the ruler contemporaneous with the origin in the output frame showing a certain value.
If you are measuring the "length" of one particular ruler in its own frame and in the observer's frame, then if one event occurs on the left end of the ruler at the origin of both frames, B must be an event on the worldline of the right end of this ruler that's simultaneous with the first event in the ruler's own frame, and C must be an event on th worldline of the right end of the same ruler that's simultaneous with the first event in the observer's frame.
As far as I can see, calling the reverse questions from the traditional ones
What equations are you talking about? Can you write them out in words as I did in my previous post, and do you understand the distinction I made there between "the temporal analogue for length contraction" and the "inverse time dilation equation"?
"time contraction" and "length dilation" would be less ambiguous than, for example, Wolfram Alpha's "stationary time" and "stationary length".
"Stationary time" and "stationary length" don't refer to a comparison between two frames, they just refer to the time between events on a clock's own worldline as measured by that clock (i.e. as measured in the clock's rest frame) or the length of ruler as measured by the ruler itself (as measured in the ruler's rest frame). Since you're only talking about the value in a single frame, it would make little sense to talk about "dilation" or "contraction".
After all, to compare times, there have to be two notional clocks (one stationary and one moving in each frame).
You can just talk about the time coordinates of events in the observer's frame without worrying about how he assigns them. But if you do want to think about that, then you really need two synchronized clocks at rest in the observer's frame, since in SR a clock only assigns time coordinates to events that occur on its own worldline, and the events in question are ones that occur on the worldline of the moving clock so they'll happen at different positions in the observer's frame.
And when we compare lengths, we're comparing two rulers (one stationary and one moving in each frame).
Again, easier to just talk about coordinates in the observer's frame. But if you do want details of how the observer assigns coordinates, then when measuring length you really need to bring in clocks too, so that the observer can make sure he's measuring the position of the ends of the moving ruler relative to his own ruler at a single moment in time in his own frame.
In the case of length, because real objects persist in time, we think naturally of an object having a certain length when stationary being contracted when seen as moving. So why not use the same convention for time, and many authors do verbally, and use the appropriate formula to match the common statement that "a moving clock ticks slow" (and therefore shows a shorter time period), just as we think of a moving object as having a shorter length?
I don't understand what you mean here. Are you talking about a formula which explicitly deals with rate of ticking rather than with periods of time between a pair of events? In that case you might write:

(clock's rate of ticking relative to time coordinate in observer's frame) = (clock's rate of ticking relative to time coordinate in clock's frame) / gamma

You can see that this is physically distinct from both the "temporal analogue for length contraction" and the "inverse time dilation" equations I wrote in my previous post, in spite of the fact that they all involve dividing by gamma on the right-hand side.

Rasalhague

Jun9-09, 01:33 PM

Huh? No it won't, if a clock is ticking at a slower rate it shows a longer period. For example, if a clock is slowed down by a factor of 3 in my frame, it will take a period of 30 seconds of time in my frame to tick forward by 10 seconds.

We seem to be talking at cross-purposes here somehow. In everyday language, if two clocks are synchronised at 00:00 and one runs slower by a factor of three, the slow clock will show 10 when the faster one shows 30. The slower clock shows the smaller number, i.e. a shorter period than the faster clock.

Likewise in your example, the clock that's slow shows 10. It's ticking at a slower rate in the sense that only ten ticks have occured by the time a clock at rest in the other frame has ticked 30 times (with the end of the period defined in the rest frame of the clock that ticked 30 times). And 10 < 30, so the clock that shows 10 is showing a shorter period, right? Smaller number = contraction (shrinking). Bigger number = dilation (stretching). In this sense, the moving clock runs slow (as the motto goes), and shows a contracted period of time (time contraction). In the same sense, a moving ruler can be said to show a contracted interval of space (length contraction).

Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is \Delta t' = 1\,meter. Show that the time lapse observed in the laboratory frame is given by the expression \Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).

Or do you take dilation in some other sense, for example (if not an increase in the rate) a stretching of the size of each unit?

But then you'd be adopting the convention that dilation/contraction refers to whether a quantity measured in the instrument's own frame is greater or smaller than the corresponding quantity measured in the observer's frame. This is simply not the convention that has been adopted, dilation/contraction always refers to whether the quantity in the observer's frame is greater or smaller.

Perhaps the source of the contradiction here is "corresponding quantity". It's a different quantity that's treated as "corresponding" in the case of time compared to space. In the case of time dilation, the convention is that dilation refers to the greater time (bigger quantity) shown by the clock at rest in the output frame, compared to the clock at rest in the input frame, at an instant defined in the output frame (here's my clock ticking a second: which event on your clock, collocal for you with the origin, do you consider simultaneous to this event on mine). In the case of length contraction, however, the convention is that contraction refers to the shorter length (smaller quantity) shown by the ruler at rest in the output frame, compared to the ruler at rest in the input frame, at a position defined in the input frame (here's my ruler reading a meter: which event on your ruler, instantaneous for you with the origin, do I consider level with it). Only if the position in the length equation and the instant in the timeequation were both defined in the output frame, or both defined in the input frame, would we be comparing like with like. And when that's done, the asymmetry vanishes.

It isn't enough to say whether the quantity is greater or smaller in "the observer's frame" (output frame); there will be some quantity greater and smaller in each. We need to define which frame's "now" or "here" we're using, and when we do, we see that it's a different definition being used for time compared to space.

No, it isn't. Length contraction follows the normal convention that you're talking about the value in the observer's frame.

We need to know which "value in the observer's frame" we're talking about. When you say "the observer's frame", do you mean what I defined in post #355 as the output frame?

No, it certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:

(time in observer's frame) = (time in clock's frame)/gamma

Whereas the normal time dilation equation is:

(time in observer's frame) = (time in clock's frame)*gamma

You can take the normal time dilation equation and divide both sides by gamma, but this doesn't give you the TAFLC above, instead it gives you what I called the "inverse time dilation equation":

(time in clock's frame) = (time in observer's frame)/gamma

See the difference?

Not yet. Since there is no formal difference - exacly the same equation is used - and since, by the principle of relativity, there can be no asymmetry between the two inertial frames, except that they're moving in opposite directions, whatever the difference is, I'm guessing it must be a subjective difference: something about how the frames are conceived? You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify? Presumably the "observer" also has a clock to compare with the clock at rest in the other frame. In a more general sense, each clock is a kind of observer, observing/recording the passage of time, regardless of whether it's consciously observed.

The way I'm looking at it is in terms of frames identical in every way possible so as not to introduce the impression that one is favoured in some way, and thereby risk introducing some false asymmetry into the example. So I'm imagining identical clocks and rulers in each frame, and (unphysically) conceiving of the rulers as somehow only existing for one moment (simultaneous with the origin in their respective rest frames), so as to make them more exactly correspond to the clocks which are restricted in space to the location of the origin in their respective rest frames.

In #355 I tried to define a few terms that could be used to distinguish between frames that would make explicit what it was about the frames that marked them out.

input : output (synonymously: source : target)
left : right

It's potentially confusing if you don't make clear whether you're talking about the distance between a set pair of events in two frames or about the length of a physical object in two frames. The latter is what length contraction is dealing with, and in this case "moving" and "stationary" has an obvious meaning, it just refers to the object whose length is being measured in two frames.

This is a source of asymmetry: physical objects persist in time, both clocks and rulers. We intuitively think of the stationary condition of an object as more fundamental. For that reason, it seems more natural to talk of length contraction than length dilation. But that doesn't preclude asking the same question of time, as people do when they say "a moving clock runs slow".

Doc Al

Jun9-09, 01:57 PM

Rasalhague

Jun9-09, 03:11 PM

In going from A to B, a moving clock measures 10 seconds. According to laboratory clocks, 30 seconds have passed. 30 > 10, thus time dilation.

A moving stick is 3 meters long in its own frame. According to laboratory measurements, it is 1 meter long. 1 < 3, thus length contraction.

What's the problem?

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples? Could we not just as well say 1 < 3, thus time contraction? That sounds simpler to me. Surely there isn't some fundamental property of time that it always has to be "greater than" ;-)

JesseM

Jun9-09, 03:32 PM

We seem to be talking at cross-purposes here somehow. In everyday language, if two clocks are synchronised at 00:00 and one runs slower by a factor of three, the slow clock will show 10 when the faster one shows 30. The slower clock shows the smaller number, i.e. a shorter period than the faster clock.
As I keep saying, the convention is that contraction/dilation is defined in terms of the measurement in the observer's frame. Do you disagree that in this case the period in the observer's frame is 30, and that 30 is a longer period than 10?
Likewise in your example, the clock that's slow shows 10. It's ticking at a slower rate in the sense that only ten ticks have occured by the time a clock at rest in the other frame has ticked 30 times (with the end of the period defined in the rest frame of the clock that ticked 30 times). And 10 < 30, so the clock that shows 10 is showing a shorter period, right? Smaller number = contraction (shrinking). Bigger number = dilation (stretching).
Both 10 and 30s are "numbers", so if 10 < 30 then you can also say 30 > 10, and in your own words, "Bigger number = dilation". In order to avoid confusingly referring to every difference as both a contraction and a dilation, we need to pick a convention about which frame's number to use (so that if that frame's number is smaller than the other frame's number we call it 'contraction' and if it's bigger we call it 'dilation'). The convention is to pick the measurement in the observer's frame, not the frame of the instrument which is used to define the "proper" quantity (proper time or proper length).
Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is \Delta t' = 1\,meter. Show that the time lapse observed in the laboratory frame is given by the expression \Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).
Yes, and note that they are using exactly the convention I described--since the time lapse between the two events in the observer's frame is more than the proper time measured by the moving clock, they call this time dilation.
Or do you take dilation in some other sense, for example (if not an increase in the rate) a stretching of the size of each unit?
Time dilation is not defined in terms of rates (the ratio of clock time to coordinate time), it's defined in terms of time intervals. The interval between two events on the moving clock's worldline is larger in the observer's frame than as measured by the clock, so that's why they call it "dilation".
Perhaps the source of the contradiction here is "corresponding quantity". It's a different quantity that's treated as "corresponding" in the case of time compared to space. In the case of time dilation, the convention is that dilation refers to the greater time (bigger quantity) shown by the clock at rest in the output frame, compared to the clock at rest in the input frame, at an instant defined in the output frame
As I said in the first section of my last post, time dilation does not refer to readings at a particular instant, but to intervals between a pair of events. For example, there might be a clock moving at 0.6c in my frame which reads 12 seconds at an event on its worldline that I assign a time coordinate t'=80 seconds, and then a little later the clock reads 20 seconds at an event on its worldline that I assign a time coordinate t'=90 seconds. Now take a look at the time dilation equation, which should really be written like so:

delta-t' = delta-t * gamma

If I try to plug in 80 and 12 it doesn't work, and it also doesn't work if I plug in 90 and 20. But if I plug in delta-t'=90-80=10 and delta-t=20-12=8, then with gamma=1.25 it does work.

Of course, if you make the assumption that the first event corresponds to the moving clock reading 0, and that the moving clock was synchronized so that it read 0 at t'=0 in the observer's frame, then the intervals will just be equal to the time-coordinates in each frame of the second event on the clock's worldline, so this is probably what you were doing implicitly. Still it's important to understand that the time dilation equation is fundamentally about intervals.
In the case of length contraction, however, the convention is that contraction refers to the shorter length (smaller quantity) shown by the ruler at rest in the output frame, compared to the ruler at rest in the input frame, at a position defined in the input frame (here's my ruler reading a meter: which event on your ruler, instantaneous for you with the origin, do I consider level with it).
In much the same way as time dilation doesn't deal with the times of individual events but with time-intervals between a single pair of events, length contraction doesn't deal with the positions of individual events but with the distance between the two endpoints of a ruler (though of course this is still not quite analogous to time dilation because we aren't talking about the distances between a single pair of events in both frames). In time dilation we could replace intervals with coordinates of a single event only in the very specific case where the first event was assigned a time coordinate of 0 in both frames; with length the only way to replace lengths with position coordinates of a single event is to have it so that the back end of the ruler reaches the origin of the observer's (output) frame at t'=0 in the observer's frame, and then let the event E in question be the event on the front end of the ruler that also occurs at t'=0 in the observer's frame. The position coordinate of this event E in the observer's frame will of course be equal to the length of the ruler in the observer's frame (since length involves simultaneous measurements of either end of an object in whatever frame you're using), and it works out that the position coordinate of event E in the ruler's own rest frame is also equal to its length in its own frame. The reason this works is that the Lorentz transformation tells us that x'=0, t'=0 in the observer's frame coincides with x=0, t=0 in the ruler's frame, and we set things up so that in the observer's frame the left end of the ruler would be at x'=0 (the spatial origin) at t'=0 in the observer's frame, so the left end of the ruler must also be at position x=0 at t=0 in its own frame, and since the ruler is at rest in its own frame this means the left end is at x=0 at all times in its frame, including the time of event E (which does not occur at t=0 in this frame).

With all this said, I'm confused by your above quote, especially the meaning of "at a position defined in the input frame". Position of what, exactly? I think it would be easier if we defined length contraction in terms of the distance between endpoints of the object rather than the position of some single event, but if you want to define it in terms of a single event you have to do it the way I described above, which I'm not sure you're doing. You go on to elaborate this by saying "here's my ruler reading a meter: which event on your ruler, instantaneous for you with the origin, do I consider level with it"; this is rather confusing because you haven't defined which of us is meant to be the "input frame" and which is meant to be the "output frame", but the combination of "position defined in the input frame" and "which event ... do I consider level with it" makes me think you're defining yourself as the input frame and me as the external observer in the output frame. So, in terms of my definition of length contraction in terms of a single event E, you'd be saying that E occurs at x=1 meter in the input frame...but in this case I just want to know the x' coordinate of the same event E on my own ruler, I don't understand the significance of the business about my having to worry about which event on my ruler you "consider level with it", or even what you mean by "level" in this context. (Do you just mean what reading on my ruler lines up with the reading on your ruler of x=1 meter at the moment the event E happens? Or does 'level' refer to spacetime, so you're talking about simultaneity with some distant event? When I measure the length of your ruler I certainly don't have to worry about how you define simultaneity, if that's what you're implying...)
Only if the position in the length equation and the instant in the timeequation were both defined in the output frame, or both defined in the input frame, would we be comparing like with like.
Again we are not normally referring to the position or time coordinates of a single event in these equations, but rather to the time intervals between a single pair of events in two frames, or to the distance between two endpoints of an object at a single moment (which is different from the distance between a single pair of events as in my 'spatial analogue for time dilation') in two frames. You can think of special cases where we are just referring to coordinates of a single event, but in that case contraction vs. dilation is just defined in terms of whether the coordinate of this one event is smaller or larger in the output frame. For example, in the above scenario involving an event E at the front end of a ruler whose back end was at the origin at t'=0 in the output frame, the position coordinate x' of E in the output frame would be smaller than the position coordinate x of E in the input frame. Likewise, if you set things up so the moving clock in the input frame reads t=0 at t'=0 in the output frame, and pick some later event E on the input clock's worldline, then the time coordinate t' of E in the output frame would be greater than the time coordinate t of E in the input frame.
It isn't enough to say whether the quantity is greater or smaller in "the observer's frame" (output frame); there will be some quantity greater and smaller in each.
Sure, but we know the specific quantity we're dealing with for time dilation (the time intervals between the same pair of events in each frame) and for length contraction (the lengths of the same object in each frame). In both cases the quantity is a "proper" quantity for an instrument at rest in the input frame--in the first case it's the proper time between events on the worldline of a clock at rest in the input frame, in the second case it's the proper length of a ruler at rest in the input frame.
No, it certainly isn't. I specifically defined the "temporal analogue for length contraction" to mean this:

(time in observer's frame) = (time in clock's frame)/gamma

...

You can take the normal time dilation equation and divide both sides by gamma, but this doesn't give you the TAFLC above, instead it gives you what I called the "inverse time dilation equation":

(time in clock's frame) = (time in observer's frame)/gamma

See the difference?
Not yet. Since there is no formal difference - exacly the same equation is used - and since, by the principle of relativity, there can be no asymmetry between the two inertial frames, except that they're moving in opposite directions, whatever the difference is, I'm guessing it must be a subjective difference: something about how the frames are conceived?
There's only "no formal difference" only if you choose to use the same notation for quantities with a different physical interpretation. This is the same issue I criticized neopolitan for--when doing physics, you have to keep in mind the physical meaning of the symbols, just because two equations can be written the same way doesn't mean they have the same meaning! For example, if I made the weird choice to define the time interval between two events in the output frame using the notation "E", and the time interval between the same two events in the input frame (where they are colocated) using the notation "m", and the square root of the gamma factor using the notation "c", then there would be "no formal difference" between the time dilation equation and the equation E=mc^2 where the symbols are interpreted in the more conventional manner. Do you therefore conclude that the difference between the time dilation equation and the relativistic energy/mass relation is only a "subjective difference"?
You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify?
It signifies that we are talking about the time intervals in each frame between events which have been specifically selected to occur on the clock's worldline (so they are colocated in the clock's frame but not the observer's). In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame. Perhaps it would be clearer if I added even more words to my way of writing out the time dilation equation:

(time interval in observer's frame between a pair of events colocated in clock's frame) = (time in clock's frame between same pair of events)*gamma

Then of course the "inverse time dilation equation" obtained by just dividing both sides by gamma is:

(time interval in clock's frame between a pair of events colocated in clock's frame) = (time in observer's frame between same pair of events)/gamma

Whereas the point of the "temporal analogue for length contraction" is meant to keep the convention of the original time dilation equation that the frame in which the quantity we're looking at takes a "special" value stays on the right side of the equation. To make it so that this is true and that the right side is divided by gamma rather than multiplied by it, the quantity in question cannot just be the time in each frame between a pair of events. My suggestion was to consider two spacelike planes which represent surfaces of simultaneity (surfaces of constant t) in the input frame, and let the quantity be the time between these two planes in either frame (i.e. the time between the points where a line of constant x in a given frame will pierce each plane). This is analogous to length contraction where we consider two timelike paths which are lines of constant x in the input frame (these paths are just the worldlines of either end of a ruler at rest in the input frame), and define length as the distance between these two lines in either frame (i.e. the distance between the points where a line of constant t in a given frame will pierce each of these lines of constant x). So, you can write the "temporal analogue for length contraction" as:

(time interval in output frame between two spacelike surfaces that are surfaces of simultaneity in the input frame) = (time interval in the input frame between same spacelike surfaces) / gamma

Here you can see the "special" frame for this quantity is the input frame, and that we have kept it on the right side just as with the original time dilation equation.
The way I'm looking at it is in terms of frames identical in every way possible so as not to introduce the impression that one is favoured in some way, and thereby risk introducing some false asymmetry into the example.
There is no asymmetry in the laws of physics, but it's crucial to understand that all of these equations--time dilation, length contraction, and the "analogues" I defined--all assume that one of the frames is "special" in regards to the quantity that's being measured. If you don't want to make that sort of assumption, just use the full Lorentz transformation equations! For example, if I have two events and I do not assume they are colocated in either frame, then if I know the coordinate separations delta-x and delta-t between them in the input frame, the time interval in the output frame is given by:

delta-t' = gamma*(delta-t - v*delta-x/c^2)

You can see that in the special case where delta-x=0 in the input frame (i.e. they are colocated in the input frame), this reduces to the time dilation equation.

JesseM

Jun9-09, 03:38 PM

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples?
Because he wanted to stick to the convention that dilation/contraction is consistently defined in terms of the observer's frame (the non-'special' frame as I discussed above).

Doc Al

Jun9-09, 03:51 PM

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples? Could we not just as well say 1 < 3, thus time contraction? That sounds simpler to me.
You must be consistent, else you render the comparison meaningless. It's always lab frame ("stationary" frame) measurements compared to moving frame measurements. There's no argument here, you just need to understand how the terms "time dilation" and "length contraction" are used.

What might be throwing you off is the apparent lack of symmetry. Going back to the example of a clock (in frame S') moving from A to B while measuring 10 seconds of elapsed time. According to laboratory clocks (frame S), the time elapsed is 30 seconds. Moving clocks run slow: time dilation.

How are things viewed from frame S'? According to frame S', the clocks in frame S are moving and therefore unsynchronized. According to frame S', during the time that S' moves from A to B the clocks in frame S have only recorded an elapsed time of 10/3 seconds. As observed by S', the moving clocks in frame S run slow by that same factor of 3; thus S' measures the time interval to be 10 seconds. 10 > 10/3. Moving clocks run slow: time dilation.

The "time dilation" effect is completely symmetric. All frames observe moving clocks to run slow.

Rasalhague

Jun9-09, 04:03 PM

Time dilation deals with the time interval between a pair of events on the clock's worldline, not just the reading at a single moment.

Yes, in this case, I should have made clear that I was still referring to the set-up described in #357, in which the clocks each have zero x coordinate indefinitely, and are synchronised (set to time = zero) at the intersection of the origins of their rest frames.

So you should say:

(1) For a given pair of events on the clock L's worldline, frame R measures this multiple of the time interval measured between the events by clock L.

Yes, that would be another way of putting it.

Just as you weren't comparing readings at a single time in time dilation, "length" does not refer to position coordinates at a single location, it refers to the distance between the endpoints of the thing being measured at a single moment of time in whatever frame you use.

Again, I should have made explicit that this was the same example I outlined in #357, and that the other end of all intervals involved here is the spacetime coincidence of the zero end of both rulers being level at time = 0.

Clocks "naturally" measure the time between events that take place on their own worldline, and rulers "naturally" measure the distance between points along the ruler, like the distance between their own endpoints. So in both cases we are starting from something that is naturally measured in the instrument's own frame, and figuring out whether the corresponding quantity in the observer's frame is smaller or larger, and if it's smaller we say "contraction" and if it's larger we say "dilation".

Okay, but this doesn't explain why the pedagogical pairing of a contraction equation for one coordinate and a dilation equation for the other.

Time dilation involves only two events which occur on the clock's worldline. Length contraction can be thought of two involve three events if you like; just pick one event A on the worldline of the ruler's left end, then the distance in the ruler's rest frame between this event and the event B on the worldline of the right side that is simultaneous with event A in the ruler's rest frame[i] can be understood as the "rest length" of the ruler, while the distance in the observer's frame between event A and the event C on the worldline of the right side [i]that is simultaneous with A in the observer's frame can be understood as the "length" of the ruler in the observer's frame.

Alternatively, you could conceptualise time dilation as involving three events: (1) the coincidence of the clocks being synchronised at the origin, (2) the first clock at x = 0 reading one value (its proper time), (3) the second clock at x' = 0 reading another value (its proper time) at the instant simultaneous in the second clock's rest frame with event two. This is equivalent to asking what is the time component of the separation between events one and two in some frame where they're not collocal. Or maybe that's a needless complication.

Alternatively, you could use the three events which correspond to time in the way that the three events of the length contraction relation correspond to distance.

The time dilation equation is not restricted to dealing with cases where one of the events is at the origin.

How would you define the restriction? Could we say: the equation converts the interval of a separation with no space component into the time coordinate of that same interval in an inertial frame moving at some speed relative to the frame in which the events happen at the same place?

If the first event occurs at the origin, why would you be interested in an event colocal with the origin in the observer's frame?

I guess only for the sake of comparison of the interval between it and the origin, on the one hand, with the interval between the origin and some other event simulateous with in one frame or the other - or, equivalently, to find out how its coordinates change when viewed according to a different frame.

You're only interested in the difference in time coordinates between two events on the clock's worldline, so naturally since the clock is moving in the observer's frame, if the first event occurred at the origin then the second event did not.

Okay.

If you are measuring the "length" of one particular ruler in its own frame and in the observer's frame, then if one event occurs on the left end of the ruler at the origin of both frames, B must be an event on the worldline of the right end of this ruler that's simultaneous with the first event in the ruler's own frame, and C must be an event on th worldline of the right end of the same ruler that's simultaneous with the first event in the observer's frame.

The purpose of visualising two rulers was to remind me that there is no physical difference between the frames, they're interchangeable (apart from the difference in the direction of movement) in that any conversion you can make from one to the other, you can make from the other to the one and get the same result. I made the rulers indefinitely long so as to emphasise the parallel with the clocks. Alternatively, if we wanted to think of meter rulers, we could visualise a pair of timebombs, or candle clocks, or hourglasses. I was thinking in this way to emphasise that you can chose either frame as your input frame, and either as your output frame, and can ask contraction and dilation questions of each.

What equations are you talking about? Can you write them out in words as I did in my previous post, and do you understand the distinction I made there between "the temporal analogue for length contraction" and the "inverse time dilation equation"?

I wrote them in words and symbols in #357, alongside the traditional ones. I described them there in terms of a left frame and a right frame, visualising a clock L and a ruler L at rest in the former, and a clock R and ruler R at rest in the latter. If we reworded them in terms of input (source) and output (target) frames, then the output frame, in each case, would be the rest frame of the measuring device said to "show this multiple" or "show this fraction". I'm still unclear about what distinguishes your "temporal analogue for length contraction" from the "inverse time dilation equation". To me they both sound like "time contraction". Do they match any of the situations I described in #357?

"Stationary time" and "stationary length" don't refer to a comparison between two frames, they just refer to the time between events on a clock's own worldline as measured by that clock (i.e. as measured in the clock's rest frame) or the length of ruler as measured by the ruler itself (as measured in the ruler's rest frame). Since you're only talking about the value in a single frame, it would make little sense to talk about "dilation" or "contraction".

This isn't the way Wolfram Alpha is using these terms. If there was no change of frame involved, there would be no change of coordinate, so they would just give you back whatever value you entered! Rather, Wolfram Alpha uses "stationary time" and "stationary length" to refer to what I'd call time contraction and length dilation, and what I think you'd call either the analogue or the inverse equations of the time dilation and length contraction.

