The importance of velocity in simultaneity

[PeterDonis]

So then... what causes the clocks to go out of sync..?

As A.T. said, gravitational time dilation is off topic for this thread. However, we can consider the flat spacetime (no gravity) example I gave earlier: two atomic clocks, with some separation in the $x$ direction, both accelerating in the $x$ direction with exactly the same proper acceleration. They will get out of sync. As to why, see below.

I had always assumed that if the exact same forces worked in exactly the same way on two objects, causing them to undergo non-uniform motion in exactly the same way, they would perform in exactly the same way (i.e still be in sync).

The problem is with those word "exactly the same". You are assuming they have an absolute meaning, but they don't. Even though the acceleration profiles of both clocks are the same to the clocks themselves (i.e., the acceleration profile of clock A, as seen by clock A, is the same as the acceleration profile of clock B, as seen by clock B), and even though they are the same as seen in a particular fixed inertial frame (i.e., in the frame in which both clocks start out at rest, and start accelerating at the same time, both acceleration profiles look the same), they are not the same when each clock looks at the other (i.e., the acceleration profile of clock A, as seen by clock B, is not the same as the acceleration profile of clock B, as seen by clock A). It's this last fact that causes the clocks to get out of sync.

 

[Micheth]

You're asking about gravitational time dilation now? This is really a derail of this SR thread, and basically what is already discussed in the other thread, about the distortion of the time dimension:
https://www.physicsforums.com/threa...t-on-space-thus-causing-gravity.803213/page-2

No, I'm still very much talking about accelerating the three objects (it was originally a train with clocks at either end and a rider in the middle, but we just got rid of the train).
(Which I now understand is referred to as "proper acceleration".)
But it was being argued that accelerating all three in exactly the same way would cause the clocks to go out of sync, and I'm still asking why...

 

[Micheth]

As A.T. said, gravitational time dilation is off topic for this thread.

(No, I'm not talking about gravitational time dilation. You stated "It's not really correct to say that proper acceleration causes clocks to go out of sync", so I asked what is it that causes them to go out of sync.)

even though they are the same as seen in a particular fixed inertial frame (i.e., in the frame in which both clocks start out at rest, and start accelerating at the same time, both acceleration profiles look the same), they are not the same when each clock looks at the other (i.e., the acceleration profile of clock A, as seen by clock B, is not the same as the acceleration profile of clock B, as seen by clock A). It's this last fact that causes the clocks to get out of sync.

Hmmm. I don't see why the clocks have to "look at each other"? They undergo the same acceleration profiles, right? And they arrive at their final destinies (the lightning bolts) at exactly the same time, and each will register the time they get struck, right?
For example, as a thought experiment, we could just do one clock, accelerate it to the lightning bolt point, and strike it.
Then, resetting the thought experiment, we do the other clock, accelerating it in the same way, striking it.
(One would have nothing to do with the other.)
So, in my example were are merely doing those two though experiments simultaneously so that they get hit by the bolts simultaneously, and thus register the same times.

 

[PeterDonis]

I don't see why the clocks have to "look at each other"?

Because that's the criterion for whether they are "in sync": whether a given clock reading for clock A is simultaneous with the same clock reading for clock B. Simultaneity is relative, so each clock has to "look at the other" to see whether this criterion is satisfied.

They undergo the same acceleration profiles, right?

Once again, you are using "the same" as if it had an absolute meaning. It doesn't.

d they arrive at their final destinies (the lightning bolts) at exactly the same time

Once again, you are using "the same" as if it had an absolute meaning. It doesn't.

Rather than continue to wave your hands, I strongly suggest that you do the math. Assign coordinates to events, draw a spacetime diagram, look at the lines of simultaneity for each clock, and see how they don't line up.

 

[A.T.]

I don't see...

As Peter noted, you fail to specify the frame for all your frame dependent statements. You will never get it, until you learn to be precise about this.

 

[Micheth]

Because that's the criterion for whether they are "in sync".

