The importance of velocity in simultaneity

[PeterDonis]

at events A1 and B1, respectively, which were simultaneous in their rest frame AB

No, frame AB is not the rest frame of A and B at events A1 and B1. That's the whole point. The inertial frame AB is their rest frame at events A0 and B0, but not at events A1 and B1. At events A1 and B1, the rest frame of clocks A and B (respectively) is frame CD, not frame AB.

You could try to define a non-inertial frame N in which A and B are always at rest, even though they have nonzero proper acceleration for some period of time. (Note that there is no unique way to do this, and all of the possible ways have significant limitations.) But if you do that, you can no longer assume that A and B will stay synchronized if they both stay at rest in the frame. That assumption is only valid for inertial frames.

 

[Micheth]

No, frame AB is not the rest frame of A and B at events A1 and B1. That's the whole point. The inertial frame AB is their rest frame at events A0 and B0, but not at events A1 and B1. At events A1 and B1, the rest frame of clocks A and B (respectively) is frame CD, not frame AB.

In SR I had always thought we could consider any frame to be at rest? i.e., AB decelerating to CD which are "at rest" would be in SR equivalent to CD decelerating toward AB which are "at rest"...?

 

[A.T.]

AB decelerating to CD which are "at rest" would be in SR equivalent to CD decelerating toward AB which are "at rest"...?

Only inertial frames are equivalent.

 

[PeterDonis]

In SR I had always thought we could consider any frame to be at rest?

It depends on what properties you want a "frame" to satisfy. If you want clocks at rest in the frame to stay synchronized, then the frame must be an inertial frame, as I said before. The frame in which C and D are always at rest is an inertial frame, but the "frame" in which A and B are always at rest is not, because A and B have nonzero proper acceleration for a period of time. (I put "frame" in quotes for A and B because, as I said before, there is no unique way to define a non-inertial frame in which A and B are always at rest.)

More generally, as A.T. said, in SR, only inertial frames are all equivalent. SR certainly does not say that all frames, inertial or not, are equivalent. GR makes a more general claim of that sort, but the sense of "equivalence" being used, in terms of what properties all the "equivalent" coordinates can be assumed to satisfy, is much weaker.

 

[Nugatory]

In SR I had always thought we could consider any frame to be at rest? i.e., AB decelerating to CD which are "at rest" would be in SR equivalent to CD decelerating toward AB which are "at rest"...?

Frames aren't at rest or not, they're inertial or not. "At rest" applies to objects, and we say that an object is at rest in a frame if its spatial coordinates in that frame are constant. (Do not allow yourself to be confused by the frequently used term "rest frame of <something>" - that's just a convenient shorthand for "frame in which <something> is at rest").

The rest frame of an object experiencing no forces (let's not get into gravitational forces right now) is inertial, and all inertial frames are equivalent. However, the rest frame of an accelerating/decelerating object is not inertial.

 

[Micheth]

It depends on what properties you want a "frame" to satisfy. If you want clocks at rest in the frame to stay synchronized, then the frame must be an inertial frame, as I said before. The frame in which C and D are always at rest is an inertial frame, but the "frame" in which A and B are always at rest is not, because A and B have nonzero proper acceleration for a period of time. (I put "frame" in quotes for A and B because, as I said before, there is no unique way to define a non-inertial frame in which A and B are always at rest.)

Because A and B were accelerated, they are no longer inertial, correct? I feel I am back to square one in my logical disconnect. :-)
(I know I keep coming back to the same thing: it is not that I doubt that the math works out that way, it is the seeming logical contradiction I keep running into, i.e:
Set A to time 0, distance A0 to A1 is fixed, run experiment, at experiment end it always records a fixed time (deltaT)
Set B to time 0, distance B0 to B1 is fixed, run experiment, at experiment end it always records (the same) fixed time (deltaT)
But according to SR, if you synchronize A&B and program both to begin the experiment when they both read time 0, then when they reach A1/B1, and they are out of sync! (record different deltaT's)

Is it simply necessary to conclude, as we do in quantum weirdness, that "it doesn't necessarily make logical sense but doing the math leads us unescapably to that conclusion, so it must simply be accepted"?

