# Clock synchronization and relativity of simultaneity

1. May 14, 2015

### Fantasist

I addressed already recently in he issue of defining the synchronicity of clocks moving relatively to each other (considering that the synchronization by Einstein's method implies clocks at rest), but it occurred to me now that even for clocks at rest relatively to each other there is a problem with the practical definition as far as its frame dependence is concerned. On the basis of his two-way light signal propagation thought experiment, Einstein concludes at the end of paragraph 2 in http://www.fourmilab.ch/etexts/einstein/specrel/www/

So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.

Let us look a little bit closer at the thought experiment on which this conclusion is based. For this purpose, let us slightly modify it for clarity and assume each of the two clocks (stationary with regard to each other) sends out a light signal at the same time (according to each of the clocks). Halfway between the two clocks is a detector that registers both signals. We can now define that the two clocks in question are synchronized with each other if the detector registers them simultaneously (according to its own time). Let's further assume that in this case both signals are completely absorbed by the detector. On the other hand, if the signals do not arrive simultaneously (within a defined window) they are not absorbed but carry on to the other clock (where they can be subsequently detected).
Now with this practical definition of simultaneity, how can this possibly be frame dependent? The two signals are either absorbed or not absorbed. All observers would have to agree about this physical fact. So evidently, this setup could not experimentally define the relativity of simultaneity. The question is how do we have to change/generalize the setup so that it is consistent with Einstein's conclusion?

2. May 14, 2015

### A.T.

According to any frame. The detection of the signals is one event. the way you set it up. Relativity of simultaneity requires spatial separation of the events.

3. May 14, 2015

### Staff: Mentor

The length of the optical path to and from the detector is frame dependent. The simultaneous arrival and unblocking will be proof that the signals were not emitted simultaneously in other frames.

4. May 14, 2015

### Staff: Mentor

All observers in all frames will agree about whether the two light signals met at the central detector or didn't meet at the central detector.

What is frame-dependent here is whether the two light signals were emitted at the same time. If the two light signals do meet at the central detector an observer who is at rest relative to the central detector will find that the emission events were simultaneous, but observers moving relative to it will find that they were emitted at different times.

5. May 15, 2015

### Fantasist

The events here are the ticks of each of the clocks, so they are spatially separated. It is only that the according timestamps are analyzed in one place as to determine whether they are actually simultaneous or not.

Note that this setup is essentially the same as Einstein's, only that the light signal is not reflected but the second clock emits its own signal and the two are then timed at a single third location (thus avoiding the problem of getting the time at the reflector back to the originating clock).

6. May 15, 2015

### Fantasist

OK, but it was our definition that the original events are simultaneous if the signals meet at the central detector. How would you change this to make this observer dependent?

7. May 15, 2015

### Staff: Mentor

(You say "observer-dependent", although "frame-dependent" would be more accurate)

We don't need to. You've specified that the two signals are emitted simultaneously according to two clocks that are at rest with one another and synchronized in the frame in which they are at rest. This condition is equivalent to saying that the signals will meet at a centrally located detector also at rest relative to the clocks; so we have the frame-independent but uninteresting tautology that if two events are simultaneous in a given frame then they are simultaneous in that frame.

8. May 15, 2015

### A.T.

The tick events aren't synchronized in other frames. Only the detection events, because they a identical or just one event.

9. May 15, 2015

### Staff: Mentor

It is not the standard definition. Furthermore, it is a bad definition because the speed of light becomes frame dependent and anisotropic.

See my previous reply. The optical path lengths are frame variant. So if you fix the time by definition then you get a variable speed of light.

Last edited: May 15, 2015
10. May 16, 2015

### Fantasist

I did not say the two signals are emitted simultaneously but only 'at the same time' i.e when each of the clocks shows a given reading. The corresponding events are only simultaneous if the clocks are synchronized. And you need some procedure to define simultaneity/synchronization, like the one described by Einstein (comparing the timings of a reflected light signal) or my version with comparing the reception time of the two light signals by a detector halfway between the two clocks (you could make up any number of other ways of doing it).

It is not a tautology. The conclusion is (given that we have defined simultaneity as suggested above)

The light signals meet at the central detector <=> The clocks are synchronized

Now you said yourself that in this case the light signals meet at the central detector in any reference frame, but if the clocks are not synchronized in any reference frame, the above conclusion would become obviously false. Which means that we require a different procedure by means of which a moving observer can decide whether the clocks are synchronized or not.

11. May 16, 2015

### Fantasist

The question is how do you practically define the speed of light for a moving observer/reference frame? In the rest frame of the two clocks it is easy: if the clocks are synchronized and you know their distance, you can get the speed of light from the recorded timestamps of the emission and reception. But how would the timestamps be recorded in a different reference frame?

