Simultaneity of accelerated clocks

[hutchphd]

I am only looking at clocks attached to me "rigidly". I only look at them when I feel no acceleration.
What you say is true but why are you telling me this? Why should I care about these other clocks?

 

[hutchphd]

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.

Even after the acceleration stops and the dust settles?

 

[PeterDonis]

I am only looking at clocks attached to me "rigidly". I only look at them when I feel no acceleration.

In other words, you refuse to do exactly the kind of test you would need to do for the kind of "causal" reasoning you are trying to do. See below.

Why should I care about these other clocks?

Because you are making a causal claim; you are saying that X causes Y, where X is proper acceleration and Y is desynchronization. But there is also a Z happening in your scenario: a change of frame.

So before you can make your causal claim, you first need to test whether Z could cause Y. And the way you would do that is to test Y for a pair of clocks for which Z happens (the clocks stay in your original frame but you test their synchronization in your final frame, which is different) but X does not (the other pair of clocks never accelerates). And when you run that test, what do you find? You find that Y still happens: the clocks are still desynchronized in frame B, even though they never accelerated. This contradicts your causal claim.

 

[PeterDonis]

the clocks are still desynchronized in frame B, even though they never accelerated.

And in fact there's even more than that: when you test synchronization in frame B, after you and your pair of clocks (clock 1 and clock 2) have finished accelerating, you find that clock 1 and clock 2 are desynchronized by exactly the same amount as clock 3 and clock 4, the clocks that stayed at rest in frame A the whole time. So the desynchronization can be entirely accounted for by the change of frame.

 

[hutchphd]

Now I understand your argument. Thanks and I agree.
But the original statement (by @Ibix my apologiesto you ) was
"And acceleration doesn't cause clocks to do anything"
The crux of my problem is that getting from one inertial frame to another necessarilly involves acceleration doesn'tit?. So this seems a tortured argument to me.

 

[Ibix]

"And acceleration doesn't cause clocks to do anything"

Well, the original context of my comment was about the relativity of simultaneity and I was answering someone claiming that (in the twin paradox) the traveling twin accelerating made Earth clocks jump about, which isn't accurate. It might change your opinion on what time it is "now", but the Earth clocks don't care about your opinion.

I'm afraid I haven't had a chance to catch up on this thread in any detail, but I suspect that you are thinking in a much more general sense. If you have flocks of clocks accelerating in arbitrary ways then their relative tick rates will depend on their current (however you define that) velocity and they will have varying offsets which will depend on their velocity history (or, equivalently, their initial velocity and acceleration history), yes.

 

[PeterDonis]

the original statement (by @Ibix my apologiesto you ) was
"And acceleration doesn't cause clocks to do anything"

Yes, and as I explained in an earlier post, that statement needs to be taken in context. @Ibix just explained the context again in post #41. That statement was never intended as a fully general claim that nothing having anything to do with acceleration can ever affect anything having to do with clocks. It was a particular response to a particular misconception (and you were not the one who had the misconception that @Ibix was responding to).

The crux of my problem is that getting from one inertial frame to another necessarilly involves acceleration doesn'tit?

If you are at rest in one inertial frame, and you want to be at rest in a different inertial frame, then yes, you have to undergo proper acceleration to do that.

 

[Dale]

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.

Yes, 10^18 g was measured by Bailey et al. with muons in a highly relativistic storage ring.

 

[A.T.]

But the original statement (by @Ibix my apologiesto you ) was
"And acceleration doesn't cause clocks to do anything"
The crux of my problem is that getting from one inertial frame to another necessarilly involves acceleration doesn'tit?.

You are still conflating the acceleration of a clock with the acceleration of reference frame.

 

[pervect]

If we consider the specific case of a born rigid accelerating rod, and use Rindler coordinates (which requires the methodologies of general relativity, not special), it can be seen that unadjusted clocks at different heights will tick at different rates depending on their position, the z coordinate or "height" in the acclerated frame. Thus they will not remain syncronized, they don't tick at the same rate. The metric I am using is:

$$-(1+gz/c^2)^2 c^2 dt^2 + dx^2 + dy^2 + dz^2$$

One can see that "gravitational time dilation factor in these coordinates is (1+gz/c^2), z being the "height" of the clock in the accelerated frame. g is the proper acceleration of the clock at height 0 - the proper acceleration of the clock will depend on the 'height'.

