# Speed of light postulate and clock synchronization

At the beginning of his 1905 paper 'On the Electrodynamics of Moving Bodies' Einstein defined the synchronization between two clocks using light signals, and on this basis formulates the following theorems

1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Furthermore, he assumes the following postulate holds true:

Any ray of light moves in the 'stationary' system of co-ordinates with
the determined velocity c, whether the ray be emitted by a stationary or
by a moving body.

So if a clock A at rest at the origin of the stationary system sends out a light signal with a timestamp t_A to a clock C at rest at location x in the stationary system, and this time stamp is found to synchronize with clock C (t_C = t_A +x/c), then the same timestamp t_B=t_A sent out by a clock B in motion at the origin of the stationary frame should, as per the above postulate, also synchronize with clock C.
However, this appears to contradict the Lorentz transformation, where a moving clock does not synchronize with a distant clock in the stationary system even if it synchronizes with a co-located stationary clock. How is this contradiction explained?

## Answers and Replies

Nugatory
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I can always arrange to make any single timestamp sent by B match any single timestamp sent by A - all I need to do is set B to whatever value is necessary to get the match. But when I do that, the next timestamp sent by B will not match the next timestamp sent by A and we will say that although A and B were both set to the same time, they didn't stay in sync.

A and C will maintain their synchronization over time and as we send more timestamps; B will drift out of sync with both as long as it is moving relative to them.

I can always arrange to make any single timestamp sent by B match any single timestamp sent by A - all I need to do is set B to whatever value is necessary to get the match. But when I do that, the next timestamp sent by B will not match the next timestamp sent by A and we will say that although A and B were both set to the same time, they didn't stay in sync.

A and C will maintain their synchronization over time and as we send more timestamps; B will drift out of sync with both as long as it is moving relative to them.

According to the Lorentz transformation formula, the clock synchronization does not only depend on time, but also on the location: if the moving clock B sends out a single timestamp t'=0 which synchronizes with the co-located rest frame clock A (t=0) then this time stamp should not synchronize with the rest frame clock C at location x. There should be a time difference x*v/c2 due to the 'relativity of simultaneity' term. However, this seems to contradict Einstein's initial definitions and postulate as explained above.

Dale
Mentor
2020 Award
I always read the definitions as assuming clocks mutually at rest.

I always read the definitions as assuming clocks mutually at rest.
With the synchronization procedure using the transmission of a light signal between the two clocks, and, according to the postulate, the speed of light constant (i.e. independent of the velocity of the emitting clock) in the reference frame of the target clock, should the definitions not apply both to clocks at rest and moving relatively to each other?

Nugatory
Mentor
should the definitions not apply both to clocks at rest and moving relatively to each other?

They do. The synchronization procedure allows you to say that two clocks are synchronized at one particular moment, no matter their state of motion (but note that saying that two clocks are synchronized at just one moment isn't very interesting).

The interesting bit happens when you try running the synchronization procedure again. If the two clocks are at rest relative to one another, they will still be synchronized - they will agree about how much time elapsed between the two attempts. If they are in motion relative to one another, they will not.

If I follow what you are asking, it looks to me like you are starting with A, B, and C subjected to the synchronization procedure using two way light travel in the usual way.

1] - You then are suggesting that a subsequent single one-way light signal with time stamp from A to C will confirm synchronization between A and C.

2]- Then you put B in motion suggesting a subsequent single one-way light signal with time stamp from moving B emitted when B is closely passing by resting A will confirm synchronization between B and C and match that between A and C.

@1] - Why do you think the one-way light signal from A to C will confirm synchronization previously performed between A and C using two-way light signals, knowing that light speed is postulated from two-way measurements?

@2\ - The one-way moving B signal emitted at A to C can match the time stamp from resting A to C, but why do you think the one-way light signal from moving B will confirm synchronization previously performed between A and C using two-way light signals, knowing that a return signal from C to moving B would be received when B is no longer at A?

Maybe I'm not following what you are asking?

Dale
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2020 Award
With the synchronization procedure using the transmission of a light signal between the two clocks, and, according to the postulate, the speed of light constant (i.e. independent of the velocity of the emitting clock) in the reference frame of the target clock, should the definitions not apply both to clocks at rest and moving relatively to each other?
I think that there are two separate issues here. One is how generally can Einstein's definition be applied, and the other is how generally he used it in his 1905 paper. It often happens that the first paper doesn't exploit the full generality of the concept that it pioneers.

