The importance of velocity in simultaneity

[A.T.]

You can't do that - the distance between the clocks contracts in exactly the same way that the train would if it were there.

He can keep the distance constant in the initial rest frame, but then it's a different scenario, like Bell's spaceships. And the distance will increase in the clocks's frame, so they can't be at relative rest all the time.

 

[Ibix]

He can keep the distance constant in the initial rest frame, but then it's a different scenario, like Bell's spaceships. And the distance will increase in the clocks's frame, so they can't be at relative rest all the time.

He can keep the distance constant in any frame of his choosing - but not in general, which is the way I read his post as intending.

 

[Micheth]

No, they cannot be at relative rest all the time. If their distance in the ground frame stays the same, then their distance in their final common rest frame must have increased. See.
http://en.wikipedia.org/wiki/Bell's_spaceship_paradox

I read this wiki article but it seems to support what I'm saying? (If we assume the clocks are the rockets.)
The article states "The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start".
So there has been no change in distance between them if they accelerated exactly the same.
And having gone through the exact same acceleration history, I can't see how they would not still be in synchrony.

 

[A.T.]

He can keep the distance constant in any frame of his choosing - but not in general, which is the way I read his post as intending.

Right, it cannot stay constant in both frames. But the two separate clocks don't have to behave "exactly the same way that the train would if it were there".

 

[Ibix]

Right, it cannot stay constant in both frames. But the two separate clocks don't have to behave "exactly the same way that the train would if it were there".

Fair enough. You know what - I'm ducking out of this conversation for now, as I seem to be having to correct everything I write today.

 

[Micheth]

You can't do that - the distance between the clocks contracts in exactly the same way that the train would if it were there.

Sorry my above comment was referring to Ibix's statement that the distance between the clocks contract. The wiki article stating that it does not.

No, because it's only before they start, and once they've both agreed that they've both stopped accelerating, that they agree that they're in the same state of motion. In between those two points, differences can accumulate.

Sorry I don't understand what you're saying here. If they undergo the exact same acceleration/uniform motion, then how can differences accumulate?

 

[A.T.]

The article states "The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start".

That's right, because Lorentz contraction relates lengths in different frames, not at different time points in the same frame.

So there has been no change in distance between them if they accelerated exactly the same.

If there is no distance change in the ground frame, then there must be a distance increase in their rest frame. That's why the string between the rockets breaks.

 

[Micheth]

If there is no distance change in the ground frame, then there must be a distance increase in their rest frame. That's why the string between the rockets breaks.

The article seems to argue that the string breaks not because of increased distance but because of Lorentz contraction (of the string). If there's no matter between them also being accelerated, then there's nothing to contract, whereas the article clearly states that the distance would remain the same...

 

[A.T.]

The article seems to argue that the string breaks not because of increased distance but because of Lorentz contraction (of the string).

Yes, Lorentz contraction says the moving string is longer in its rest frame, than in the ground frame. So if it keeps its length in the ground frame, it must elongate in it rest frame.

If there's no matter between them also being accelerated, then there's nothing to contract, whereas the article clearly states that the distance would remain the same...

If the distance remains the same in the ground frame, then it must increase in the frame of the rockets. That's what Lorentz contraction says, and the existence of the string is irrelevant here.

 

[Micheth]

If the distance remains the same in the ground frame, then it must increase in the frame of the rockets. That's what Lorentz contraction says, and the existence of the string is irrelevant here.

I think the distance stays the same even in the frame of the rockets (clocks in this case).
If the distance were to increase, then equally accelerating objects would move away from each other.

 

[A.T.]

I think the distance stays the same even in the frame of the rockets (clocks in this case).

No, it cannot stay the same in both frames, because Lorentz contraction relates the lengths in different frames, and says they are different.

If the distance were to increase, then equally accelerating objects would move away from each other.

Their distance increases, in their rest frames.

 

[Micheth]

Their distance increases, in their rest frames.

But then why does wiki state: "The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start."
(It doesn't say that their distance increases either)

 

[A.T.]

But then why does wiki state: "The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start."

https://www.physicsforums.com/threa...ty-in-simultaneity.789927/page-2#post-5045671

 

[Micheth]

QUOTE=A.T. said: ↑
Their distance increases, in their rest frames.[/QUOTE]

QUOTE="Micheth, post: 5045717, member: 547880"]But then why does wiki state: "The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start."
(It doesn't say that their distance increases either)[/QUOTE]Oops I missed that it did say their distance increases but from the viewpoint of an observer on the ground (the initial rest frame).
But the clocks themselves accelerate equally and at the point they pass the platform (and get hit by lightning bolts) they ought to register the same time, no?

 

[A.T.]

Oops I missed that it did say their distance increases but from the viewpoint of an observer on the ground (the initial rest frame).

It increases in the frame of the rockets. It stays constant in the ground frame.

