Simultaneity of accelerated clocks

[hutchphd]

<Moderators note: split from https://www.physicsforums.com/threa...han-just-a-fudge-factor.1012027/#post-6596771 >

And acceleration doesn't cause clocks to do anything - it just changes your inertial rest frame, so if you want to work in your current inertial rest frame then you need to deal with the changes of your interpretation of your data. That's all.

I have a (probably) stupid question. My rest frame contains myself and the usual rigid 3D array of clocks. I use Einstein's method to synchronize them. All is well.
Me and my clocks accelerate for a time but acquire no velocity.relative to each other. Are my clocks still Einstein synchronized? My understanding is that the answer is no. How does this comport with the statement above?

 

[PeterDonis]

Me and my clocks accelerate for a time but acquire no velocity.relative to each other. Are my clocks still Einstein synchronized? My understanding is that the answer is no.

Your understanding is correct. Your clocks will have gone out of synchronization during the acceleration period. However, once and your clocks stop accelerating and are inertial again, you can re-synchronize them using Einstein synchronization and they will stay synchronized.

How does this comport with the statement above?

The statement @Ibix made that "acceleration doesn't cause clocks to do anything" needs to be taken in its proper context, i.e., as a response to the OP. An expanded version of the statement that takes the context into account might be something like: "acceleration of observer A doesn't cause clocks that are spatially distant from observer A to do anything". Acceleration of observer A does change the path that observer A is taking through spacetime, and that can of course affect observer A's clock in relation to other clocks. But observer A's acceleration doesn't affect observer B's clock.

In your case, you have a whole family of observers all accelerating in a particular coordinated way, in order to maintain constant proper distance between them. That is not the kind of scenario that @Ibix was talking about.

There is also something in SR called the "clock postulate", which basically says that acceleration, in and of itself, does not affect the "tick rate" of clocks. @Ibix might have been referring to that as well. But that postulate is not violated in your scenario; the reason your clocks get out of synchronization during the acceleration period is not that their "tick rates" change due to acceleration, but that the lengths of the paths they take through spacetime during the acceleration period differ. A detailed discussion of that probably needs to be in its own thread (but a good key term to search on would be "Rindler observers" or "Rindler coordinates).

 

[hutchphd]

but that the lengths of the paths they take through spacetime during the acceleration period differ.

I understand this argument but find it a very peculiar indeed. "Paths through spacetime" indicates a calculational technique. The clocks at different places respond differently to the same change in velocity. I can hold one in my leading hand and one in my trailing hand and watch them go asynchronous. In fact I can use them to measure the acceleration. This argument begs a special definition of causality it seems to me.

 

[PeterDonis]

"Paths through spacetime" indicates a calculational technique.

No, it indicates a geometric property, namely, arc length, of curves in a 4-dimensional manifold.

The clocks at different places respond differently to the same change in velocity.

No, they don't. More precisely, if "respond" and "change in velocity" are interpreted as invariants (which they have to be to have physical meaning), then "respond" means "change tick rate as a function of arc length along their worldlines" and "change in velocity" means "proper acceleration", and clocks do not respond differently at all to different proper accelerations, let alone clocks at different places responding differently. This is the clock postulate, and even though it is called a postulate, it has been tested experimentally on subatomic particles in magnetic traps to, IIRC, about $10^{18}$ g accelerations.

I can hold one in my leading hand and one in my trailing hand and watch them go asynchronous.

"Asynchronous" is not an invariant, because it depends on a choice of simultaneity convention. Even though there happens to be a natural such choice in your particular scenario, it's still a convention, not an invariant.

To have an invariant comparison between clocks, they need to be co-located. For example, in the twin paradox, the two twins separate, then come back together again, and their aging is compared. Any difference in aging between them is an invariant, but it tells you that the arc lengths of their paths through spacetime were different; it does not tell you that there was any difference between them in terms of "rate of aging" as a function of arc length.

I can use them to measure the acceleration.

Please describe how you would do this. Include exactly how you would "watch them go asynchronous", since, as noted above, simultaneity is a convention, not an invariant.