You can just talk about the time coordinates of events in the observer's frame without worrying about how he assigns them. But if you do want to think about that, then you really need two synchronized clocks at rest in the observer's frame, since in SR a clock only assigns time coordinates to events that occur on its own worldline, and the events in question are ones that occur on the worldline of the moving clock so they'll happen at different positions in the observer's frame.

True, although both of those clocks at rest in one frame would keep the same time relative to each other (if I've understood this right), so this is just a matter of how physical we want to make the thought experiment, or how simple and abstract.

Again, easier to just talk about coordinates in the observer's frame. But if you do want details of how the observer assigns coordinates, then when measuring length you really need to bring in clocks too, so that the observer can make sure he's measuring the position of the ends of the moving ruler relative to his own ruler at a single moment in time in his own frame.

And by the same logic, we'd want rulers handy when talking about time to make sure the clocks are where they should be... I'm trying to go by the rule that whatever details are suppressed (or expressed) when visualising the situation with changes in time coordinates are also suppressed (or expressed) when visualising the situation with changes in space coordinates. My intention there is to identify or highlight any genuine, fundamental asymmetries between time and space, and to eliminate from the visualisation any arbitrarily asymmetrical details.

I don't understand what you mean here. Are you talking about a formula which explicitly deals with rate of ticking rather than with periods of time between a pair of events? In that case you might write:

(clock's rate of ticking relative to time coordinate in observer's frame) = (clock's rate of ticking relative to time coordinate in clock's frame) / gamma

You can see that this is physically distinct from both the "temporal analogue for length contraction" and the "inverse time dilation" equations I wrote in my previous post, in spite of the fact that they all involve dividing by gamma on the right-hand side.

No, that's not what I had in mind. What other people are thinking of when they use the expression "a moving clock runs slow" I can't say, but to me it suggests time contraction which you subdivide into "temporal analogue for length contraction" and "inverse time dilation". But I'm puzzled by the general use of the expression "a moving clock runs slow" as a verbal summary of the time dilation formula, t' = t * gamma, whose output is a bigger number. In everyday life, when we think of a clock a running slow, we'd expect it to show a smaller number than it would otherwise.

Doc Al

Jun9-09, 04:14 PM

But I'm puzzled by the general use of the expression "a moving clock runs slow" as a verbal summary of the time dilation formula, t' = t * gamma, whose output is a bigger number.
That's not the correct formula. The "time dilation" formula is: ΔT = gamma * ΔT0, where ΔT0 is the time elapsed on the moving clock and ΔT is the time measured in the laboratory frame doing the observing. Of course, ΔT > ΔT0.
In everyday life, when we think of a clock a running slow, we'd expect it to show a smaller number than it would otherwise.
And it does. You have the formula backwards.

Rasalhague

Jun9-09, 06:09 PM

Thanks to you both for pointing me towards the more general expression of these equations with differences as opposed to coordinates. I was just used the simplification I've seen in several introductory texts of denoting one end of each of the intervals involved as the origin.

The "time dilation" effect is completely symmetric. All frames observe moving clocks to run slow.

I'm okay with that. The thing that's been troubling me is the way that, while time and space are treated symmetrically in standard presentations of the full Lorentz transformation, when it comes to these special cases "time dilation" and "length contraction", the symmetry has gone. How much is this due to convention, how much is it a fundamental difference between time and space? It isn't as if no one ever divides a time coordinate by gamma or multiplies a space coordinate by gamma, but the terms dilation and contraction seem to attach to time and length respectively, regardless of whether the calculation is making a big number small (contracting it), as here for instance http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html , or a small number big (dilating it). To me that seems inconsistent, but obviously I have a lot to learn.

As I keep saying, the convention is that contraction/dilation is defined in terms of the measurement in the observer's frame. Do you disagree that in this case the period in the observer's frame is 30, and that 30 is a longer period than 10?

Of course, you're right: 30 is bigger than 10!

Thanks for your patience in explaining. I'll get back to you when I've had a chance to mull over what your wrote some more.

Rasalhague

Jun9-09, 07:49 PM

>You call one frame "the observer's frame" and the other "the clock's frame". But what exactly does this signify?

It signifies that we are talking about the time intervals in each frame between events which have been specifically selected to occur on the clock's worldline (so they are colocated in the clock's frame but not the observer's).

So the "clock's frame" is whichever frame it is in which the time interval between the events is equal to the proper time, and the "observer's frame" is the other (the frame where the time interval is bigger than the proper time because the separation also has some spatial component), regardless of which of these happens to be input or output?

In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame. Perhaps it would be clearer if I added even more words to my way of writing out the time dilation equation:

(time interval in observer's frame between a pair of events colocated in clock's frame) = (time in clock's frame between same pair of events)*gamma

Then of course the "inverse time dilation equation" obtained by just dividing both sides by gamma is:

(time interval in clock's frame between a pair of events colocated in clock's frame) = (time in observer's frame between same pair of events)/gamma

Whereas the point of the "temporal analogue for length contraction" is meant to keep the convention of the original time dilation equation that the frame in which the quantity we're looking at takes a "special" value stays on the right side of the equation. To make it so that this is true and that the right side is divided by gamma rather than multiplied by it, the quantity in question cannot just be the time in each frame between a pair of events. My suggestion was to consider two spacelike planes which represent surfaces of simultaneity (surfaces of constant t) in the input frame, and let the quantity be the time between these two planes in either frame (i.e. the time between the points where a line of constant x in a given frame will pierce each plane). This is analogous to length contraction where we consider two timelike paths which are lines of constant x in the input frame (these paths are just the worldlines of either end of a ruler at rest in the input frame), and define length as the distance between these two lines in either frame (i.e. the distance between the points where a line of constant t in a given frame will pierce each of these lines of constant x). So, you can write the "temporal analogue for length contraction" as:

(time interval in output frame between two spacelike surfaces that are surfaces of simultaneity in the input frame) = (time interval in the input frame between same spacelike surfaces) / gamma

Here you can see the "special" frame for this quantity is the input frame, and that we have kept it on the right side just as with the original time dilation equation.

So would you call the operation carried out in this example TAFLC?

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

The time interval in the input frame (Jack's rest frame) is 10 seconds. It's the time interval between two surfaces of simultaneity in the input frame, namely that through C1 = 0 = C2, and that through C1 = 10 = C2. This value is divided by gamma to give the time interval in the output frame (Jill's rest frame) between the same surfaces, namely 8 seconds.

Or would you call it "the inverse of time dilation"?

The time interval in the input frame is the time interval between a pair of events collocated in the clock's frame (i.e. the special one, the one where those events are separated by the minimum time interval). This value is divided by gamma to give the time interval in the clock's frame between a pair of events collocated in the clock's frame, namely C' passing C1 at the event of them both being set to 0, and the event of C' passing C2 when C2 = 10.

Does the fact that I can describe it in terms of both concepts indicate that they are basically the same thing after all, or have I misunderstood?

JesseM

Jun9-09, 08:30 PM

So the "clock's frame" is whichever frame it is in which the time interval between the events is equal to the proper time, and the "observer's frame" is the other (the frame where the time interval is bigger than the proper time because the separation also has some spatial component), regardless of which of these happens to be input or output?
Yes, exactly.
So would you call the operation carried out in this example TAFLC?

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

The time interval in the input frame (Jack's rest frame) is 10 seconds. It's the time interval between two surfaces of simultaneity in the input frame, namely that through C1 = 0 = C2, and that through C1 = 10 = C2. This value is divided by gamma to give the time interval in the output frame (Jill's rest frame) between the same surfaces, namely 8 seconds.
Yes, that actually works, provided you here treat Jill as "the observer" rather than Jack. I hadn't thought of it like this, but you're right that the measurements involved in an ordinary time dilation experiment like this can be re-interpreted as a TAFLC measurement, just by switching who we call "the observer", and by switching what defines the "special" frame from the frame where the time between two events on Jill's worldline is minimized (i.e. her own frame) to the frame where two spacelike surfaces that Jill passes through are surfaces of simultaneity (i.e. Jack's frame, since we were already considering his surfaces of simultaneity when showing how he would measure the time elapsed on Jill's clock).
Or would you call it "the inverse of time dilation"?
You've made a good point in that the difference between the two is smaller than I was making it out to be, but I still think it's worth distinguishing them by specifying in words exactly what physical quantity is being measured (i.e. whether you want to say that Jack is just using two of his own surfaces of simultaneity in order to measure the time between two events on Jill's worldline, which is also what Jill's clock is measuring, or whether you want to say that both of them are explicitly trying to measure the time between two spacelike surfaces which are surfaces of simultaneity in Jack's frame), and calling the frame where this quantity takes a non-"special" value the outside observer's frame. Then the TAFLC is the equation that has the outside observer's frame as the output of the equation (the left-hand side) just like the time dilation equation (because the time between the spacelike surfaces takes a 'special' value in Jack's frame, so Jill is defined as 'the outside observer'), whereas the "inverse time dilation equation" has the outside observer's frame as the input (because the time between events on Jill's worldline takes a 'special' value in Jill's frame, so Jack is defined as 'the outside observer').
Does the fact that I can describe it in terms of both concepts indicate that they are basically the same thing after all, or have I misunderstood?
Well, you've convinced me that it's not absolutely essential to distinguish between the "inverse time dilation" equation and the "TAFLC" equation, that the type of conceptual distinction I make above is really more of an aesthetic preference; I still think it's clearer to think in these terms but if you don't want to it's kind of a matter of taste.

neopolitan

Jun9-09, 08:38 PM

JesseM

Jun9-09, 09:02 PM

I'm still thinking about how to write the next stage in the process.

JesseM, you asked what I come up with at the end. I come up with an interval between B and an event according to B which, because B is (notionally) at the origin of the B axes corresponds with the coordinates of the event according to B.
By "interval" do you mean a spatial interval or a time interval rather than a spacetime interval? Also, what do you mean by "between B and the event"? Do you mean the interval between the spacetime origin of B's frame (i.e. x=0 and t=0 in B's frame) and this other event, or do you mean the distance between B and the event at the instant the event occurs in B's frame, or something else? Also, presumably what you derive is an equation that gives you this value on the left side and something that looks like the Lorentz transformation equation on the right side, i.e. something like gamma*(x - vt) on the right, yes? If so, what is the corresponding interpretation of x and t (or whatever symbols you use on the right side) in terms of your physical setup?
As for leading into the next stage, I've taken on board the fact that you don't like my assumption that if there is a factor or function affecting measurements in the B frame, according to A then that same factor or function affects measurements in the A frame, according to B.
Such an assumption might well be justifiable in terms of the assumption that the laws of physics should work the same way in both frames (the first postulate of SR), it's just that I would need to see a little more of a detailed justification for it if you're trying to do a rigorous proof.
Additionally, I am being more careful about extrapolations, keeping in mind that A and B can only measure times at colocation with themselves.
I'm not sure this is strictly necessary in a derivation--as long as we start from the basic postulate that A and B each assume light moves at c in their own frame, we can basically take as read that whenever they deal with time coordinates of events that don't happen along their worldlines, they calculate it by noting the time they recieve the light from the event and subtracting the travel time based on the distance the event happened from them (as measured by a ruler at rest relative to themselves). It's fine if you don't state this explicitly each time you talk about the time of events, and in fact it probably makes the derivation easier to follow if you just assume this is understood and don't worry about it (or just mention it once at the beginning).
The time (interval) I need is:

(between colocation of A and B in the A frame and when the photon from the Event passes B, in the A frame)

As you pointed out, I have used t'a to mean something else.
You used it to mean the time between an event Ea on the photon's worldline which was simultaneous with A and B being colocated in the A frame, and the event of the photon passing B, in the A frame. So, this will obviously always give the same value as your definition above, even if you selected the events differently; if you only did this because you're worried about "extrapolations" then like I said my advice would just be not to worry. But if you want to introduce this interval separate from the other one that's fine too.
My suggestion for naming this time (interval) is t'oa, where the o indicates a photon interception event.
Don't all your time intervals include a photon interception event as one of the two events they're giving the interval between? But the terminology doesn't really matter, t'oa is fine with me.

neopolitan

Jun10-09, 01:07 AM

By "interval" do you mean a spatial interval or a time interval rather than a spacetime interval?

If you talk about the "Lorentz Transforms" (plural) then it is spacetime. Otherwise I would arrive at one equation each.

Also, what do you mean by "between B and the event"? Do you mean the interval between the spacetime origin of B's frame (i.e. x=0 and t=0 in B's frame) and this other event, or do you mean the distance between B and the event at the instant the event occurs in B's frame, or something else?

I mean:

(spatial and temporal separation between B and an event, in the B frame) = (a function operating on or a factor multiplied by the spatial and temporal separation between A and an event, in the A frame)

And I am not giving away the end by saying that these will end up in the form:

(spatial separation between B and an event, in the B frame) = (a factor) . ((spatial separation between A and an event, in the A frame) - (relative velocity).(temporal separation between A and an event, in the A frame))

(temporal separation between B and an event, in the B frame) = (a factor) . ((temporal separation between A and an event, in the A frame) - (relative velocity).(spatial separation between A and an event, in the A frame)/(the speed of light squared))

Also, presumably what you derive is an equation that gives you this value on the left side and something that looks like the Lorentz transformation equation on the right side, i.e. something like gamma*(x - vt) on the right, yes? If so, what is the corresponding interpretation of x and t (or whatever symbols you use on the right side) in terms of your physical setup?

Addressed above, I think.

I'm not sure this is strictly necessary in a derivation--as long as we start from the basic postulate that A and B each assume light moves at c in their own frame, we can basically take as read that whenever they deal with time coordinates of events that don't happen along their worldlines, they calculate it by noting the time they recieve the light from the event and subtracting the travel time based on the distance the event happened from them (as measured by a ruler at rest relative to themselves). It's fine if you don't state this explicitly each time you talk about the time of events, and in fact it probably makes the derivation easier to follow if you just assume this is understood and don't worry about it (or just mention it once at the beginning).

Ok, happy with that.

You used it to mean the time between an event Ea on the photon's worldline which was simultaneous with A and B being colocated in the A frame, and the event of the photon passing B, in the A frame. So, this will obviously always give the same value as your definition above, even if you selected the events differently; if you only did this because you're worried about "extrapolations" then like I said my advice would just be not to worry. But if you want to introduce this interval separate from the other one that's fine too.

Prior to the summation, I introduced subscripts at a later point in my derivation process, which allowed me to use a subscripted x' differently. I don't want to introduce a x'a which is not the same as the summation x'a.

Don't all your time intervals include a photon interception event as one of the two events they're giving the interval between?

Yes, but not photon interception events which are separated from the observer. So to be more precise, I intend to use o to mean a reference to a non-local photon interception event. So:

t'oa = time (t), according to A a, that a photon passes B ('o) if that photon was at a distance of xa when A and B were colocated, according to A.

__________________________________________________ _____________________

At the risk of repeating myself, I want to make clear that I am talking about a single event described in two frames.

I also want to make clear that A and B won't know anything about that event before a photon from the event reaches them.

I also want to make clear that, even when photon reaches them, A and B won't know more than "I received a photon".

If they are given information about when the event took place (in their own frame), they can work out where the event took place (in their own frame). But they can't work out when and where the photon was released from the mere fact that they receive a photon at a specific time and place.

So, my derivation works on the principle that if we consider an event which was simultaneous (in the A frame) with the colocation of A and B, at t=0, then we can calculate a "where" (in the A frame) for this event. We can also work out a "when" and "where" (in the A frame) for colocation of B and the photon, from the timing of colocation of A and B and the timing of the photon's colocation with A (in the A frame).

Because A and B were colocated at t=0, and at colocation t'=0, we then have sufficient information to work out the "when" and "where" in the B frame of the event that was simultaneous with colocation of A and B in the A frame, which would be the coordinates of that same event in the B frame.

If we say that the event that is simultaneous (in the A frame) with the colocation of A and B is the event which spawns the photon, this is just for the sake of convenience. Without more information, A can't really know when or where the photon was spawned.

Are you happy for me to go on to the next stage of the derivation?

cheers,

neopolitan

JesseM

Jun10-09, 02:00 AM

If you talk about the "Lorentz Transforms" (plural) then it is spacetime. Otherwise I would arrive at one equation each.
I should have been more specific, I was wondering about the meaning of the specific variables in your final equations, like the "interval between B and an event" which you mentioned. In the Lorentz transform each variable represents a purely spatial interval or a purely temporal interval between two events (in whatever frame each variables are dealing with), so presumably the same is true for your final equations?
Also, what do you mean by "between B and the event"? Do you mean the interval between the spacetime origin of B's frame (i.e. x=0 and t=0 in B's frame) and this other event, or do you mean the distance between B and the event at the instant the event occurs in B's frame, or something else?
I mean:

(spatial and temporal separation between B and an event, in the B frame) = (a function operating on or a factor multiplied by the spatial and temporal separation between A and an event, in the A frame)

And I am not giving away the end by saying that these will end up in the form:

(spatial separation between B and an event, in the B frame) = (a factor) . ((spatial separation between A and an event, in the A frame) - (relative velocity).(temporal separation between A and an event, in the A frame))

(temporal separation between B and an event, in the B frame) = (a factor) . ((temporal separation between A and an event, in the A frame) - (relative velocity).(spatial separation between A and an event, in the A frame)/(the speed of light squared))
When you say "temporal separation between A and an event", you must mean the temporal separation between the event and some other specific event that occurs on A's worldline--can I assume this is just the event of A and B being colocated at t=0 in A's frame? And likewise for "temporal separation between B and an event"?

Also, can you tell if the derivation will be based on treating the event as one of the specific events you've already introduced, like the event of the photon crossing either A or B's worldline, or an event that's simultaneous with their being colocated in one of the frames? If so it seems to me your proof is not going to be fully general--once you've calculated the space and time intervals between this event and the event of A and B being colocated at the origin, you are of course free to move the origin if you introduce a lemma of the type I talked about in the second paragraph of post 338 (http://www.physicsforums.com/showpost.php?p=2228439&postcount=338) and earlier in post 249, but just shifting the origin wouldn't change the fact that if you used one of these four events originally, then the line between the event and the A&B colocation event will be either purely spatial (meaning the two events are simultaneous) or purely temporal (meaning the events are colocated) in one of the two frames, so this will remain true if you shift the position of the origin. This is not to say I think it's pointless to derive a special case of the Lorentz transformation, but if that's what you're doing we should at least be clear on this.
Yes, but not photon interception events which are separated from the observer. So to be more precise, I intend to use o to mean a reference to a non-local photon interception event. So:

t'oa = time (t), according to A a, that a photon passes B ('o) if that photon was at a distance of xa when A and B were colocated, according to A.
OK, makes sense.
I also want to make clear that, even when photon reaches them, A and B won't know more than "I received a photon".

If they are given information about when the event took place (in their own frame), they can work out where the event took place (in their own frame). But they can't work out when and where the photon was released from the mere fact that they receive a photon at a specific time and place.

So, my derivation works on the principle that if we consider an event which was simultaneous (in the A frame) with the colocation of A and B, at t=0, then we can calculate a "where" (in the A frame) for this event. We can also work out a "when" and "where" (in the A frame) for colocation of B and the photon, from the timing of colocation of A and B and the timing of the photon's colocation with A (in the A frame).

Because A and B were colocated at t=0, and at colocation t'=0, we then have sufficient information to work out the "when" and "where" in the B frame of the event that was simultaneous with colocation of A and B in the A frame, which would be the coordinates of that same event in the B frame.

If we say that the event that is simultaneous (in the A frame) with the colocation of A and B is the event which spawns the photon, this is just for the sake of convenience. Without more information, A can't really know when or where the photon was spawned.

Are you happy for me to go on to the next stage of the derivation?
This part is fine, but see my questions at the beginning.

Rasalhague

Jun10-09, 04:31 AM

Yes, that actually works, provided you here treat Jill as "the observer" rather than Jack. I hadn't thought of it like this, but you're right that the measurements involved in an ordinary time dilation experiment like this can be re-interpreted as a TAFLC measurement, just by switching who we call "the observer", and by switching what defines the "special" frame from the frame where the time between two events on Jill's worldline is minimized (i.e. her own frame) to the frame where two spacelike surfaces that Jill passes through are surfaces of simultaneity (i.e. Jack's frame, since we were already considering his surfaces of simultaneity when showing how he would measure the time elapsed on Jill's clock).

I'm not quite sure what you mean by "switching", and I'm confused by your redefinition of "special". How can we compare TAFLC with INV(TD) if the meaning of the terms used to define INV(TD) have to be changed to define TAFLC? I thought you'd consider Jack "the observer" whether his rest frame was the input or not, and whether we're dilating the input value or contracting it, since, of the two quantities involved (input/known and output/unknown), his rest frame is the one where the variable has the greater value (is not minimised). Going back to your definition of "the observer"...

You wrote: >>In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame.

...when you say "the quantity we are dealing with", was I right to think that this quality of being minimum is the only distinguishing feature of "the quantity we are dealing with", or is there something else that marks out one quantity in this way? There are two quantities involved, one known (the time interval in the input frame), and one unknown (the time interval in the output frame). My impression was that you were defining the "quantity we are dealing with" as whichever of these quantities is minimum (i.e. a proper time in one of the frames). The difference between your TD and INV(TD) equations is that in TD the observer's frame is the output frame, while in INV(TD) the observer's frame in the input frame, so it can't be the quality of being an input or an output that defines "oberser's" value and "clock" value.

You've made a good point in that the difference between the two is smaller than I was making it out to be, but I still think it's worth distinguishing them by specifying in words exactly what physical quantity is being measured (i.e. whether you want to say that Jack is just using two of his own surfaces of simultaneity in order to measure the time between two events on Jill's worldline, which is also what Jill's clock is measuring, or whether you want to say that both of them are explicitly trying to measure the time between two spacelike surfaces which are surfaces of simultaneity in Jack's frame), and calling the frame where this quantity takes a non-"special" value the outside observer's frame. Then the TAFLC is the equation that has the outside observer's frame as the output of the equation (the left-hand side) just like the time dilation equation (because the time between the spacelike surfaces takes a 'special' value in Jack's frame, so Jill is defined as 'the outside observer'), whereas the "inverse time dilation equation" has the outside observer's frame as the input (because the time between events on Jill's worldline takes a 'special' value in Jill's frame, so Jack is defined as 'the outside observer').

What is the defining feature of "special" that applies to both TAFLC and INV(TD) if it's not the quality of being the input, or the quality of being minimum? What is the difference between "Jack is just using" and "both of them are explicitly trying to measure", given that calculation doesn't depend on who's performing it?

Well, you've convinced me that it's not absolutely essential to distinguish between the "inverse time dilation" equation and the "TAFLC" equation, that the type of conceptual distinction I make above is really more of an aesthetic preference; I still think it's clearer to think in these terms but if you don't want to it's kind of a matter of taste.

It's been a fascinating discussion. You've taught me a lot along the way, and given me a lot to think about.

neopolitan

Jun10-09, 04:43 AM

I should have been more specific, I was wondering about the meaning of the specific variables in your final equations, like the "interval between B and an event" which you mentioned. In the Lorentz transform each variable represents a purely spatial interval or a purely temporal interval between two events (in whatever frame each variables are dealing with), so presumably the same is true for your final equations?

Yes, x'b for instance is purely spatial in the B frame.

When you say "temporal separation between A and an event", you must mean the temporal separation between the event and some other specific event that occurs on A's worldline--can I assume this is just the event of A and B being colocated at t=0 in A's frame? And likewise for "temporal separation between B and an event"?

Temporal separation between the event and colocation of A and the photon from the event, in the A frame. (I do use t=0 such that it is simultaneous with the event in the A frame.)

Temporal separation between the event and colocation of B and the photon from the event, in the B frame. (The event that A considers to be simultaneous with t=0, is not simultaneous with t'=0 in the B frame. However, by same token, the event that is the location of the photon at t'=0 in the B frame, is not simultaneous with t=0 in the A frame.)

Also, can you tell if the derivation will be based on treating the event as one of the specific events you've already introduced, like the event of the photon crossing either A or B's worldline, or an event that's simultaneous with their being colocated in one of the frames?

The event I intend to use will be simultaneous with A and B being colocated in one frame.

What I propose is to continue with what you will tell me not a general case and I will remain aware that I have a burden of proof to show that you may be wrong about it not being a general case.

Or do you want me to provide the general case arguement first? (Which is difficult, but maybe not impossible, before I have shown the derivation.)

cheers,

neopolitan

Rasalhague

Jun10-09, 06:50 AM

You must be consistent, else you render the comparison meaningless. It's always lab frame ("stationary" frame) measurements compared to moving frame measurements. There's no argument here, you just need to understand how the terms "time dilation" and "length contraction" are used.

On consistency, see the end of this post. When you say "compared to", are you defining "lab frame" as input frame (the frame for which we know the value), and "moving frame" as output frame (the frame for which we want to calculate the interval)? If not, what are the distinguishing features of "lab frame" and "moving frame"; how are they defined?

In post #357 I described various possible questions we might ask of these formulas, calling contraction whatever operation contracted the input, and dilation whatever dilated the input. (Excuse the lack of deltas; I hope any ambiguity there is removed by the description of the set-up at the beginning of that post and by the definitions I gave in #355.) Instead of "moment defined in frame X", I could have said "surface of simultaneity in frame X". Instead of "location defined in frame X", I could have said "worldline of a mark on the ruler at rest in frame X".

Maybe it would help to match up definitions to this scenario.

Could it be that we're arguing over whether 1 < 3, or 3 > 1?! Why did you reverse the inequality between the two examples?Because he wanted to stick to the convention that dilation/contraction is consistently defined in terms of the observer's frame (the non-'special' frame as I discussed above).