Wouldn't the criterion for whether or not they are in sync simply be, when the simultaneous lightning bolts (simultaneous in the ground frame) strikes them, whether or not they register or don't register the same time? There wouldn't have to be any communication between them at all, at least in this thought exp.

Rather than continue to wave your hands, I strongly suggest that you do the math. Assign coordinates to events, draw a spacetime diagram, look at the lines of simultaneity for each clock, and see how they don't line up.

Well, one would if one knew how. :-)
I'm actually only capable of dealing with the logic of it, as I should hope it would make logical sense.
You could conduct the experiment with one clock, with exactly defined accelerating conditions.
When it arrives at the lightning bolt point (a fixed distance away), it registers a time.
You do it later again with another clock, and if all the conditions were exactly the same, it ought to register exactly the same time, no?
But if you do them together, somehow they get out of sync! Do you see why I'm stumped on that?

 

[PeterDonis]

Wouldn't the criterion for whether or not they are in sync simply be, when the simultaneous lightning bolts (simultaneous in the ground frame) strikes them, whether or not they register or don't register the same time?

Once again, you're assuming that "at the same time" has an absolute meaning. It doesn't.

Please, instead of continuing to wave your hands, actually do the math. You are continuing to repeat things that you have been repeatedly told are not correct.

one would if one knew how. :-)

If you don't currently know how, then I strongly suggest learning. It's really hard to do relativity without the proper tools.

I'm actually only capable of dealing with the logic of it, as I should hope it would make logical sense.

It does. But logic by itself isn't enough. One has to be reasoning from correct premises, and you are not. Your incorrect premise has been pointed out repeatedly, yet you continue to use it.

if you do them together, somehow they get out of sync

You can't "do them together" if they are spatially separated. That's the whole point. If they are both at the exact same spatial location (joined at the hip, perhaps), they they won't get out of sync. But that wasn't your original scenario. Your original scenario was that they are spatially separated: clock A starts at $x = x_A$, and clock B starts at $x = x_B$, where $x_B \neq x_A$, and they both accelerate in the $x$ direction. In that scenario, they get out of sync, because they are not "together".

 

[Micheth]

You can't "do them together" if they are spatially separated. That's the whole point. If they are both at the exact same spatial location (joined at the hip, perhaps), they they won't get out of sync. But that wasn't your original scenario. Your original scenario was that they are spatially separated: clock A starts at x=xAx = x_A, and clock B starts at x=xBx = x_B, where xB≠xAx_B \neq x_A, and they both accelerate in the xx direction. In that scenario, they get out of sync, because they are not "together".

Are you saying that it is impossible to sync the clocks in the first place?
I'm assuming they could be synched initially, in a rest frame.
Yes, my scenario is that they are spatially separated, is that the reason they can't be initially synched?
If so, then are you saying that the problem is not that they "got out of sync" by being accelerated, but that they were never in sync in the first place?

 

[PeterDonis]

Are you saying that it is impossible to sync the clocks in the first place?

No. They can start out in sync, and like you, I am assuming that they do.

 

[Micheth]

No. They can start out in sync, and like you, I am assuming that they do.

So, being initially in sync, couldn't they be programmed to begin accelerating when each reads a specific time value, and wouldn't that be defined as "doing the experiments together"?
To be exactly clear what I'm saying:
There is a fixed distance each must accelerate through. Each could do that separately (on different days maybe) and each would presumably register the exact same time when they reached the "endpoint". Since it takes a fixed mass a fixed time to accelerate under a fixed force through a fixed distance.
This particular experiment would merely be the trivial case where they each began accelerating when they had specific times values reading the same (i.e. at the same time).
By this logic, how could they then get out of sync?

 

[PeterDonis]

couldn't they be programmed to begin accelerating when each reads a specific time value

Yes. I was assuming that that value was $t = 0$; since both are at rest in the original inertial frame at this point, there is no ambiguity about when they start accelerating, according to either one.

wouldn't that be defined as "doing the experiments together"?

Not in the sense you mean. See below.

There is a fixed distance each must accelerate through.

In the original inertial frame, yes. Remember that distance is frame-dependent, so you have to specify which frame it is relative to.