But the clocks themselves accelerate equally and at the point they pass the platform (and get hit by lightning bolts) they ought to register the same time, no?

If the lightning bolts are simultaneous in the ground frame, and the clocks had always the same speed (and thus tick rate) in the ground frame, then yes.

 

[Micheth]

It increases in the frame of the rockets. It stays constant in the ground frame.
If the lightning bolts are simultaneous in the ground frame, and the clocks had always the same speed (and thus tick rate) in the ground frame, then yes.

The article said: "the rest length between the two has increased in the frames in which they are momentarily at rest (S′)".
I interpret that as meaning the ground frame, where they were initially at rest.
(There being no increase in distance between them in their inertial frame.)

I still can't see why the platform frame is not going to see the clocks registering the same time. They both had the same acceleration history (even as viewed from the ground), in the same direction, for the same time period. Why would they show different times?

 

[A.T.]

The article said: "the rest length between the two has increased in the frames in which they are momentarily at rest (S′)".
I interpret that as meaning the ground frame, where they were initially at rest.

No, it means their rest frames (note the plural) during the acceleration.

I still can't see why the platform frame is not going to see the clocks registering the same time.

It will, if they accelerate synchronously in the ground frame, as is the case in Bell's scenario.

 

[Micheth]

No, it means their rest frames (note the plural) during the acceleration.

Then I have no idea what rest frames it's referring to...
I thought they were at rest, and then and then accelerated

It will, if they accelerate synchronously in the ground frame, as is the case in Bell's scenario.

I would think that's what happened with my clocks too.
So the platform observer observes the clocks reading exactly the same times, and the lightning bolts hitting simultaneously, right?
And they stop functioning at those identical times.

 

[A.T.]

Then I have no idea what rest frames it's referring to...

The inertial frames where the rockets are at rest at different time points during the acceleration.

So the platform observer observes the clocks reading exactly the same times, and the lightning bolts hitting simultaneously, right? And they stop functioning at those identical times.

In your modified scenario without the train, that is a possible variant.

 

[Micheth]

The inertial frames where the rockets are at rest at different time points during the acceleration.

Ah I see, which makes sense to me since as far as they're concerned, they haven't moved with respect to each other.

In your modified scenario without the train, that is a possible variant.

Ok :-) then continuing with the modified scenario, instead of a rider on the train, we have a third party accelerating between and in tandem with the clocks.
(In other words, all the previous characters just without the train)
But it has been established that the rider would experience the lightning bolts occurring out of synchrony, no? His own clock (carried with him and in synchrony with the other two) would say they hit at different times, but when everything comes to rest (or even before that), he checks the clocks that have stopped, and they register the same time, i.e. the time the platform observer claims it happened.

 

[A.T.]

Ah I see, which makes sense to me since as far as they're concerned, they haven't moved with respect to each other.

No. According to the ground frame they haven't moved apart. But according to their rest frame they have.

instead of a rider on the train, we have a third party accelerating between and in tandem with the clocks.

In the middle with same acceleration in the ground frame as the clocks?

But it has been established that the rider would experience the lightning bolts occurring out of synchrony, no?

If they are simultaneous in the ground frame, they aren't in the middle-man's frame.

His own clock (carried with him and in synchrony with the other two)

It's synchronous in the ground frame, not in his own rest frame.

would say they hit at different times, but when everything comes to rest (or even before that), he checks the clocks that have stopped, and they register the same time,

Yes, and he has no problem with that. In his frame the clocks where running with an offset, and where hit with the same offset, so they stopped showing the same time.

 

[Micheth]

If they are simultaneous in the ground frame, they aren't in the middle-man's frame.

It's synchronous in the ground frame, not in his own rest frame.

Yes, and he has no problem with that. In his frame the clocks where running with an offset, and where hit with the same offset, so they stopped showing the same time.

Hmmm. I would think two (or three clocks) set to be synchronous, and then all undergoing exactly the same accelerations, would be synchronous especially in their own frames?
Otherwise wouldn't atomic clocks running next to each other quickly run out of synch? They're constantly undergoing identical acceleration (earth rotation+revolution, not to mention galactic, cluster, etc.)

 

[A.T.]

Hmmm. I would think two (or three clocks) set to be synchronous, and then all undergoing exactly the same accelerations, would be synchronous especially in their own frames?

Their accelerations aren't synchronous in their own frames, so there is no reason to expect the clock times to remain synchronous in their own frames.

 

[Micheth]

Their accelerations aren't synchronous in their own frames, so there is no reason to expect the clock times to remain synchronous in their own frames.

I thought they were, being identically accelerated.
If not, then wouldn't three clocks lined up in the same order on earth, and being hurled through space toward the Great Attractor, have asynchronous accelerations and thus we couldn't trust synchronized "stationary" atomic clocks on the earth?