 

[PeterDonis]

This argument begs a special definition of causality it seems to me.

Not at all. I strongly suggest that you spend some time with a relativity textbook that covers the geometric interpretation of spacetime. The difference in aging between the twins in the standard twin paradox, for example, is no different than the difference in odometer readings between two cars that take different routes between the same two points; the two odometers don't "tick off distance" at different rates, they are just traveling on paths that have different lengths.

 

[Ibix]

The clocks at different places respond differently to the same change in velocity.

No they don't, but there are an awful lot of subtleties here.

If all of your clocks are equipped with identical rockets which activate simultaneously in their initial rest frame and accelerate with identical proper acceleration profiles then they remain synchronised in their original frame. They will not show the same time as clocks at rest because of time dilation, and the disagreement will grow while they accelerate, but they will all show the same time. This is because they are all experiencing the same thing at the same time, according to this frame. According to the final rest frame, however, the clocks were initially not synchronised and they didn't start accelerating at the same time, so have different velocity profiles, so their offsets change.

I can hold one in my leading hand and one in my trailing hand and watch them go asynchronous. In fact I can use them to measure the acceleration. This argument begs a special definition of causality it seems to me.

This is where the fun begins. If you are literally holding the clocks in your hands, they aren't accelerating the same. This is easiest to see from the initial rest frame where, if the clocks are accelerating the same then they stay the same distance apart - but your arms are length contracting as you speed up. So either your arms are being stretched or the clocks aren't accelerating the same.

Depending how you actually resolve that, yes youmay be able to use the clock rates to measure your acceleration. If you go with the varying acceleration solution, so your arms remain the same length from your perspective, then yes you can. The differing tick rates are the same as gravitational time dilation, via the equivalence principle. But this will depend on how you choose to accelerate the clocks.

This argument begs a special definition of causality it seems to me.

The details depend on your initial choice of synchronisation and how you do the acceleration and how well you avoid being bitten by assumptions about synchronisation. I don't think there are any causal issues.

 

[PeterDonis]

there are an awful lot of subtleties here.

Oh, most definitely.

If all of your clocks are equipped with identical rockets which activate simultaneously in their initial rest frame and accelerate with identical proper acceleration profiles then they remain synchronised in their original frame.

I have bolded the crucial qualifier here. This original frame (i.e., the inertial frame in which the clocks all started out being at rest) is not the same as the (non-inertial) frame that @hutchphd is considering in the OP of this thread, when he talks about accelerating all his clocks while keeping the distance between them constant (which is what I take him to mean when he says that his clocks "acquire no velocity relative to each other").

Which of course brings up another subtlety here, that in the scenario you describe, which is not the same as the scenario the OP is envisioning (your scenario is the Bell spaceship paradox scenario, whereas the OP's scenario is the Rindler observer scenario), the distance between the clocks as seen by the clocks themselves (i.e., in the momentarily comoving inertial frame of any of the clocks) will not remain constant, it will increase.

Of course, the difference between the Bell spaceship paradox scenario and the Rindler observer scenario is itself another of the subtleties involved; there is no such difference in non-relativistic Newtonian physics, and the OP is probably not aware that there is such a difference in relativity.

 

[PeterDonis]

Me and my clocks accelerate for a time but acquire no velocity.relative to each other.

As I noted just now in response to @Ibix, you might not be aware that in order to meet the "acquire no velocity relative to each other" specification, at least as I am interpreting you to intend it (that you mean "velocity" in the non-inertial frame in which the clocks remain at rest), your clocks must have different proper accelerations; the clocks in front must have lower proper acceleration than the clocks in the rear. So if you were imagining that they would all have the same proper acceleration, that is not correct.

 

[PeroK]

I understand this argument but find it a very peculiar indeed. "Paths through spacetime" indicates a calculational technique. The clocks at different places respond differently to the same change in velocity. I can hold one in my leading hand and one in my trailing hand and watch them go asynchronous. In fact I can use them to measure the acceleration. This argument begs a special definition of causality it seems to me.

To get you started, consider this.