 

[PeterDonis]

Because A and B were accelerated, they are no longer inertial, correct?

Yes. More precisely, they are not moving inertially while they are accelerating; before the acceleration, they are at rest in one inertial frame, but after the acceleration, they are at rest in a different inertial frame. There is no single inertial frame in which they are at rest for the entire time.

according to SR, if you synchronize A&B and program both to begin the experiment when they both read time 0, then when they reach A1/B1, and they are out of sync! (record different deltaT's)

They do not "record different deltaT's". The time elapsed on clock A between events A0 and A1 is the same as the time elapsed on clock B between events B0 and B1. The "disconnect" here is your unstated assumption that, if the time elapsed on both clocks is the same between the start and end of the acceleration, then if they were in sync before the acceleration, they must still be in sync after the acceleration. That assumption is false.

Is it simply necessary to conclude, as we do in quantum weirdness, that "it doesn't necessarily make logical sense but doing the math leads us unescapably to that conclusion, so it must simply be accepted"?

It does make logical sense, but only if you reason from correct assumptions. SR does violate assumptions that seem very natural to our pre-relativistic intuitions; the one I described above is one of them.

 

[Micheth]

They do not "record different deltaT's". The time elapsed on clock A between events A0 and A1 is the same as the time elapsed on clock B between events B0 and B1. The "disconnect" here is your unstated assumption that, if the time elapsed on both clocks is the same between the start and end of the acceleration, then if they were in sync before the acceleration, they must still be in sync after the acceleration. That assumption is false.

I could accept them reading the same times but being out of sync, if they moved with respect to each other (where there would have been time dilation between them), but in the final analysis all I see that has happened is that they had their coordinates translated by equal amounts.
We're ... not talking about time dilation (or are we?)

 

[PeterDonis]

I could accept them reading the same times but being out of sync

Um, what? "Out of sync" means what I have repeatedly said: that the two clocks have different readings at simultaneous events. Once the clocks are both at rest in frame CD (after the acceleration is complete), they have different readings at events which are simultaneous in that frame. You seem to agree that this is the case, so I really don't understand why this thread has gone on so long.

all I see that has happened is that they had their coordinates translated by equal amounts

The elapsed times on the two clocks are not coordinates; they are direct observables. The fact that the clocks read different times at events which are simultaneous in frame CD is also a direct observable. The fact that the distance each clock travels during its acceleration, in frame AB, is the same, is true, but irrelevant; that's not what determines whether the clocks are in sync.

We're ... not talking about time dilation (or are we?)

No. After the acceleration, both clocks are at rest in the same inertial frame, so neither one is time dilated relative to the other. I've already said that they both run at the same rate once they stop accelerating. But I've also explained that running at the same rate is not sufficient for them to be in sync. I get the feeling that you are not really reading my posts.

 

[Micheth]

No. After the acceleration, both clocks are at rest in the same inertial frame, so neither one is time dilated relative to the other. I've already said that they both run at the same rate once they stop accelerating. But I've also explained that running at the same rate is not sufficient for them to be in sync. I get the feeling that you are not really reading my posts.

Well, my attempts to articulate why this seems such a paradox aren't being too successful, and you're evidently finding it somewhat exasperating, so perhaps we should just let this close.
I appreciate your efforts, though.
Cheers.

 

[A.T.]

Well, my attempts to articulate why this seems such a paradox aren't being too successful

It seems like a paradox because of imprecise reasoning, which comes from imprecise language.