To me, the intuitive way would be to define dilation/contraction in terms of the input frame in the sense that the value of the input variable is dilated (made bigger) by the operation, or contracted (made smaller), to give the result. But apparently this isn't the convention. When you say "in terms of the observer's frame", this suggests that the variable would take a smaller value in the observer's frame, in the case of dilation, and a dilated value in what you called the clock frame. But that's the opposite of your definition of the terms "observer's frame" and "clock frame"; you defined the clock frame as the one where the variable takes its minimum value, and the observer's frame as other other frame.

In going from A to B, a moving clock measures 10 seconds. According to laboratory clocks, 30 seconds have passed. 30 > 10, thus time dilation.

A moving stick is 3 meters long in its own frame. According to laboratory measurements, it is 1 meter long. 1 < 3, thus length contraction.

What's the problem?

The problem is that I'd have expected "dilation" to refer to the process of dilating the input to produce the output. Dilation seems to imply a starting point at which something is small, and an endpoint in the process at which it's bigger. I'm aware that this isn't the convention, but I don't understand why not. So when we're given a value of 10 seconds, and calculate a value of 30 seconds from it, it seem only natural to call this operation dilation. But when the same term, dilation, is used of the inverse operaion, as here

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

that sounds bizarre to me. Our input is a big number, our output is a smaller number, and yet we're supposed to call this operation dilation as well (or multiplication by the "time dilation factor").

In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.

* Time dilation. In the case of clocks, the time measured in the output frame between two events on the worldline of a clock at rest in the input frame is greater than the time measured by the clock between those two events.

(We take as our standard the variable of the input, in comparison to which the output variable is bigger. But when we say "a moving clock runs slow", we're conceptualising it as the inverse, taking as our standard the variable with the bigger variable: given a certain input, the output will be smaller, contracted. Hence...)

* Time contraction. The time measured in the output frame between the ends of that period (defined as surfaces of simultaneity in the input frame) is smaller than the time measured by the clock at rest in the input frame.

* Length dilation. In the case of rulers, the distance measured in the output frame between two events whose separation has no time component in the rest frame of some ruler is greater than the distance measured by that ruler itself.

* Length contraction. The distance measured in the output frame between the ends of the a ruler at rest in the input frame (defined as the worldlines of marks on the ruler, lines of collocality) is smaller than the distance measured by the ruler at rest in the input frame.

The only inconsistency, it seems to me, is in talking as if there was something inherently dilatory about time, and something inherently contractory about space, in spite of the fact that we can and do dilate or contract either, depending on the context and what we want to find out from the equations. An example of this inconsistency is the way that Wolfram Alpha is obliged to reverse its definitions of moving and stationary depending on whether you want to transform a time interval or a space interval.

matheinste

Jun10-09, 07:42 AM

Rasalhague

Jun10-09, 08:38 AM

Is there perhaps some confusion here between number of ticks and duration between ticks. When we say that clocks moving relative to a given frame run slow when compared with that frame we mean that the moving clock ticks slower, that is the time between ticks is dilated ( longer or greater or bigger ). However the number of ticks will be decreased ( contracted, smaller , less in number), it records less time. I think the standard and accepted usage is that time dilates for a moving clock, that is the time between ticks is longer. I have never seen it used any other way.

Here is an example of dilation used in the opposite sense.

Taylor/Wheeler: "Let the rocket clock read one meter of light-travel time between the two events [...] so that the lapse of time recorded in the rocket frame is \Delta t' = 1\,meter. Show that the time lapse observed in the laboratory frame is given by the expression \Delta t' = \Delta t\, cosh \theta_{r} = \Delta t \,/ \left(1 - \beta^{2}\right)^{\frac{1}{2}}. This time lapse is more than one meter of light-travel time. Such lengthening is called time dilation ("to dilate" means "to stretch")." (Spacetime Physics, Ch. 1, Ex. 10, p. 66).

My impression so far as been that this is the standard interpretation, especially given other people's comments in this thread, although I've wondered whether some people interpreted it as you do. Many texts I've read don't make any explicit statement on how the word "dilation" is to be interpreted.

After I posted this in #365, Doc Al wrote: "Exactly! Laboratory clocks measure a greater time interval than the moving clock, thus time dilation."

JesseM wrote: "Yes, and note that they are using exactly the convention I described--since the time lapse between the two events in the observer's frame is more than the proper time measured by the moving clock, they call this time dilation."

Doc Al

Jun10-09, 08:42 AM

Rasalhague

Jun10-09, 01:23 PM

No, just the opposite. (Assuming I understand what you mean by "input frame".)

In the so-called "time dilation" formula, you start with a time interval measured on a moving clock (the "input", I suppose) and use the formula to compute what the lab frame would measure for that time interval (the "output" of the formula). The "output" is always bigger than the "input".

The "lab frame" is the frame of the observer whose measurements we want to calculate; the "moving frame" is the frame in which the clock in question is at rest.

By input I mean the value you start with (what's given, known), and by output the value you want to calculate (what's unknown). But if your lab frame is what I was calling input frame (the frame in which the known value is a coordinate), and your moving frame what I was calling output frame, what happens when we have a question of the inverse type and want to calculate a smaller output, as in Michael Fowler's example? Well, we've seen what happens (output < input), but what would you call this operation, how would you labels the frames and values involved in that operation?

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

There the output is smaller than the input. Although he doesn't explicitly call this operation time dilation, he does refer to 1 / gamma as "the time dilation factor", even though multiplication by this number results in a smaller number. If making a number bigger is called dilating it, then I'd expect making a number smaller to be called contracting it, whatever that number represents. But here, and generally, the convention seems to be that whatever is done to a number is called dilation if the number happens to represent a portion of time, and whatever is done to a number is called contraction if the number happens to represent a portion of distance.

For example, I observe a clock moving past me as it goes from position A to position B. My rest frame is the lab frame; my cohorts and I in our frame have measured the time interval (on our lab frame clocks, of course) for the clock to pass from A to B. Call that time interval ΔT. What time interval does the clock itself record (the "moving" frame, to us)? Call that time interval ΔT0. The time dilation formula relates those two time intervals: ΔT = gamma*ΔT0. This is just a precise statement of the loose phrase "moving clocks run slow".

Here you've reversed the labels you proposed above where you defined the movng clock as the input, and the lab as the output. Was that intentional? If distinguishing input from output isn't the defining feature of your labels lab and moving, what is? In this example, ΔT is lab time = input (what we're given). ΔT0 is moving clock time = output (what we want to calculate). The output will be smaller than the input, as in Michael Fowler's example (with positions A and B corresponding to C1 and C2). To solve the equation in the form you give it for ΔT0 we have to divide both sides by gamma. ΔT0 = ΔT / gamma. Since the number has been made smaller, it makes more sense to call what we did contraction than dilation. If this is the sense behind the motto "moving clocks run slow", then apparently it does refer to the operation of contracting (making smaller) a quantity, and matches the everyday idea that if a clock runs slow it will show a smaller number than it would otherwise, or than a clock relative to which which it's running slow.

Where do you get the idea that "dilation" means anything other than it does in normal usage? To "dilate" means to expand--get bigger.

Apart from the convention I mentioned above, and your own use of the term "time dilation" in the example for an operation whose effect was to shrink a number, there seems to be a further ambiguity about the term, as applied to a measurement. Taylor and Wheeler in Spacetime Physics, you and Jesse all - unless I'm mistaken - take it to refer to the operation of making the number bigger (t' = t * gamma; using the prime symbol here simply to denote output value), that is increasing the quantity/number/amount of time units, whereas Matheinste in #383, and perhaps others, interpret it in the sense of the individual units getting bigger and therefore it taking less of them to fill up a specified interval. Thus Matheinste wrote:

"Is there perhaps some confusion here between number of ticks and duration between ticks. When we say that clocks moving relative to a given frame run slow when compared with that frame we mean that the moving clock ticks slower, that is the time between ticks is dilated ( longer or greater or bigger ). However the number of ticks will be decreased ( contracted, smaller , less in number), it records less time. I think the standard and accepted usage is that time dilates for a moving clock, that is the time between ticks is longer. I have never seen it used any other way."

I don't know if Matheinste would, on the basis of this, call t' = t / gamma time dilation, and t' = t * gamma time contraction. Wolfram Alpha, like Michael Fowler, treats both equations under the heading "time dilation". They use the term "moving time" for the (output of?) t' = t * gamma, and "stationary time" for t' = t / gamma. On the other hand, it seems that they reverse the meanings of moving and stationary for "length contraction", so as to maintain the traditional pairing t' = t * gamma and l' = l / gamma.

http://www01.wolframalpha.com/input/?i=time+dilation

I'm familiar with that site. I don't see anything there that would contradict the usual usage of the term "time dilation".

Realize that the "time dilation" formula applies to time readings on a single moving clock. You cannot take a time interval measured in the moving frame using multiple clocks, blindly apply the time dilation formula, and expect it to to give the correct time interval measured in another frame*. When multiple clocks are involved you must also include the effects of clock desynchronization (the relativity of simultaneity). All of this is factored in automatically when you use the full Lorentz transformations.

*I suspect that this is at the root of your confusion.

Aren't the effects of desynchronisation all part and parcel of time dilation/contraction anyway, however we look at it, hence the symmetry between frames that you referred to in #370? The Taylor/Wheeler book begins with a visualisation of an orthogonal grid filling space, made of meter sticks with clocks at the vertices. Each frame is conceptualised as such a grid. So multiple clocks are implied in any measurement. The "time dilation" formula could be conceptualised as making one clock explicit and suppresses the rest. But in order to visualise the way the time shown by this clock relates to times that are shown by clocks moving relative to it, or the way it would relate to the time shown by a notional clock moving at some speed relative to it, if there was such a clock, I've been tending to explicitly imagine comparisons between two physical clocks. Part of my motivation was to make it more concrete. Part was because I feared it would be all to easy, as a beginner, to slip into thinking of one frame as privileged. But comparison between two clocks is made in plenty of the examples and explanations of "time dilation" that I've read: be they clocks with rotating hands, or light clocks or short-lived muons or identical twins. Of course, we don't need to picture all these clocks to make the calculation, and we could think of it in more abstract geometric terms: input = the proper time of some separation; output = the time component of that separation in some frame (or the other way around, in the case of time contraction).

Doc Al

Jun10-09, 01:46 PM

By input I mean the value you start with (what's given, known), and by output the value you want to calculate (what's unknown). But if your lab frame is what I was calling input frame (the frame in which the known value is a coordinate), and your moving frame what I was calling output frame, what happens when we have a question of the inverse type and want to calculate a smaller output, as in Michael Fowler's example? Well, we've seen what happens (output < input), but what would you call this operation, how would you labels the frames and values involved in that operation?
Obviously, when you are dealing with an equation the "input" and "output" are entirely arbitrary. You can take ΔT = gamma*ΔT0 and rewrite it as ΔT0 = ΔT/gamma. If you choose to call ΔT your "input", then of course your "output" (ΔT0) will be smaller. So what???

As I said many times now, the reason it's called time dilation is that a lab frame observes a larger time interval than recorded by a moving clock. That's what "moving clocks run slow" means.

Note that even though you can reverse the equation to solve for either quantity given the other, that doesn't change the meaning of the quantities. ΔT0 is always the time interval recorded by the moving clock, thus ΔT > ΔT0.

JesseM

Jun10-09, 01:48 PM

On consistency, see the end of this post. When you say "compared to", are you defining "lab frame" as input frame (the frame for which we know the value), and "moving frame" as output frame (the frame for which we want to calculate the interval)? If not, what are the distinguishing features of "lab frame" and "moving frame"; how are they defined?
Again, the moving frame is just the one where the variable being measured takes the "special" value--in the case of time dilation and length contraction, it's the frame where the time interval and length in that frame are equal to the proper time and rest length. The lab/observer's frame is the one where they don't, because the object in question (a clock or a ruler) is moving. The equation is usually written under the assumption that we know the proper time/rest length (so that's the input) and want to find the time interval/length in the lab frame, but of course with any equation you can rearrange it to solve for whatever variable you don't know.
In post #357 I described various possible questions we might ask of these formulas, calling contraction whatever operation contracted the input, and dilation whatever dilated the input.
But that just isn't the convention. The convention is that it's based on whether the value is bigger or smaller in the non-special frame.
(Excuse the lack of deltas; I hope any ambiguity there is removed by the description of the set-up at the beginning of that post and by the definitions I gave in #355.) Instead of "moment defined in frame X", I could have said "surface of simultaneity in frame X".
How is "moment" different from "surface of simultaneity"? Again, time dilation isn't defined in terms of readings at one particular moment/surface of simultaneity, it's defined by the interval of time between two specified events.
Instead of "location defined in frame X", I could have said "worldline of a mark on the ruler at rest in frame X".
...and length contraction isn't defined by a single worldline, it's defined by the distance between two worldlines.
To me, the intuitive way would be to define dilation/contraction in terms of the input frame in the sense that the value of the input variable is dilated (made bigger) by the operation, or contracted (made smaller), to give the result. But apparently this isn't the convention.
But that would mean that in the exact same physical scenario you could call it either contraction of dilation depending on the whims of what variable your teacher gave you first. In any case, the convention is also that the time dilation and length contraction equations are written in a form where the observer's frame is the output of the equation, although of course you can rearrange to solve for the proper time/proper length if you wish.
When you say "in terms of the observer's frame", this suggests that the variable would take a smaller value in the observer's frame, in the case of dilation
Dilation means an increase, so why do you say it suggests the variable would take a smaller value in the observer's frame? I said several times that when I said it was defined "in terms of the observer's frame", I meant that you used the word "dilation" if the value was bigger in the observer's frame, and "contraction" if the value was smaller in the observer's frame. If you think it's better to describe this as defining it "in terms of the moving frame", I would find that very confusing, but go ahead and do so as long as we're clear on the previous sentence.
The problem is that I'd have expected "dilation" to refer to the process of dilating the input to produce the output. Dilation seems to imply a starting point at which something is small, and an endpoint in the process at which it's bigger.
And again, the convention is that the moving frame is the "input", and in fact most problems will give you this first. But as I said I think it would be confusing to have the phrase dilation and contraction depend on the whims of which value a textbook or teacher provided you with first.
I'm aware that this isn't the convention, but I don't understand why not. So when we're given a value of 10 seconds, and calculate a value of 30 seconds from it, it seem only natural to call this operation dilation. But when the same term, dilation, is used of the inverse operaion, as here

http://galileoandeinstein.physics.virginia.edu/lectures/time_dil.html

that sounds bizarre to me. Our input is a big number, our output is a smaller number, and yet we're supposed to call this operation dilation as well (or multiplication by the "time dilation factor").
I understand the concern, but see above for why I think this alternate convention would be confusing. In any case, the issue of the convention is already decided for us.
The only inconsistency, it seems to me, is in talking as if there was something inherently dilatory about time, and something inherently contractory about space, in spite of the fact that we can and do dilate or contract either, depending on the context and what we want to find out from the equations.
But there is something inherently dilatory about the proper time between two events and something inherently contractory about the length of an object in its rest frame--do you agree? On the other hand, there is also something inherently dilatory about the proper distance between two spacelike-separated events, where "proper distance" refers to the distance between the events in the inertial frame where they're simultaneous (the distance in other frames will always be greater).
An example of this inconsistency is the way that Wolfram Alpha is obliged to reverse its definitions of moving and stationary depending on whether you want to transform a time interval or a space interval.
What question did you give to Wolfram Alpha, exactly?

Rasalhague

Jun10-09, 03:18 PM

Obviously, when you are dealing with an equation the "input" and "output" are entirely arbitrary. You can take ΔT = gamma*ΔT0 and rewrite it as ΔT0 = ΔT/gamma. If you choose to call ΔT your "input", then of course your "output" (ΔT0) will be smaller. So what???

The only reason I'm harping on about input and output is that I'm trying to identify some unambiguous way to label the frames that can be used whatever objects happen to be visualised in the interaction. Textbooks use a variety of symbols and names, and sometimes differ in their conventions. As a beginner, I've often found it hard to work out what are the fundamental properties of the process being exemplified, and which details are just accidental, arbitrary features of a particular example. Often we seemed to be talking at cross-purposes in these discussions for want of clear terms, although a lot of the confusion might just be due to my ignorance. Given that one of main themes of special relativity is the symmetry between inertial frames - each is moving from the perspective of the other, and physical laws operate identically in each - I thought of left and right as possible labels (if we want to define two frames and stick to the same names for them), and of input and output as another possible way of labelling the frames (if we want to refer to frames in terms of what role they're playing in a particular calculation or question). I thought these might be more general than terms like "lab frame" and "rocket frame" (which Wheeler and Taylor used), and more explicit as to what property of the frame was being used to identify it, and less ambiguous than "primed frame" and "unprimed frame". I chose to avoid terms like "lab" and "rocket" that suggest stasis and movement in order to wean myself off the intuitive, innate feeling we tend, as humans, to start out with that movement is an absolute property - or to avoid lapsing back into that way of thinking without realising it. Of course, I want to know about any more standard terms and what exactly they mean. If I can understand a better labelling system, I'll gladly use it. I'm just trying to feel my way into the subject and understand it as best I can from the diverse presentations of it that I've found.

As I said many times now, the reason it's called time dilation is that a lab frame observes a larger time interval than recorded by a moving clock. That's what "moving clocks run slow" means.

What makes a lab frame a lab frame, or a moving clock a moving clock, given that each of these frames is static from its own perspective, and moving from the perspective of the other frame? In other words, what are the defining features of a lab frame and a moving frame? When I asked whether they meant what I've been calling input and output frame, respectively, you said: "No, just the opposite." Accordingly, you went on to characterise the moving frame as input and the lab frame as output in the time dilation equation. You then gave an example of an application of what you called the "time dilation" equation in which the value recorded by clocks at rest in the lab frame was the input, and the value recorded by a clock at rest in the moving frame as the output. So my question stands: how are you defining these labels of the frames?

Note that even though you can reverse the equation to solve for either quantity given the other, that doesn't change the meaning of the quantities. ΔT0 is always the time interval recorded by the moving clock, thus ΔT > ΔT0.

Well, if the meaning of the quantities changed every time we inverted the equation, that really would be confusing... Then we could never ask the inverse question, whatever we called it!

Doc Al

Jun10-09, 03:34 PM

What makes a lab frame a lab frame, or a moving clock a moving clock, given that each of these frames is static from its own perspective, and moving from the perspective of the other frame? In other words, what are the defining features of a lab frame and a moving frame?
The terms "lab frame" and "moving frame" are relative terms, of course, since each observer views him or herself as stationary in his or her own frame. From your frame as an observer, if you see a clock moving with respect to you, then from your frame it is a moving clock. Simple as that! And from your frame you can observe the "time dilation" effect expressed as "moving clocks run slow".

Of course, observers in that other frame moving along with that clock can just as well observe a clock at rest in your frame. And to them your clock is a moving clock so "time dilation" applies; to them, your clock "runs slow".

Rasalhague

Jun10-09, 05:22 PM

How is "moment" different from "surface of simultaneity"

They're synonymous as far as I know. That's how I intended them, anyway.

Again, time dilation isn't defined in terms of readings at one particular moment/surface of simultaneity, it's defined by the interval of time between two specified events.

...and length contraction isn't defined by a single worldline, it's defined by the distance between two worldlines.

Okay, perhaps I should have use the more general, more explicit forms of the equations with delta symbols, rather than incorporating an event at the mutual origin of the two frames into the definition (clock's being synchronised as they pass; rulers aligned as their zero ends pass).

But that would mean that in the exact same physical scenario you could call it either contraction of dilation depending on the whims of what variable your teacher gave you first. In any case, the convention is also that the time dilation and length contraction equations are written in a form where the observer's frame is the output of the equation, although of course you can rearrange to solve for the proper time/proper length if you wish.

Dilation means an increase, so why do you say it suggests the variable would take a smaller value in the observer's frame? I said several times that when I said it was defined "in terms of the observer's frame", I meant that you used the word "dilation" if the value was bigger in the observer's frame, and "contraction" if the value was smaller in the observer's frame. If you think it's better to describe this as defining it "in terms of the moving frame", I would find that very confusing, but go ahead and do so as long as we're clear on the previous sentence.

When I asked what does it signify to call one frame the "observer's frame" and the other the "clock frame", you said:

"It signifies that we are talking about the time intervals in each frame between events which have been specifically selected to occur on the clock's worldline (so they are colocated in the clock's frame but not the observer's). In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame."

But in #375, in response to my description of Michael Fowler's example in terms of your definition of the "temporal analogue for length contraction", you redefined special:

"Yes, that actually works, provided you here treat Jill as "the observer" rather than Jack. I hadn't thought of it like this, but you're right that the measurements involved in an ordinary time dilation experiment like this can be re-interpreted as a TAFLC measurement, just by switching who we call "the observer", and by switching what defines the "special" frame from the frame where the time between two events on Jill's worldline is minimized (i.e. her own frame) to the frame where two spacelike surfaces that Jill passes through are surfaces of simultaneity (i.e. Jack's frame, since we were already considering his surfaces of simultaneity when showing how he would measure the time elapsed on Jill's clock)."

I'm a bit confused by this switching. When you use "observer's frame" and "clock frame" now are you going by your original first definition, or should I take them to have the second meaning sometimes, depending on the problem to be solved?

Yes the same physical scenario could be described with value A as input and value B as output, or vice versa, but what's so whimsical about that? I thought this is just what you advised me the convention was with the terms unprimed and primed. Input and output are more explicit names than unprimed and primed, given the various different uses that prime symbols are put to in this context by different textbooks. I thought they might be handy terms to distinguish between kinds of frames when talking in the abstract about the kinds of questions that can be asked of the formulas, but I agree it would be impractical to switch back and forth in the middle of working on a complex problem. If we want to use fixed labels for frames that don't vary depending on the question, we need use some other names, like left and right, or something that expressed the idea of - would it be correct to say - "the rest frame of the (spacetime) interval"? (I.e. what you were describing in your first definition of "special frame".)

And again, the convention is that the moving frame is the "input", and in fact most problems will give you this first. But as I said I think it would be confusing to have the phrase dilation and contraction depend on the whims of which value a textbook or teacher provided you with first.

So what do you make of Doc Al's example in #385, equivalent to Michael Fowler's with Jack and Jill, where the moving frame is the output frame? Is that unconventional? How would you express the problem in conventional terms? Would you just swap the names of the frames? What if you were making a series of calculations of various qualities back and forth between too frames; would you switch labels every time you needed to divide by gamma in moving from a frame that you'd previously multiplied by gamma in order to find a time value for? That sounds even more complicated to me than continually switching which frame we call the primed or output frame.

Does your observer's frame equate with Doc Al's lab frame, and your clock frame equate with Doc Al's moving frame? And if so, is that your observer's frame and clock frame as originally defined, or as redefined in the example that involved dividing my gamma which we described in terms of "temporal analogue for length contraction"?

I understand the concern, but see above for why I think this alternate convention would be confusing. In any case, the issue of the convention is already decided for us.

But there is something inherently dilatory about the proper time between two events and something inherently contractory about the length of an object in its rest frame--do you agree? On the other hand, there is also something inherently dilatory about the proper distance between two spacelike-separated events, where "proper distance" refers to the distance between the events in the inertial frame where they're simultaneous (the distance in other frames will always be greater).

Yes and, to complete the picture, something inherently contractory about the time period (whatever we call it) which bears the same relation to time as the length of an object does to space. The convention of matching up time dilation with length contraction, as somehow representative of time and space respectively, seems like someone holding up a whole apple and the core of an eaten pear and saying, "Look, apples are a whole fruit, but pears are eaten." Of course, being whole is no more a defining feature of apples than being eaten is a defining feature of pears; they aren't a whole kind of fruit and an eaten kind; either can be whole or eaten. Okay, that's an absurd analogy: no one's going to think of fruit that way. But because relativity is so counterintuitive when we first meet it, we don't know what the distinguishing properties of time and space might be. We don't have everyday experience of passing macroscopic objects at "relativistic" speed. So when we meet this combination of equations and their associated names, it's easy get confused or jump to the (mistaken) conclusion that the pairing directly embodies some fundamental difference or asymmetry between how time and space behave in special relativity, when really it's a matter of convention (albeit there might be reasons motivating that convention). A different pairing (a different convention), one that compared like with like as the full Lorentz transformation does, might avert that problem and make a better mnemonic. There would be no loss in calculating convenience, since we could carry on - as now - inverting either equation as required.

What question did you give to Wolfram Alpha, exactly?

I didn't ask a specific question. I just typed "time dilation" then toggled between "moving time" and "stationary time" in the "calculate" menu directly under the input field. Likewise with "length contraction" ("moving length", "stationary length"). It takes as its default input 1 second, in the case of time, and 1 meter, in the case of length.

Rasalhague

Jun10-09, 05:41 PM

The terms "lab frame" and "moving frame" are relative terms, of course, since each observer views him or herself as stationary in his or her own frame. From your frame as an observer, if you see a clock moving with respect to you, then from your frame it is a moving clock. Simple as that! And from your frame you can observe the "time dilation" effect expressed as "moving clocks run slow".

Of course, observers in that other frame moving along with that clock can just as well observe a clock at rest in your frame. And to them your clock is a moving clock so "time dilation" applies; to them, your clock "runs slow".

So are you saying that your lab frame and moving frame are completely arbitrary terms, and that we're free to choose which frame to call lab, and which moving, according to taste or convenience: either can be the input or the output frame? Or is the rule that we have to choose which frame is lab and which moving in such a way that the time value being transformed will always be bigger in the frame labelled lab? Would you say that your lab frame and moving frame are equivalent to Jesse's terms "observer's frame" and "clock frame" respectively? If not, how do they differ?

Suppose you were working on a complicated problem which involved taking inputs first from one frame, then from the other - would you switch the labels of the frames if need be to avoid having to reverse the convention that the time value in any calculation is bigger in the lab frame? Wouldn't that be potentially confusing? Or have I got this all wrong?