There are two clocks at rest in a given inertial frame. As measured in that frame, both clocks accelerate identically (in the direction of the separation between them). They remain synchronised in that frame (identical velocity profiles) and remain the same fixed distance apart (identical velocity profiles).

Once the acceleration stops, therefore, the clocks are moving at the same velocity, synchronised and the original distance apart (as measured in the original rest frame).

The clocks themselves, however, share an inertial rest frame moving with respect to their original rest frame. This implies they are now further apart in that frame (as there is length contraction) and cannot be synchronised in that frame (as there is relativity of simultaneity).

The question now is to precisely analyse the motion of each accelerating clock, as measured by the other accelerating clock. We know in advance that we must find an asymmetry, as relative to one another they have drifted further apart and gone out of sync.

This analysis, I suggest, is a good exercise to carry out before you despair of causality.

 

[Ibix]

(which is what I take him to mean when he says that his clocks "acquire no velocity relative to each other").

Depends which frame he's using for simultaneity when measuring the clocks' velocities. But I agree with you here, and I should have made clearer that I was (probably) starting with a different experimental plan. Failing to clearly draw that particular distinction is exactly the trap in Bell's spaceship paradox.

 

[hutchphd]

Please describe how you would do this. Include exactly how you would "watch them go asynchronous", since, as noted above, simultaneity is a convention, not an invariant.

I (Einstein) synchronize them before acceleration. After the acceleration stops I (Einstein) resynchronize them, having them record their asynchrony. Is this somehow fraught ?

This original frame (i.e., the inertial frame in which the clocks all started out being at rest) is not the same as the (non-inertial) frame that @hutchphd is considering in the OP of this thread, when he talks about accelerating all his clocks while keeping the distance between them constant (which is what I take him to mean when he says that his clocks "acquire no velocity relative to each other").

As the observer I am comparing before the acceleration to after the acceleration. At each observation I am at rest relative to my test clocks and in an inertial frame. So I disagree with this characterization of events.

So if you were imagining that they would all have the same proper acceleration, that is not correct.

Your statement was "And acceleration doesn't cause clocks to do anything" so I don't understand why you are now worrying about whether the accelerations were the same (?). I believe we are foundering here on the definition of "cause" in the space-time context. If every time I do X (acceleration), and I note later that Y (synchrony) has changed, then I conclude that X causes Y. I believe you are arguing that an unseen common mechanism actually causes both X and Y. Is this other than dogma (this is a serious question)?

 

[PeroK]

Your statement was "And acceleration doesn't cause clocks to do anything" so I don't understand why you are now worrying about whether the accelerations were the same (?).

To try to be precise. The instantaneous value of acceleration is not an additonal factor in measurements of time dilation. The only relevant factor is the instantaneous speed. I.e. the proper time $\tau$ for any particle relative to the coordinate time in the inertial reference frame in which its measured is:
$$\frac{d\tau}{dt} = \sqrt{1- \frac{v(t)^2}{c^2}}$$However, acceleration of a particle may have the effect of changing a particle's speed! You could, in fact, argue that if you knew a particle's initial velocity and acceleration profle, then you could calculate its time dilation without explicity using its velocity as per the above formula. But, that's a bit like saying that you get a speeding ticket for accelerating too quickly!

The point is that if two particles have different acceleration profiles, then they may have different velocity profiles and different rates of time dilation as measured in a given IRF.

 

[hutchphd]

But, that's a bit like saying that you get a speeding ticket for accelerating too quickly!

The appropriate turn of phrase would be "having the wrong acceleration profile" I think.
But it would (IMHO) be equally foolish to argue that your speeding ticket "was not caused by acceleration"!
We have been at this semantic impasse before! I believe the issues are (merely) dogmatic.
And you can get a ticket for accelerating too quickly...but that is a different story.

 

[PeroK]

PS

The appropriate turn of phrase would be "having the wrong acceleration profile" I think.
But it would (IMHO) be equally foolish to argue that your speeding ticket "was not caused by acceleration"!
We have been at this semantic impasse before! I believe the issues are (merely) dogmatic.
And you can get a ticket for accelerating too quickly...but that is a different story.