 

[PeterDonis]

my attempts to articulate why this seems such a paradox aren't being too successful

The word "seems" is key. I understand why it seems like a paradox to you: because you are reasoning from assumptions that, however intuitively plausible they seem, are not valid in a relativistic universe. But we do live in a relativistic universe; experiments demonstrate that. So those assumptions are simply not valid, however intuitively plausible they seem. Part of learning relativity is learning to give up intuitively plausible but invalid assumptions; we've all had to go through that at some point in the learning process.

 

[Micheth]

Once the clocks are both at rest in frame CD (after the acceleration is complete), they have different readings at events which are simultaneous in that frame. You seem to agree that this is the case, so I really don't understand why this thread has gone on so long.

(Actually I said I could agree with them being out of sync if there was time dilation (which you've pointed out there is not).

I assure you I'm reading your posts. How much I understand is a different issue, but if I went through and read your posts again anyway.

As I understand it:
AB are separated (let's say by 10 meters e.g.), and synchronized in a rest frame.
There are two points in AB's original rest frame (far, far away) that are C & D, also separated by 10 meters in that same frame.
AB then accelerate to some near-light speed toward CD.
I was originally thinking you meant that AB's clocks would simply get out of sync because they accelerated.
That was mistaken, right?
Instead, then what's really happening is that the distance between AB has increased (from CDs viewpoint) as they pass CD, and therefore they cannot respectively meet CD at the same time, correct? (because the distance AB is now different from distance CD). From AB's viewpoint, CD are no longer separated by 10 meters but by some shorter distance as they measure it.
Hence the non-simultaneity of the meeting of AC vs. BD.
If AB's clocks are fine, I'm happy with that :-)

Now, to be honest I still can't understand why distance AB would have to increase (in anybody's frame) since AB merely translated their coordinates in space (underwent identical movements in the same direction), but not to open another can of worms. :-)

 

[PeterDonis]

AB are separated (let's say by 10 meters e.g.), and synchronized in a rest frame.

Yes.

There are two points in AB's original rest frame (far, far away) that are C & D, also separated by 10 meters in that same frame.

No. C and D are two clocks that are moving, in A and B's original rest frame, with some speed $v$. C and D never accelerate; they remain in inertial motion the whole time.

AB then accelerate to some near-light speed toward CD.

No. A and B accelerate in such a way that they just match speeds with C and D, respectively, at the instants when C and D are spatially co-located with A and B, respectively. In other words, C and D are behind A and B, respectively, when A and B start accelerating; but C and D are moving faster, so they catch up to A and B, respectively (all of this is as seen by A and B), just as A and B match speeds with C and D, respectively.

I was originally thinking you meant that AB's clocks would simply get out of sync because they accelerated.
That was mistaken, right?

No. A and B's clocks do get out of sync because they accelerated. Their clocks stay in sync in their original rest frame (meaning the inertial frame in which they were originally at rest and synchronized), but A and B do not remain at rest in their original rest frame. They accelerate, hence they start moving in that frame. When they finish accelerating, they are both moving relative to that original frame, so the fact that they are synchronized in that frame is no longer relevant. See further comments below.

what's really happening is that the distance between AB has increased (from CDs viewpoint) as they pass CD

Please stop using the word "really". You keep on applying it to things that are frame-dependent. That's a very bad habit in relativity.

The correct way to state how things look in the frame in which C and D is at rest is basically what you say: A and B are moving towards C and D in this frame; then B starts decelerating; then A, a bit later, starts decelerating; then B comes to a stop right at D; then A comes to a stop right at C. Because B started decelerating first, the distance between A and B, in this frame, increases. But this is all relative to C and D's rest frame, so the word "really" is inappropriate.

(There are ways of describing the motion of A and B, and C and D, in frame-invariant form. For example, we could say that the worldlines of A and B, during the period when both are accelerating, have positive expansion; whereas the worldlines of C and D always have zero expansion. "Expansion" here is a technical term, referring to the frame-invariant mathematical description of these motions. In this sense, you could say that the distance between A and B "really" does increase as they accelerate. But then you would have to remove the phrase "from CDs viewpoint", because the expansion is frame-invariant.)