Each could do that separately (on different days maybe) and each would presumably register the exact same time when they reached the "endpoint".

Yes, but that doesn't mean the two clocks will be in sync when they do it on the same day, starting at the same time in the original rest frame, but separated along their direction of acceleration (the $x$ direction).

how could they then get out of sync?

Because, as you have been told repeatedly, "at the same time" does not have an absolute meaning. Simultaneity is relative.

Once again, clock A starts at $x = x_A$ at time $t = 0$ in the original inertial frame (I will use $t$ to denote time according to this frame). Clock A also reads $\tau_A = 0$ (I will use $\tau$ to denote that actual times read on each clock) when it starts. Clock B starts at $x = x_B$ at time $t = 0$, and clock B reads $\tau_B = 0$ when it starts.

Now clocks A and B each accelerate for a fixed time $\Delta \tau$, i.e., clock A stops accelerating when it reads $\tau_A = \Delta \tau$, and clock B stops accelerating when it reads $\tau_B = \Delta \tau$. We assume they each had the same (constant) proper acceleration $a$ while they were accelerating. Then they will both stop at the same time $t = \Delta t$ according to the initial rest frame, and in that time, they will each have traveled the same distance $\Delta x$ according to that frame. So in the initial rest frame, clock A will stop accelerating when it reaches $x_A + \Delta x$, and clock B will stop accelerating when it reaches $x_B + \Delta x$.

Now, when you say the two clocks stop accelerating "at the same time", this is frame-dependent. In the initial rest frame, yes, they do; they both stop accelerating at $t = \Delta t$. But they are spatially separated when they stop; the events at which they stop accelerating are given by $x = x_A + \Delta x, t = \Delta t$, and $x = x_B + \Delta x, t = \Delta t$. This pair of events is simultaneous according to the initial rest frame; they both happen "at the same time" according to that frame.

But simultaneity is relative; those two events are not simultaneous in any other inertial frame. That is, only an observer (or a clock) which is at rest in the original rest frame will see those two events as simultaneous. Clock A and clock B are not at rest in that frame when they stop accelerating; they are both moving in the $x$ direction with some nonzero speed $v$. So the two events at which the two clocks stop accelerating are not simultaneous according to either clock. And at those two events, the two clocks read the same time, $\Delta \tau$. So the event at which clock A reads $\Delta \tau$ is not simultaneous, according to either clock A or clock B, with the event at which clock B reads $\Delta \tau$. This is why the clocks get out of sync; more precisely, it is what "the two clocks are out of sync" means.

 

[Micheth]

Now clocks A and B each accelerate for a fixed time Δτ\Delta \tau, i.e., clock A stops accelerating when it reads τA=Δτ\tau_A = \Delta \tau, and clock B stops accelerating when it reads τB=Δτ\tau_B = \Delta \tau. We assume they each had the same (constant) proper acceleration aa while they were accelerating. Then they will both stop at the same time t=Δtt = \Delta t according to the initial rest frame, and in that time, they will each have traveled the same distance Δx\Delta x according to that frame. So in the initial rest frame, clock A will stop accelerating when it reaches xA+Δxx_A + \Delta x, and clock B will stop accelerating when it reaches xB+Δx

This is essentially what I'm assuming, yes. In the initial rest frame (where we also we have the lightning bolts and the platform observer).
(I don't necessarily need to have them stop accelerating, just get hit by lightning bolts after equal distances have been traversed.)
So that whatever they do after that, they will continue to record the time they were hit.

The logical disconnect (for me) is that, we could have separately accelerated them on different days, the same distance, same acceleration, and doing it a hundred times, each time the experiment was conducted with either clock separately, it would show the same time elapsed, even if we did the individual experiments on different sides of the universe (the individual experiments being separated by distance), since the laws of physics apply everywhere. Each time the experiment would show the same time elapsed on the clock, right?

But somehow doing the two experiments when both clocks happen to read the same time (simultaneously in their rest frame), why only in that case would they read different times (= having gotten out of sync) at the very same points where they have traversed the very same distance as in each of the individual experiments?