 

[PeterDonis]

wouldn't atomic clocks running next to each other quickly run out of synch? They're constantly undergoing identical acceleration

Careful. You are conflating two different concepts of acceleration. Also, you are now talking about curved spacetime (because you said the clocks were at rest on Earth), which brings in additional complications that I would recommend avoiding for this discussion.

An object sitting at rest on Earth experiences nonzero proper acceleration--this is just what we normally call "weight". Similarly, an object on the train in your scenario experiences proper acceleration while the train is starting up; it feels "weight" as the train accelerates. Once the train is at a constant speed, the weight goes away; if we imagine the train out in free space somewhere, it and everything in it will be weightless once it's moving at a constant speed.

The "acceleration" of the object relative to the Sun, the galaxy, the galactic cluster, etc. is not proper acceleration; it's not felt as weight. It's just coordinate acceleration; in coordinates in which the Sun is at rest, the Earth and everything on it are accelerated.

Now suppose we have two atomic clocks, far out in free space somewhere (to eliminate any complications involved with curved spacetime), lined up along the $x$ axis. They start accelerating--in the sense of proper acceleration--in the $x$ direction. As they accelerate, their clocks will indeed get out of sync, even if they are both feeling identical proper acceleration. But if they both accelerate in the $y$ direction (i.e., perpendicular to their separation), they will not get out of sync.

(You can indeed carry this over to curved spacetime, with some caveats. For example, two atomic clocks sitting at the same altitude on Earth, separated by a small distance horizontally, will not get out of sync; the acceleration they feel is perpendicular to their separation. But two atomic clocks sitting at slightly different altitudes, with no separation horizontally, only vertically, will get out of sync. However, other aspects of this scenario will not work the same as the corresponding scenario in flat spacetime, so once again, I recommend avoiding any scenario in which gravity is present for this discussion.)

 

[A.T.]

I thought they were, being identically accelerated.

Only in the ground frame. But if they stop accelerating simultaneously in the ground frame, then they do not stop accelerating simultaneously in their frame.

atomic clocks on the earth?

See Peter's post. You should try to understand it in flat space time first, without gravity.

 

[Micheth]

Careful. You are conflating two different concepts of acceleration. Also, you are now talking about curved spacetime (because you said the clocks were at rest on Earth), which brings in additional complications that I would recommend avoiding for this discussion.
...
(You can indeed carry this over to curved spacetime, with some caveats. For example, two atomic clocks sitting at the same altitude on Earth, separated by a small distance horizontally, will not get out of sync; the acceleration they feel is perpendicular to their separation. But two atomic clocks sitting at slightly different altitudes, with no separation horizontally, only vertically, will get out of sync. However, other aspects of this scenario will not work the same as the corresponding scenario in flat spacetime, so once again, I recommend avoiding any scenario in which gravity is present for this discussion.)

Thanks for the detailed response. I was under the impression that GR saw gravity and acceleration as identical, but now I see there is difference between proper & coordinate acceleration (which incidentally I find intuitive as well) (I might shorthand it as: gravity "pulls" while proper acceleration "pushes")
Anyway, so to understand this correctly, proper acceleration (PA) WOULD cause the clocks to get out of sync whereas coordinate acceleration (CA) would not?

But as I continue to think about it, it seems that CA would produce at least as much non-synchrony in the clocks because any gravitational source you placed (in the direction of their alignment) is going to affect the closer one more (earlier) and the farther one less (later), and I can't imagine a scenario where CA could be produced in exactly the same manner to all 3 clocks lined up. Maybe there would be a way.
Whereas it still seems PA could (theoretically) be simultaneously produced in all 3... (if all conditions (forces, vectors, etc.) were reproduced in all 3)

 

[PeterDonis]

proper acceleration (PA) WOULD cause the clocks to get out of sync whereas coordinate acceleration (CA) would not?

It's not that simple. First, consider flat spacetime. In flat spacetime, any object with nonzero proper acceleration will also have nonzero coordinate acceleration in any inertial frame. So you can't separate the two.

In a non-inertial frame, in either flat spacetime or curved spacetime (in curved spacetime, there are no global inertial frames), it's possible to have proper acceleration without coordinate acceleration. However, gravitational time dilation is not a function of acceleration; it's a function of position--how deep you are in the gravity well. So even in this case, it's not really correct to say that proper acceleration causes clocks to go out of sync.

 

[Micheth]

However, gravitational time dilation is not a function of acceleration; it's a function of position--how deep you are in the gravity well.

That's interesting. I never considered it but now that you mention it a body in free fall in a gravitational field would have double time dilation, that of it's motion toward the grav. center and that of the grav. field itself (how far it's getting into the grav. well)...

So even in this case, it's not really correct to say that proper acceleration causes clocks to go out of sync.

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

 

[A.T.]

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

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