Benefits of time dilation / length contraction pairing?

[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.

 

[JesseM]

neopolitan

In post #146 I gave the answer

$$t_B = t_A \sqrt{\frac{c-v}{c+v}}$$​

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

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

And, JesseM, do you follow my logic and agree with my conclusion?

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

 

[neopolitan]

JesseM,

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

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

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

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

cheers,

neopolitan

 

[JesseM]

JesseM,

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

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

Yes, 10 s was the time between the light from each event hitting B as measured in A's frame, 8 s was the time between the light from each event hitting B in B's own frame. Here we are dealing with two events that are colocated in B's frame (the events of the light hitting B's worldline, not the events that emitted the light in the first place).

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

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

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

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

The trick with the TAFLC is that if you write it as t' = t / gamma to contrast with the standard time dilation equation t' = t * gamma, then in order for the notation to be consistent you have to be assuming the same thing about the unprimed frame being the one where the events are colocated in both cases.

 

[neopolitan]

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

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

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

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

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

I feel that we could circle forever on this.

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

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

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

cheers,

neopolitan

 

[JesseM]

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

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

You mean the question "If A and B have identical rods and identical clocks, what sort of conclusions will B come to" from post #133? I thought I already answered that. If it wasn't that, which question/post are you referring to?

I feel that we could circle forever on this.

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

I don't see how this has anything to do with visualizations, terms like "TAFLC" and "inverse time dilation" are ultimately supposed to refer to specific equations where the terms have specific physical meanings, if you aren't clear on this then any "visualizations" you come up with will be too ill-defined to have any real meaning. In the original diagram you reposted in post #5 of this thread, I specifically defined the TAFLC as t' = t / gamma, in contrast to the normal time dilation which I wrote as t' = t * gamma. What you're talking about is just dividing both sides of the normal time dilation equation by gamma, a simple algebraic operation which gives t = t' / gamma, which is different than the TAFLC above because if you want the notation to be consistent the unprimed frame has to be the one where the events separated by a time interval of t are colocated. If you just want a change in terminology from what I wrote in that diagram, and want to call the equation t = t' / gamma the "TAFLC", then this is just a semantic issue, I'm fine with calling that equation whatever you want to call it as long as you agree that it's nothing more than the ordinary time dilation equation with both sides divided by gamma. But if you think the TAFLC as I defined it, t' = t / gamma (with t the frame in which whatever events we're dealing with are colocated), is somehow applicable to the problem with the light hitting B, you have to explain what t and t' are supposed to represent physically. And if you just aren't willing to pin down what you mean by terms like "TAFLC" and "inverse time dilation" in terms of specific equations where the terms have specific meanings, then the whole discussion is pointless because pictures in physics are essentially just ways of illustrating the math (as is true of all Minkowski diagrams, for example).

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

When did I have trouble coming to a result? I didn't immediately give an answer because, as I think you now acknowledge, you misunderstood my question about dealing with multiple possibilities and thus gave me a misleading answer which made me unclear what you could be asking. Once this was cleared up I gave a pretty detailed numerical example.

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

All you need to do is give the equations which are supposed to correspond to these words (preferably side-by-side with how you'd write the normal time dilation equation so I can check that the notation is consistent), and what the terms in the equations represent physically. In the meantime, happy Easter!

 

[neopolitan]

Jesse,

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

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

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

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

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

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

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

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

Now the thing that concerns me is that if I were to plug your figures into my drawing, I don't think I would arrive at 16 seconds between the arrival of the photons for A and 8 seconds between the arrival of the photons for B. I'd arrive at 16s and 12.8s.

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

cheers,

neopolitan

 

[JesseM]

Jesse,

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

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

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

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

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

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

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

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

Now the thing that concerns me is that if I were to plug your figures into my drawing, I don't think I would arrive at 16 seconds between the arrival of the photons for A and 8 seconds between the arrival of the photons for B. I'd arrive at 16s and 12.8s.