 

[PeterDonis]

Each time the experiment would show the same time elapsed on the clock, right?

Yes. But that is not sufficient to show that the clocks are in sync. "In sync" means that a given reading on clock A is simultaneous, according to clock A and clock B, with the same reading on clock B.

If you do the experiment with different clocks on different days, you can't even judge whether they are "in sync", because neither clock can even compare its reading with the other's. You have to do the experiment with both clocks at once for "in sync" to even have any meaning. It makes no sense to say that your clock on Monday is "in sync" with my clock on Tuesday.

when both clocks happen to read the same time (simultaneously in their rest frame)

They don't have the same "rest frame" once they start moving.

why only in that case would they read different times (= having gotten out of sync) at the very same points

They are NOT at "the very same points". They are spatially separated. You keep on ignoring this.

 

[Micheth]

If you do the experiment with different clocks on different days, you can't even judge whether they are "in sync", because neither clock can even compare its reading with the other's. You have to do the experiment with both clocks at once for "in sync" to even have any meaning. It makes no sense to say that your clock on Monday is "in sync" with my clock on Tuesday.

Sorry if was confusing. Certainly performing on different days would not be in sync! But subjecting them to exactly defined acceleration processes through exactly defined distances, wherever they were in the universe, would cause them to register a deltaT that would always be the same, correct?
This deltaT would seem to have to be the same whether you performed the experiment separately, or when they just happened to read identical times.

They are NOT at "the very same points". They are spatially separated. You keep on ignoring this.

Perhaps the nature of the experiment got lost in the conversation. Here's how I'm proposing they are lined up:

ClockA initial position (CA)
|
(distanceD between them)
|
Clock B initial position (CB)
|
(distanceC to the platform)
|
LightningStrikeA position (LSA)
|
(distanceD)
|
LightningStrikeB position (LSB)

On one day you accelerate ClockA from CA to LSA. It registers a time elapse.
Another time you accelerate ClockB from CB to LSB. It registers exactly the same time elapse, because the distance is the same (staggered though it is).
This would have to be the case because the laws of physics don't change just because you become translated in space.
You do both a hundred times and always the same time elapse is registered.

But then you conduct both experiments not on different days but on the same day when they just happen to read the same time (synchronized in their rest frame)
Yes, they are spatially separated but they each start at the same points as before, that's what I mean by them being at the very same points.
Their times are noted when they start and when they arrive at LSA and LSB, respectively. Since they are still the same distances (as in the previous experiments) shouldn't they register the same time differences as in the individual experiments?

 

[A.T.]

Since they are still the same distances (as in the previous experiments) shouldn't they register the same time differences as in the individual experiments?

Sure, they will get stuck showing the same times, as explained here:
https://www.physicsforums.com/threa...ty-in-simultaneity.789927/page-3#post-5045831

 

[PeterDonis]

subjecting them to exactly defined acceleration processes through exactly defined distances, wherever they were in the universe, would cause them to register a deltaT that would always be the same, correct?

Yes, but that is NOT what "in sync" MEANS. Please read what I'm posting; you keep on skipping right over the key point. I'll quote it and bold the key part:

"In sync" means that a given reading on clock A is simultaneous, according to clock A and clock B, with the same reading on clock B.

In other words, to even judge whether clocks are "in sync", you need to know what events clock A and clock B see as simultaneous. None of what you have said addresses that at all. You haven't even talked about simultaneity. At most, you have implicitly used the notion of simultaneity in the original rest frame; but that is the wrong notion of simultaneity to use if you want to know whether the clocks are in sync. You have to use the clocks' own notion of simultaneity, and you have not addressed that at all.

It's useless to keep posting the same things about how your experiment is set up and what each clock reads. We don't disagree about any of that. You need to specifically address the simultaneity issue, as I have described it above.

 

[Micheth]

Yes, but that is NOT what "in sync" MEANS.
"In sync" means that a given reading on clock A is simultaneous, according to clock A and clock B, with the same reading on clock B.
In other words, to even judge whether clocks are "in sync", you need to know what events clock A and clock B see as simultaneous. None of what you have said addresses that at all. You haven't even talked about simultaneity.

