Benefits of time dilation / length contraction pairing?

[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, 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 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 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 realize 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? ()

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, traveling 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, traveling 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 traveled 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 traveled 660km in 2.2ms. According to the laboratory, the photon in the laboratory frame's clock has traveled 660km in 2.2ms. According to the laboratory, both photons have traveled 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 L_o and T_o are.

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

T = T_o * gamma
L = L_o / gamma

so:

T_o / T = 1 / gamma
L / L_o = 1 / gamma

so:

(T_o / T)(L / L_o) = 1 /(gamma)²

Which is partly why I question it.

Rearranging (T_o / T)(L / L_o) = 1 gives you:

(T_o / L_o)(L / T) = 1

or

L / T = L_o / T_o

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 traveling 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 L_o or L' in time T_o or T' (so that L_o/T_o=c or L'/T'=c).

cheers,

neopolitan

 

[bernhard.rothenstein]

It depends a little on what L_o and T_o are.

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

T = T_o * gamma
L = L_o / gamma

so:

T_o / T = 1 / gamma
L / L_o = 1 / gamma

so:

(T_o / T)(L / L_o) = 1 /(gamma)²

Which is partly why I question it.

Rearranging (T_o / T)(L / L_o) = 1 gives you:

(T_o / L_o)(L / T) = 1

or

L / T = L_o / T_o

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 traveling 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 L_o or L' in time T_o or T' (so that L_o/T_o=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 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.

 

[Dale]

I find myself in the (for me) very odd position of reconciler or mediator or some similar "kum-ba-ya" campfire nonsense. I am traveling 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 L_o and T_o are.

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

T = T_o * gamma
L = L_o / gamma

so:

T_o / T = 1 / gamma
L / L_o = 1 / gamma

so:

(T_o / T)(L / L_o) = 1 /(gamma)²

Which is partly why I question it.

Rearranging (T_o / T)(L / L_o) = 1 gives you:

(T_o / L_o)(L / T) = 1

or

L / T = L_o / T_o

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 traveling 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 L_o or L' in time T_o or T' (so that L_o/T_o=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 traveling 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 L_o or L' in time T_o or T' (so that L_o/T_o=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 L_o.

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 L_o.

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 x_o=x'_o=0 and t_o=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 x_o=x'_o=0 and t_o=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, traveling 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 traveled 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 traveled 660km in 2.2ms. According to the laboratory, the photon in the laboratory frame's clock has traveled 660km in 2.2ms. According to the laboratory, both photons have traveled 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

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 L_A and a time interval of t_A, 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

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 t_A apart by her clock. B (Bob) receives signals from the two events at a time t_B 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.