Because the clocks tick at different rates, it's not really sensible to talk about syncrhonizing them, though one could imagine a syncrhonization error that grows with time.

Note that the whole idea of clock synchronization requires some more detailed specification to have any physical meaning, that's why I specified the use of specific coordinates (Rindler coordinates) for the accelerating observer. The failure to specify this needed information suggests to me that the posters in question don't realize that it's necessary, because they don't fully understand that simultaneity is realtive.

This can also be explained in the language of special relativity, but it requires more work. The way I'd go about it is to describe in detail the worldlines of two clocks at different positions on the rigid rod, and write the trajectory.

"The Relativistic Rocket", https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html, has most, but not all, of the necessary equations to specify the trajectories of the points on the rod. What's missing is the the proper accleration $\alpha$ of a point on the rod at height z. I think it's something like $c^2 g / (c^2 + gz)$ in the particular coordinates I've suggested, but I could be making an error.

 

[PeterDonis]

Because the clocks tick at different rates, it's not really sensible to talk about syncrhonizing them

You can still send light signals between them and establish a simultaneity convention (indeed, the Rindler coordinates you use assume a particular such convention which is the "natural" one you would get if you sent light signals between the clocks); but the elapsed time on the clocks between two successive round-trip light signals will be different.

 

[PeroK]

The metric I am using is:

$$-(1+gz/c^2)^2 c^2 dt^2 + dx^2 + dy^2 + dz^2$$

It's worth emphasising that this metric represents flat spacetime, which can be seen by applying an appropriate coordinate transformation (Kottler-Moller):

https://en.wikipedia.org/wiki/Rindler_coordinates#Variants_of_transformation_formulas

 

[cianfa72]

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.

It take it as 'the same light source and clocks that have accelerated in between' (call them objects A) are actually now 'at rest' in the inertial frame defined by light source and clocks that move inertially with the 'final' velocity that objects A after the acceleration have w.r.t their original rest frame (i.e. w.r.t the inertial frame they were at rest before the acceleration).

 

[PeterDonis]

It take it as 'the same light source and clocks that have accelerated in between' (call them objects A) are actually now 'at rest' in the inertial frame defined by light source and clocks that move inertially with the 'final' velocity that objects A after the acceleration have w.r.t their original rest frame (i.e. w.r.t the inertial frame they were at rest before the acceleration).

I am describing the scenario @hutchpd proposed. I am including a "light source" in addition to clocks (which are all @hutchpd included in his original proposed scenario) because of how @hutchpd described Einstein clock synchronization in post #28.

I can't tell if what you are saying in the quote above is the same as the scenario @hutchpd described or not.

 

[cianfa72]

I am describing the scenario @hutchpd proposed. I am including a "light source" in addition to clocks (which are all @hutchpd included in his original proposed scenario) because of how @hutchpd described Einstein clock synchronization in post #28.

Yes, my point was to have a clear understanding of which are the inertial frames involved before and after the clocks have accelerated.

At the end of 'proper acceleration' process, the two clocks will be 'at rest' w.r.t. the inertial frame in which they have the same velocity w.r.t the original rest frame (i.e. w.r.t the inertial frame from which they started proper accelerating).

 

[PeterDonis]

At the end of 'proper acceleration' process, the two clocks will be 'at rest' w.r.t. the inertial frame in which they have the same velocity w.r.t the original rest frame

Yes, that is stipulated in the scenario. Note that it requires that, according to the original inertial frame, the clocks stop accelerating at different times.

 

[cianfa72]

Note that it requires that, according to the original inertial frame, the clocks stop accelerating at different times.

Yes, my 'envision' of this is to image a full 'grid' of standard clocks at rest and Einstein synchronized in the original inertial frame. According this 'set' of standard clocks, the two accelerating clocks stop at different times (as shown by the clock spatially near the event in which each accelerating clock stop).