Benefits of time dilation / length contraction pairing?

[neopolitan]

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

[ZapperZ]

Would it surprise you to learn that these "time dilation" and "length contraction" are merely consequences of the Lorentz transformation coupled with Einstein's SR postulates?

So do you really have a problem with the fundamental theory, or do you simply have a problem with the consequences of the theory? If you do not have a problem with the fundamental theory (because you didn't even mention it), then where in the logical derivation of the consequences did we lose you and they become "inconsistent"? For example, start with a standard derivation of time dilation as presented in standard physics textbooks. Point out exactly where such a thing becomes logically inconsistent.

What is so "great" about time dilation? It explains a whole bunch of empirical observations. What more can you ask for?

Zz.

[JesseM]

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.
They are consistent. Unprimed is used in these equations to represent times and lengths in the frame where the ruler/clock is at rest; primed is used to represent the corresponding times and lengths in the frame where the ruler/clock is moving. Can you specify what alternative you're proposing?

[Saw]

where in the logical derivation of the consequences did we lose you and they become "inconsistent"? For example, start with a standard derivation of time dilation as presented in standard physics textbooks. Point out exactly where such a thing becomes logically inconsistent.

I would put it the other way round, ZapperZ. I've looked at the transversal light clock example and the longitudinal light clock example by themselves and wondered how one can derive from those examples the consequence that there is RECIPROCAL TD and LC , if you consider those derivations isolatedly.

A different thing is that, if you consider jointly RECIPROCAL TD, LC and RS, the whole system of SR really seems to make sense. Thus the "standard derivation of time dilation as presented in standard physics textbooks" may have a limited use as a way to derive the quantitative aspect of those elements, which however only make sense (as a RECIPROCAL measurement) when they are put together and integrated in a "global system", which by the way does not contemplate any duality of opinions about reality, but, on the contrary, full agreement on what has really happened.

But maybe I am imagining something you are not saying. In your view, the "standard derivation of time dilation as presented in standard physics textbooks" (for instance, the light clock example)..., does it logically prove that there is RECIPROCAL time dilation?

[neopolitan]

Would it surprise you to learn that these "time dilation" and "length contraction" are merely consequences of the Lorentz transformation coupled with Einstein's SR postulates?

No, if L=x' and L'=x (or perhaps L = \Delta x' and L' = \Delta x).

Is that what you are saying?

But then where is the consistency of prime notation that JesseM claims in his post?

I do understand that, with the awkward interpretation, time dilation does explain a lot of empirical observations. No problems there. But so would a time equation in the same form as the length equation, and it would not lead to the problem with people thinking "speed of light in another inertial frame is contracted length divided by dilated time ... hang on, that's not invariant!" Then they visit here and have someone metaphorically yelling at them "you are mixing frames!" without actually explaining why you can't use time dilation and length contraction that way.

I know you can't, I just want to know what advantages exist with having time dilation and length contraction expressed the way they are?

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?

cheers,

neopolitan

[ZapperZ]

I would put it the other way round, ZapperZ. I've looked at the transversal light clock example and the longitudinal light clock example by themselves and wondered how one can derive from those examples the consequence that there is RECIPROCAL TD and LC , if you consider those derivations isolatedly.

A different thing is that, if you consider jointly RECIPROCAL TD, LC and RS, the whole system of SR really seems to make sense. Thus the "standard derivation of time dilation as presented in standard physics textbooks" may have a limited use as a way to derive the quantitative aspect of those elements, which however only make sense (as a RECIPROCAL measurement) when they are put together and integrated in a "global system", which by the way does not contemplate any duality of opinions about reality, but, on the contrary, full agreement on what has really happened.

But maybe I am imagining something you are not saying. In your view, the "standard derivation of time dilation as presented in standard physics textbooks" (for instance, the light clock example)..., does it logically prove that there is RECIPROCAL time dilation?

Er.. come again?