Keep in mind that if you draw 1-second ticks along A's time axis and 1-light second ticks along A's space axis, then also draw similar ticks along B's space and time axis, then when drawn from the perspective of A's frame, B's space and time axes are not merely rotated versions of A's, in A's frame the distances (in the diagram) between ticks on B's axes are greater than the distances between ticks on A's axes. You can sort of see this if you look at the diagram from post #5, where in the diagram from the unprimed perspective you can see that the distance between pink events on the light blue space axis of the unprimed frame is just 4 light-seconds, and the distance between pink events on the yellow time axis of the unprimed frame is 4 seconds; but in the diagram from the primed frame, the light blue and yellow axes are not only slanted, but the distance along these axes between the same pink events also looks stretched (and the location of the three pink events in both frames was calculated using the Lorentz transform so I know they're correct).

 

[neopolitan]

You mean like this?

 

[JesseM]

You mean like this?

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

 

[neopolitan]

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

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




(The following is off-track somewhat.)

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

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

cheers,

neopolitan

 

[JesseM]

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

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

 

[neopolitan]

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

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

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

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

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

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

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

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

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

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

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

Is that right?

cheers,

neopolitan

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

 

[JesseM]

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

In the same sense that dots on 2D surface like a sheet of paper or a globe have a unique location on that surface even though they can be assigned different coordinates in different coordinate systems drawn on that surface, right.

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

In A's frame, A is moving along the t_A axis at a rate of one second per second while B is moving at an angle to the t_A axis such that the gradient of the line is given by 1/v.

With the "gradient" defined as $$\frac{\Delta t}{\Delta x}$$, yes.

I have to admit that I initially thought of B moving along what B takes to be the x_B axis. I see clearly that this is wrong, since that would equate to a speed greater than c. (I was thinking of an F(x) graph, where you plot F(x) on the vertical axis and x on the horixontal axis. This diagram shows an F(t) graph, where F(t) is on the horizontal axis and the gradient is therefore t/F(t) = 1/v, rather than v as I had in mind.)

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

Right, the t_B axis represents B's worldline.

 

[neopolitan]

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

I'm just wondering, where is length contraction?

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

cheers,

neopolitan

 

[JesseM]

I'm just wondering, where is length contraction?

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

 

[neopolitan]

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

Damn,

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

Anyways, here it is.

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

Is that right?

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

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

cheers,

neopolitan

 

[JesseM]

Diagram looks good...

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

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

2. non-simultaneously, they are further apart), but the rod appears contracted to A.

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

As for me, it seems that maybe SAFTD might be a simpler feature to champion rather than TAFLC (although there are still benefits attached to TAFLC in my way of thinking).

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

A benefit to SAFTD manifests in the derivation of the Lorentz Transformations from the Gallilean boosts, where you remove the assumption of instantaneous transmission of information (so you are discussing an event at a distance of x where x=ct and at t=0, t'=0 and the origins of the K and K' frames are colocated).

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

 

[neopolitan]

Diagram looks good...

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

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

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

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

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

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

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

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

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

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

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

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

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

cheers,

neopolitan

 

[JesseM]

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

The observers consider themselves to be at rest and the other to be moving. Both assume that the event that spawned the photon occurred when they were colocated with the other observer.

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

Each observer will come to the conclusion that the spawning event occurred at x=ct (x and t in their own frame).

OK.

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

In Galilean physics they don't disagree about simultaneity, so in this case they could both be right that the event occurred simultaneously with their being colocated. However, the meaning of your terms is a little ambiguous. For example, say the event occurs at t=0 and x=-10 light-seconds in A's frame, and they are colocated at t=0 and x=0 in A's frame, and B is moving in the +x direction at 0.6c in this frame. Suppose also that the signal from the event moves at 1c in the +x direction in A's frame (which will mean it moves at a different speed in B's frame). In this case the signal will catch up with A at t=10, x=0, and will catch up with B at t=25, x=15 (again in A's frame). In B's own frame the time will be the same but the light was only moving at 0.4c in his frame, so the initial event was at x'=-10 and t'=0, and it hit B at x'=0 and t'=25. So, if x is the distance between A and the event in A's frame, and x' is the distance between B and the event in B's frame, in which case x=x'=10, meaning your equation isn't right. On the other hand, you may have meant that x is supposed to be the distance between the original event and the event of the light hitting B as measured in A's frame, while x' is the distance between the original event and the event of the light hitting B as measured in B's frame, in which case x=25, t=25 and x'=10, so this does work because 10 = 25 - 0.6*25. So it only works if both frames are talking about the distance x and x' between the same two events.