12. May 16, 2015

### A.T.

The other reference frame (where your emitters are moving) has its own resting synchronized clocks, which record the timings of the emission/reception events.

13. May 16, 2015

### Staff: Mentor

In any inertial frame it is easy. The speed of light is c.

I think what you really want to know is how can you define a simultaneity convention which is compatible with the speed of light being c. Einstein already did that.

14. May 16, 2015

### Fantasist

The only problem is that these events have nothing to do with the events we are interested in (which are related to the clocks/detectors defining the original reference frame).

15. May 16, 2015

### A.T.

They are the same physical events. Different frames merely assign different space-time coordinates to them.

16. May 16, 2015

### Staff: Mentor

@Fantasist Let me see if I can restate what I think you are trying to do.

Given a reference frame O and two clocks a and b both moving at speed v in O, you want to find some simple experiment to determine if those clocks are synchronized in O, not in their rest frame.

17. May 16, 2015

### Fantasist

The events we are talking about here are tied in to the clocks/detectors defining the reference frame (namely when these register the emitted light signal), and as you pointed out above yourself, these are physically different sets of clocks. So how could the associated detection events be physically the same?

18. May 16, 2015

### A.T.

Only the clocks resting in the original frame emit signals. The clocks resting in the other frame merely time stamp these emission events for the other frame.

Is DaleSpam's suggestion in post #16 what you are trying to ask?

19. May 16, 2015

### Fantasist

Essentially, yes. If we have an experimental procedure that defines the synchronization of the clocks a and b in their rest frame, then we must also have an experimental procedure that defines the synchronization of the same clocks a and b in O. Otherwise we would not be entitled to make any statement about their synchronization in the latter frame.

20. May 16, 2015

### Fantasist

That would mean you couldn't determine the speed of light in one frame without knowing the events defined by the clocks of the other frame.

21. May 16, 2015

### A.T.

You know their distance in O, so just place two clock resetters at rest in O that distance apart.

22. May 16, 2015

### Staff: Mentor

Sure. A new experimental procedure is not needed. Simply determine the "the time of the stationary system" O, as defined by Einstein, and then a and b are synchronized in O if they read the same value at the same time in O.

Remember, this is how we concluded the previous discussion.

Last edited: May 17, 2015
23. May 18, 2015

### Fantasist

Well, obviously, if the clocks read the same value they synchronize; the question is do they read the same value or not (given the assumptions)? Since the discussion seems to have lost focus of this question, let me address the problem again using Einstein's own definitions:

in paragraph 1 of his 1905 paper, he defines the synchronization condition in the rest frame of the clocks A and B with the help of the emission, reflection and return timings of a light signal in the form

(1) tA2 - tB = tB - tA1

( where I have used tA1 and tA2 instead of tA and tA' )

As one can imagine these timings being logged at each clock, they are therefore unambiguous and frame independent (as was agreed by everybody here).

Now in paragraph 2, Einstein considers the situation from a frame O moving with velocity v relatively to the clocks A and B, and claims that in this reference frame the timings would be

(2) tB - tA1 = rAB/(c-v) ; tA2 - tB = rAB/(c+v)

(where rAB is the distance of the clocks A and B in the frame where they appear to be moving).

Now as written, Eqs (1) and (2) are obviously algebraically inconsistent (unless v=0), so the timing variables in (2) can actually not be the same as those in (1). Strictly speaking he should have therefore written for instance

(2a) tB' - tA1' = rAB/(c-v) ; tA2' - tB' = rAB/(c+v)

where the prime indicates that timings are taken in frame O.

The question is where are these primed timings taken and logged? It can obviously not be the clocks A and B themselves as we have agreed they record the unprimed timings. But if they are recorded elsewhere, we have no right to associate these timings with the clocks A and B anymore. So one way or another one appears to run into contradictions here.

Last edited: May 18, 2015
24. May 18, 2015

### Staff: Mentor

So your complaint is about his notation? OK. Yes, in (1) the times refer to the proper time on clocks A and B whereas in (2) the times refer to the coordinate time in the stationary system at clocks A and B.

Since he clarifies what each means in the text this does not seem to be a substantive complaint.

He very clearly describes that in the text. The clocks A and B in section 2 display "time of the stationary system".

They are synchronous in the stationary system by definition, but the experiment of section 1 shows that they are not synchronous in the moving system.

Last edited: May 19, 2015
25. May 23, 2015

### Fantasist

Sorry, but I don't see how you can get to this conclusion. The timings in the stationary system are based on stationary clocks/detectors and how the path of the moving clock and the light signal is registered by these. The synchronization of the moving clocks in their own system does not come into it at all. They could be synchronized or de-synchronized as you want, the timings in the stationary system are not affected by this (as you agreed above, the corresponding timing variables should actually be different variables to those representing the timings in the moving frame).