To me the paper has always seemed like he used it assuming the clocks at A and B were at rest. I could be wrong in my reading, but he makes it clear that A and B are "points in space" and so I always thought the associated clocks stayed at their respective points.

That said, I think the definition applies if the clock at B is moving in the reference frame of interest, but not clock A. I.e. the definition is a definition of simultaneity in A's frame. Of course you can always change the frame of interest and thus swap the roles of A and B.

They do. The synchronization procedure allows you to say that two clocks are synchronized at one particular moment, no matter their state of motion (but note that saying that two clocks are synchronized at just one moment isn't very interesting).

The interesting bit happens when you try running the synchronization procedure again. If the two clocks are at rest relative to one another, they will still be synchronized - they will agree about how much time elapsed between the two attempts. If they are in motion relative to one another, they will not.
It seems you missed my reply to your previous post (post #3): my argument aims at the location dependence of the synchronization, not on the time dependence .

@1] - Why do you think the one-way light signal from A to C will confirm synchronization previously performed between A and C using two-way light signals, knowing that light speed is postulated from two-way measurements?

@@1) Why do you think a two-way transmission is required for synchronization? If I send a time stamp t_A from clock A to clock B and if I know the travel time of the signal t_AB, then the two clocks are synchronized by setting clock B to the time t_B = t_A + t_AB on arrival of the signal. Nothing more is required.

I think that there are two separate issues here. One is how generally can Einstein's definition be applied, and the other is how generally he used it in his 1905 paper. It often happens that the first paper doesn't exploit the full generality of the concept that it pioneers.

To me the paper has always seemed like he used it assuming the clocks at A and B were at rest. I could be wrong in my reading, but he makes it clear that A and B are "points in space" and so I always thought the associated clocks stayed at their respective points.

Initially Einstein defines the synchronization for clocks at rest relatively to each other, and then argues that two clocks synchronized in their rest frame do not appear as synchronized anymore in a moving reference frame (and even in the latter, the clocks are then obviously still at rest relatively to each other). Yet later when he derives the time dilation from the Lorentz transformation, he says (near the end of §4) "if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize". This statement clearly requires a concept of synchronization between moving clocks, which however he has not defined at all before (note that, as you pointed out already, Einstein uses A and B here to designate the points where the clocks are located, rather than the clocks themselves; so this would correspond here to my scenario of clock B moving between the stationary clocks A and C).

PeterDonis
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2020 Award
if I know the travel time of the signal t_AB

How do you know that?

Fantasist said:
if I know the travel time of the signal t_AB
How do you know that?

You can encode for instance the location of clock A also into the signal, together with the time stamp. Since clock B knows its location as well, it can calculate the travel time from the corresponding distance and the speed of light.

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Dale
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Initially Einstein defines the synchronization for clocks at rest relatively to each other,
It sounds like you read it the same way that I do.

So I am not sure what your question is. Are you simply saying "gotcha" to Einstein for not writing the definition as generally as he could have or should have?

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It sounds like you read it the same way that I do.

So I am not sure what your question is. Are you simply saying "gotcha" to Einstein for not writing the definition as generally as he could have or should have?

My question is how Einstein can come to the conclusion that the synchronization depends on the relative velocity between the two clocks (as implied by the Lorentz transformation t'=γ(t-vx/c2)), when a) he hasn't even defined the synchronization procedure for two clocks in relative motion, and b) this seems to contradict his postulate regarding the invariance of the speed of light (which should imply that the relative velocity between the two clocks should have no impact on the light signal propagation time between them (and thus the synchronization)) .

Dale
Mentor
2020 Award
My question is how Einstein can come to the conclusion that the synchronization depends on the relative velocity between the two clocks (as implied by the Lorentz transformation t'=γ(t-vx/c2)), when a) he hasn't even defined the synchronization procedure for two clocks in relative motion,
I don't think that is necessary. Towards the end of section 1 he defines "the time of the stationary system" purely in terms of clocks at rest in the stationary system. With that you can follow all of his reasoning in the remaining sections without separately or explicitly defining synchronization between relatively moving clocks.

b) this seems to contradict his postulate regarding the invariance of the speed of light (which should imply that the relative velocity between the two clocks should have no impact on the light signal propagation time between them (and thus the synchronization)) .
That isn't what the second postulate implies at all.