As regards simultaneity, the purpose of my experiment was in fact all about simultaneity; that there stubbornly seems to be an objective simultaneity even though the rider experiences it differently.
As far as I can comprehend, the fact that the rider experiences the front lightning bolt first and the rear one later, is simply the trivial fact that the lights reach him at different times, whereas as the clock readings show, they really happened simultaneously, as can be objectively shown by the readings on the broken clocks, each of which would have traversed the same distance with identical accelerations.
(For me that's what this is all about.)
Of course I understand that the rider would not experience them in sync, my point is that, since the separate experiments always take the same amount of time, and in this experiment they are trivially simply being conducted in tandem (being started when both clocks read the same time), they "really did" get broken after an equal elapse of time, the rider simply not seeing them to happen simultaneously, but upon stopping the system, looks at the clocks' readings, knowing they performed exactly as in the previous separate experiments, and concludes (ah, well I saw them occur at different times, but the record shows it was merely an illusion since I was moving at the time).
That's my problem with the "relative simultaneity" issue.

It's not that I don't agree the moving entities experience the "out of synchness" (if the rider got shot from the front and behind he would feel them at different times, even though they were shot simultaneously (from the ground view)). It's that it seems that one of the perspectives must actually be correct given the recorded clock times and their equal acceleration histories.

I understand that SR claims differently, but isn't that merely an interpretation, between saying "it really was non-simultaneous, depending on the reference frame", as opposed to "the rider merely experienced the illusion of non-simultaneity"?

 

[PeterDonis]

the fact that the rider experiences the front lightning bolt first and the rear one later, is simply the trivial fact that the lights reach him at different times

Different times according to his clock, yes. That's what "experiences one first and the other one later" means.

I'm going to phrase things in terms of the "two clocks accelerating" version of your scenario for the rest of this post, because it's cleaner.

whereas as the clock readings show, they really happened simultaneously

No, the clock readings being the same shows that those events happened simultaneously in a particular reference frame--the frame in which the clocks were originally at rest. The clock readings being the same does not show that those events happened simultaneously in the rest frame of either of the clocks after they accelerated. (Those are two different frames, btw, because the clocks are spatially separated.) There is nothing that makes one particular frame's notion of simultaneity the "real" one.

I understand that SR claims differently, but isn't that merely an interpretation, between saying "it really was non-simultaneous, depending on the reference frame", as opposed to "the rider merely experienced the illusion of non-simultaneity"?

No, the relativity of simultaneity is not an "interpretation", it's a physical fact. To see why, consider a variation on your scenario:

Clock A and clock B are as in the previous version of the scenario: they start off both at rest and synchronized in a particular frame; they both start accelerating at $t = 0$ in that frame, at which instant both of the clocks read $\tau = 0$; each clock accelerates until it reads $\Delta \tau$, and then stops accelerating. After each clock stops accelerating, it is moving at speed $v$ in the original rest frame (the same speed for both clocks). Each clock continues to advance after it stops accelerating.

Clock C and clock D start out synchronized and at rest in a frame that is moving with speed $v$ relative to the frame in which clock A and clock B are originally at rest. Their motion is arranged so that clock C is co-located with clock A at the instant that clock A stops accelerating, and clock D is co-located with clock B at the instant that clock B stops accelerating.

Now the physical meaning of relativity of simultaneity is simple: if we suppose that clock C's "zero" of time is chosen so that it reads the same as clock A at the instant when clock C meets clock A, i.e., that clock C reads $\Delta \tau$ at that instant, then clock D will not read $\Delta \tau$ at the instant at which it meets clock B. Clock D will read an earlier time than $\Delta \tau$ (how much earlier depends on the speed $v$ and the spatial separation between the clocks). So clocks C and A will end up reading the same time from $\Delta \tau$ on (since they are now spatially co-located and at rest relative to each other), but clocks D and B will not end up reading the same time; they will remain out of sync by the same amount forever.