Doc Al

Jun10-09, 06:26 PM

So are you saying that your lab frame and moving frame are completely arbitrary terms, and that we're free to choose which frame to call lab, and which moving, according to taste or convenience: either can be the input or the output frame?
Forget about the terms "input" and "output"--they just add to the confusion. "Lab frame" and "moving frame" are relationship terms. If you are doing the measuring, then your frame is the lab frame; if a frame is moving with respect to you, then that frame is the "moving" frame.
Or is the rule that we have to choose which frame is lab and which moving in such a way that the time value being transformed will always be bigger in the frame labelled lab?
"Time value" is too vague a term.
Would you say that your lab frame and moving frame are equivalent to Jesse's terms "observer's frame" and "clock frame" respectively?
Absolutely. Calling the frame of the moving clock the "clock frame" makes it kind of easy to remember, doesn't it?
Suppose you were working on a complicated problem which involved taking inputs first from one frame, then from the other - would you switch the labels of the frames if need be to avoid having to reverse the convention that the time value in any calculation is bigger in the lab frame? Wouldn't that be potentially confusing? Or have I got this all wrong?
Generally, one calls the "moving" frame the primed frame (S') and the "lab" frame the unprimed frame (S). But the main thing is that each frame is moving with respect to the other. And you can, using the Lorentz transformations, transform measurements made in one frame to measurements made in the other. No need to "switch labels".

Careful with vague terms like "time value". There's no rule that all time intervals measured in the lab frame must be greater than the time interval in the moving frame. The rule is moving clocks run slow. A time interval recorded by a single moving clock will be smaller than the time interval measured in any other frame. It is certainly possible to choose events such that the time interval between them is greater in the moving frame--but such an interval does not correspond to an interval measured on a single clock (as JesseM might say, it does not represent a proper time).

Rasalhague

Jun10-09, 09:03 PM

Forget about the terms "input" and "output"--they just add to the confusion. "Lab frame" and "moving frame" are relationship terms. If you are doing the measuring, then your frame is the lab frame; if a frame is moving with respect to you, then that frame is the "moving" frame.

I found input and output helpful in digging myself out of some of the confusion I was in. If they're no use to anyone else, that's fine. They may not be convenient labels to give to a pair of frames in practice if we want to calculate back and forth between them, but at least they allow us to name that concept without going round in circles. I'm not saying there aren't better labels to use to distinguish frames on the basis of some other feature.

In the example we looked at with Jill in her rocket (carrying a clock) and Jack outside the rocket, also carrying a clock, and moving relative to Jill, we could label either of these as lab frame and moving frame, and when we do so, presumably we need to specify who or what they are the lab frame / moving frame of? I get the impression that by lab frame you mean the same as rest frame, is that right? Is Jill's lab frame synonymous with Jill's rest frame, and would the latter a more precise and self-explanitory term (given that a particular example might happen to involve an actual, physical laboratory at rest in some observer's "moving frame", in which case the term lab frame would become rather confusing)?

"Time value" is too vague a term.

What would you suggest? I left it vague because I didn't know whether your definition was precisely this inexact in its requirement. By time value I meant precisely: whatever the time values involved represent. But perhaps your definition of lab frame depends on some more specific kind of time value being bigger in the frame labelled lab.

Absolutely. Calling the frame of the moving clock the "clock frame" makes it kind of easy to remember, doesn't it?

I suppose it will be once I've worked out what exactly it is that it's making so easy to remember ;-)

By "the frame of the moving clock" do you mean "the rest frame of a clock which is moving in the rest frame of some other specified object or person, their rest frame being labelled the observer's frame of that object or person (or synonymously their lab frame)". Or do you mean it the other way around: a frame in which the clock that it's named after is moving, i.e. not that clock's rest frame. Maybe it would help if you could apply these labels to an example such as the one with Jack and Jill that we're already familiar with.

I imagine this label could get very confusing if there are clocks explicity visualised in both frames, or if the visualisation only involves one explicit clock altogether and that clock is at rest in the lab frame, or is it always possible to chose the terms so that this doesn't happen?

Generally, one calls the "moving" frame the primed frame (S') and the "lab" frame the unprimed frame (S). But the main thing is that each frame is moving with respect to the other. And you can, using the Lorentz transformations, transform measurements made in one frame to measurements made in the other. No need to "switch labels".

Ah, so here's yet another different usage of primed and unprimed frames to add to my list. So, in this system, we'd begin by making an arbitrary choice such as "let the lab frame in this example denote Jill's rest frame", and then stick to talking about "the lab frame", with this definition implicit, rather than talking variously about Jill's lab frame and Jack's lab frame?

Careful with vague terms like "time value". There's no rule that all time intervals measured in the lab frame must be greater than the time interval in the moving frame. The rule is moving clocks run slow. A time interval recorded by a single moving clock will be smaller than the time interval measured in any other frame. It is certainly possible to choose events such that the time interval between them is greater in the moving frame--but such an interval does not correspond to an interval measured on a single clock (as JesseM might say, it does not represent a proper time).

All time intervals represent the possible proper time of some notional clock though, don't they, in the sense that you could always imagine a clock following a certain trajectory whose proper time would be equal to that time interval - or is that the wrong way to look at it? Be that as it may, yes, I agree with what you say, and we have formulas to calculate from this proper time its time component in some other frame, which will be bigger than the proper time, and likewise from proper distance to the spatial component of some spacelike separation.

Why then is this operation, time dilation, conventionally paired not with its spatial equivalent, but with the inverse operation for space? Why not compare like with like?

To extend the metaphor I introduced in my previous post, it's as if someone was trying to teach people who've never seen apples and pears before about the nature of these fruit by holding up a whole apple and a mostly eaten pear and saying, "Today we're going to learn about the wholeness of apples and the eatenness of pears." Sure, this apple is whole, and this pear is eaten, but being whole or eaten isn't among the properties which distinguish apples from pairs, so as an introduction to apples and pairs it introduces an added, arbitrary complication which obscures both their similarity and the real differences between them.

I grant it's perfectly possible and conventional to avoid such a conceptualisation, but we match up time dilation with length dilation in the full Lorentz transformation, so why not match up with length contraction a time example of the sort we looked at conceptualised as time contraction, for aesthetic, pedagogic and mnemonic purposes?

As will be obvious, I'm new to this subject, and may well be missing something. I thank you all for your efforts at explaining this to me.

JesseM

Jun10-09, 10:17 PM

When I asked what does it signify to call one frame the "observer's frame" and the other the "clock frame", you said:

"It signifies that we are talking about the time intervals in each frame between events which have been specifically selected to occur on the clock's worldline (so they are colocated in the clock's frame but not the observer's). In any of these equations, the quantity we are dealing with takes a "special" value in one of the two frames--for example, if the quantity is the time interval between two events with a timelike separation, then this time interval is minimized in the frame where the two events are colocated, making that the "special" frame. The frame where the quantity does not take a special value is the one we have been calling the "observer's" frame."

But in #375, in response to my description of Michael Fowler's example in terms of your definition of the "temporal analogue for length contraction", you redefined special:

"Yes, that actually works, provided you here treat Jill as "the observer" rather than Jack. I hadn't thought of it like this, but you're right that the measurements involved in an ordinary time dilation experiment like this can be re-interpreted as a TAFLC measurement, just by switching who we call "the observer", and by switching what defines the "special" frame from the frame where the time between two events on Jill's worldline is minimized (i.e. her own frame) to the frame where two spacelike surfaces that Jill passes through are surfaces of simultaneity (i.e. Jack's frame, since we were already considering his surfaces of simultaneity when showing how he would measure the time elapsed on Jill's clock)."
I wasn't redefining "special", I was just saying that which frame is treated as the "special" one depends on conceptually what it is you say that you want to know the value of in both frames. If you want to know the value in both frames of the time interval between two events on Jill's worldline, in this case Jill's frame is the special one. If you want to know the time in both frames between two spacelike surfaces that happen to be surfaces of simultaneity in Jack's frame, then it's Jack's frame that's special. However, the idea of wanting to know "the time between two spacelike surfaces" is sort of a contrived idea that doesn't really ever come up in normal problems, it's much more natural to want to know the time between two particular events, like the events of Jill passing the two different clocks on that webpage. In any case the only two equations of this type this that have standard names are the time dilation equation and the length contraction equation, the other "equivalent" equations I introduced have no standard name so no one will understand what you mean if you try to refer to them by name.
Yes the same physical scenario could be described with value A as input and value B as output, or vice versa, but what's so whimsical about that?
It's not whimsical that you could be given either as input, it's whimsical that you would change the name of the equation you use to get the answer based on which you happen to know first. Communication is much less confusing if you adopt a naming convention that allows you to call the equation by the same name all the time. As an analogy, we call E=mc^2 the "mass-energy equivalence equation", it would be confusing if there were two separate names for it depending on whether you knew the mass and wanted to find the energy or if you knew the energy and wanted to find the mass.
I thought this is just what you advised me the convention was with the terms unprimed and primed.
No, the usual convention is that unprimed is the frame where the time or length is proper time or rest length, primed is the frame where the clock or object is in motion.
Input and output are more explicit names than unprimed and primed, given the various different uses that prime symbols are put to in this context by different textbooks.
But "input" and "output" don't refer to anything physical, they just refer to which quantity you happen to have been given first. If the goal is communication, don't you want words for equations and the symbols that appear in them to refer to physical details?
If we want to use fixed labels for frames that don't vary depending on the question, we need use some other names, like left and right, or something that expressed the idea of - would it be correct to say - "the rest frame of the (spacetime) interval"? (I.e. what you were describing in your first definition of "special frame".)
I don't think it makes sense to talk about an interval having a rest frame since it isn't an object that persists over time, but when talking about an interval between a pair of events perhaps you could say something like "the co-location frame" (though this isn't a standard term). Normally in these types of problems the two events in question are events on the worldline of some object like a clock or Jill's ship, so you can tailor the description to the problem and say things like "the clock rest frame" or "Jill's rest frame".
So what do you make of Doc Al's example in #385, equivalent to Michael Fowler's with Jack and Jill, where the moving frame is the output frame? Is that unconventional?
No, with any equation of two variables you're free to take either variable as input and use it to find the value of the other variable as output.
What if you were making a series of calculations of various qualities back and forth between too frames; would you switch labels every time you needed to divide by gamma in moving from a frame that you'd previously multiplied by gamma in order to find a time value for? That sounds even more complicated to me than continually switching which frame we call the primed or output frame.
In a problem with multiple objects you could again just denote different frames based on which object's rest frame they were.
Does your observer's frame equate with Doc Al's lab frame, and your clock frame equate with Doc Al's moving frame?
Yes.
And if so, is that your observer's frame and clock frame as originally defined, or as redefined in the example that involved dividing my gamma which we described in terms of "temporal analogue for length contraction"?
No, because again, the name you use for the equation has nothing to do with which variable you put in as input. Doc Al was just calculating the time between a single pair of events in two different frames, so that means he was using the time dilation equation.
Yes and, to complete the picture, something inherently contractory about the time period (whatever we call it) which bears the same relation to time as the length of an object does to space. The convention of matching up time dilation with length contraction, as somehow representative of time and space respectively
I don't think there is any such convention, I've never seen anyone say they are "representative of time and space" or anything along those lines. They are just representative of what they actually give you, namely the time between two events in different frames and the length of an object in two different frames. Both of these are quantities that actually come up regularly in ordinary SR problems, whereas there are fewer situations where you'd want to know the distance between two spacelike-separated events in two frames, and I can't think of any non-contrived situations where you'd be directly interested in finding out the time between two parallel spacelike surfaces in two frames.
But because relativity is so counterintuitive when we first meet it, we don't know what the distinguishing properties of time and space might be. We don't have everyday experience of passing macroscopic objects at "relativistic" speed. So when we meet this combination of equations and their associated names, it's easy get confused or jump to the (mistaken) conclusion that the pairing directly embodies some fundamental difference or asymmetry between how time and space behave in special relativity, when really it's a matter of convention (albeit there might be reasons motivating that convention).
Yes, I do agree with you there--when presenting the two equations it'd be a good idea to point out that one shouldn't jump to the sort of conclusion you describe that they reflect some basic difference between time and space, since this is probably not an uncommon misunderstanding.
I didn't ask a specific question. I just typed "time dilation" then toggled between "moving time" and "stationary time" in the "calculate" menu directly under the input field. Likewise with "length contraction" ("moving length", "stationary length"). It takes as its default input 1 second, in the case of time, and 1 meter, in the case of length.
OK, but I don't understand what you meant when you talked about an "inconsistency" here:
An example of this inconsistency is the way that Wolfram Alpha is obliged to reverse its definitions of moving and stationary depending on whether you want to transform a time interval or a space interval.
What do you mean "reverse its definitions"? If the time dilation equation is understood to give you the time in two frames between events that occur on the worldline of a clock, and the length contraction equation is understood to give you the length of a ruler in two frames, then in both cases Wolfram Alpha uses "stationary" to refer to the frame in which the clock/ruler is at rest and "moving" to refer to the frame where the clock/ruler is in motion.

atyy

Jun10-09, 10:43 PM

Communication is much less confusing if you adopt a naming convention that allows you to call the equation by the same name all the time. As an analogy, we call E=mc^2 the "mass-energy equivalence equation", it would be confusing if there were two separate names for it depending on whether you knew the mass and wanted to find the energy or if you knew the energy and wanted to find the mass.

There's a cute story about this in Wilczek's Nobel lecture:
"My friend and mentor Sam Treiman liked to relate his experience of how, during World War II, the U.S. Army responded to the challenge of training a large number of radio engineers starting with very different levels of preparation, ranging down to near zero. They designed a crash course for it, which Sam took. In the training manual, the first chapter was devoted to Ohm’s three laws. Ohm’s first law is V = IR. Ohm’s second law is I = V/R. I’ll leave it to you to reconstruct Ohm’s third law.
Similarly, as a companion to Einstein’s famous equation E = mc2 we have his second law, m = E/c2."
http://nobelprize.org/nobel_prizes/physics/laureates/2004/wilczek-lecture.pdf

neopolitan

Jun11-09, 04:38 AM

Rasalhague

Jun11-09, 07:55 AM

I wasn't redefining "special", I was just saying that which frame is treated as the "special" one depends on conceptually what it is you say that you want to know the value of in both frames. If you want to know the value in both frames of the time interval between two events on Jill's worldline, in this case Jill's frame is the special one. If you want to know the time in both frames between two spacelike surfaces that happen to be surfaces of simultaneity in Jack's frame, then it's Jack's frame that's special. However, the idea of wanting to know "the time between two spacelike surfaces" is sort of a contrived idea that doesn't really ever come up in normal problems, it's much more natural to want to know the time between two particular events, like the events of Jill passing the two different clocks on that webpage. In any case the only two equations of this type this that have standard names are the time dilation equation and the length contraction equation, the other "equivalent" equations I introduced have no standard name so no one will understand what you mean if you try to refer to them by name.

You want to know the interval between two events on Jill's worldline, given that you already know its time component in a frame where Jill is not at rest. The time component is minimised in Jill's rest frame. Jill's rest frame is special. Time dilation.

You want to know the time component of the spacetime interval between two events on Jill's worldline, given that you already know the interval. The time component is minimised in Jill's rest frame. Jill's rest frame is special. Time dilation inverse. ...Unless you chose to conceptualise the same relation in a subjectively different way, as "the temporal analogue of length contraction" (time/period/while contraction), in which case Jack's frame is special.

I guess what's changed is that, in the original concept, special frame (=moving frame =clock frame?) was defined as: the frame where the value is minimum if we're talking about a time coordinate; the frame where the value is maximum if we're talking about a space coordinate. In the revised, or expanded, definition, we add the proviso that the space definition will apply to a time calculation if-and-only-if we make an arbitrary decision to view that calculation in this certain subjective way (which doesn't affect the calculation itself). That seems like a sort of redefnition to me. Or have I misunderstood?

Do you switch labels for frames if you have a situation involving a clock and ruler at rest in the same frame if, having first performed a calculation of the sort we call time dilation (according to the conventional definition of time dilation), you then want to perform a calculation of the sort we call length contraction (according to the conventional definition of length contraction). And might that not get confusing, given that it's the same physical situation we're looking at?

E.g. suppose you follow the convention you outlined for a time calculation (time dilation) and label Jill's rest frame the clock/moving frame, and Jack's rest frame the observer/lab frame. Having got the answer to that, suppose you then want to know how long Jill's rocket is in Jill's rest frame from Jack's measurement of it, or you want to know how long Jack will measure Jill's rocket to be, given that at has such-and-such a length in Jill's rest frame. Does the convention dictate that you reverse the labels you gave to the frames when you were performing a time calculation simply because now Jill's rest frame is special, because special means "where the coordinate is maximum" in the case of space?

It's not whimsical that you could be given either as input, it's whimsical that you would change the name of the equation you use to get the answer based on which you happen to know first. Communication is much less confusing if you adopt a naming convention that allows you to call the equation by the same name all the time. As an analogy, we call E=mc^2 the "mass-energy equivalence equation", it would be confusing if there were two separate names for it depending on whether you knew the mass and wanted to find the energy or if you knew the energy and wanted to find the mass.

Then why do we have two different names for the same equation (dilation, contraction) that depend on which coordinate is being computed? It's as if we had a convention when using Cartesian coordinates of only calling multiplication of the y component of a vector "multiplication", and insisting on calling the same operation "inverse division" when we perform it on an x component.

No, the usual convention is that unprimed is the frame where the time or length is proper time or rest length, primed is the frame where the clock or object is in motion.

But what, in general, defines a clock as "the clock"?

But "input" and "output" don't refer to anything physical, they just refer to which quantity you happen to have been given first. If the goal is communication, don't you want words for equations and the symbols that appear in them to refer to physical details?

The whole point of them is that they don't refer to anything physical. I chose them to explicitly communicate that. Lab and clock and rocket and observer are all physical objects. I appreciate that they may be used by convention to refer to a frame defined by the role it plays in the calculation, or by a free choice of how to visualise the situation, but until I've understood, absorbed and become familiar with that convention, they seem cumbersome in that they inevitably conjure up images of labs and rockets and clocks and observers which may either not correspond directly to the way a particular problem is worded, or - even worse - actually clash with the way the problem is worded, e.g. if the problem happened to involve an actual physical lab in the frame you've defined as the moving frame, or two physical labs, or a whole bunch of explicit, physical clocks, in which case we need a rule to define which clock is going to be called "the clock" and which lab "the lab" and which observer "the observer". That's why I'm looking for some general, abstract terms that would express what we're doing mathematically in any such problem, to see which kinds of operation are really the same mathematically, underneath all the varied trappings of clocks and labs and rockets and identical twins that change from problem to problem, so that I'll have a general terminology with which to view any problem of this kind, no matter what the parochial details are.

I don't think it makes sense to talk about an interval having a rest frame since it isn't an object that persists over time, but when talking about an interval between a pair of events perhaps you could say something like "the co-location frame" (though this isn't a standard term). Normally in these types of problems the two events in question are events on the worldline of some object like a clock or Jill's ship, so you can tailor the description to the problem and say things like "the clock rest frame" or "Jill's rest frame".

I'd certainly like to have so way to to express the concept: collocation frame and contemporary frame, or something like that. Is there just no standard term at all? I think we're touching on something really interesting here which is related to a genuine difference between time and space, or our ways of relating to them, namely that the term "rest frame" doesn't bear the same relation to time as it does to space. With time dilation and length contraction, in both cases, we're talking about physical object with well defined spatial limits, and in both cases we're talking about a physical object that persists indefinitely in time. So when we define dilation or contraction in terms of an object's rest frame, we naturally find that time and space seem to be behaving differently.

No, with any equation of two variables you're free to take either variable as input and use it to find the value of the other variable as output.

In a problem with multiple objects you could again just denote different frames based on which object's rest frame they were.

So when we use labels like "lab/observer's" or "moving/clock's/special", these being as Doc Al said relative terms, we always have to explicitly state "Jill's lab frame" = "Jill's rest frame" - is that right? Does this not become confusing if the problem is such that we're forced to talk about "this clock's clock frame" and "that clock's lab frame", or "that observer's observer's frame", "this observer's clock's clock frame", etc.? Well, I suppose if the two names are settled on at the outset and stated in the definition, it might not get that bad - but you'd still be thinking implicitly in those terms, wouldn't you, and if you got confused, you might find yourself trying to work it out in such terms in order to "clarify" how the elements of the problem relate to each other, or else revert to unambiguously, explicitly relative terms like rest frame.

No [in answer to "[...] is that your observer's frame and clock frame as originally defined, or as redefined in the example that involved dividing my gamma which we described in terms of "temporal analogue for length contraction"?], because again, the name you use for the equation has nothing to do with which variable you put in as input. Doc Al was just calculating the time between a single pair of events in two different frames, so that means he was using the time dilation equation.

It does depend on "which variable" in a different sense though: namely it depends on whether it's a temporal or spatial variable, hence "calculating the time [...] so that means he was using the time dilation equation". If he'd been calculating the same relation with respect to length, I suppose he'd have used the "length contraction" equation, which is the same equation!

I don't think there is any such convention, I've never seen anyone say they are "representative of time and space" or anything along those lines. They are just representative of what they actually give you, namely the time between two events in different frames and the length of an object in two different frames. Both of these are quantities that actually come up regularly in ordinary SR problems, whereas there are fewer situations where you'd want to know the distance between two spacelike-separated events in two frames, and I can't think of any non-contrived situations where you'd be directly interested in finding out the time between two parallel spacelike surfaces in two frames.

But didn't you agree in #375 that any perfectly regular situation that involves what you call "inverse time dilation" can also be conceptualised as "the temporal analogue of length contraction" and that "the type of conceptual distinction" between the two "is really more of an aesthetic preference [...] kind of a matter of taste"? So it's not that there are a set of contrived situations that are the only application of this, and no non-contrived situations. Rather it's idea of conceptualising the relation as time contraction that seems contrived to you, and what seems contrived to me is the idea of conceptualising the same equation for transforming coordinates as two different operations with different names that depend only on which coordinate is being transformed. But maybe I'm exaggerating the contrivedness of that, given the genuine differences between time and space that might motivate us not to pair like with like.

OK, but I don't understand what you meant when you talked about an "inconsistency" here:

What do you mean "reverse its definitions"? If the time dilation equation is understood to give you the time in two frames between events that occur on the worldline of a clock, and the length contraction equation is understood to give you the length of a ruler in two frames, then in both cases Wolfram Alpha uses "stationary" to refer to the frame in which the clock/ruler is at rest and "moving" to refer to the frame where the clock/ruler is in motion.

I'm not saying that it's inconsistent within those (conventional) terms. It's internally consistent. It's the correct terminology, given those definitions. I'm just saying that the pairing of those defenitions seems arbitrary. If we wanted to emphasise the interchangeability of time and space, we could define which clock is moving so that clicking on "moving" coordinate resulted in multiplication by gamma, in each case, and selecting "stationary" coordinate resulted in division by gamma, in each case. Or substitute for moving and stationary whatever pair of terms is suitable (minimally ambiguous, maximally general, associated with the same mathematical operation).

But maybe there is a flaw in that plan. It's easy to talk about a moving length because we think of length as an inherent, persistent attribute of an object. Because of this, we can easily imagine a "moving length". We use length in two ways: in the abstract as a measurement, and more concretely as a persistant, physical property of a specified object. On the other hand, what would it mean to say "a moving while" or "a moving period"? We have no intuition. That creates a hurdle for anyone trying to make a neat, mnemonic pairing of the sort "a moving [INSERT SPATIAL PROPERTY HERE] does this, a moving [INSERT TEMPORAL PROPERTY HERE] does the same". Then again, at least the lack of an existing intuition makes a clear mental gap to fill with the appropriate definition.

JesseM

Jun11-09, 04:59 PM

You want to know the interval between two events on Jill's worldline, given that you already know its time component in a frame where Jill is not at rest. The time component is minimised in Jill's rest frame. Jill's rest frame is special. Time dilation.