Hence the non-simultaneity of the meeting of AC vs. BD.

Non-simultaneity in frame CD. The two meetings are simultaneous in frame AB (the frame in which A and B were originally at rest). But once A and B are no longer at rest in frame AB, that frame's definition of simultaneity is no longer the right one to use to judge A and B's clock synchronization.

In other words, the reason A and B's clocks get out of sync is that they have changed inertial frames: they start out at rest in one inertial frame, and end up at rest in a different inertial frame. (They have to accelerate to do this, which is why I said above that they get out of sync because they accelerated.) That changes which definition of simultaneity is the one that determines whether their clocks are synchronized.

If all that is too abstract, consider how, physically, A and B would check that their clocks are synchronized. (This will also show how they can check the distance between them and verify that, as far as they are concerned, it has increased.) In the original rest frame, AB, before any acceleration has taken place, A and B can exchange light signals, and determine two things: (1) that the distance between them is in fact 10 meters (or whatever it turns out to be), based on the round-trip travel time of light signals between them; and (2) that their clocks are in fact synchronized, based on the fact that, when A receives a light signal from B, it shows B's clock reading exactly what A's clock read one light-travel time ago (so if they are 10 meters apart, when A receives a light signal from B, B's clock will read what A's clock read 33 nanoseconds ago, since it takes light about 33 nanoseconds to travel 10 meters). That is, the reading shown on B's clock in the light signal will be 33 nanoseconds before the reading shown on A's clock when the signal is received.

Now, once A and B have completed their acceleration, and are now co-located with C and D and moving inertially, at rest in frame CD instead of frame AB, they can repeat the above process. And when they do, they will find: (1) that the distance between them has increased, based on the round-trip travel time of light signals between them; and (2) that their clocks are no longer synchronized: when A receives a light signal from B, it shows B's clock reading something later than what A's clock read one light travel-time ago, and when B receives a light signal from A, it shows A's clock reading something earlier than what B's clock read one light travel-time ago.

For example, if A and B are now 20 meters apart (corresponding to a relative speed of about 0.87c between frames AB and CD), then when A receives a light signal from B, it will show B's clock reading, not a time 66 nanoseconds before A's (which would be one light-travel time before), but a time about 15 nanoseconds before A's, indicating that B's clock is about 51 nanoseconds ahead of A's. And when B receives a light signal from A, it will show A's clock reading, not a time 66 nanoseconds before B's, but a time about 117 nanoseconds before B's, indicating that A's clock is about 51 nanoseconds behind B's. So the two measurements are consistent (they both show the same offset between the clocks), and they clearly show the clocks not synchronized.

I still can't understand why distance AB would have to increase (in anybody's frame)

Because, as I noted above, in frame CD, B starts decelerating before A does. This has to be true because of relativity of simultaneity; A and B both start accelerating simultaneously in frame AB, therefore they cannot start accelerating (or decelerating, depending on how you view it) simultaneously in any other frame, including frame CD. Therefore, the distance between them must change in any other frame besides frame AB. (And, as the discussion above shows, they can verify that the distance has increased by exchanging light signals.)

Once again, I think part of the issue is that you are trying to reason from the wrong assumptions. Instead of starting with the known properties of relativistic spacetime (the main one here being relativity of simultaneity), and deducing what must happen in the scenario based on those properties, you are trying to start with your assumptions about how things "ought" to work, then wondering why relativity says they don't work that way.

 

[Micheth]

No. C and D are two clocks that are moving, in A and B's original rest frame, with some speed vv. C and D never accelerate; they remain in inertial motion the whole time.

Wow! That's a very long and detailed response which will take me much time to forge through since this is all very difficult to visualize (for me at least).
But first, I see we're not talking about the same thing from the start. I am talking about my original scenario as described above (where C&D are merely the points where lightning bolts will strike when A&B reach them. A&B having their clocks synchronized in that rest frame where A, B, C & D and whatever produces the lightning bolts, are all in the same frame.
(That is, where A&B are not moving with respect to C&D at the start.)
Then A&B accelerate to CD...