So now, if you insist on an absolute notion of simultaneity, you have the following paradox: clocks A and B started out synchronized, and clocks C and D started out synchronized. When clock C meets clock A, they read the same time, and continue reading the same time forever after that; when clock D meets clock B, they do not read the same time, and they stay out of sync by the same amount forever after that. So we have two clocks, C and A, both reading the same time, but they are supposedly "synchronized" with two clocks, D and B, that are reading different times. So which clock are C and A "really" synchronized with at the end--D or B?

The answer SR gives is "neither, because there is no absolute meaning to simultaneity". According to the simultaneity convention of the original rest frame of A and B, clock A (and therefore C) and clock B are synchronized; according to the simultaneity convention of the rest frame of C and D, clocks C (and therefore A) and D are synchronized. There is no contradiction because both senses of simultaneity are just conventions; but the difference in readings between clocks B and D when they are co-located is not a convention, it's a physical fact, and it directly illustrates relativity of simultaneity.

 

[Micheth]

Clock C and clock D start out synchronized and at rest in a frame that is moving with speed vv relative to the frame in which clock A and clock B are originally at rest. Their motion is arranged so that clock C is co-located with clock A at the instant that clock A stops accelerating, and clock D is co-located with clock B at the instant that clock B stops accelerating.

Now the physical meaning of relativity of simultaneity is simple: if we suppose that clock C's "zero" of time is chosen so that it reads the same as clock A at the instant when clock C meets clock A, i.e., that clock C reads Δτ\Delta \tau at that instant, then clock D will not read Δτ\Delta \tau at the instant at which it meets clock B. Clock D will read an earlier time than Δτ\Delta \tau (how much earlier depends on the speed vv and the spatial separation between the clocks).

I'm not sure i understand: Are C & D mutually accelerated in the same way and separated by the same distance as A & B?
If so then I would intuitively think that all 4 are in sync at the point where A meets C and D meets B.
But if not, that is if C & D were accelerated differently, C in such a way that it reads \Delta \tau when it meets A, but not necessarily D in the same way, then I would guess that D would not match any of the other 3 clocks anymore because of its different acceleration history and hence different time dilation with respect to the others.

 

[PeterDonis]

Are C & D mutually accelerated in the same way and separated by the same distance as A & B?

No, C and D are moving inertially the whole time. They just happen to be moving inertially in such a way that, when A and B stop accelerating, they are co-located with, and moving at the same speed as, C and D respectively.

if C & D were accelerated differently

C and D are never accelerated at all. As above, they are moving inertially the whole time. And they are synchronized in the inertial frame in which they are always at rest.

If it helps, here's how the experiment looks in the inertial frame in which C and D are always at rest: initially, A and B are moving at speed $- v$, but then A and B decelerate until they stop moving; and when they stop moving, they are co-located with C and D, respectively. In this frame, A and B do not start decelerating at the same time, or stop at the same time; B starts decelerating before A, and stops before A. Note that this does not change the time that clocks A and B display at the instants they start and stop decelerating; they both display time zero at the instant they start decelerating, and they both display time $\Delta \tau$ at the instant they stop decelerating. (This means, of course, that in this frame clocks A and B are never synchronized.)

 

[Micheth]

No, C and D are moving inertially the whole time. They just happen to be moving inertially in such a way that, when A and B stop accelerating, they are co-located with, and moving at the same speed as, C and D respectively.

If it helps, here's how the experiment looks in the inertial frame in which C and D are always at rest: initially, A and B are moving at speed $- v$, but then A and B decelerate until they stop moving; and when they stop moving, they are co-located with C and D, respectively. In this frame, A and B do not start decelerating at the same time, or stop at the same time; B starts decelerating before A, and stops before A. Note that this does not change the time that clocks A and B display at the instants they start and stop decelerating; they both display time zero at the instant they start decelerating, and they both display time $\Delta \tau$ at the instant they stop decelerating. (This means, of course, that in this frame clocks A and B are never synchronized.)