You want to know the time component of the spacetime interval between two events on Jill's worldline, given that you already know the interval. The time component is minimised in Jill's rest frame. Jill's rest frame is special. Time dilation inverse.
Yeah, but "time dilation inverse" just means rearranging terms in the time dilation equation without changing their physical meaning, so I wouldn't really consider it a physically distinct equation, "inverse" is just an adjective to indicate that the equation has been rearranged in this way.
...Unless you chose to conceptualise the same relation in a subjectively different way, as "the temporal analogue of length contraction" (time/period/while contraction), in which case Jack's frame is special.
Yeah, but as you pointed out the difference is kind of cosmetic, any given problem involving clocks can be thought of in either way. And in practice no one ever thinks in terms of the time between spacelike surfaces as opposed to the time between events, so if you find it confusing to talk about "the temporal analogue of length contraction" it might be better to just forget the whole thing and assume by default that in any problem involving clocks the thing you're interested in is the time between events on some clock's worldline.
I guess what's changed is that, in the original concept, special frame (=moving frame =clock frame?) was defined as: the frame where the value is minimum if we're talking about a time coordinate; the frame where the value is maximum if we're talking about a space coordinate.
Not "a space coordinate". A length. Remember that you can also talk about the distance between two spacelike-separated events, in which case the value is minimum in the frame where they're simultaneous.
In the revised, or expanded, definition, we add the proviso that the space definition will apply to a time calculation if-and-only-if we make an arbitrary decision to view that calculation in this certain subjective way (which doesn't affect the calculation itself). That seems like a sort of redefnition to me. Or have I misunderstood?
The point about "specialness" was just to explain the conceptual similarity between time dilation and length contraction, since in time dilation the time between events is greater (dilated) in the non-special frame, and in length contraction the length is smaller (contracted) in the non-special frame. "Special" is not some officially defined term with a rigorous meaning, it's just my way of saying why I think the terminology is consistent. It's true that a calculation involving clocks can in principle be conceptualized in terms of the time between two spacelike surfaces in both frames rather than the time between two events in both frames, but this is a very contrived-seeming way of conceptualizing it that no one ever does in practice. Likewise you can conceptualize the ordinary length contraction equation in terms of the distance between two events on the front and back worldline of an object that are simultaneous in the frame where the object is moving, making this frame the "special" one, but in practice most people conceptualize it just as the length of the object in both frames since that's an a more natural way to think about it. It makes sense that the terminology would naturally reflect the way that's most natural to conceptualize what it is that's being calculated, doesn't it? Perhaps you're confusion is that you're trying to understand a mere naming convention for two equations as having some super-rigorous justification, when it's really just an aesthetic choice that arises from how the equations are normally used and conceptualized in practice, you could call them the "time whatsit equation" and the "length whosit equation" if you preferred.
Do you switch labels for frames if you have a situation involving a clock and ruler at rest in the same frame if, having first performed a calculation of the sort we call time dilation (according to the conventional definition of time dilation), you then want to perform a calculation of the sort we call length contraction (according to the conventional definition of length contraction).
Keep in mind there are no official "labels" for frames, just ones that I've introduced for the purpose of explaining the thought processes behind the names of the equations. But if we stick to the convention of special/non-special or moving frame/observer's frame, why do you think we'd have to switch labels? If the clock and ruler are at rest in the same frame, then their rest frame would be the special or moving frame according to the labels I've introduced, and the frame where they were moving would be the non-special or observer's frame.
E.g. suppose you follow the convention you outlined for a time calculation (time dilation) and label Jill's rest frame the clock/moving frame, and Jack's rest frame the observer/lab frame. Having got the answer to that, suppose you then want to know how long Jill's rocket is in Jill's rest frame from Jack's measurement of it, or you want to know how long Jack will measure Jill's rocket to be, given that at has such-and-such a length in Jill's rest frame. Does the convention dictate that you reverse the labels you gave to the frames when you were performing a time calculation simply because now Jill's rest frame is special, because special means "where the coordinate is maximum" in the case of space?
Again I don't see why there'd be a need to switch labels. Just as the frame where the time between events on Jill's worldline is minimized would be Jill's rest frame, so the frame where the length of Jill's rocket is maximized would also be Jill's rest frame. In each case I would therefore call Jill's frame the special/moving frame.
Then why do we have two different names for the same equation (dilation, contraction) that depend on which coordinate is being computed?
Not coordinates. What's being computed is either a time interval or a length. And conceptually, the difference in terms is just based on whether the thing being a computed increases in the observer's "non-special" frame or whether it decreases in this frame.
It's as if we had a convention when using Cartesian coordinates of only calling multiplication of the y component of a vector "multiplication", and insisting on calling the same operation "inverse division" when we perform it on an x component.
Is there anything in this analogy corresponding to the notion of their being one particular frame where the quantity in question takes a special value? If not I don't see the relevance.
But what, in general, defines a clock as "the clock"?
That the pair of events you're calculating the time between both occur on this particular clock's own worldline.
The whole point of them is that they don't refer to anything physical. I chose them to explicitly communicate that. Lab and clock and rocket and observer are all physical objects. I appreciate that they may be used by convention to refer to a frame defined by the role it plays in the calculation, or by a free choice of how to visualise the situation, but until I've understood, absorbed and become familiar with that convention, they seem cumbersome in that they inevitably conjure up images of labs and rockets and clocks and observers which may either not correspond directly to the way a particular problem is worded, or - even worse - actually clash with the way the problem is worded, e.g. if the problem happened to involve an actual physical lab in the frame you've defined as the moving frame, or two physical labs, or a whole bunch of explicit, physical clocks, in which case we need a rule to define which clock is going to be called "the clock" and which lab "the lab" and which observer "the observer". That's why I'm looking for some general, abstract terms that would express what we're doing mathematically in any such problem, to see which kinds of operation are really the same mathematically, underneath all the varied trappings of clocks and labs and rockets and identical twins that change from problem to problem, so that I'll have a general terminology with which to view any problem of this kind, no matter what the parochial details are.
As I said, if you're trying to make it so the conventions about terminology can be justified in a super-rigorous way then you're going up a blind alley. But I don't see it as particularly confusing; it doesn't matter if multiple clocks may be present if you're calculating the time between events on the worldline of one specific clock, and likewise it doesn't matter if multiple rulers or other physical objects are present if you're calculating the length of a specific one. If you don't like that, what about the alternate special/non-special terminology I introduced for the two frames? If you're calculating the time between two events, isn't it unambiguous which frame is the "special" one where the time between them is minimized?
I don't think it makes sense to talk about an interval having a rest frame since it isn't an object that persists over time, but when talking about an interval between a pair of events perhaps you could say something like "the co-location frame" (though this isn't a standard term). Normally in these types of problems the two events in question are events on the worldline of some object like a clock or Jill's ship, so you can tailor the description to the problem and say things like "the clock rest frame" or "Jill's rest frame".
I'd certainly like to have so way to to express the concept: collocation frame and contemporary frame, or something like that. Is there just no standard term at all?
Not that I know of.
I think we're touching on something really interesting here which is related to a genuine difference between time and space, or our ways of relating to them, namely that the term "rest frame" doesn't bear the same relation to time as it does to space. With time dilation and length contraction, in both cases, we're talking about physical object with well defined spatial limits, and in both cases we're talking about a physical object that persists indefinitely in time. So when we define dilation or contraction in terms of an object's rest frame, we naturally find that time and space seem to be behaving differently.
But again, although this may be a confusion sometimes experienced by students when they see the two equations, I don't think any textbook author intends any sort of implication that the equations prove that "time and space seem to be behaving differently".
So when we use labels like "lab/observer's" or "moving/clock's/special", these being as Doc Al said relative terms, we always have to explicitly state "Jill's lab frame" = "Jill's rest frame" - is that right?
It depends on the context. If only one observer or lab is mentioned in the problem, in that case it's fine to just say "the lab frame" or "the observer's frame" and the meaning will be clear.
Does this not become confusing if the problem is such that we're forced to talk about "this clock's clock frame" and "that clock's lab frame", or "that observer's observer's frame", "this observer's clock's clock frame", etc.?
"Clock frame" and "lab frame" and "observer's frame" are not official terms that you have to use in any situation, and they make very little sense the way you use them above (I don't know what a 'clock's clock frame' even means, for example). If there are multiple clocks, then presumably they are given different names, and you can just talk about "[NAME X]'s rest frame" or even just "[NAME X]'s frame". All that's important is that you talk in a way that it's clear from the context of the problem what frame you're referring to.
It does depend on "which variable" in a different sense though: namely it depends on whether it's a temporal or spatial variable, hence "calculating the time [...] so that means he was using the time dilation equation". If he'd been calculating the same relation with respect to length, I suppose he'd have used the "length contraction" equation, which is the same equation!
Again, conceptually it makes sense to me to use this terminology, since in the case of time intervals between events on the worldline of an object (which is usually a clock though it doesn't have to be) the time interval will be larger in the non-special frame where the object is moving, and in the case of the length of some object the length will be smaller in the non-special frame where the object is moving. But it is just a naming convention, it just has to do with how people tend to conceptualize the equations and is not meant to have an ultra-rigorous justification.
But didn't you agree in #375 that any perfectly regular situation that involves what you call "inverse time dilation" can also be conceptualised as "the temporal analogue of length contraction" and that "the type of conceptual distinction" between the two "is really more of an aesthetic preference [...] kind of a matter of taste"?
Yes, and naming conventions are based on how things are actually conceptualized in practice, and I imagine there are very few people who regularly conceptualize time dilation problems in terms of the time between two spacelike surfaces.
So it's not that there are a set of contrived situations that are the only application of this, and no non-contrived situations. Rather it's idea of conceptualising the relation as time contraction that seems contrived to you,
More specifically, conceptualizing what the equation is telling you as "the time between two spacelike surfaces" which just happen to pass through whatever events are mentioned in the problem, rather than the time between the two events themselves, seems contrived to me.
and what seems contrived to me is the idea of conceptualising the same equation for transforming coordinates as two different operations with different names that depend only on which coordinate is being transformed.
But neither equation "transforms coordinates" of single events, not unless you introduce the artificial condition that the event in question is either on one system's time axis, or that it's on one system's space axis. The time dilation equation is ordinarily understood to transform time intervals between pairs of events, and the length contraction is ordinarily understood to transform lengths (i.e. the spatial distances between two worldlines in each frame). If you want to make them more analogous you can either conceptualize the time equation in a weirder way where it's transforming the time between two spacelike surfaces in each frame, or conceptualize the spatial equation in a weirder way where it's transforming the distance between two specific events on the two worldlines in the frame where those worldlines have a constant position. But in practice people don't normally conceptualize them in either of these last two ways.
I'm not saying that it's inconsistent within those (conventional) terms. It's internally consistent. It's the correct terminology, given those definitions. I'm just saying that the pairing of those defenitions seems arbitrary.
What do you mean by "pairing"? It's not like they are introduced as analogous to one another, it's just that in combination they are two useful equations that make it possible to deal with many types of relativity problems that would otherwise require you to use the full Lorentz transformation equations. Personally I like to mention them along with a third equation that deals with the relativity of simultaneity, saying that if two clocks are a distance d apart and synchronized in their own rest frame, then in a frame where they are moving at speed v along the axis between them, they will be out-of-sync by vd/c^2.
If we wanted to emphasise the interchangeability of time and space, we could define which clock is moving so that clicking on "moving" coordinate resulted in multiplication by gamma, in each case, and selecting "stationary" coordinate resulted in division by gamma, in each case.
"Moving" and "stationary" are not official terms for the two frames dealt with by the time dilation and length contraction equations, they are just understood in terms of the context of a particular problem. If you just have one observer who is trying to calculate the length of a ruler moving relative to him and the time interval between two readings on a clock moving relative to him, it would be confusing to switch whether the observer or the ruler/clock is called "stationary" or "moving".

Perhaps it would help if I say that pedagogically, the point of introducing these equations has nothing at all to do with "emphasizing the interchangeability of time and space", the point is that they are helpful when actually doing calculations about specific word-problems. Switching the terminology in the manner you suggest would make it more confusing to try to apply them to specific word-problems.

neopolitan

Jun12-09, 01:26 AM

A comment on post #397 (http://www.physicsforums.com/showpost.php?p=2232756&postcount=397) which may go some way to explaining why I think my derivation holds generally.

The final two equations I arrive at (spatial and temporal) are:

x'_b = \gamma.( x_a - v . t_a)

and

\Delta t' = \gamma . ( \Delta t - v.x_a / c^2 )

In words:

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) )

and

(how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame) = gamma * ((how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened), in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

I want to highlight that in each case there is an interval, explicit or implied, between an event about which there is agreement in all frames and one other event.

We discuss the scenario in terms of there being a colocation of A and B, which is going to be agreed by both A and B - at least at the level that "A and B were colocated where they were colocated at the time at which they were colocated". Really, that is all they know.

Each of A and B are likely to have a coordinate system established before they are colocated which does not result in their colocation being at (0,0). Making their colocation (0,0) is handy, but by no means essential.

The point is that this colocation of A and B is an event which A and B have some level of agreement about. And there must be some such event - and it doesn't have to be a colocation.

Say A and B bump into each other in the street, literally, in their cars.

The coordinate system they use could be such that this collision is (0,0,0,0), but in reality we know it is not likely to be.

Much more likely it will be something like (t=number of days, hours and minutes since a notional event, h=ground level, N/S=degrees from the equator, E/W=degrees from the Greenwich Meridian) or (t=number of days, hours and minutes since a notional event, h=ground level, x=distance along a road from a specific junction, y=distance from the edge of the road on one side)

Hopefully this is so blatantly obvious that it doesn't really need more emphasis.

So, we have an interval between one event for which the coordinates are agreed (nominally (0,0)) and another event for which the coordinates are not agreed.

If the event for which there is agreement is the nominal (0,0), then the intervals are also coordinates.

In my scenario, the event which is agreed is colocation of A and B (0,0).

In the final equations the (possibly implied) intervals are:

(spatial interval between where A and B were colocated and where the photon was when A and B were colocated, in the B frame) = gamma * ((spatial interval between where A and B were colocated and where the photon was when A and B were colocated, in the A frame) - (speed of A in the B frame) * (time interval between colocation of A and B and colocation of A and photon in the A frame) )

and

(time interval between when A and B were colocated and when the event took place, in the B frame) = gamma * (time interval between when A and B were colocated and when the event took place, in the A frame) - (speed of A in the B frame) * (spatial interval between where A and B were colocated and where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

To make this general, A and B just need to agree on a different event. Conceptually, I know this works but while proving it mathematically won't be impossible, it might be messy.

cheers,

neopolitan

PS I am aware that to be totally consistent, I should express the spatial equation in delta format, but I think you can understand that this would be trivial mathematically.

Rasalhague

Jun12-09, 10:24 AM

Not "a space coordinate". A length. Remember that you can also talk about the distance between two spacelike-separated events, in which case the value is minimum in the frame where they're simultaneous.

And yet the way we express length is in terms of spatial coordinates, albeit not events that bear the same relationship to each other as the events in "time dilation", as normally conceived. But this led me to thinking: we now have abstract geometrical descriptions of length contraction and its temporal analogue, and a colloquial description of length contraction. To complete the picture, we'd need a colloquial way of expressing "time contraction". Of course, it may say something about the difference between space and time that we don't have such a description, or find it less intuitive, or less intuitively necessary - but still I'd like to have a try.

A physical object like a clock doesn't bear the same realtion to time, in all its particlars, as a ruler bears to space. Clock and ruler are both sharply bounded in space; both persist indefinitely in time. No wonder the symmetry between time and space is obscured if we treat them (or inadvertently let them appear by convention) as if a clock is, in all relevant respects, to time as a ruler is to space. The length of a ruler in different frames is determined by the changing relationship, in their different coordinate systems, between two worldlines (those of its ends), whereas, in the traditional conceptualisation of time dilation, we're instead talking simply about the changing relationship between one pair of points as we change the frame use to describe them. But what if we were to conceptualise this same situation in terms of the duration of a journey (as the temporal equivalent of the length of an object)?

Just as understanding of length in special relativity requires additional definitions beyond our naive intuitions about length, so too any definition of the "duration of a journey" would involve some additional convention to be defined. In fact, I've wondered at times whether the very naturalness of the idea of length is, in some sense, beguilingly natural. That is, it's all too easy as beginners to see that familiar word and think we know what it means, which can lead to paradoxes until we realise that the relativistic definition of length depends on concepts such as the relativity of simultaneity, for which we have no naive intuition. We're used to the idea of objects shrinking in everyday life, but length contraction in relativity isn't quite the same thing. Of course, the same criticism could be levelled at "duration contraction" or "travel-time contraction", which, aside from definitions, is probably every bit as ambiguous a name as the alternatives.

That said, here's my attempt at parallel geometric and colloquial definitions:

*Edit: I got a bit muddled with these next two paragraphs: see #403 for revised definitions. I'll leave these here though for the sake of continuity.

Length Contraction. The spacelike interval covered by the segment of a line of synchrony/simultaneity/now between its intersection with two worldlines in a frame where the worldlines are oblique compared to the unique frame where they're parallel to the x axis. (Colloquially: the length of an object is greatest in the unique frame where its ends are at rest. Restriction: in the frame where the ends of the object are moving, we must measure the position of both ends at the same time.)

Duration Contraction (travel-time/journey-time contraction). The timeline interval covered by the segment of a line of syntopy/collocality/here (a worldline) between two hypersurfaces of synchrony in a frame where the hypersurfaces of synchrony are oblique compared to the unique frame where they're parallel to the t axis. (Colloquially: the duration of a journey is greatest in the unique frame where its ends are at rest. Restriction: in the frame where the ends of the journey are moving, we must meaure the time of both ends in the same place.)

*

For example, last night, I worked through problem 29 in Taylor/Wheeler: Spacetime Physics, the purpose of which is to demonstrate the relativity of simultaneity, and how that relates to the problem of synchronising clocks. A pair of clocks are synchronised at the spacetime coincidence of their passing. One clock they call Big Ben, the other is being carried by a Mr Engelsberg. After some time Mr Engelsberg comes to, a third clock, called Little Ben, at rest with respect to Big Ben, and synchronises Little Ben to the time shown by the clock he's carrying. The question asks how much will Little Ben lag behind Big Ben once it's been set to the time shown by Mr Engelberg's clock as he passes.

They show multiple ways of solving the problem, one of which begins by thinking of the time shown on Mr Engelsberg's clock as he passes Little Ben (journey's end) as the interval between this event and his passing Big Ben. We calculate this interval from its time coordinate in the rest frame of Big Ben and Little Ben. In terms of "duration/travel-time contraction", this is the frame in which the journey takes place, since by definition Mr Engelsberg does no travelling in his own rest frame. The duration of Mr Engelsberg's journey between Big Ben showing a certain time and Little Ben showing some other time is biggest in the unique frame where that journey takes place. We can define this frame, as above, in a way that bears the same relation to time as the rest frame of an object bears to space.

Does the concept work? If so, are there more standard names that I could have used for any of the entities these definitionsm, and - where there are no standard terms - can we think of better, more descriptive, less ambiguous names for any of these ideas?

Perhaps it would help if I say that pedagogically, the point of introducing these equations has nothing at all to do with "emphasizing the interchangeability of time and space", the point is that they are helpful when actually doing calculations about specific word-problems. Switching the terminology in the manner you suggest would make it more confusing to try to apply them to specific word-problems.

The purpose of my attempt above to define (geometrically and colloquially) a relation that is to time what length contraction is to space is to understand the symmetry between space and time and how they relate to each other. It may be an arcane way of putting it, needlessly complicated, or unnecessary for solving word-problems, but it's often said that there's more to understanding than the ability to plug in numbers and get the right answer. Even if this duration idea turns out to be impractical or irrelevant to solving textbook excercises, without exploring the issues in these ways, I wouldn't feel confident of having really grasped what was going on, and which technique it's appropriate to apply where. Part of my motivation is that, having read some introductory texts and been confused by the accidental suggestion of asymmetry in the apparent pairing of T.D. and L.C., I suspected it would be all too easy to phrase a problem in some unconventional way that would throw me. But thanks to this discussion and your explanations, I hope I've taken a few small steps towards unravelling what confused me when I first began. Of course, I make have to retrace a few steps along the way...

Rasalhague

Jun12-09, 11:21 AM

Or would it clash too much with the convention to call them both "length contraction"? Thus: contraction of the length (in space) of an object (when measured in any frame other than the unique frame where its ends are at rest), and contraction of the length (in time) of a journey (when measured in any frame other than the unique frame where its ends are at rest).

Rasalhague

Jun12-09, 08:44 PM

Hmm, I got a bit muddled with my geometric definitions. For one thing, I got x and t the wrong way round. Obviously no worldline can be parallel to the x-axis! Here's a second attempt.

LENGTH CONTRACTIONS FOR SPACE AND TIME.

x_{i}' = \frac{x_{i}}{\gamma} = \frac{x_{i}}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}

1. Spacelike. Consider two events at either end of a spacelike interval. Input: the spacelike interval between two parallel lines of syntopy (worldlines) which intersect the events, that is shortest in the unique frame where these lines are parallel to the t-axis. Output: the interval between these events.

Meaning: a length of space, such as the linear extent of a physical object, is greatest in the unique frame where the locations of the object's ends are at rest.

Restriction: in a frame where the positions of the object's ends are moving, we must locate them both at the same time.

2. Timelike. Consider two events at either end of a timelike interval. Input: The timelike interval between two hypersurfaces of synchrony which intersect the events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval between these events.

Meaning: a length of time, such as the duration of a journey, is greatest in the unique frame where the locations of the journey's ends are at rest.

Restriction: in a frame where the positions of the journey's ends are moving, we must time them both at the same location.

How could this definition of the meaning of conventional spacelike length contraction be reworded in a more general way, so as to eliminate the reference to an object, and could a corresponding generalisation be made to this timelike version?

JesseM

Jun12-09, 10:10 PM

LENGTH CONTRACTIONS FOR SPACE AND TIME.

x_{i}' = \frac{x_{i}}{\gamma} = \frac{x_{i}}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}

1. Spacelike. Consider two events at either end of a spacelike interval. Input: the spacelike interval between two parallel lines of syntopy (worldlines)
What do you mean "spacelike interval between two parallel lines of syntopy"? Do you mean the distance between the points that the lines intersect a surface of constant t in whatever frame you're using? And on that point, what frame are you using for the input? The frame where both "events at either end of a spacelike interval" happen simultaneously, or the frame where the "two parallel lines of syntopy" are parallel to the t axis? Are you choosing the two parallel lines so that these frames are one and the same? If not, is the idea just to pick two arbitrary (not necessarily simultaneous) events in this first frame, draw two parallel lines parallel to the t-axis which intersect the events, and then use as input the distance between the two lines in this frame where they're parallel to the t-axis?
which intersect the events, that is shortest in the unique frame where these lines are parallel to the t-axis.
What is shortest? The distance between the two lines of syntopy, or the distance between the two events? Note that if you are defining the distance between lines as "the distance between the points that the lines intersect a surface of constant t in whatever frame you're using" as I suggested above, then the distance between them is longest in the unique frame where the lines are parallel to the t-axis, not shortest. On the other hand, the distance between a single pair of events is shortest in the frame where the events are simultaneous.
Output: the interval between these events.
Interval between the events, or between the lines of syntopy? And in what frame? Since you include gamma, which is a function of v, it's presumably one moving at v relative to the first frame, but I'm not clear on what the first frame is. If the first frame was the one where the lines of syntopy are parallel to the t-axis, but the events were not simultaneous in that frame, and the input was the distance between the lines in that frame, then in order for the output to be the distance between the events themselves in another frame, the output frame would have to be the one where the events are simultaneous.
Meaning: a length of space, such as the linear extent of a physical object, is greatest in the unique frame where the locations of the object's ends are at rest.
If your output is the distance between two events on the lines of syntopy, then the only way this would be the same as the "length" of an object with those lines as its boundary worldlines in the output frame is, again, to impose the rule that the output frame must be the one where the events are simultaneous, since "length" always represents a simultaneous measurement of the positions of the boundaries of an object.
2. Timelike. Consider two events at either end of a timelike interval. Input: The timelike interval between two hypersurfaces of synchrony which intersect the events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval between these events.
Similar questions as above...are you assuming the output is the time between the events in the frame where they were co-located?
Meaning: a length of time, such as the duration of a journey, is greatest in the unique frame where the locations of the journey's ends are at rest.
I don't think that verbal summary really makes sense. What is greatest is the time between the two spacelike hypersurfaces in the frame where they are parallel to the t-axis. However, the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located, and there is no upper limit on how large the time between these events can be in other frames.

JesseM

Jun13-09, 12:05 AM

#381 (http://www.physicsforums.com/showpost.php?p=2231347&postcount=381) was my last post where I asked if I should move on, I'm assuming a yes.

We've discussed a scenario in which a photon passes B then A, where A and B have a relative separation speed of v. When A and B were colocated, t=0 and t'=0, ie this event serves as the origin of the t axis for both A and B.

Once a photon reaches A, A can work out when (t'oa B and that photon were colocated, in the A frame, and the separation between where the photon was at t=0 and where (x'oa) B was when B and the photon were colocated, in the A frame.

The values A has are:

time of colocation of A and B, t=0
time of colocation of A and photon, t = ta
speed of B towards where the photon originated, v
location of photon at t=0, xa = c.ta

Therefore, we have (all in the A frame):

(location of colocation of B and photon) = (location of photon at t=0) - (speed of B) * (time of colocation of B and photon) = (speed of light) * (time of colocation of B and photon)

x'oa = xa - v.t'oa = c.t'oa
Why would the location of the photon passing B be equal to (location of photon at t=0) - (speed of B) * (time of colocation of B and photon)? Seems like the leftmost side should rather be (the separation between the position of the photon at t=0 and the position of B when the photon was colocated with B), or xa - x'oa. For example, if we go back to the numbers from the old numerical example, at t=0 seconds the photon is at position xa=8 light-seconds, then at t'oa=5 s the photon passes B at position x'oa=3 ls, with B moving at v=0.6c. So you can see here that (the separation between the position of the photon at t=0 and the position of B when the photon was colocated with B) = xa - x'oa = 8 - 3 = 5, and likewise (location of photon at t=0) - (speed of B) * (time of colocation of B and photon) = xa - v*t'oa = 8 - 0.6*5 = 8 - 3 = 5, and finally it also works that (speed of light) * (time of colocation of B and photon) = c*t'oa = 5.
or

c.ta - v.t'oa = c.t'oa

so

ta = t'oa + v.t'oa / c . . . . . . (1)
OK, I see you didn't actually need to make use of the left hand side of the equation above so this is fine.
We can follow the same procedure for B to reach (all in the B frame):

(time of colocation of photon with B) = (time of colocation of A and photon) - (speed of A) * (time at which A and photon will be colocated) / (speed of light)

t'b = tob - v.tob / c . . . . . . (2)
OK
We've already concluded that since

ta = time interval between Event and when a photon from the Event reaches A, in A's frame

and

t'b = time interval between Event and when a photon from the Event reaches B, in B's frame

we have no expectation that ta = t'b

We can now test the hypothesis that:

(time of colocation of B and photon in the A frame) = (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (time of colocation of A and photon in the A frame)

That would mean:

t'oa = t'b . . . . . . (3)

and

tob = ta . . . . . . (4)

Substituting (3) into (1):

ta = t'b + v.t'b / c = t'b ( 1 + v / c ) . . . . . . (5)

Substituting (4) into (2):

t'b = ta - v.ta / c = ta (1 - v / c ) . . . . . . (6)

Substituting (6) into (5):

ta = ta (1 - v / c ) (1 + v / c)

So (3) and (4) are not valid.
OK, looks like a good proof by contradiction.
This indicates that:

(time of colocation of B and photon in the A frame) does not = (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the A frame) does not = (time of colocation of A and photon in the B frame).
Well, it proves that they can't both be true. I don't think it proves that neither could be true...what if one was true and the other false? Your proof-by-contradiction above made use of the assumption that both were true.
If we make an alternative hypothesis that:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)
You can make this hypothesis, but you could equally well make the hypothesis that the factors in the two equations were different. If the derivation is meant to be rigorous you need a justification for the idea that the factors must be the same in both equations.
That would mean:

t'oa = G.t'b . . . . . . (7)

and

tob = G.ta . . . . . . (8)

Substituting (7) into (1):

ta = G.t'b + v.G.t'b / c = G.t'b ( 1 + v / c ) . . . . . . (9)

Substituting (4) into (2):

t'b = G.ta - v.G.ta / c = G.ta (1 - v / c ) . . . . . . (10)

Substituting (10) into (9):

ta = G.G.ta (1 - v / c ) (1 + v / c)

so:

G2 = 1/(1 - v2 / c2)

so G = \gamma

Therefore:

t'oa = \gamma . t'_b . . . . . . (11)

and

tob = \gamma . t_a . . . . . . (12)

Substituting (11) into (1):

ta = \gamma . t'_b + v . \gamma . t'_b / c . . . . . . (13)

Substituting (12) into (2):

t'b = \gamma . t_a - v. \gamma.t_a / c . . . . . . (14)

or in words:

(time of colocation of A and photon in the A frame) = gamma * ((time of colocation of B and photon in the B frame) + (speed of B in the A frame) * (time of colocation of B and photon in the B frame) / (speed of light) )

and

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) / (speed of light) )

I've put speed in bold to highlight that it is not a velocity.