Nevertheless, the formula for time dilation involves only the instantaneous velocity and not acceleration. This is important as often students want to add a pseudo-gravirational term involving acceleration, based on a misunderstanding of the equivalence principle.

 

[A.T.]

I have a (probably) stupid question. My rest frame contains myself and the usual rigid 3D array of clocks. I use Einstein's method to synchronize them. All is well.
Me and my clocks accelerate for a time but acquire no velocity.relative to each other. Are my clocks still Einstein synchronized? My understanding is that the answer is no. How does this comport with the statement above?

There is no contradiction, if you pay attention to distinguish acceleration of the clock from acceleration of the reference frame. And between clock rate and accumulated time.

The statement by @Ibix means that clock rate doesn't depend in the proper acceleration of the clock:
https://en.wikipedia.org/wiki/Time_dilation#Clock_hypothesis

What you describe is that the clock rate can depend on the proper acceleration of the reference frame and the position of the clock.

Note that in your scenario the clock rates still don't depend on proper acceleration of the clocks. A free falling clock that is instantaneously stationary right next to one of your accelerating clocks will tick at the same rate as the accelerating clock. So there is no contraction to the clock hypothesis.

 

[hutchphd]

A free falling clock that instantaneously stationary right next to one of your accelerating clocks will tick at the same rate as the accelerating clock.

Again I am in the same experimental quandary. This would not necessarilly be true over any finite interval of time (how do you actually measure the instantaneous rate of the clock). I see that the Wikipedia article addresses this explicitly:
The clock hypothesis is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity.
As an additional hypothesis, this gives me no particular heartburn. I just have trouble extracting it without such an hypothesis. Thanks to all.

 

[FactChecker]

The clocks at different places respond differently to the same change in velocity.

"the same change in velocity" is a very complicated concept. Suppose the acceleration at each clock is done in regularly timed pulses. Suppose the clocks are initially synchronized and then some acceleration occurs so that they are on a frame of a different velocity. Is the next pulse of acceleration going to happen at the same time according to the original clocks at the higher velocity? Or same time according to the synchronized clocks of the original frame? Or same time according to a set of newly synchronized clocks in the higher velocity frame?
If the next pulse of acceleration at each clock occurs at the time of synchronized clocks at the higher velocity, then it does not occur at the same time according to the original clocks synchronized at the original velocity. So the definition of "same change in velocity" is relative.

 

[A.T.]

Again I am in the same experimental quandary. This would not necessarilly be true over any finite interval of time (how do you actually measure the instantaneous rate of the clock).

Experimentally measurements of instantaneous values of changing quantities are often just averages over a finite interval of time. The smaller that time interval, the more instantaneous is the measurement.

 

[hutchphd]

"the same change in velocity" is a very complicated concept.

Repeating

I (Einstein) synchronize them before acceleration. After the acceleration stops I (Einstein) resynchronize them, having them record their asynchrony. Is this somehow fraught ?

Why is this complicated? (These are clocks built onto a rigid Cartesian support and the asynchrony is the difference between what they read at the second spherical light pulse.). Inarticulate perhaps. Sorry.

 

[A.T.]

The clocks at different places respond differently to the same change in velocity.

This is gravitational time dilation, and has nothing to do with the proper acceleration of the individual clocks. It applies to free falling clocks at different positions in an accelerating frame in the same manner, as it applies to accelerating clocks that are at rest in the accelerating frame.

 

[PeterDonis]

I (Einstein) synchronize them before acceleration. After the acceleration stops I (Einstein) resynchronize them, having them record their asynchrony.

In other words, you don't "watch them go asynchronous" during acceleration. But you could, at least indirectly. You could have them exchange round-trip light signals during the acceleration process, and the clocks towards the front would show more elapsed time during a round trip than the clocks towards the rear.

As the observer I am comparing before the acceleration to after the acceleration. At each observation I am at rest relative to my test clocks and in an inertial frame.