Hmm... I am having trouble visualizing this as 4 things are going on at the same time, but if i understand correctly, in C and D's frame, they are always at rest and A/B decelerate to their rest position. (correct?)
In this scenario, C and D are not moving with respect to each other? They are both initially and finally separated by the same distance separated by A and B?

 

[PeterDonis]

in C and D's frame, they are always at rest and A/B decelerate to their rest position. (correct?)

Yes.

They are both initially and finally separated by the same distance separated by A and B?

C and D are always separated by the same distance, since they are always at rest in this frame. A and B are not, because B starts decelerating before A does, so A and B end up further apart than they started in this frame. They end up as far apart as C and D (since they end up co-located with C and D), so they start out closer together than C and D are.

 

[Micheth]

Yes.
C and D are always separated by the same distance, since they are always at rest in this frame. A and B are not, because B starts decelerating before A does, so A and B end up further apart than they started in this frame. They end up as far apart as C and D (since they end up co-located with C and D), so they start out closer together than C and D are.

Well in this case, yes I totally agree and accept that A and B would be out of sync, because they moved with respect to each other.
(they end up further apart than when they started).
So either A or B experienced time dilation with respect to the other, so I agree they couldn't be in sync any longer.

But surely this is a different case from my original scenario where they accelerated in the same way and therefore never moved with respect to each other?

 

[PeterDonis]

surely this is a different case from my original scenario

No; it is exactly the same as your original scenario, just with C and D added. The motion of A and B is identical to the motion of A and B in your original scenario. My description of A and B's motion with respect to C and D's rest frame is a description of the same motion of A and B that's in your original scenario, just viewed from a different inertial frame. That's the whole point.

 

[Micheth]

No; it is exactly the same as your original scenario, just with C and D added. The motion of A and B is identical to the motion of A and B in your original scenario. My description of A and B's motion with respect to C and D's rest frame is a description of the same motion of A and B that's in your original scenario, just viewed from a different inertial frame. That's the whole point.

But, if it's the same as my original scenario, then B doesn't in fact start decelerating before A does (it just looks that way to C and D who are at rest as A&B come towards them).
I mean, if A and B each took photographic records of their clock times at the moment they began decelerating, and later gave these photos to C and D when they met, if I were C or D I would conclude, "ah, now I see that the times read the same, and A and B also report that they were originally synchronized in a rest frame, so although it looked to us like B decelerated first, the facts record that they decelerated together, so that's what happened."

 

[A.T.]

...doesn't in fact start decelerating before ... the facts record...

There is no "in fact" here, only "in frame ...". There is no "facts record" here, just "record of frame ...".

 

[PeterDonis]

f it's the same as my original scenario, then B doesn't in fact start decelerating before A does (it just looks that way to C and D who are at rest as A&B come towards them).

As A.T. said, there is no "in fact". No inertial frame is "privileged" over any other.

if A and B each took photographic records of their clock times at the moment they began decelerating, and later gave these photos to C and D when they met, if I were C or D I would conclude, "ah, now I see that the times read the same, and A and B also report that they were originally synchronized in a rest frame, so although it looked to us like B decelerated first, the facts record that they decelerated together, so that's what happened."

You're missing a key point. Before A and B decelerate, they are both at rest in one frame, and are synchronized in that frame. After they decelerate, they are both at rest in another frame (the C/D rest frame), and they are not synchronized in that frame. If, once they are at rest in the C/D frame, A and B try to perform the same procedure that they originally performed to synchronize themselves in their original rest frame, if they were still synchronized, that procedure would do nothing; it would just tell them "no adjustment needed, you're still synchronized". But in fact, they will find that they are no longer synchronized: that same procedure will now tell them that B's clock is "ahead" of A's, so one of the two will have to be adjusted to put them back in sync.

 

[Micheth]

You're missing a key point. Before A and B decelerate, they are both at rest in one frame, and are synchronized in that frame. After they decelerate, they are both at rest in another frame (the C/D rest frame), and they are not synchronized in that frame. If, once they are at rest in the C/D frame, A and B try to perform the same procedure that they originally performed to synchronize themselves in their original rest frame, if they were still synchronized, that procedure would do nothing; it would just tell them "no adjustment needed, you're still synchronized". But in fact, they will find that they are no longer synchronized: that same procedure will now tell them that B's clock is "ahead" of A's, so one of the two will have to be adjusted to put them back in sync.