Now those equations are not Lorentz transformations. I grant you that, but multiply through by c.

x_a = \gamma.(x'_b + v.t'_b)

and

x'_b = \gamma.( x_a - v . t_a)

(where the photon was when A and B were colocated, in the A frame) = gamma * ((where the photon was when A and B were colocated, in the B frame) + (speed of B in the A frame) * (time of colocation of B and photon in the B frame) )

and

(where the photon was when A and B were colocated, in the B frame) = gamma * ((where the photon was when A and B were colocated, in the A frame) - (speed of A in the B frame) * (time of colocation of A and photon in the A frame) )

Making A the unprimed frame, and B the primed frame, then this latter equation (in A, the unprimed frame is at rest) is, at the very least, a spatial Lorentz Transform analogue.
I don't think it's very closely analogous. Remember that the spatial Lorentz transform takes either the coordinates of a single event in one frame and finds the spatial coordinates of the same event in the other frame, or else it takes the coordinate intervals between a single pair of events in one frame and finds the spatial interval between the same pair of events in the other frame. But just looking at the first of the two equations above, if you're talking about coordinates rather than coordinate intervals, you're dealing with three separate events: the event on the photon's worldline that occurred at t=0 in the A frame, the event on the photon's worldline that occurred at t=0 in the B frame, and the event of the photon passing B. There's no way to re-interpret this so all the variables represent intervals between a single pair of events, either.
Substituting xa = c.ta into (14) gives us:

t'_b = \gamma.t_a - v . \gamma . x_a / c^2

or, in words

(time of colocation of B and photon in the B frame) = gamma * ((time of colocation of A and photon in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )

This is not quite what we want, since the event we are talking about was back when A and B were colocated (in the A frame), but this equation does express an interval of note:

(how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame) = gamma * ((how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened), in the A frame) - (speed of A in the B frame) * (where the photon was when A and B were colocated, in the A frame) / (speed of light squared) )
I don't understand the phrase "how long it took a photon to get from the event to B minus (when colocation of A and B happened minus when the event happened), in the B frame". You seem to be mentioning three events in this phrase--the event of the photon reaching B, the event of the colocation of A and B, and "when the event happened" (which I assume means the event on the photon's worldline that occurred at t=0 in B's frame)? So how can you have an interval between three events? Likewise with "how long the photon took took to get from the event to A minus (when colocation of A and B happened minus when the event happened)".
\Delta t' = \gamma . ( \Delta t - v.x_a / c^2 )

which is, at the very least, a temporal Lorentz Transform analogue.
Again, not very analogous since all the terms don't refer to the coordinates of a single event or to intervals between a single pair of events (in fact, you seem to be mixing time intervals with spatial coordinates here).

neopolitan

Jun13-09, 12:49 AM

Would it upset things to reword your timelike like this (changes look like this):

2. Timelike. Consider two events at either end of a timelike interval. Input: The timelike interval between two hypersurfaces of synchrony which intersect the events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval between these events.

Meaning: a length of time, such as the interval between causally related events, like the ticks of a clock, is greatest in the unique frame where the causally related events are spatially colocated (or the clock is at rest).

Restriction: in a frame where causally related events are not spatially colocated (such as the ticks of a moving clock), we must consider alternative events which are simultaneous with the causally related events but at a single location or alternative events which are causally related to the first set of events such that they maintain the same temporal separation, such as the spawning and receipt of photons.

An alternative to this, given that the temporal restriction is quite complex, is to consider a specially designed "rod clock" such that the spatial and temporal intervals are intertwined.

The specifications of the rod clock are such that is has a length of L, it has two photon tubes in it with two photons and two sets of mirrors. The photons bounce between the mirrors in phase (at least while the clock is at rest) so that when one photon hits the mirror at one end of the rod, the other photon hits the mirror at the other end of the rod.

It makes sense to count ticks, doesn't it?

The number of ticks on the moving clock will be fewer than for a rest clock.

It also makes sense to measure a rod at a single moment in time, doesn't it?

The interval between where one end of a moving rod is at a single moment in a rest frame and the other end of the same moving rod at the same single moment in the same rest frame is going to be less than the interval between one end of the moving rod and the other end in the moving frame.

The thing that is all screwed up (in our basic perception of these things, but not our sophisticated SR perception), is that the moving rod is at rest in the moving frame.

That means that the moving length (in the rest frame) is less than the rest length (in the moving frame).

Going back to time to try to express it in similar terms:

The number of "moving ticks" (in the moving frame) is less than the number of "rest ticks" (in the rest frame).

We have a couple of options for consistency here: compare "in the rest frame" with "in the moving frame" in each of the pairs, ie priming the "in the moving frame" values; or consider a single rod clock which is in motion relative to a notional rest frame, ie priming the rod clock values.

Comparing these approaches, the rod clock despite having a single construction has two natures for our purposes "rod" and "clock". If we have "rod clock frame" and "observer frame", then the "rod clock frame" is the rest frame for considering the rod clock's rod-nature, but the "observer frame" is the rest frame for considering the rod clock's clock-nature.

I think it is this inconsistency that Rasalhague is getting at (and which I have touched on once or twice).

Anyways, Rasalhague might want to use the rod clock approach to frame "space-like" and "time-like" in an internally consistent way, since using the terminology "moving ends of a journey" is fraught with danger :)

cheers,

neopolitan

JesseM

Jun13-09, 01:19 AM

Would it upset things to reword your timelike like this (changes look like this):
Meaning: a length of time, such as the interval between causally related events, like the ticks of a clock, is greatest in the unique frame where the causally related events are spatially colocated (or the clock is at rest).
You should say it's smallest in that unique frame, not greatest.
The thing that is all screwed up (in our basic perception of these things, but not our sophisticated SR perception), is that the moving rod is at rest in the moving frame.

That means that the moving length (in the rest frame) is less than the rest length (in the moving frame).
It seems like this is unnecessarily confusing because of your use of the phrase "moving frame" to refer to the rod's rest frame. If you used a less ambiguous pair of names for the frames, like "the rod's rest frame" and "the observer's rest frame" (or the rest frame of whatever object you want to see the rod in motion), then this problem of terminology wouldn't arise.
Comparing these approaches, the rod clock despite having a single construction has two natures for our purposes "rod" and "clock". If we have "rod clock frame" and "observer frame", then the "rod clock frame" is the rest frame for considering the rod clock's rod-nature, but the "observer frame" is the rest frame for considering the rod clock's clock-nature.
How do you figure? We say "time dilation" because the time between two events on the worldline of one of the clocks is greater in the observer's frame than the time between the same two events in the rod/clock frame. Likewise, we say "length contraction" because the length of the rod/clock is shorter in the observer's frame than its length in the rod/clock frame. So the terminology is consistent, if that's what you're talking about (it's what Rasalhague was talking about). If you're talking about something else, can you elaborate on what you mean by the phrase "the rest frame for considering the rock clock's ___ nature"?

neopolitan

Jun13-09, 01:22 AM

Rasalhague

Jun13-09, 01:24 AM

What do you mean "spacelike interval between two parallel lines of syntopy"? Do you mean the distance between the points that the lines intersect a surface of constant t in whatever frame you're using?

Yes. Specifically the frame you mention is a frame where the object is moving. These lines are the worldlines of the ends of the object in the object's rest frame. This interval is meant to represent its length contracted in a frame where the object is moving. Syntopy = collocation = same place = constant x (in some frame).

I should have made it explicit that two different intervals are involved in each case:

1. Spacelike. Consider two events with spacelike interval \sigma_{small}. Input: the spacelike interval \sigma_{big} between two parallel lines of syntopy (worldlines) which intersect the aforementioned events, that is shortest in the unique frame where these lines are parallel to the t-axis. Output: the interval \sigma_{small}.

2. Timelike. Consider two events with timelike interval \tau_{small}. Input: The timelike interval \tau_{big} between two hypersurfaces of synchrony which intersect the aforementioned events, that is shortest in the unique frame where these hypersurfaces are parallel to the x-axis. Output: the interval \tau_{small}.

And on that point, what frame are you using for the input? The frame where both "events at either end of a spacelike interval" happen simultaneously, or the frame where the "two parallel lines of syntopy" are parallel to the t axis? Are you choosing the two parallel lines so that these frames are one and the same? If not, is the idea just to pick two arbitrary (not necessarily simultaneous) events in this first frame, draw two parallel lines parallel to the t-axis which intersect the events, and then use as input the distance between the two lines in this frame where they're parallel to the t-axis?

They're not the same frame. The input is characterised as a spacelike interval \sigma_{big} defined in terms of two different frames. One of these frames is that in which the events with interval \sigma_{small} are intersected by the worldlines of the object's ends (the two parallel lines of syntopy). The other frame is that in which those worldlines are parallel to the t-axis. The interval of the events (the only events explicitly mentioned) is the output. Which events these are (i.e. their position in spacetime) is determined by the interval \sigma_{big} and the relative velocity between the frames, represented by the slope of the separation between the events in the rest frame of the object.

What is shortest? The distance between the two lines of syntopy, or the distance between the two events?

The distance between the two lines of syntopy, \sigma_{big}. It's the shortest possible interval between these lines in the frame where they're parallel to the t-axis, and thus orthogonal to them (and the t-axis), and thus parallel to the x-axis. The frame where the lines of syntopy (worldlines of the ends of the object) are parallel to the t-axis is the object's rest frame, so \sigma_{big} represents the length of the object in its rest frame.

Note that if you are defining the distance between lines as "the distance between the points that the lines intersect a surface of constant t in whatever frame you're using" as I suggested above, then the distance between them is longest in the unique frame where the lines are parallel to the t-axis, not shortest. On the other hand, the distance between a single pair of events is shortest in the frame where the events are simultaneous.

I'm aware of that. My definition wasn't clear enough. I should have made it explicit that these are two different intervals I'm talking about.

Output: the interval between these events.

Interval between the events, or between the lines of syntopy? And in what frame? Since you include gamma, which is a function of v, it's presumably one moving at v relative to the first frame, but I'm not clear on what the first frame is.

The interval between the events (the only events mentioned explicitly, those refered to at the beginning of the definition), labelled in the revised definition \sigma_{small}. By "interval" (without qualification) I mean "spacetime interval", which is constant in all frames.

If the first frame was the one where the lines of syntopy are parallel to the t-axis, but the events were not simultaneous in that frame, and the input was the distance between the lines in that frame, then in order for the output to be the distance between the events themselves in another frame, the output frame would have to be the one where the events are simultaneous.

If your output is the distance between two events on the lines of syntopy, then the only way this would be the same as the "length" of an object with those lines as its boundary worldlines in the output frame is, again, to impose the rule that the output frame must be the one where the events are simultaneous, since "length" always represents a simultaneous measurement of the positions of the boundaries of an object.

The events have a spacelike separation, so there will be some frame in which there're simultaneous. That they lie on the worldlines of the object's ends ensures that they will represent the length of the object as measured in the frame where they're simultaneous. Which frame it is that they're simultaneous in is determined by the relative speed, represented geometrically by the hypersurface of synchrony (simultaneity, constant t). I hope my clarifications above have answered your other questions in this section, but let me know if there's anything still unclear (or that I've just got plain wrong). Perhaps it would help to make explict all of the events involved, rather than just naming two of them as "the events".

Similar questions as above...are you assuming the output is the time between the events in the frame where they were co-located?

Yes.

I don't think that verbal summary really makes sense. What is greatest is the time between the two spacelike hypersurfaces in the frame where they are parallel to the t-axis.

I guess you mean parallel to the x-axis?

However, the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located,

Yes.

and there is no upper limit on how large the time between these events can be in other frames.

I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame of the its start and finish (point of departure and point of arrival). The faster the traveller goes from start to finish, the shorter the duration, as speed increases arbitrarily close to c.

The definition is intended to parallel the way that the upper limit of the length of an object is its linear extent, the distance it spans, from one end to the other. The faster the object travels, the shorter its length, as speed increases arbitrarily close to c.

JesseM

Jun13-09, 01:46 AM

JesseM,

Just to focus in on one issue for once, does the fact that:

the selection of A and B
their separation velocity v
the direction selected as positive, and
the event under consideration

are all arbitrary, not mean that I'd have to prove a contention that one of A and B somehow have primacy? I'd think that the contention that neither have primacy could be taken as read given the selection process.
Not exactly sure what you mean by "primacy" here, I'm just saying you haven't proved that you haven't given any real reason to believe that the factor here:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

must be the same as the factor here:

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

And as long as we entertain the possibility that they're different, it's also possible the factor in one of the equations could be 1--is that what you meant by "primacy", in response to my comment that you hadn't disproved the possibility that "one was true and the other false"? Of course, intuitively it seems pretty unlikely that the factor would always be 1 in one of these equations but not the other regardless of your choice of v and separation of A and B at the moment the photon passes either of them. Still,"intuitively it seems pretty unlikely" is not a rigorous argument, and in any case for the sake of your derivation it's not the possibility that the factor could be 1 that you have to worry about, but rather the more general possibility that the factors could be different. I bet if we tried hard enough we could even come up with an example of a coordinate transformation where the factors would in general be different and the speed of the photon would still be c in both frames, although presumably the transformation would violate the first postulate in the sense that the coordinate transformation from A's frame to B's frame would look different from the transformation from B's frame to A's frame.

neopolitan

Jun13-09, 02:06 AM

You should say it's smallest in that unique frame, not greatest.

Clocking ticking here on earth, causally related events are twin travels away, twin arrives back.

On the earth clock the time elapsed is greater in the earth frame (for this clock, all the intermediate events between the departure event and arrival home are colocated with those events). For the travelling twin (for whom all the intermediate causally related events were not colocated with the departure event and the arrival home event event), the time elapsed is less than for the earth clock.

See what I mean?

Apart from that, you broke up my post before you read it all. Otherwise you would have seen that I addressed your second question later.

As for "rod clock's ____ nature", the rod clock is both a rod, and a clock. I described that.

So there is a spatial interval associated with the rod clock (length of rod clock = L' and a temporal interval associated with the rod clock (time between ticks = dt').

Describe those in the observer's rest frame, where an identical rod clock in the observer's rest frame has length = L and time between ticks = dt.

You'll arrive at time dilation and length contraction.

But I wasn't talking about "time between ticks", I was talking about "number of ticks" on the clocks (notionally as the ticking end of the clock travels between two locations in the observer frame). I've talked often about the fact that our normal use of clocks is to look at how many ticks have taken place, rather than how long the period between ticks is. Using number of ticks, we arrive at something other than time dilation. Call it what you will.

number of ticks on rod clock in motion is fewer than number of ticks on observer's clock

trod clock = tobserver's clock/gamma

length of rod clock in motion is less than it's rest length

Lrod clock = Lobserver's clock/gamma

I agree that

interval between ticks on rod clock in motion is greater than interval between ticks on observer's clock (according to the observer)

cheers,

neopolitan

JesseM

Jun13-09, 02:11 AM

Yes. Specifically the frame you mention is a frame where the object is moving. These lines are the worldlines of the ends of the object in the object's rest frame. This interval is meant to represent its length contracted in a frame where the object is moving. Syntopy = collocation = same place = constant x (in some frame).

I should have made it explicit that two different intervals are involved in each case:

1. Spacelike. Consider two events with spacelike interval \sigma_{small}. Input: the spacelike interval \sigma_{big} between two parallel lines of syntopy (worldlines) which intersect the aforementioned events, that is shortest in the unique frame where these lines are parallel to the t-axis. Output: the interval \sigma_{small}.
Still a little unclear on "shortest in the unique frame where these lines are parallel to the t-axis". Do you mean 1) that out of all frames, the interval between these parallel lines is shortest in the frame where where the lines are parallel to the t-axis? Or 2) that considering only the frame where the lines are parallel to the t-axis, you want to look at the shortest interval between a pair of events on each line? If 2), then maybe a less confusing way of stating it is that you just want to talk about the distance between two events on either line which are simultaneous in this frame where the lines are parallel to the t-axis. But yeah, if it is 2) I think I understand what you're saying overall here.
I don't think that verbal summary really makes sense. What is greatest is the time between the two spacelike hypersurfaces in the frame where they are parallel to the t-axis.
I guess you mean parallel to the x-axis?
Yeah, my mistake.
However, the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located,
Yes.
and there is no upper limit on how large the time between these events can be in other frames.
I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame of the its start and finish (point of departure and point of arrival). The faster the traveller goes from start to finish, the shorter the duration, as speed increases arbitrarily close to c.
I don't understand, didn't you just agree the time between two events (point in spacetime of departure and point in spacetime of arrival) is greater in other frames besides the traveler's frame, because "the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located"? Just as an example, suppose in the traveler's rest frame the departure point passes next to him at coordinates x=0 light years, t=0 years and the arrival point passes next to him at x=0 light years, t=10 years. So, in his rest frame the time is 10 years. Do you agree that in any other frame where the traveler is moving, the time will be greater than 10 years, not smaller? If so, what do you mean when you say "I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame"? This should be the lower limit, not the upper limit.
The definition is intended to parallel the way that the upper limit of the length of an object is its linear extent, the distance it spans, from one end to the other. The faster the object travels, the shorter its length, as speed increases arbitrarily close to c.
If you want to make a parallel with length contraction, you should use your idea above about talking about the time between two parallel spacelike surfaces (which is how I have conceptualized the 'temporal analogue of length contraction' equation), not the time between two distinct events (which is how we ordinarily conceptualize the time dilation equation). Out of all frames, the time between two events is minimized in the frame where they are colocated (so if they are both events on the worldline of a traveler, this time is minimized in the traveler's rest frame). On the other hand, out of all frames, the time between two parallel spacelike surfaces is maximized in the frame where these spacelike surfaces are surfaces of simultaneity (parallel to the x-axis). This is more like with length contraction, where the distances between two parallel timelike worldlines is maximized in the frame where these lines are parallel to the t-axis.

JesseM

Jun13-09, 02:34 AM

Clocking ticking here on earth, causally related events are twin travels away, twin arrives back.

On the earth clock the time elapsed is greater in the earth frame (for this clock, all the intermediate events between the departure event and arrival home are colocated with those events). For the travelling twin (for whom all the intermediate causally related events were not colocated with the departure event and the arrival home event event), the time elapsed is less than for the earth clock.
But in this case one of the twins is not inertial. When you rewrote Rasalhague's statement to read "Meaning: a length of time, such as the interval between causally related events, like the ticks of a clock, is greatest in the unique frame where the causally related events are spatially colocated (or the clock is at rest)", I assumed "frames" referred to inertial frames, since this is what all the discussion had been about so far.
Apart from that, you broke up my post before you read it all. Otherwise you would have seen that I addressed your second question later.
Please don't jump to uncharitable conclusions like this. In fact I did read the whole post before responding, and I saw nothing (and still see nothing) in it that allows me to understand what you meant by phrases like "the rod clock's clock nature". I'm sure it's clear to you since you wrote it, but you aren't communicating your ideas in a way that allows me (or others reading, I'd wager) to follow.
As for "rod clock's ____ nature", the rod clock is both a rod, and a clock. I described that.
I understood that part, of course.
So there is a spatial interval associated with the rod clock (length of rod clock = L' and a temporal interval associated with the rod clock (time between ticks = dt').

Describe those in the observer's rest frame, where an identical rod clock in the observer's rest frame has length = L and time between ticks = dt.

You'll arrive at time dilation and length contraction.

But I wasn't talking about "time between ticks", I was talking about "number of ticks" on the clocks (notionally as the ticking end of the clock travels between two locations in the observer frame). I've talked often about the fact that our normal use of clocks is to look at how many ticks have taken place, rather than how long the period between ticks is. Using number of ticks, we arrive at something other than time dilation. Call it what you will.
You didn't mention this distinction in your post, so why did you conclude that my failure to infer your meaning (via mind-reading?) proved I didn't read your post? In any case the time dilation equation doesn't deal with "time between ticks", it deals with the time period between some specific pair of events that occurs on the worldline of a clock (preferably events separated by a large number of ticks if we want to measure the time interval fairly accurately), and how much time the clock itself measures between these events (or the 'number of ticks' between them as measured by that clock) vs. the amount of time between the same events in the observer's frame where the clock is moving (which could be measured by 'number of ticks' on a pair of synchronized clocks at rest in the observer's frame if you like). The time between these events in the observer's frame is greater than the time between them in the clock's rest frame, no? And measuring time intervals between specific physical events of interest (like the beginning and ending of a race) is part of "our normal use of clocks", no?

neopolitan

Jun13-09, 03:32 AM

Not exactly sure what you mean by "primacy" here, I'm just saying you haven't proved that you haven't given any real reason to believe that the factor here:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

must be the same as the factor here:

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

And as long as we entertain the possibility that they're different, it's also possible the factor in one of the equations could be 1--is that what you meant by "primacy", in response to my comment that you hadn't disproved the possibility that "one was true and the other false"? Of course, intuitively it seems pretty unlikely that the factor would always be 1 in one of these equations but not the other regardless of your choice of v and separation of A and B at the moment the photon passes either of them. Still,"intuitively it seems pretty unlikely" is not a rigorous argument, and in any case for the sake of your derivation it's not the possibility that the factor could be 1 that you have to worry about, but rather the more general possibility that the factors could be different. I bet if we tried hard enough we could even come up with an example of a coordinate transformation where the factors would in general be different and the speed of the photon would still be c in both frames, although presumably the transformation would violate the first postulate in the sense that the coordinate transformation from A's frame to B's frame would look different from the transformation from B's frame to A's frame.

By primacy I mean having privilege, which would in turn violate the first postulate.

If I explicitly state the first postulate, which I take to follow from Galilean relativity, would that satisfy you or do you demand that I do a proof by elimination for the hypothesis that one transformation has one form and the other transformation has another form?

I just can't see what the justification would be for uneven transformations given that I should be able swap B for A without affecting the result (because of the way they were selected and because everything about them was expressed in general terms).

The only addition I can see which would make sense would be to make v a velocity (with a potentially negative value in one frame and a positive value in the other frame, but the same magnitude).

cheers,

neopolitan

neopolitan

Jun13-09, 03:34 AM

The number of ticks on the moving clock will be fewer than for a rest clock.

Reading the whole post means you don't need to read minds. (Remember to check from which post this quote was taken.)

neopolitan

Jun13-09, 03:39 AM

I feel like we are going over old ground unprofitably, really we don't disagree on the time dilation thing other than whether there is utility in thinking about an inverse function or a temporal analogue for the spatial function. I'd prefer not to waste effort on going over it again so is it possible that we focus on the derivation (and I won't intrude on the separate strand you have going with Rasalhague).

cheers,

neopolitan

JesseM

Jun13-09, 09:29 AM

Reading the whole post means you don't need to read minds. (Remember to check from which post this quote was taken.)
Sorry, read it all again for a third time, nothing there that would indicate that you were talking about your own weird notion that "number of ticks" contracts rather than dilates like "time between ticks". You did use the phrase "number of ticks" but did not indicate that you were using this in anything other than the normal way, where we are talking about the "number of ticks" of both the observer's time and the clock's time between a specific pair of events, where naturally the number of ticks of the observer's time is greater, so it makes sense that we use the word "dilation" to be consistent with the fact that in the case of length, the length of the object is smaller for the observer. So clearly in this (standard) sense, both "time between ticks" on the moving clock and "number of ticks" between two events on the moving clock's worldline are larger for the observer than they are for the moving clock. If you meant something else you needed to actually explain it.