There are two aspects to this. One is the logical derivation, i.e. mathematical derivation, based on SR's postulates. It has to based on that because there's nothing mathematically that can derive c being a constant in all frames.

The second is the experimental verification. A logical derivation of anything in physics is no guarantee that it is valid. It is the experimental observation consistent with such result that elevates its validity. It somehow appears as if you want to start from the tail end of it to justify itself, which is fine if you are trying to formulate a new theory. But considering that SR is such a well-established theory with solid foundation, and every one of the consequences can be derived from such foundation, it would be logical to start from there. And that's where I do not get the OP. Is there a problem with the foundation in the first place or is he/she only do not get the consequences? I don't think that is such an unreasonable query.

Zz.

[ZapperZ]

No, if L=x' and L'=x (or perhaps L = \Delta x' and L' = \Delta x).

Is that what you are saying?

But then where is the consistency of prime notation that JesseM claims in his post?

I do understand that, with the awkward interpretation, time dilation does explain a lot of empirical observations. No problems there. But so would a time equation in the same form as the length equation, and it would not lead to the problem with people thinking "speed of light in another inertial frame is contracted length divided by dilated time ... hang on, that's not invariant!" Then they visit here and have someone metaphorically yelling at them "you are mixing frames!" without actually explaining why you can't use time dilation and length contraction that way.

I know you can't, I just want to know what advantages exist with having time dilation and length contraction expressed the way they are?

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?

cheers,

neopolitan

I don't see it.

Forget SR/Lorentz transformation. Start with Galilean transformation. Do you have a problem with that as well? Write down the velocity and displacement of a moving object in two different inertial frames via Galilean transformation. My guess is that you have a problem with that as well, because fundamentally, none of what you wrote above really has anything to do with SR.

At what point do you acknowledge about the empirical verification of these things?

Zz.

[Mentz114]

(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.)
Hard to believe. Can't you see that this is an irrelevance. How else could you express time dilation except as an increase the gap between ticks ? Counting ticks still means you have to multiply the number of ticks by the time between ticks.

When you talk about primes, are you making a point about notation ? You could use some other way to distiguish two frames, but it wouldn't make any difference !

[JesseM]

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?
Yes, and never did you offer any convincing argument that there was something "inconsistent" about the standard definitions. My diagram shows that they are consistent, given the usual definition of "length" and "time interval"--the "spatial analogue of time dilation" and the "temporal analogue of of length contraction" in the diagram don't refer to any commonly-used or intuitive physical quantities (the 'spatial analogue of time dilation' refers to the distance in the primed frame between two events that are simultaneous in the unprimed frame; the 'temporal analogue of length contraction' is even weirder, it refers to the time in the primed frame between two surfaces of simultaneity that cross through the events in the unprimed frame).

[neopolitan]

Hard to believe. Can't you see that this is an irrelevance. How else could you express time dilation except as an increase the gap between ticks ? Counting ticks still means you have to multiply the number of ticks by the time between ticks.

When you talk about primes, are you making a point about notation ? You could use some other way to distiguish two frames, but it wouldn't make any difference !

Yes, I know that is what time dilation is (and must be to keep everything right).

What is difficult for people new to SR, the very ones who are being taught time dilation, is that the equation involving t, ostensibly the same t which they may well be used to in Gallilean boosts (x'=x-vt) and the kinetics equations (v = (si-so)/t and so on), is using a quite different definition of t.

Jesse mentions "time interval", which is fine, I just wonder why we don't use a different symbol for time dilation (\tau perhaps) to highlight the difference between "time interval" - time between ticks - and "(measured) time elapsed" - number of ticks.

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

(It seems the only responses so far are "you can't do it any other way" and "you are confused". Is there really no other way?)

cheers,

neopolitan

[ZapperZ]

Yes, I know that is what time dilation is (and must be to keep everything right).

What is difficult for people new to SR, the very ones who are being taught time dilation, is that the equation involving t, ostensibly the same t which they may well be used to in Gallilean boosts (x'=x-vt) and the kinetics equations (v = (si-so)/t and so on), is using a quite different definition of t.