But your frame is not inertial during acceleration, so you are not in the same inertial frame before and after. In the inertial frame in which your clocks are at rest after the acceleration, the clocks are out of sync before the acceleration, so the fact that they are also out of sync in this frame after the acceleration (before you resynchronize them) does not tell you anything useful.

If you try to construct a single frame in which your clocks are always at rest, that frame must be non-inertial during the acceleration.

Your statement was "And acceleration doesn't cause clocks to do anything"

It was originally a statement by @Ibix, not me. I explained in more detail what that statement meant in post #2 of this thread.

I don't understand why you are now worrying about whether the accelerations were the same (?).

I'm not worrying about it, I'm pointing it out because you said the clocks respond differently to "the same change in velocity", but the clocks have different proper accelerations so they do not have "the same change in velocity".

If every time I do X (acceleration)

"I do X" is an inaccurate description of your scenario. Let's limit it to two clocks for simplicity. Then clock A accelerates, and clock B accelerates. So there is not one single "I" doing "X". There are two clocks, each doing X, for different numerical values of X.

I note later that Y (synchrony) has changed

Y is not a property of either clock on its own. It is a property of both clocks together, plus a choice of simultaneity convention. When you changed inertial frames, you changed simultaneity conventions. That will change the synchrony of the clocks even if the clocks themselves don't accelerate at all. Your analysis does not take this into account at all.

Note, btw, that "Einstein clock synchronization" is in fact a definition of one particular family of simultaneity conventions, the one that is used to define inertial frames in SR.

I believe you are arguing that an unseen common mechanism actually causes both X and Y.

No.

X, the proper acceleration of the clocks, is obviously caused by whatever is making them accelerate--let's say rocket engines.

Y involves a simultaneity convention, and as I remarked above, if you change conventions in the middle of a scenario, you should expect that to change Y all on its own. Also, since the clocks have different proper accelerations, they do not have "the same change in velocity", so you do have a difference in X that could contribute to a difference in Y.

 

[PeterDonis]

The clock hypothesis is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity.

It's called a "hypothesis", but it's been verified experimentally for accelerations up to, IIRC, about $10^{18}$ g for subatomic particles in the lab.

As an additional hypothesis, this gives me no particular heartburn. I just have trouble extracting it without such an hypothesis.

Since the hypothesis has been confirmed experimentally, there is no reason to not use it.

 

[hutchphd]

It's called a "hypothesis", but it's been verified experimentally for accelerations up to, IIRC, about 1018 g for subatomic particles in the lab.

Absolutely. But it is an hypothesis separate from (no preferred frame) and (speed c).

 

[hutchphd]

No.

X, the proper acceleration of the clocks, is obviously caused by whatever is making them accelerate--let's say rocket engines.

Y involves a simultaneity convention, and as I remarked above, if you change conventions in the middle of a scenario, you should expect that to change Y all on its own. Also, since the clocks have different proper accelerations, they do not have "the same change in velocity", so you do have a difference in X that could contribute to a difference in Y.

Why would you change measurement conventions in the middle of an experiment? And I thought I made it clear that the me and the clocks were the same for each measurement within our inertial frame.

 

[hutchphd]

When you changed inertial frames,

Again I am rigidly attached to all the clocks I mention. So the clocks go with me and after the acceleration I am again at rest wrt to every clock.

 

[PeterDonis]

Why would you change measurement conventions in the middle of an experiment?

I wouldn't. But you did. You changed inertial frames. You had to, since the inertial frame in which the clocks are at rest after acceleration is different from the inertial frame in which the clocks are at rest before acceleration.

I thought I made it clear that the me and the clocks were the same for each measurement within our inertial frame.

There is no such thing as "our inertial frame" unchanged throughout the experiment. The "before acceleration" inertial frame is different from the "after acceleration" inertial frame. So you cannot say that anything "changed" during acceleration just because of the lack of synchronization in the "after acceleration" inertial frame; the frame itself has changed.

 

[PeterDonis]

I am rigidly attached to all the clocks I mention. So the clocks go with me and after the acceleration I am again at rest wrt to every clock.