JesseM

Jun13-09, 09:47 AM

By primacy I mean having privilege, which would in turn violate the first postulate.
But why should we think that one of the frames having a different value for the constant indicates that one frame has privilege (for example, if the first equation had the constant 2 and the second equation had the constant 3, which frame would be priveleged? Your two equations each involve quantities from both frames anyway) or that the first postulate has been violated? The first postulate says the laws of physics must be the same in both frames, it doesn't say that specific physical scenarios will look the same in both frames. For example, if I had a pair of objects traveling in opposite directions with the same speed in one frame, they would not have equal speeds in a different frame, but this wouldn't violate the first postulate. The physical scenario you are considering does look different in the two frames--the photon starts at a different distant from the origin at t=0 in each frame, and A's frame B is moving towards the photon while in B's frame A is moving away from the photon--so I don't see how you can assume a priori that the first postulate says the constant in one equation must be the same as the constants in the other, although we know in retrospect that this does turn out to be true.
If I explicitly state the first postulate, which I take to follow from Galilean relativity, would that satisfy you or do you demand that I do a proof by elimination for the hypothesis that one transformation has one form and the other transformation has another form?
Yes, it needs to be proved. And those equations are not "transformation" equations, the first equation is a relationship between the time of one event in the A frame and the time of a second event in the B frame, and the second equation is the relationship between the time of the first event in the B frame and the time of the second event in the A frame. Do you agree that the constants in the equations would not necessarily be the same if we picked two totally arbitrary events, rather than the events being ones where a single photon crossed the time axis of each frame? If so maybe you can see the need for an explanation as to what specific property of the two events you chose ensures that the constants in those two equations will be the same.

Rasalhague

Jun13-09, 10:41 AM

Still a little unclear on "shortest in the unique frame where these lines are parallel to the t-axis". Do you mean 1) that out of all frames, the interval between these parallel lines is shortest in the frame where where the lines are parallel to the t-axis? Or 2) that considering only the frame where the lines are parallel to the t-axis, you want to look at the shortest interval between a pair of events on each line? If 2), then maybe a less confusing way of stating it is that you just want to talk about the distance between two events on either line which are simultaneous in this frame where the lines are parallel to the t-axis. But yeah, if it is 2) I think I understand what you're saying overall here.

It was 2 that I meant.

I don't understand, didn't you just agree the time between two events (point in spacetime of departure and point in spacetime of arrival) is greater in other frames besides the traveler's frame, because "the time between the event of the journey beginning and the event of the journey ending is smallest in the frame where these events are co-located"?

That's right, but the faster the traveller goes, the smaller the (spacetime) interval, the proper time, of the separation between the event of departure and that of arrival. So greater speed results in a smaller output. That's to say, a greater contraction of the input. The input is the time component of this separation in the rest frame of the place the traveller left and the place they're going to. A faster traveller arrives at the same point in space as a slower traveller, but not the same point in spacetime (which is to say: not the same event). So I didn't mean to give the impression that this operation consists of calculating the time component of a given separation (whose interval is the proper time between two fixed events) in some arbitrary frame other than the traveller's rest frame. Rather, I meant that a faster speed determines which separation (between which pair of events) it will be whose interval is the output. The duration of the journey through space (as I'm trying to define it) is smaller the faster the journey, even though the time component of whichever separation is selected will be greater than the interval in any frame other than the traveller's rest frame.

Just as an example, suppose in the traveler's rest frame the departure point passes next to him at coordinates x=0 light years, t=0 years and the arrival point passes next to him at x=0 light years, t=10 years. So, in his rest frame the time is 10 years. Do you agree that in any other frame where the traveler is moving, the time will be greater than 10 years, not smaller? If so, what do you mean when you say "I'm defining the upper limit for the duration of the journey as the time it takes in the rest frame"? This should be the lower limit, not the upper limit.

The full phrase I used was "in the rest frame of its start and finish (point of departure and point of arrival)". By "its", I meant "the journey's". This it not the same frame as the traveller's rest frame; if the traveller remained at rest with respect to their destination, they'd never get there! (Unless it was a trivial journey, with the place of origin and the destination being identical, making leaving and arrival the same event, which is no journey at all, just a rod whose ends are collocated would have no length, and thus be no rod.)

I think something Neopolitan mentioned in #406 sheds light on one possible source of confusion, namely the danger of talking about the "moving ends of a journey". I can see that "start" and "finish" are potentially ambiguous as to whether what's meant is the events of departure and arrival, or the points in space where these events take place (at whatever time).

I'm trying to depict the journey itself as the entity that relates to time as a ruler relates to space. My motivation for this is that the pairing of clock and ruler is, I suspect, the source of this apparent asymmetry. Clocks and rulers are both objects with limited spatial extent and indefinite (arbitrary) temporal extent. The time dilation operation, as tratitionally conceived, is defined in terms of finding the time component for a given proper time. The length contraction operation, as traditionally conceived, could be defined in terms of finding the proper distance with a given x component (i.e. the inverse of time dilation), although it's more often expressed in terms of three explicitly defined measuring events. The ruler (or rod or rocket) in these thought experiments is a more complex entity than the clock in that it has both temporal and spatial extent. In talking about a journey, I'm trying to define some entity that would be to time what a ruler is to space. Thus the journey is a more complex entity than a single, pointlike clock (conceived of as locatable at a single point in space, and with single worldline for its trajectory). Like a ruler, a journey extends through space as well as time; as a finite ruler has a sharply defined pair of ends, a finite journey has a sharply defined beginning and end in time.

If you want to make a parallel with length contraction, you should use your idea above about talking about the time between two parallel spacelike surfaces (which is how I have conceptualized the 'temporal analogue of length contraction' equation), not the time between two distinct events (which is how we ordinarily conceptualize the time dilation equation).

This is still my intention. From what you've said, it seems that I need to come up with a clearer way of wording the definition.

Out of all frames, the time between two events is minimized in the frame where they are colocated (so if they are both events on the worldline of a traveler, this time is minimized in the traveler's rest frame). On the other hand, out of all frames, the time between two parallel spacelike surfaces is maximized in the frame where these spacelike surfaces are surfaces of simultaneity (parallel to the x-axis). This is more like with length contraction, where the distances between two parallel timelike worldlines is maximized in the frame where these lines are parallel to the t-axis.

I suppose I was trying to define the operations in terms of invariant intervals without reference to components. I'm not sure why I chose to do that (maybe just exploring the concepts, maybe looking for a definition that incorporated the idea of space and time components of a separation without naming them, to break it down into the most basic concepts), and I might have another go at that and try to do it more clearly, but reverting the language of components for now, would the following work?

1. Spacelike. Consider two events, each on a different one out of two timelike lines, these events having a separation with spacelike interval \sigma. Input: the x component of the separation of these events in a frame where the lines are parallel to the t-axis. Output: the interval \sigma.

Meaning: a length of space, such as the linear extent of a physical object, is greatest in the unique frame where the locations of the object's ends are at rest.

Restriction: in a frame where the locations of the object's ends are changing, we must locate them both at the same time.

2. Timelike. Consider two events, each on a different one out of two spacelike hypersurfaces, these events having a separation with timelike interval \tau. Input: the time component of the separation of these events in a frame where the hypersurfaces are parallel to the x-axis. Output: the interval \tau.

Meaning: a length of time, such as the duration of a journey, is greatest in the unique frame where the locations of the journey's ends are at rest.

Restriction: in a frame where the locations of the journey's ends are changing, we must time them both at the same location.

neopolitan

Jun13-09, 12:15 PM

Yes, it needs to be proved. And those equations are not "transformation" equations, the first equation is a relationship between the time of one event in the A frame and the time of a second event in the B frame, and the second equation is the relationship between the time of the first event in the B frame and the time of the second event in the A frame. Do you agree that the constants in the equations would not necessarily be the same if we picked two totally arbitrary events, rather than the events being ones where a single photon crossed the time axis of each frame? If so maybe you can see the need for an explanation as to what specific property of the two events you chose ensures that the constants in those two equations will be the same.

It seems to me that we have two stopping points here.

First, the need to prove that there must be one single factor (as in my derivation shown earlier), rather than possibly two factors.

Second, generality, in that you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames, and that you feel that there is a "need for an explanation as to what specific property of the two events (I) chose ensures that the constants in those two equations will be the same".

Is this correct?

Do you further agree that, if I were to convince you that there being one single factor is the default, that feat would negate the need for a proof? I'm going to think about the proof angle anyway, but I am not abandoning my argument that what you are asking me to prove is actually the default.

cheers,

neopolitan

JesseM

Jun13-09, 02:12 PM

It seems to me that we have two stopping points here.

First, the need to prove that there must be one single factor (as in my derivation shown earlier), rather than possibly two factors.
Yes.
Second, generality, in that you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames, and that you feel that there is a "need for an explanation as to what specific property of the two events (I) chose ensures that the constants in those two equations will be the same".
How is that different from the above? I'm just asking for a demonstration that the specific properties of the events you chose ensure that the factor will be the same in both equations, and pointing out that since you presumably agree the factor wouldn't be the same for an arbitrary pair of events, then your demonstration will have to somehow make use of those specific properties rather than just making use of more general facts like the first postulate of SR.
Do you further agree that, if I were to convince you that there being one single factor is the default, that feat would negate the need for a proof?
I would say any totally airtight argument for why they must be the same would constitute a "proof", so I don't really understand this distinction. Unless by "default" you just mean "the assumption that seems most plausible a priori even if we aren't sure it's actually correct", in which case I don't think that would negate the need for a proof.

Keep in mind, also, my later criticisms involving the final equations you derived, which I say have a physical meaning that's totally unlike the Lorentz transformation equations...it seems in a way a bit pointless to spend a lot of time thinking about how to justify this one step in your demonstration (which I agree happens to be true even if we haven't found a way to justify it) if the final endpoint of your demonstration is just an equation that looks superficially like the Lorentz transformation but really has almost nothing to do with it, and has no real utility outside of relating quantities in the specific physical scenario you describe where a light ray crosses path with two observers (so it cannot even really be understood as a special case of the Lorentz transformation, since the variables that appear in the equations aren't all defined in terms of the same single event or pair of events)

neopolitan

Jun13-09, 02:56 PM

How is that different from the above?

I must be getting tired.

My second point was supposed to be that generality seems to be a stopping point, in that you think my selection of event has a significant bearing on the final equation (to the extent that it is not actually the Lorentz Transform or even equivalent to it, but something less general).

I had meant to finish the phrase "you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames" with "then my derivation would not stand". I did mean to end with the ending I had, for reasons which might become clear when, at a time when I am not so dog tired, I will try to explain that my starting scenario is so general that A should be totally interchangeable with B. Generality in, generality out.

cheers,

neopolitan

JesseM

Jun13-09, 03:08 PM

I had meant to finish the phrase "you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames" with "then my derivation would not stand".
OK, yes, I think your derivation seems to depend on the fact that both events lie on the path of a photon. What's more, even with this restriction, your final equations don't even end up measuring "the interval between them" (i.e. the variables in your final equations don't all refer to the space and time intervals between a single pair of events).
I did mean to end with the ending I had, for reasons which might become clear when, at a time when I am not so dog tired, I will try to explain that my starting scenario is so general that A should be totally interchangeable with B. Generality in, generality out.
Showing that A could be exchanged with B might show that the constants in those equations would be the same, although it wouldn't address the issue of the final equations you derive. In any case it doesn't seem to me A can be exchanged with B here, since after all both frames agree the photon passes B before it passes A.

neopolitan

Jun13-09, 11:47 PM

In any case it doesn't seem to me A can be exchanged with B here, since after all both frames agree the photon passes B before it passes A.

Surely you see that that depends on v and the x values being positive. If we plug in a negative v, the photon passes A before it passes B. Similarly if we start with a negative displacement for the event (while v stays positive), then the photon passes A before it passes B. To complete the picture, if we put in a negative v and a negative starting displacement, then the photon passes B first.

We're not constrained to using positive values.

Can you understand that if I say "t time units before it passes A, the photon passes B" and the scenario makes t negative, then that is the same as saying "|t| time units after it passes A, the photon passes B" where |t| is the magnitude of t.

Does this help you see that A and B are interchangeable?

If so, that will leave us with what the derived equations describe.

I do have one other question, you've not said anything about it but do you have issue with using (0,0) as an agreed location (conceptually the same as a colocation, if not an actual colocation, of A and B)? If you do have a problem with it, I will have to talk exclusively about intervals (even though I think we have agreed that coordinates are implicitly intervals anyway).

cheers,

neopolitan

Rasalhague

Jun14-09, 09:13 AM

We have a couple of options for consistency here: compare "in the rest frame" with "in the moving frame" in each of the pairs, ie priming the "in the moving frame" values; or consider a single rod clock which is in motion relative to a notional rest frame, ie priming the rod clock values.

Comparing these approaches, the rod clock despite having a single construction has two natures for our purposes "rod" and "clock". If we have "rod clock frame" and "observer frame", then the "rod clock frame" is the rest frame for considering the rod clock's rod-nature, but the "observer frame" is the rest frame for considering the rod clock's clock-nature.

I think it is this inconsistency that Rasalhague is getting at (and which I have touched on once or twice).

Anyways, Rasalhague might want to use the rod clock approach to frame "space-like" and "time-like" in an internally consistent way, since using the terminology "moving ends of a journey" is fraught with danger :)

That's true. Partly I wanted to take advantage of the ambiguity in words like length that could refer to a length of time or a length of space. I was wondering if we could come up with a rigorous definition of a length of travel time that would intuitively match our idea of the length of an object, but I don't know how practical this would be, or whether the risk of confusion would outweigh the benefits.

Your "rod clock" is an interesting idea, although it's taken me a while to get my tongue around "rod clock rest frame" - maybe our next example can involve a red lorry and a yellow lorry... I'd been thinking along similar (albeit more primative) lines with candle clocks, or a burning fuse, but I always seemed to come back to this apparent asymmetry, as if even an object that's both ruler and clock, using the same physical markings to measure time along its length as it does to measure distance, leads to this strange feeling that we have to treat time and space as behaving differently from each other with respect to the object, or else switch which frame we take as out starting point.

How would the situation change, I wonder, if we replace the photons in the rod clock with a massive particle? Then we could think in terms of the particle's rest frame. I was looking at the way the idea of frame dependent distance is introduced in Fishbane et al.: Physics for Scientists and Engineers with a muon formed in the Earth's atmosphere and travelling from its point of creation to the ground. It lives just long enough to reach the ground. Of all the possible values they could take in all possible reference frames, the following two variables are minimum in the muon's rest frame: the time t_{\mu} that passes between the events of the muon's birth and death, and the distance x_{\mu} that extends between the receding bit of air that marks the spot where the muon was created and the approaching ground that marks the spot where it will be annihilated. So in the ground's rest frame (where the muon is moving), both the t component and the x component of the separation of these events (muon birth and death) will be greater than they were in the muon's rest frame.

But in that example, the muon is the clock and the Earth is the ruler. Although I subscripted the space coordinate with a mu there, this seems at odds with the fact that it's the Earth that's actually doing the measuring of the muon's journey, since in the muon's rest frame, the muon makes no journey. Will this always be the case, due to the way the distortions of time and space complement each other, even when we construct an object that incorporates the roles of clock and ruler: will the part or aspect of it that's used to determine time need to be moving in the rest frame of the part or aspect of it that's used to measure distance?

JesseM

Jun14-09, 10:06 AM

Surely you see that that depends on v and the x values being positive. If we plug in a negative v, the photon passes A before it passes B. Similarly if we start with a negative displacement for the event (while v stays positive), then the photon passes A before it passes B. To complete the picture, if we put in a negative v and a negative starting displacement, then the photon passes B first.

We're not constrained to using positive values.

Can you understand that if I say "t time units before it passes A, the photon passes B" and the scenario makes t negative, then that is the same as saying "|t| time units after it passes A, the photon passes B" where |t| is the magnitude of t.

Does this help you see that A and B are interchangeable?
I don't understand what you mean by "interchangeable" here, or how it proves the constant should be the same in your equations. It's true that whatever xa and v we pick, then whatever is seen in A's frame vs. B's frame, we could then pick a different xa and v so that B saw what A formerly saw and vice versa. Still, for any particular choice of xa and v, their viewpoints are asymmetrical in that they both agree the photon passed one of them first. And your equations both assume we have picked a particular choice of xa and v, and then assert the constants relating the times will be the same.

As an analogy, suppose that instead of the two events being the photon crossing the x=0 axis of A's frame and the event of the photon crossing the x=0 axis of B's frame, we instead chose Event #1 to be the event of the photon crossing the x=5 axis of A's frame, and Event #2 to be the event of the photon crossing the x=5 axis of B's frame. Now suppose that someone said that since the two frames are "interchangeable", we should expect the constant in the following pair of equations to be the same:

(time of Event #2 in B's frame) = (constant)*(time of Event #1 in A's frame)
(time of Event #2 in A's frame) = (constant)*(time of Event #1 in B's frame)

Without doing any new calculations to check the validity of this, are you confident enough in the meaning of "interchangeability" that you would bet a large sum of money that this guy's argument is correct or incorrect? If you feel any uncertainty here that perhaps shows why the use of "interchangeability" in your own argument is not really rigorous, and is more of what physicists would call a "handwavey" argument. You really need to provide a step-by-step proof where each statement clearly follow from the last, and you show specifically where the two postulates of relativity come into play (since I think you'd agree that if we came up with a coordinate transformation that didn't respect these postulates the constants in your equations would not be the same), and what the relevance is of the fact that the two events in question are the events of a photon's worldline intersecting each frame's x=0 axis.
I do have one other question, you've not said anything about it but do you have issue with using (0,0) as an agreed location (conceptually the same as a colocation, if not an actual colocation, of A and B)? If you do have a problem with it, I will have to talk exclusively about intervals (even though I think we have agreed that coordinates are implicitly intervals anyway).
What do you mean "an agreed location"? Are you just referring to the idea that whenever we talk about the space and time coordinates of an event, we're implicitly talking about the space and time intervals between that event at the event at (0,0)? Is so I have no problem with that, again my problem with your final equations is that all the variables don't refer to intervals between a single pair of events.

neopolitan

Jun14-09, 09:34 PM

I started to reply to this last night, what I wrote it probably sitting on the home computer in "Preview Post" form rather than being submitted. Oh well.

I'm not going to be able to reply in depth to your first chunk because you seem to have gone off on a tangent. I've highlighted the phrases which are of concern.
I don't understand what you mean by "interchangeable" here, or how it proves the constant should be the same in your equations. It's true that whatever xa and v we pick, then whatever is seen in A's frame vs. B's frame, we could then pick a different xa and v so that B saw what A formerly saw and vice versa. Still, for any particular choice of xa and v, their viewpoints are asymmetrical in that they both agree the photon passed one of them first. And your equations both assume we have picked a particular choice of xa and v, and then assert the constants relating the times will be the same.

As an analogy, suppose that instead of the two events being the photon crossing the x=0 axis of A's frame and the event of the photon crossing the x=0 axis of B's frame, we instead chose Event #1 to be the event of the photon crossing the x=5 axis of A's frame, and Event #2 to be the event of the photon crossing the x=5 axis of B's frame. Now suppose that someone said that since the two frames are "interchangeable", we should expect the constant in the following pair of equations to be the same:

(time of Event #2 in B's frame) = (constant)*(time of Event #1 in A's frame)
(time of Event #2 in A's frame) = (constant)*(time of Event #1 in B's frame)

Without doing any new calculations to check the validity of this, are you confident enough in the meaning of "interchangeability" that you would bet a large sum of money that this guy's argument is correct or incorrect? If you feel any uncertainty here that perhaps shows why the use of "interchangeability" in your own argument is not really rigorous, and is more of what physicists would call a "handwavey" argument. You really need to provide a step-by-step proof where each statement clearly follow from the last, and you show specifically where the two postulates of relativity come into play (since I think you'd agree that if we came up with a coordinate transformation that didn't respect these postulates the constants in your equations would not be the same), and what the relevance is of the fact that the two events in question are the events of a photon's worldline intersecting each frame's x=0 axis.

Here's what you are referring back to:
If we make an alternative hypothesis that:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

I never said it was a constant. You are correct in that the factor is dependent on a particular value of v, and therefore not a constant. I don't call it a "function" because within the scenario v is fixed (if we changed v, then we would be talking about a different relationship between a different pair of frames a different relative speed with respect to each other). It would be a function if it varied as a result of changing our input values from within the pair of frames to which v applies (ie x and t values).

The selection of a different xa in the scenario does not change the factor and the factor does not vary with time.

By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values).

I'm going to redo the equations with v as a velocity, rather than a speed. This may help.

What do you mean "an agreed location"? Are you just referring to the idea that whenever we talk about the space and time coordinates of an event, we're implicitly talking about the space and time intervals between that event AND the event at (0,0)? Is so I have no problem with that, again my problem with your final equations is that all the variables don't refer to intervals between a single pair of events.

I made a correction, I think that is what you meant.

You are part of the way there. I mean what you wrote in the corrected sentence, plus the fact that both A and B see that (0,0) event as (0,0), ie x=0,t=0 is the same event as x'=0,t'=0. There's agreement between A and B as to that event.

They can agree on another event rather than (0,0) if it is convenient to them as well, it will just make the coordinate transformation a little different.

When I redo the equations with v as a velocity rather than a speed, I will do them all with only intervals involved, rather than aiming for a coordinate transformation. I will also choose an event which is not necessarily simultaneous with a colocation of A and B.

At the same time, I will attempt to maintain enough rigour to satisfy you that "some factor" is in fact the same in both frames. If I can't manage that, then I might have to try a proof by elimination, eliminating the possibility of two different factors. I've already eliminated a single factor which is equal to 1.

I'll give it some thought and get back to you.

cheers,

neopolitan

JesseM

Jun14-09, 10:29 PM

I'm not going to be able to reply in depth to your first chunk because you seem to have gone off on a tangent. I've highlighted the phrases which are of concern.

Here's what you are referring back to:

I never said it was a constant.
All I meant was that for any given choice of xa and v it's just a number, but I probably shouldn't have used the word "constant" since it could vary as you vary xa and v (although in fact it turns out that it doesn't). However, I wasn't actually assuming it remained constant as you varied these, so my argument in no way depends on this assumption, nor is it a "tangent". Please just mentally replace "constant" with "factor" in the block of text you quoted and reread it.
By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values).
What "end result" are you referring to? The end result that the same factor appears in both equations? In this case you are simply asserting what you are supposed to prove, but perhaps you mean something else. In any case, my point is that you haven't made any coherent argument as to why your ill-defined notion of "interchangeability" proves that for a specific choice of xa and v, the factor in your two equations must be the same. The actual values of the variables that appear in the equations are certainly not invariant under a switching of labels--for example, the value of (the time that the photon is colocated with B, as measured in the A frame) is not equal to (the time that the photon is colocated with A, as measured in the B frame) for most specific choices of xa and v.

Note that since A and B's worldlines are assumed to lie along the x=0 axis of their respective frames, the event of the photon being colocated with A is equivalent to the event of the photon crossing A's x=0 axis, and likewise for B...this means your two equations can be written as:

(time of photon crossing B's x=0 axis in A's frame) = (factor)*(time of photon crossing B's x=0 axis in B's frame)

(time of photon crossing A's x=0 axis in B's frame) = (factor)*(time of photon crossing A's x=0 axis in A's frame)

So maybe now you can see why it wasn't a tangent when I asked in the previous post if you'd agree with a hypothetical guy making the argument that because of the "interchangeability" of the two frames, we could also assume the same factor will appear in the following two equations (same as one another, not same as in the previous two equations):

(time of photon crossing B's x=5 axis in A's frame) = (factor)*(time of photon crossing B's x=5 axis in B's frame)

(time of photon crossing A's x=5 axis in B's frame) = (factor)*(time of photon crossing A's x=5 axis in A's frame)

All that's changed in these two equations is that x=0 has been replaced with x=5. Please tell me whether you'd be confident about whether this guy's argument is right or wrong, without doing any SR calculations to check explicitly what factor appears in each. If you're unsure, that suggests that you don't really have good justification for being confident that "interchangeability" means the factor should be the same in your own two equations.
You are part of the way there. I mean what you wrote in the corrected sentence, plus the fact that both A and B see that (0,0) event as (0,0), ie x=0,t=0 is the same event as x'=0,t'=0. There's agreement between A and B as to that event.

They can agree on another event rather than (0,0) if it is convenient to them as well, it will just make the coordinate transformation a little different.
Agree on another event for what purposes? Do you mean some of the variables in equations in your derivation would no longer represent the interval between 0,0 and some specific event like the photon crossing B's worldline (i.e. the coordinate of the latter event), but would instead represent the interval between a different event other than 0,0 on one side of the interval, but the same event on the other side? I think I'd need to see you go through at least some of the same steps of the derivation as in post 397, but with whatever altered assumptions about "another event" you want to make, in order to follow what you're talking about here.

neopolitan

Jun15-09, 02:18 AM

I just meant that for any given choice of xa and v it's just a number, but I probably shouldn't have used the word "constant" since it could vary as you vary xa and v (although in fact it turns out that it doesn't). However, I wasn't actually assuming it remained constant as you varied these, so my argument in no way depends on this assumption, nor is it a "tangent". Please just mentally replace "constant" with "factor" in the block of text you quoted and reread it.

It will vary with v, but not xa as long as you don't do something weird.

What "end result" are you referring to? The end result that the same factor appears in both equations?

The end result is the the final pair of equations. The same factor in both equations is a middle stage in the process. But I am going to have a go at redoing it. We can continue to go over what you don't understand here though, until you are satisfied.

Note that since A and B's worldlines are assumed to lie along the x=0 axis of their respective frames, the event of the photon being colocated with A is equivalent to the event of the photon crossing A's x=0 axis, and likewise for B...this means your two equations can be written as:

(time of photon crossing B's x=0 axis in A's frame) = (factor)*(time of photon crossing B's x=0 axis in B's frame)

(time of photon crossing A's x=0 axis in B's frame) = (factor)*(time of photon crossing A's x=0 axis in A's frame)

So maybe now you can see why it wasn't a tangent when I asked in the previous post if you'd agree with a hypothetical guy making the argument that because of the "interchangeability" of the two frames, we could also assume the same factor will appear in the following two equations:

(time of photon crossing B's x=5 axis in A's frame) = (factor)*(time of photon crossing B's x=5 axis in B's frame)

(time of photon crossing A's x=5 axis in B's frame) = (factor)*(time of photon crossing A's x=5 axis in A's frame)

All that's changed in these two equations is that x=0 has been replaced with x=5. Please tell me whether you'd be confident about whether this guy's argument is right or wrong, without doing any SR calculations to check explicitly what factor appears in each. If you're unsure, that suggests that you don't really have good justification for being confident that "interchangeability" means the factor should be the same in your own two equations.