Jesse mentions "time interval", which is fine, I just wonder why we don't use a different symbol for time dilation (\tau perhaps) to highlight the difference between "time interval" - time between ticks - and "(measured) time elapsed" - number of ticks.

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

(It seems the only responses so far are "you can't do it any other way" and "you are confused". Is there really no other way?)

cheers,

neopolitan

This complain has nothing to do with SR. Look at your kinematics problem. You use the SAME thing there! This simply re-enforces my earlier assertion that this isn't about "time dilation" at all. You are simply confused on what we call 'time' in any dynamical system.

Zz.

[Mentz114]

Neopolitan,
But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?
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'.

[neopolitan]

This complain has nothing to do with SR. Look at your kinematics problem. You use the SAME thing there! This simply re-enforces my earlier assertion that this isn't about "time dilation" at all. You are simply confused on what we call 'time' in any dynamical system.

Zz.

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.

Therefore, t= (number of ticks on my stopwatch).

Not (time between each tick on my stopwatch).

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).

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.

I can get around that by changing the definition of time in my dynamical system. I don't think that is such a fabulous idea, but I could do it.

Am I simply confused here?

cheers,

neopolitan

[JesseM]

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?
What would be the pair of equations you're proposing, that wouldn't involve quantities which are far more unintuitive to students than "length" and "time interval", and which wouldn't be much more difficult to actually apply to the types of introductory problems found in textbooks? Please write them down.

[JesseM]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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.
neopolitan
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.

[neopolitan]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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

[JesseM]

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]

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]

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]

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'.

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.

If I want to work out something different, specifically how much time has passed on my clock while an observed t has passed on my buddy's clock, I have to use a different equation to get:

T' = t * gamma

cheers,

neopolitan

[JesseM]

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]

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]

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]

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]

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.

Shouldnt muon decays be additionally slowed down because of very strong centrifugal forces affecting the muon? (GR time dilation)?

[neopolitan]

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]

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]

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]

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]

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]

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)

There are a lot of things in the standard presentation of SR that I think are sub-optimal from a pedagogical perspective.

[JesseM]

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]

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]

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]

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]

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

[JesseM]

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]

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]

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]

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]

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]

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. Admittedly there'd be multiple possible states of the rest of space that could be responsible for the state of your surface of constant x, but then you could likewise have different histories that lead to the same state for some surface of constant t. So is it something about the future specifically, and its predictability, that makes all the difference? If that's even a meaningful question...

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.)

[JesseM]

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]

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]

"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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

(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]

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]

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]

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]

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]

"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]

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]

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]

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]

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]

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

[JesseM]

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]

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]

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.

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

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.

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.

(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

So long as no matter what frame you view it from, the photon travels a distance of ct in t and a distance of ct' in t', I am happy - irrespective of how you want to link t and t'.

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).

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.)

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 am focussed very much on the effects of on something which is in motion relative to me or some impartial observer.

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.

You seem to be unable to put that second part aside for a moment, perhaps because you think I think it isn't true. I do think it is true, just not currently helpful (as was the fact that muons have mass as Mentz will have us know, true but not actually helpful).

Again, I hope this helps.

cheers,

neopolitan

[bernhard.rothenstein]

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]

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]

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]

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
Thank you for your answer. The last case you mention is very interesting, because length and time intervals are related by the Doppler factor in an electromagnetic wave. The light signal generates in I the event (x;ct) whereas in I' the event (x';ct'). The cortresponding Lorentz transformations lead to
x'=g(x-Vt)=gx(1-V/c)
t'=g(t-Vx/cc)=gt(1-V/c)
g standing for the Lorentz factor.
Kind regards

[Mentz114]

\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]

\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]

\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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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:

transformations in which L/t=L'/t' becase L and t transform via the same Doppler factor.