(I know part way through you proposed seeing A&B moving from the start and decelerating to rest at C&D, but I managed to simply become more confused, so may I suggest the original scenario.)

 

[PeterDonis]

I am talking about my original scenario as described above (where C&D are merely the points where lightning bolts will strike when A&B reach them.

Ok; but then I can just define two clocks, E and F, which happen to pass by C and D, respectively, at the exact same instants that clocks A and B reach them and the lightning bolts strike, and which are moving at the exact same speed as A and B are at those instants. Then just substitute E and F for C and D in everything I wrote.

The point I was trying to make with C and D (or E and F in the new nomenclature) was not to construct a new scenario; it was to make it easier to see explicitly aspects of your original scenario that you were not considering. The frame in which E and F are at rest exists, and can be used to analyze your scenario, regardless of whether clocks E and F are actually there. Putting them there just helps to give an actual physical realization of the frame. And A and B end up at rest in that frame--the one I was calling frame CD, and will now call frame EF--regardless of whether clocks E and F are there to compare with. And the procedure I described by which A and B can exchange light signals after they have accelerated and confirm that their clocks are now out of sync and the distance between them has increased, can be done regardless of whether clocks E and F are there.

You can't make the non-synchronization of A and B at the end of your scenario go away by sticking to your original frame (what I call frame AB); once A and B start accelerating, they are no longer at rest in that frame, there's no way around that, so the fact that various events happen at the same time in that frame is no longer relevant for assessing the synchronization of clocks A and B. You can analyze the procedure A and B can follow to exchange light signals after they accelerate in frame AB if you like; it will still tell you that A's and B's clocks are out of sync, in the precise sense I described (the light signals they receive from each other show clock readings that are different than what a synchronized clock would be sending, given the light travel time delay), and that the distance between them has increased (as measured by light travel time).

 

[Micheth]

Now, once A and B have completed their acceleration, and are now co-located with C and D and moving inertially, at rest in frame CD instead of frame AB, they can repeat the above process. And when they do, they will find: (1) that the distance between them has increased, based on the round-trip travel time of light signals between them; and (2) that their clocks are no longer synchronized

I think I understand what you're saying here. At least I hope so.
A and B will find their measurements between them to be actually different in their new frame.
I may not like it, but if SR is correct it'll just have to be accepted, AND in fact I think I can retract any of my logical "objections" if (and i guess only if) the answer to the following question is "yes".
(Please tell me the answer is yes! :-)
If A and B are the only objects in the universe (they carry out the process you describe to synchronize their clocks). Then A and B (persons with them) go to sleep, during which time both vessels are accelerated by the same degree).
When they wake up (not knowing they have accelerated, or having any way of knowing that they are in a different frame than before), they carry out the same process but surprisingly find that, as you describe above, now (1) they are further apart than before, based on the round-trip light signal times, and (2) their clocks are no longer synchronized.
And they would find this strange - strange enough to conclude that they must have accelerated, correct?
(If this is what they would find then I am satisfied, and I have no further questions.)

 

[PeterDonis]

A and B will find their measurements between them to be actually different in their new frame.

Yes.

If A and B are the only objects in the universe (they carry out the process you describe to synchronize their clocks). Then A and B (persons with them) go to sleep, during which time both vessels are accelerated by the same degree).
When they wake up (not knowing they have accelerated, or having any way of knowing that they are in a different frame than before), they carry out the same process but surprisingly find that, as you describe above, now (1) they are further apart than before, based on the round-trip light signal times, and (2) their clocks are no longer synchronized.

Yes.

And they would find this strange - strange enough to conclude that they must have accelerated, correct?

Yes.

 

[Micheth]

Yes.

Thank you :-)