Actually no, it convinces me more that you have gone off on a tangent.

A and B are colocated at (0,0) ie (x=0,t=0) and (x'=0,t'=0) is the same event, so there is agreement about that event. There is no agreement about (5,0).

So I am pretty confident that "this guy's argument is wrong". (It might be right if the scenario is changed enough, but then it would be another scenario.)

Agree on another event for what purposes? Do you mean some of the variables in equations in your derivation would no longer represent the interval between 0,0 and some specific event like the photon crossing B's worldline (i.e. the coordinate of the latter event), but would instead represent the interval between a different event other than 0,0 on one side of the interval, but the same event on the other side? I think I'd need to see you go through at least some of the same steps of the derivation as in post 397, but with whatever altered assumptions about "another event" you want to make, in order to follow what you're talking about here.

Well, agree on another event for the purposes that your friend "this guy" could make the claim he did above about (5,0). Not that I know why they would do that.

However, as I said before: "When I redo the equations with v as a velocity rather than a speed, I will do them all with only intervals involved, rather than aiming for a coordinate transformation."

cheers,

neopolitan

JesseM

Jun15-09, 07:55 PM

The end result is the the final pair of equations. The same factor in both equations is a middle stage in the process. But I am going to have a go at redoing it. We can continue to go over what you don't understand here though, until you are satisfied.
We seem to be going in circles here. You've been saying that the magic word "interchangeable" somehow justifies the claim that the factors in your two equations should be the same, but then you said "By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values)." So here again it sounds like you're asserting what you're supposed to be proving, that "interchangeability" somehow will lead to the same "end result" (keeping in mind that what you got for the 'end result' itself depended on the earlier step where you were supposed to show that the factors would be the same in the two equations you wrote). Can you define "interchangeability" in a way that doesn't make reference to any steps in your derivation that happened after the step where you assume the factors are in fact the same in both equations?
Actually no, it convinces me more that you have gone off on a tangent.

A and B are colocated at (0,0) ie (x=0,t=0) and (x'=0,t'=0) is the same event, so there is agreement about that event. There is no agreement about (5,0).
Huh? The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=ta in A's frame) and the colocation of B and the photon (at x'=0, t=t'b in B's frame). Of course the time coordinate of either event in a given frame can be understood as the time interval between that event and (0,0), but then exactly is true about the time coordinate of either event of the photon crossing the x=5 axis of one of the frames. Perhaps there is something in what you mean by "interchangeable" that equations involving the time coordinates of a pair of events in both frames can only be considered interchangeable if the two events happened on the worldlines of observers at rest in each frame who crossed paths at (0,0), but if so nothing you have written about interchangeability so far even hinted at such a requirement. And what about observers who crossed paths at (0,0) but who are not at rest? What if we considered two observers who did cross paths there, with the first observer traveling at velocity V in the A frame and the second traveling at the same velocity V in the B frame? Then if we defined our two events in terms of where the light crossed each of these observer's paths, then without doing any numerical calculations do you think "interchangeability" means the factors in the equations relating the time coordinates of these two events in each frame would be the same?

Even if your answer is once again no, I say that these questions are not "tangential" because they help pin down exactly what you do or do not mean by the slippery word "interchangeable" and how it's supposed to be relevant to showing the factors are the same in those equations--this would be unnecessary if you would be willing to actually spell this stuff out, but so far you've been exceedingly vague. Remember my other request from the "tangent" post which you just ignored:
You really need to provide a step-by-step proof where each statement clearly follow from the last, and you show specifically where the two postulates of relativity come into play (since I think you'd agree that if we came up with a coordinate transformation that didn't respect these postulates the factors in your equations would not be the same), and what the relevance is of the fact that the two events in question are the events of a photon's worldline intersecting each frame's x=0 axis.
Here's another tack: if physicists say that certain things are "interchangeability", they are usually referring to some sort of symmetry where we see that something (like a dynamical equation or the values of some variables) remains the same under a certain transformation of labels (like which charge you call positive and which you call negative, or which of two frames you label A and B, etc.) Is this the sort of thing you're referring to here? If so, what is the specific thing that remains unchanged under a switch of which frame we call A and B, and how does this relate to proving that the factors will be the same in both equations?
You are part of the way there. I mean what you wrote in the corrected sentence, plus the fact that both A and B see that (0,0) event as (0,0), ie x=0,t=0 is the same event as x'=0,t'=0. There's agreement between A and B as to that event.

They can agree on another event rather than (0,0) if it is convenient to them as well, it will just make the coordinate transformation a little different.
Agree on another event for what purposes? Do you mean some of the variables in equations in your derivation would no longer represent the interval between 0,0 and some specific event like the photon crossing B's worldline (i.e. the coordinate of the latter event), but would instead represent the interval between a different event other than 0,0 on one side of the interval, but the same event on the other side? I think I'd need to see you go through at least some of the same steps of the derivation as in post 397, but with whatever altered assumptions about "another event" you want to make, in order to follow what you're talking about here.
Well, agree on another event for the purposes that your friend "this guy" could make the claim he did above about (5,0). Not that I know why they would do that.
I have no idea what this comment has to do with the questions in my post above--what "purposes" are you talking about? It's difficult to communicate with you when you a) often seem to conceptualize things in very idiosyncratic ways, and likewise seem to have your own idiosyncratic vocabulary for describing your own concepts, and b) don't seem to be aware of the fact that these ways of thinking/talking are idiosyncratic, or just not make allowances for the fact that they are, and thus you often act as though you expect others to pick up on your meaning from a few enigmatic keywords, that somehow you think the meaning you infer from these words is the "natural" one that everyone else should infer by default too (much like with your use of the word 'interchangeable' and how you immediately dismissed my questions as a 'tangent' when I didn't read your mind and detect all the subtle nuances of what you meant by that, or like when you expected me to immediately pick up on exactly what you meant by the phrase 'number of ticks' in that other recent post...this seems to happen time and time again in my discussions with you).

neopolitan

Jun15-09, 09:02 PM

We seem to be going in circles here. You've been saying that the magic word "interchangeable" somehow justifies the claim that the factors in your two equations should be the same, but then you said "By interchangeable, I mean that I could assign either of two observers, one in each frame, with the label A, and the other B. It won't affect the end result in any meaningful way (we might as a result prime A values)." So here again it sounds like you're asserting what you're supposed to be proving, that "interchangeability" somehow will lead to the same "end result" (keeping in mind that what you got for the 'end result' itself depended on the earlier step where you were supposed to show that the factors would be the same in the two equations you wrote). Can you define "interchangeability" in a way that doesn't make reference to any steps in your derivation that happened after the step where you assume the factors are in fact the same in both equations?

Ok, you'll probably not like it, but I am going to have to jump past the factors stage to the end result which, for me, is:

x_b'=\gamma . (x_a - vt_a)

and

x_a=\gamma . (x_b' + vt_b')

(There is a similar pair for time, but the same argument will apply to that pair as to this pair.)

In this pair, can you see that all b terms are also primed and all a terms are unprimed. So we could, if we wanted to, drop the subscripts leaving us with:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')

Alternatively, we could swap the A and B terms (what we called A before is now B and what we called B before is now A). Our v was (implicitly) defined as positive in the direction that B moves away from A, and negative in the direction that A moves away from B. We keep the same priming notation, since we have shifted focus from the erstwhile A to the new A.

This gives us:

x_a=\gamma . (x_b' - vt_b')

and

x_b'=\gamma . (x_a + vt_a)

Again all our b terms are primed and all our a are unprimed. So we could drop our subscripts, giving:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')

A and B are interchangeable labels. The possible confusion here is that in my idiosyncratic way, I have considered A and B to be labels. You seem to want to have more concrete (and thus less general) determinations of what A and B are.

Huh? The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=ta in A's frame) and the colocation of B and the photon (at x'=0, t=t'b in B's frame). Of course the time coordinate of either event in a given frame can be understood as the time interval between that event and (0,0), but then exactly is true about the time coordinate of either event of the photon crossing the x=5 axis of one of the frames. Perhaps there is something in what you mean by "interchangeable" that equations involving the time coordinates of a pair of events in both frames can only be considered interchangeable if the two events happened on the worldlines of observers at rest in each frame who crossed paths at (0,0), but if so nothing you have written about interchangeability so far even hinted at such a requirement. And what about observers who crossed paths at (0,0) but who are not at rest? What if we considered two observers who did cross paths there, with the first observer traveling at velocity V in the A frame and the second traveling at the same velocity V in the B frame? Then if we defined our two events in terms of where the light crossed each of these observer's paths, then without doing any numerical calculations do you think "interchangeability" means the factors in the equations relating the time coordinates of these two events in each frame would be the same?

I've mentioned a few times that there are only three colocation events. I honestly thought that I didn't need to make explicit that the colocation of A and B was at (0,0). I do think I have done it anyway - posts #397 (http://www.physicsforums.com/showpost.php?p=2232756&postcount=397), #381 (http://www.physicsforums.com/showpost.php?p=2231347&postcount=381), #378 (http://www.physicsforums.com/showpost.php?p=2231215&postcount=378), #342 (http://www.physicsforums.com/showpost.php?p=2228520&postcount=342), #336 (http://www.physicsforums.com/showpost.php?p=2228402&postcount=336) - sadly I don't have time to go on but I think there are more references where I have stated that A and B are colocated at t=0,t'=0. In at least one of those I have even made explicit that that event is (0,0).

As for the tangent, your asking questings for clarifications doesn't constitute a tangent, but your assertion that events associated with photons crossing the x=5 axis can be profitably used in my scenario indicates that you have gone off on a tangent.

The difference is like between these:

"I don't understand what you are saying, what are you saying"

and

"You are claiming this wrong thing, you are wrong"

when I never actually claimed the wrong thing you asserted that I claimed. In the real life example where I said it seems you were going off on a tangent, it might not have been so clear that you were asserting that I claimed something that I had not claimed. That's why I said you were going off on a tangent rather than saying you were making unfounded assertions.

But yes, we are going around in circles.

cheers,

neopolitan

JesseM

Jun15-09, 09:28 PM

Ok, you'll probably not like it, but I am going to have to jump past the factors stage to the end result which, for me, is:

x_b'=\gamma . (x_a - vt_a)

and

x_a=\gamma . (x_b' + vt_b')

(There is a similar pair for time, but the same argument will apply to that pair as to this pair.)

In this pair, can you see that all b terms are also primed and all a terms are unprimed. So we could, if we wanted to, drop the subscripts leaving us with:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')

Alternatively, we could swap the A and B terms (what we called A before is now B and what we called B before is now A). Our v was (implicitly) defined as positive in the direction that B moves away from A, and negative in the direction that A moves away from B. We keep the same priming notation, since we have shifted focus from the erstwhile A to the new A.

This gives us:

x_a=\gamma . (x_b' - vt_b')

and

x_b'=\gamma . (x_a + vt_a)

Again all our b terms are primed and all our a are unprimed. So we could drop our subscripts, giving:

x'=\gamma . (x - vt)

and

x=\gamma . (x' + vt')

A and B are interchangeable labels. The possible confusion here is that in my idiosyncratic way, I have considered A and B to be labels. You seem to want to have more concrete (and thus less general) determinations of what A and B are.
Sigh. Of course I understand that once you have the final Lorentz transformation (which is not what you actually derive, but never mind that for now), A and B are interchangeable labels, "interchangeable" in the sense that the transformation equations still work fine if you switch the subscripts for A and B and also switch which frame v is defined in terms of. How does this have anything whatsoever to do with the question of how you intend to show that the factor relating the two specific events you chose--and only those two events, not two other events like the ones I mentioned--should be the same in the two equations you wrote down, without assuming the Lorentz transformations to begin with? Do you really not understand what it means to "show" something in a derivation without engaging in circular reasoning?
I've mentioned a few times that there are only three colocation events. I honestly thought that I didn't need to make explicit that the colocation of A and B was at (0,0).
I'm not talking about how many colocation events appear in your overall derivation, I'm just talking about the two events that appear in these two equations you wrote earlier:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

Can you please stop your mind from wandering all over the place and imagining I am talking about all these other aspects of your derivation, when I think I made it pretty clear I'm just talking about the specific question of how in these two equations you intend to show that the factor must be the same? If you don't in fact believe that your notion of "interchangeability" is sufficient to show this specific thing, and you haven't actually thought of a good way to show this yet, then that's fine, just say so. But if you do, then please stick to this specific topic for now.
As for the tangent, your asking questings for clarifications doesn't constitute a tangent, but your assertion that events associated with photons crossing the x=5 axis can be profitably used in my scenario indicates that you have gone off on a tangent.
I didn't say it "can be profitably used" in your scenario. I was saying that your argument about "interchangeability" is so vague that there's no apparent reason it shouldn't apply to those two events just as much as the two events you chose, and thus it "proves too much". You could respond by actually elaborating on what you meant by "interchangeability" (as I keep asking you to do but you keep not doing), giving an expanded definition which would show why interchangeability demands the factor be the same in your two equations but not in my closely analogous equations. Likewise for my question about another fairly analogous pair of events in the last post:
And what about observers who crossed paths at (0,0) but who are not at rest? What if we considered two observers who did cross paths there, with the first observer traveling at velocity V in the A frame and the second traveling at the same velocity V in the B frame? Then if we defined our two events in terms of where the light crossed each of these observer's paths, then without doing any numerical calculations do you think "interchangeability" means the factors in the equations relating the time coordinates of these two events in each frame would be the same?
Again, if your "proof by interchangeability" is supposed to make any sense, you should have some clear explanation as to why "interchangeability" does or doesn't imply that the factors in equations relating these two events would be the same, and if it doesn't what the crucial difference is that makes it imply that for your pair of events but not this pair of events.
The difference is like between these:

"I don't understand what you are saying, what are you saying"

and

"You are claiming this wrong thing, you are wrong"

when I never actually claimed the wrong thing you asserted that I claimed.
I never asserted that you claimed anything about the x=5 example, my point was that if you didn't have a clear idea of why interchangeability is supposed to imply one thing about your equations and something different about the equations I wrote, then the concept is too vague to make for a convincing demonstration that the factors should in fact be the same in your own pair of equations.

neopolitan

Jun16-09, 12:47 AM

Going back to this point:

It seems to me that we have two stopping points here.

First, the need to prove that there must be one single factor (as in my derivation shown earlier), rather than possibly two factors.

Second, generality, in that you think that if I chose two totally arbitrary events in order to to measure the interval between them in two frames, and that you feel that there is a "need for an explanation as to what specific property of the two events (I) chose ensures that the constants in those two equations will be the same".

Is this correct?

Do you further agree that, if I were to convince you that there being one single factor is the default, that feat would negate the need for a proof? I'm going to think about the proof angle anyway, but I am not abandoning my argument that what you are asking me to prove is actually the default.

I am going to assume that I have not convinced you that there being one single factor is the default.

That means we still have two stopping points, and I have to address both of them.

From here I suggest that I provide a general scenario, rather than one which is limited by having an event which is simultaneous with colocation of A and B in one frame, and work from there, with as much rigour as I can muster.

Included in that rigour is a requirement to prove, by elimination of other options if necessary, that a single factor applies in the intermediate step which we have been discussing.

Is that right?

cheers,

neopolitan

JesseM

Jun16-09, 02:34 AM

I am going to assume that I have not convinced you that there being one single factor is the default.
Not sure what you mean by "default". Certainly if someone had no familiarity with SR and saw your two equations, if forced to guess, they might by default choose the same factor in each. But if this is supposed to be a derivation you need to establish that it must be the same factor.
From here I suggest that I provide a general scenario, rather than one which is limited by having an event which is simultaneous with colocation of A and B in one frame, and work from there, with as much rigour as I can muster.
Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.
Included in that rigour is a requirement to prove, by elimination of other options if necessary, that a single factor applies in the intermediate step which we have been discussing.
Yes, that's what must be proved.

neopolitan

Jun16-09, 04:30 AM

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

This statement means that you clearly have not grasped something. It's good to know this now, rather than later.

The light beam is totally arbitrary, because I intended a general proof.

Take a totally arbitrary light beam, let it pass A and B (order not really important), then select an event on that light beam (I didn't make it totally arbitrary, I restricted it to being simultaneous with colocation of A and B in one frame).

You then have four events:

colocation of A and B, at (0,0) - ie x=0,t=0 and x'=0,t'=0
the event I just talked about selecting
colocation of the light beam and A
colocation of the light beam and B

With these four events (plus the laws of physics) you have enough information to derive the Lorentz Transforms (or analogues thereof).

To make my proof more general, I need to select from the events along the light beam an event which is not necessarily simultaneous with (0,0) in either frame.

You don't seem to like me using the intervals (0,0)-event in one frame and (0,0)-event in the other frame - which are the coordinates of the event in one frame and the coordinates of the event in the other frame (perhaps I haven't managed to make it abundantly clear that that is what I am doing, despite having done a bit of prepartory work on the issue of coordinates basically being intervals - latest attempt was here). Therefore, I will select from the events along another arbitrary light beam another event which is also not necessarily simultaneous with (0,0) in either frame. Then I will do the derivation based on a totally arbitrary spacetime interval.

cheers,

neopolitan

JesseM

Jun16-09, 10:16 AM

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.
This statement means that you clearly have not grasped something. It's good to know this now, rather than later.

The light beam is totally arbitrary, because I intended a general proof.
What do you think I didn't grasp? Isn't that what I just said, "a totally arbitrary light beam"?
You then have four events:
Yes, in your overall derivation you have four events. But in those two equations where you say the factor is the same, only two events appear. Can we please try to stick to the subject of those two equations and the factor in them, unless you are willing to actually acknowledge that your previous "interchangeability" argument isn't sufficient to establish that the factor should be the same?
You don't seem to like me using the intervals (0,0)-event in one frame and (0,0)-event in the other frame
Where did you like the idea that I don't like that? In post 430 I wrote:
The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=ta in A's frame) and the colocation of B and the photon (at x'=0, t=t'b in B's frame). Of course the time coordinate of either event in a given frame can be understood as the time interval between that event and (0,0), but then exactly is true about the time coordinate of either event of the photon crossing the x=5 axis of one of the frames. Perhaps there is something in what you mean by "interchangeable" that equations involving the time coordinates of a pair of events in both frames can only be considered interchangeable if the two events happened on the worldlines of observers at rest in each frame who crossed paths at (0,0), but if so nothing you have written about interchangeability so far even hinted at such a requirement.
If you're talking more generally about the final equations you derive, my objection is not that you're talking about intervals from (0,0) to some other event, it's that for different variables in those equations the "other event" is different, whereas in the Lorentz transformation equations every variable is supposed to refer to the space and time intervals between the same pair of events in different frames (or the space and time coordinates of the same single event in different frames).

neopolitan

Jun16-09, 10:33 AM

What do you think I didn't grasp? Isn't that what I just said, "a totally arbitrary light beam"?

I'm sorry, I was taking your comment to have been meant dismissively. If that was not your intention then ok.

Look back at the whole post (I know you have read it all before responding). I said:

Take a totally arbitrary light beam, let it pass A and B (order not really important), then select an event on that light beam (I didn't make it totally arbitrary, I restricted it to being simultaneous with colocation of A and B in one frame).

You then have four events:

colocation of A and B, at (0,0) - ie x=0,t=0 and x'=0,t'=0
the event I just talked about selecting
colocation of the light beam and A
colocation of the light beam and B

With these four events (plus the laws of physics) you have enough information to derive the Lorentz Transforms (or analogues thereof).

To make my proof more general, I need to select from the events along the light beam an event which is not necessarily simultaneous with (0,0) in either frame.

I start with "the event", the other (implied) event which allows me to obtain the coordinates of this event is both frames is what I call an "agreed" event, ie the colocation of A and B at (0,0).

The other two events (colocation of A and the photon) and (colocation of B and the photon) are used for the information we can glean from them, namely the time elapsed when they occur since A and B were colocated.

I'm not probably going to get this across properly right now, but I am glad to have a little better insight into what has not been conveyed correctly.

cheers,

neopolitan

JesseM

Jun16-09, 11:02 AM

Look back at the whole post (I know you have read it all before responding).
Ugh, please don't make comments like this, it comes across as patronizing. Especially when you have a history of imagining I didn't get some point of yours because I "didn't read the whole post" or "broke it up into pieces" when in fact the issue was your idiosyncratic communication style where you seem to have expected others to interpret some random phrase in exactly the way you were thinking of without your having to actually explain your meaning (again think of the 'number of ticks' thing I responded to at the end of post 413)
I start with "the event", the other (implied) event which allows me to obtain the coordinates of this event is both frames is what I call an "agreed" event, ie the colocation of A and B at (0,0).

The other two events (colocation of A and the photon) and (colocation of B and the photon) are used for the information we can glean from them, namely the time elapsed when they occur since A and B were colocated.
Here you just seem to be throwing out random statements about your derivation (statements which I think I already understood, unless I am misinterpreting your words in some way), without explaining their relevance to what we are actually discussing. Is this supposed to somehow address either my point that only two events appear in the pair of equations where you said the factor is the same, or my point that the variables in your final equations don't all refer to a consistent pair of events?

neopolitan

Jun16-09, 08:53 PM

Here you just seem to be throwing out random statements about your derivation (statements which I think I already understood, unless I am misinterpreting your words in some way), without explaining their relevance to what we are actually discussing. Is this supposed to somehow address either my point that only two events appear in the pair of equations where you said the factor is the same, or my point that the variables in your final equations don't all refer to a consistent pair of events?

I took one section of your post #434 and addressed it in post #435. Here:

Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.

In that section, you specifically refer to two events twice (I accept that I may be reading "neither" a little too strictly).

In second part of post #437, I go over what two events I am talking about, which is (one event on the path of a totally arbitrary light beam) and (the colocation of A and B). I also point out that I use information associated with two more events which are also on the path of the totally arbitrary light beam.

All up that makes three events on the path of the totally arbitrary light beam.

Since I am not talking about two events on the path of the light beam, it is clear to me that something is amiss if you think I am talking about two events on the light beam. I could agree with one or three and I might even understand what you were talking about if you said four.

I will do my best to provide a derivation in which the consistency of the pair of events is very clear. Are we ready to proceed to that?

cheers,

neopolitan

JesseM

Jun16-09, 09:23 PM

I took one section of your post #434 and addressed it in post #435. Here:
Neither of the events in your two equations is simultaneous with colocation of A and B. They're just two events on the path of a totally arbitrary light beam that crosses paths with A and B.
Question: what specific equations do you imagine I was referring to when I said "in your two equations" in that post? (see below if it wasn't clear)
In that section, you specifically refer to two events twice (I accept that I may be reading "neither" a little too strictly).

In second part of post #437, I go over what two events I am talking about, which is (one event on the path of a totally arbitrary light beam) and (the colocation of A and B). I also point out that I use information associated with two more events which are also on the path of the totally arbitrary light beam.

All up that makes three events on the path of the totally arbitrary light beam.

Since I am not talking about two events on the path of the light beam, it is clear to me that something is amiss if you think I am talking about two events on the light beam. I could agree with one or three and I might even understand what you were talking about if you said four.
Let me just repeat my comment from post 432 again:
I'm not talking about how many colocation events appear in your overall derivation, I'm just talking about the two events that appear in these two equations you wrote earlier:

(time of colocation of B and photon in the A frame) = (some factor) * (time of colocation of B and photon in the B frame)

and

(time of colocation of A and photon in the B frame) = (some factor) * (time of colocation of A and photon in the A frame)

Can you please stop your mind from wandering all over the place and imagining I am talking about all these other aspects of your derivation, when I think I made it pretty clear I'm just talking about the specific question of how in these two equations you intend to show that the factor must be the same? If you don't in fact believe that your notion of "interchangeability" is sufficient to show this specific thing, and you haven't actually thought of a good way to show this yet, then that's fine, just say so. But if you do, then please stick to this specific topic for now.
Are you claiming that more than two events appear in those two specific equations? I only see "colocation of B and photon" on both sides of the first equation, and "colocation of A and photon" on both sides of the second (and these equations are quoted from your own post 427).

Note that I had said something similar in the earlier post 430:
Huh? The event (0,0) is not referred to in the two equations where you assert the factors will be the same. The two events referred to in those equations of yours are the colocation of A and the photon (at x=0,t=ta in A's frame) and the colocation of B and the photon (at x'=0, t=t'b in B's frame).
...but you didn't seem to get it in your response which was why I elaborated in post 432. Then in post 436 I again repeated the fact that I was just talking about those two specific equations where you said the factor was supposed to be the same:
Yes, in your overall derivation you have four events. But in those two equations where you say the factor is the same, only two events appear. Can we please try to stick to the subject of those two equations and the factor in them, unless you are willing to actually acknowledge that your previous "interchangeability" argument isn't sufficient to establish that the factor should be the same?
And then in post 438 I said:
Is this supposed to somehow address either my point that only two events appear in the pair of equations where you said the factor is the same, or my point that the variables in your final equations don't all refer to a consistent pair of events?
...so I don't think the problem here is that I've failed to make clear I'm just talking about the events that appear in those specific equations.
I will do my best to provide a derivation in which the consistency of the pair of events is very clear. Are we ready to proceed to that?
Does "the consistency of the pair of events" mean the same thing as your earlier talk of "interchangeability"? If so please address my questions from post 430. If not, please tell me whether you are abandoning your earlier claims that "interchangeability" was sufficient to rigorously establish that the factor should be the same, or whether you stick by that claim.