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

[bernhard.rothenstein]

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]

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]

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
< Previous article | Next article * | This volume ^^ | This issue ^ | Content finder *

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]

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]

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

cheers,

neopolitan

[neopolitan]

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).

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:

Note that there are no givens for what the events actually are, where they are or when they took place. The experimental controllers may even have lied about the events being "together" at all.

cheers,

neopolitan

(I am leading somewhere, by the way. I think the problem may be easier to solve geometrically, but numbers are fine too.)

[DrGreg]

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]

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]

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]

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]

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]

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]

.... 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]

...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]

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]

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]

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.

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.)

However, I was not concerned about whether A and B confer and I didn't assume that the experiment is set up so that events are arranged to make the events happen "together" for both, in different fashions. Pretend it is a psychology experiment (where test subjects are routinely misled).

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.

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.

cheers,

neopolitan

[neopolitan]

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]

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]

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]

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]

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?

[JesseM]

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]

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]

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]

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]

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]

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]

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.

Clearly something is wrong somewhere, either with my understanding, my explanation or your assumptions.

cheers,

neopolitan

[JesseM]

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]

You mean like this?

[JesseM]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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).

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).

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. You may want to check that they are referring to the right sort of things in your terminology (ie in which frame are things simultaneous and in which frame are things collocated), but I think you will find that while saying "time and space are measured oddly" might sound less than rigorous, it is actually more intuitive. To work out the simultaneity/collocation issues, you really have to already understand so I think it is a stage that should come after you have given a derivation.

cheers,

neopolitan


PS I had another thought about the simultaneity/collocation thing. I agree wholeheartedly that you can work out TD and LC from taking into consideration which frames have co-local ticks of a clock and which frames a rod is simultaneous along its length (or, in other words, frames which have non-simultaneous but collocated events and frames which have simultaneous non-collocated events). But using the derivation shown in #174, this can be worked out retrospectively, rather than having to be known a priori. So, it is not just a case of confusing the new learner, I think that my derivation requires less a priori understanding (and I'm willing to accept that this might not be the case).

[neopolitan]

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

You are right, they don't need to assume that the event that spawned the photon occurred when they were collocated with the other observer.

This was very poorly worded (particularly the words I have made red) and I apologise for the confusion it created.

It is better to say that when the photon passes B at tB' in my later diagram (my poor phrasing quoted above was in post #168, diagrams were post #174), B could assume that one of the possible spawning events could have been at ctB' when A and B were collocated.

When the photon passes A at tA, A could assume that one of the possible spawning events could have been at ctA when A and B were collocated.

I do believe that neither A nor B are wrong in being able to make these assumptions. Being objectively right never comes into it.

Where I wrote "Both assume", I should have written "Both could assume". It made sense to me at the time :smile:

cheers,

neopolitan

[neopolitan]

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]

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]

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]

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. 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).

That means that, according to A, tB = t' and tA = t, therefore:

t' = gamma.t

which is time dilation.

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.

Application of the gamma of 1.25 is time dilation and the stretching of the tB axis.

This would mean that the darker orange line is time dilation for A and the dark blue line is time dilation for B.

I'm happy with that.

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.

Are you happy with that?

What you can't really do though, is measure these values directly. Even to show it on the diagram is difficult, I would have to animate it or show a rotation around the origin of the axis (which may introduce problems). You can measure the time dilation indirectly in either of the ways described above (relativistic doppler or some variant of the twin's experiment).

You might say they have already tested it and showed that it works, and I would say "yes, they have done so, indirectly". What they have not done is something equivalent to directly measuring length contraction which could be done by simply passing two rods past each other - and both would measure the other to be shorter.

If you can come up with an experiment in which both observers can directly observe the other be time dilated, I'd be curious to know about it. I think that relativistic doppler is as close as you can get.

I've posted updated TD and SAFTD diagrams (Attempt 2) here (http://www.geocities.com/neopolitonian/index.htm).

cheers,

neopolitan

[neopolitan]

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]

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]

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]

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]

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]

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]

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]

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