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

I wouldn't say Galilean boosts assume instantaneous transfer of information, you could have a Galilean physics where information can travel no faster than c in both directions in one preferred frame (the aether frame, perhaps), so in a frame moving at speed v relative to the first, signals could travel at a max of c-v in one direction and c+v in another.

Since information is not transmitted instantaneously, both observers will work out that the other observer must be affected by some temporal and spatial skewing (this is required to reconcile a single photon apparently coming from two different locations).

I don't understand what you mean by that--there is no way to reconcile a single photon coming from two different locations in spacetime, that's a physical contradiction and only one of them can be correct if they both make this assumption. If the event that created the photon was simultaneous with A and B being colocated in A's frame, then it wasn't simultaneous in B's frame, and vice versa.

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

The TD and SAFTD are meant to apply to a single well-defined pair of events, not to a case where different frames disagree about where in spacetime one of the events occurred (which is different from disagreeing about the coordinates assigned to a single well-defined event). It is true that you get equations that look like the TD and SAFTD with your assumptions though. For example, let's use the same numbers for A's frame, so in this light from the event reaches B at t=25, x=15. Because of time dilation in A's frame, B's clock will only read 0.8*25 = 20 when the light hits it, so if B assumes the event happened at t'=0 in his own frame, he must conclude it happened 20 light-seconds away at x'=-20. So A thinks the original event happened 25 light-seconds away from where B was when the light struck him, and B thinks it happened 20 light-seconds away from where B was when the light struck him. But again, conceptually this has nothing to do with TD or SAFTD because they are each making totally incompatible assumptions about the spacetime location of the original event, whereas TD and SAFTD are derived using the assumption that we are talking about two events with well-defined locations in spacetime (and also using the assumption that the events have a spacelike separation and are simultaneous in one frame in the case of TD, or a timelike separation and are colocated in one frame in the case of SAFTD).

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

 

[neopolitan]

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

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

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

Plus I reiterate:

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

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

cheers,

neopolitan

 

[JesseM]

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

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

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

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

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

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

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

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

Plus I reiterate:

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

There is no objective right or wrong here, unless you can uniquely identify individual photons. Note that I am using thought-experiment magic here, so the photon is detected by both observers without being absorbed by the first.

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

 

[neopolitan]

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

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

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

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

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

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

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

You are still looking at it the wrong way.

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

Really there are three events:

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

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

Does this help?

cheers,

neopolitan

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

 

[JesseM]

You are still looking at it the wrong way.

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

Really there are three events:

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

As I say, I don't care that much about the first event and the Lorentz Transformations only uses the second two events.

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

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

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

 

[neopolitan]

See the three diagrams, numbered:

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

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

$$G = \gamma$$ ... (7)

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

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

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

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

cheers,

neopolitan

PS You can call the result of (7) into (1) SAFTD, or inverse length contraction, or length dilation, or relativistic effect on lengths, or the spatial relativistic effect. Or whatever you like. Irrespective of what you call it, can you see the physical significance of it?

 

[neopolitan]

Some clarifications to the last post (done in the early hours of the morning).

G to E 01 - (Galileo to Einstein) - shows the standard Galilean boost. There is an assumption of instantaneous transfer of information (or god-like powers to see everything at once).

G to E 02 - shows what happens when you remove the assumption of instantaneous transfer of information. E is an "event". I stress that it could be something that causes the emission of a photon, or it just could be an event along the path of the photon. A and B see the same photon (thought experiment magic).

G to E 03 - shows the reconciliation, with subscripts to show "according to ..."

In this process, it becomes explicit what sort of physical things your x, t, x', t' and, if you like, L and L' refer to. You may want to check that they are referring to the right sort of things in your terminology (ie in which frame are things simultaneous and in which frame are things collocated), but I think you will find that while saying "time and space are measured oddly" might sound less than rigorous, it is actually more intuitive. To work out the simultaneity/collocation issues, you really have to already understand so I think it is a stage that should come after you have given a derivation.

cheers,

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