None of this changes the fact that you and the clocks are not at rest in a single inertial frame throughout the experiment: the "before acceleration" inertial frame is different from the "after acceleration" inertial frame.

 

[hutchphd]

I wouldn't. But you did. You changed inertial frames. You had to, since the inertial frame in which the clocks are at rest after acceleration is different from the inertial frame in which the clocks are at rest before acceleration.

Perhaps I am using the terms incorrectly. By "Einstein" I mean using a spherical pulse of light to infer t=0 for all clocks in my inertial frame. The "our" in each case refers to me and my clocks. The procedure is not frame dependent but the result obviously is or we wouldn't be having this colloquy.

 

[PeroK]

Again I am rigidly attached to all the clocks I mention. So the clocks go with me and after the acceleration I am again at rest wrt to every clock.

You're hiding absolute simultaneity in there. In Minkowski spacetime you cannot have everything that you want. Look up Born Rigidity:

https://en.wikipedia.org/wiki/Born_rigidity

Something's got to give!

 

[PeterDonis]

By "Einstein" I mean using a spherical pulse of light to infer t=0 for all clocks in my inertial frame.

For all clocks at rest in your inertial frame (in which the light source is also at rest) during the procedure, yes.

The procedure is not frame dependent

Yes, it is. More precisely, it is dependent on the state of motion of the light source and the clocks receiving the light pulse. The light source and the clocks, all moving inertially and all at rest relative to each other, define a frame; and a light source and clocks that are moving relative to the original light source and clocks (which includes the same light source and clocks if they have accelerated in between, as in your scenario) define a different frame. And the procedures they follow for Einstein synchronization, while they look the same relative to each frame, are still relative to each frame. Each procedure only validly synchronizes clocks at rest in the frame in which it is being done.

 

[hutchphd]

ou're hiding absolute simultaneity in there. In Minkowski spacetime you cannot have everything that you want. Look up Born Rigidity:

You may be correct but do you see it? I just have two clocks on a stick and I read them (in an approved fashion) twice with an acceleration event in between. I will nibble at that knot tomorrow. Thanks

Each procedure only validly synchronizes clocks at rest in the frame in which it is being done.

Yes I understand that and knowingly have said nothing contradictory to this

 

[PeterDonis]

I just have two clocks on a stick and I read them (in an approved fashion) twice with an acceleration event in between.

And the "acceleration event in between" means that the two synchronization procedures are done in different inertial frames. Do you see that?

 

[hutchphd]

Yes. I wrote it.
But if I am in a lab on, say, the space shuttle in free space, I perform the same procedure twice (Einstein ).
If the lab has changed velocity, along the interclock axis, during that interval, it will reveal that the clocks needed to be reset relative to each other. The procedure does not change.

 

[Nugatory]

Again I am rigidly attached to all the clocks I mention.

You are not, because there are no rigid objects in relativity. At best, you and your clocks are undergoing Born rigid motion, and there’s a world of simultaneity pitfalls there.

 

[PeterDonis]

If the lab has changed velocity, along the interclock axis, during that interval, it will reveal that the clocks needed to be reset relative to each other.

But it will also reveal this for clocks that are moving to start with.

In your scenario you and your clocks, call them clock 1 and clock 2, start out at rest in inertial frame A, and are Einstein synchronized in that frame. Then you and your clocks undergo proper acceleration, after which you and your clocks are at rest in inertial frame B. When you re-do the synchronization procedure, it tells you that the clocks are desynchronized.

However, now suppose there are two other clocks, call them clock 3 and clock 4, which are always at rest in inertial frame A (i.e., they never undergo any proper acceleration) and are Einstein synchronized in that frame. If you do the synchronization procedure on them in frame B (i.e., you check their synchronization after you and your clocks have finished accelerating), you will find them to be desynchronized.

So you have two clocks, clock 3 and clock 4, which are desynchronized in frame B, as an obvious consequence of them being synchronized in frame A, i.e., the desynchronization in frame B is obviously due to the change of frame. How do you know that the desynchronization of clocks 1 and 2 in frame B is not due to the same thing?