That's exactly it, I don't see why A/B would not be synchronized in C/D's rest frame, once A/B reaches the point where A/B meets C/D.
I fully agree that C/D would see B's clock running ahead as A/B approach (B being in front of A), but as they got closer certainly B would appear to run less and less ahead of A until, when they reached the meeting point (side by side in the same rest frame where light from them would reach C/D essentially in tandem), light from B would no longer be reaching them earlier, and C/D would seem them as synchronized.

 

[PeterDonis]

I don't see why A/B would not be synchronized in C/D's rest frame, once A/B reaches the point where A/B meets C/D.

That's why I suggested learning the math; it makes it a lot easier to see why counterintuitive results like this are in fact true.

However, the repeated mention of relativity of simultaneity in this thread should give you a clue. Consider the relevant events:

* Event A0 is the event at which clock A starts accelerating (or decelerating, depending on which frame you are using).

* Event B0 is the event at which clock B starts accelerating (or decelerating).

* Event A1 is the event at which clock A stops accelerating (or decelerating).

* Event B1 is the event at which clock B stops decelerating.

For conciseness, we'll also label the frame in which A and B are originally at rest frame AB, and the frame in which C and D are at rest, and in which A and B are both at rest after they finish accelerating/decelerating, frame CD.

Now, by hypothesis, events A0 and B0 are simultaneous in frame AB, and events A1 and B1 are also simultaneous in frame AB. By relativity of simultaneity, that means events A0 and B0 are not simultaneous in frame CB, nor are events A1 and B1; in frame CD, event B0 happens before A0, and event B1 happens before A1. Hence, A and B are never synchronized in frame CD. This is a simple, obvious consequence of relativity of simultaneity, and it is not at all in doubt; it is an unambiguous prediction of SR.

Mathematically, you could verify the above by assigning coordinates in frame AB to all four events, and then Lorentz transforming all four sets of coordinates into frame CD, and seeing that the time coordinates of A0 and B0, and A1 and B1, are not the same in frame CD, even though they were the same in frame AB.

as they got closer certainly B would appear to run less and less ahead of A until, when they reached the meeting point (side by side in the same rest frame where light from them would reach C/D essentially in tandem), light from B would no longer be reaching them earlier, and C/D would seem them as synchronized.

Light travel time alone is not sufficient to determine clock synchronization. All it can tell you by itself is that, once clocks A and B are both at rest in frame CD, they are both running at the same rate as clocks C and D, which is true. But running at the same rate is not enough for clock synchronization; the "zero points" of both clocks also have to be the same, and they're not. They were in frame AB, but they're not in frame CD. That's what relativity of simultaneity tells you.

Once again, learning the math, or even better, learning how to draw and interpret spacetime diagrams, will make it a lot easier to see how all this works.

 

[Micheth]

Event A0 is the event at which clock A starts accelerating (or decelerating, depending on which frame you are using).
* Event B0 is the event at which clock B starts accelerating (or decelerating).
* Event A1 is the event at which clock A stops accelerating (or decelerating).
* Event B1 is the event at which clock B stops decelerating.
For conciseness, we'll also label the frame in which A and B are originally at rest frame AB, and the frame in which C and D are at rest, and in which A and B are both at rest after they finish accelerating/decelerating, frame CD.
Now, by hypothesis, events A0 and B0 are simultaneous in frame AB, and events A1 and B1 are also simultaneous in frame AB. By relativity of simultaneity, that means events A0 and B0 are not simultaneous in frame CB, nor are events A1 and B1;

(You are correct that I would have to learn to do the math to properly converse with you on this. I appreciate your efforts, though.)

What happens, though, when A and B (who have taken photographic records of their clock readings at A0 and B0, respectively, and also at events A1 and B1, respectively, which were simultaneous in their rest frame AB (they merely saw the rest of the universe accelerate toward them), and show them to C&D?