# The definition of synchronized clocks

1. Jan 5, 2014

### WannabeNewton

The "definition" of synchronized clocks

This is a purely terminological question. Consider the following setup: we have two inertial observers A and B at rest with respect to one another each equipped with a clock (clock A and clock B respectively), a means of exchanging light signals between one another, and each equipped with a profuse of meter sticks. Using the meter sticks, the distance between A and B is established as $L$. Furthermore, let $\gamma_A$, $\gamma_B$ be the worldlines of A and B respectively, with $t_A:\gamma_A \rightarrow \mathbb{R}$, $t_B:\gamma_B \rightarrow \mathbb{R}$ the local time (i.e. proper time) indicated by the hands of clocks A and B along the respective worldlines of A and B. We assume that the clocks are ideal i.e. isochronous.

Let A emit a light signal at an event $p$ on $\gamma_A$ with local time $t_A$ that is received at an event $q$ in space-time whereupon the signal is reflected and arrives back to A at an event $p'$ on $\gamma_A$ with local time $t'_A$.

We define clocks A and B to be synchronized if, when $q$ is an event on $\gamma_B$ with local time $t_B$, we have $t_B = t_A + \frac{L}{c}$. If $t_B \neq t_A + \frac{L}{c}$ then B can instantaneously readjust the hand of clock B so as to read $t_B = t_A + \frac{L}{c}$. Thus clocks A and B can always be synchronized (that this is a consistent method of synchronization was proven by Weyl, where consistent means symmetric and transitive). This is of course Poincare's operational procedure of clock synchronization. See the following paper: http://www.dma.unifi.it/~minguzzi/salamanca.pdf

But to what extent is this operational procedure of clock synchronization congruent to the intuitive notion of synchronized clocks? What I mean by this is, we normally think of clock A and clock B as being synchronized if whenever an event $q$ on $\gamma_B$ is simultaneous with an event $p''$ on $\gamma_A$, clock B's hand at $q$ is in the same position as clock A's hand at $p''$ i.e. $t_B = t''_A$. But this presupposes a notion of simultaneity that Poincare's operational procedure of clock synchronization does not. So if A and B use Poincare's synchronization procedure then in what sense are clocks A and B "synchronized" if no convention of simultaneity is defined?

Consider for example the following passage from the text "Concepts of Simultaneity: From Antiquity to Einstein and Beyond"-Jammer:

"As stated in the beginning of this chapter, we define 'simultaneity' as the 'temporal coincidence of events.' Assuming we have two clocks we regard the coincidence of their hands with certain numbers on their dials as an event. Then we can say that the clocks are 'synchronized at a certain moment of time' if and only if at that moment of time the hands of the two clocks are in the same position, that is, they indicate the same time... If the positions of the hands of the two clocks are the same at a certain moment of time they constitute two, generally separated, simultaneous events. It is therefore clear that the notion of 'synchronism' involves or presupposes the concept of 'simultaneity.'" (p.13)

Immediately we see a problem because the phrases "at a certain moment of time" and "at that moment of time" have no meaning whatsoever as given in the above passage; there is no absolute global time function available for us to make any sense of such phrases. Therefore at best one can interpret the definition of synchrony given in the passage for clocks A and B as stating that clocks A and B are synchronized if whenever an event $q$ on $\gamma_B$ is simultaneous with an event $p''$ on $\gamma_A$, we have $t_B = t''_A$ which is exactly the "intuitive" definition of synchrony stated above. Therefore the "intuitive" definition of synchrony only makes sense if we have a convention of simultaneity to begin with.

Now if we adopt the $\epsilon = \frac{1}{2}$ convention of simultaneity, that is, $t_A'' = \frac{1}{2}(t_A + t'_A)$, and consider the "intuitive" notion of clock synchronization then clocks A and B are synchronized if clock B's hand at $q$ reads the same time as clock A's hand at $p''$ meaning $t_B = t_A + \frac{L}{c}$ which follows trivially from the two-way speed of light. In other words, if we adopt the $\epsilon = \frac{1}{2}$ simultaneity convention and A and B want to synchronize their clocks in the sense that clock A's local time and clock B's local time are equal at simultaneous events, then A and B can arrange for this explicitly by using Poincare's operational method of synchronization.

But if we have no notion of simultaneity available then, as asked above, what does it even mean for clocks A and B to be "synchronized" after A and B employ Poincare's operational method of synchronization? What does it even mean to call this a synchronization convention?

I ask this terminological question partly because of the paper I linked above and partly because of the way some relativity texts make it seem like simultaneity and clock synchronization are equivalent from a physical (intuitive) standpoint.

For example, Friedman states in his text "Foundations of Space-time Theories: Relativistic Physics and Philosophy of Science" that "Our problem is to synchronize the clock at A with the clock at B, to say when, according to A-time, the light signal arrives at B. In other words, we must determine which event between $t_A$ and $t'_A$ at A is simultaneous (in F) with $t_B$." (p.166)

Clearly a definition of simultaneity allows for us to adopt the "intuitive" definition of synchronized clocks and subsequently derive an operational means of actually synchronizing clocks in accordance with this "intuitive" definition of synchronized clocks.

But what about the converse? That is, does it even make sense to define some convention of clock synchronization, such as Poincare's, and then use this convention to define simultaneity of events, when it doesn't even make sense to talk about synchrony intuitively without simultaneity being defined in the first place? How should one interpret the term "synchronization" if such a procedure is adopted?

Sorry for being verbose and desultory but this terminological detail has me confused. Thanks in advance.

Last edited: Jan 5, 2014
2. Jan 5, 2014

### PAllen

To me, it seems given a definition of simultaneity, the corresponding notion of clock synchronization is a consequence: synchronized clocks must read the same time for a pair of events that are simultaneous.

Equally reasonable to me is the reverse framework: given a defined clock synchronization, we define events to be simultaneous per that synchronization if synchronized clocks read the same time for them.

I don't see any reason to favor one direction of definition over another. I also don't think you can plausibly define both independently - either defines the other.

3. Jan 5, 2014

### ghwellsjr

I don't see why we need to include the length of a rod in our understanding of synchronization of remote inertial clocks when the standard for length is based itself on the speed of light. What's wrong with the simple concept that two clocks are synchronized if one sends a signal to the other and the other responds immediately with the time on its clock and when the first receives that signal it takes the average of the sent time and the received time and if it matches the others responded time message, then they are synchronized? Isn't that Einstein's convention that everyone already agrees with?

4. Jan 5, 2014

### The_Duck

I'm not sure what your worry is exactly. Why wouldn't it make sense to define simultaneity in terms of a clock synchronization procedure? Poincare's procedure defines both what it means for two clocks to be synchronized and what it means for two events to be simultaneous. You don't have to have any preexisting intuitive notions to carry out the procedure. You just do it. Then you decide that what you *mean* by "clocks A and B are synchronized" is "I carried out the Poincare procedure on clocks A and B." Then you decide that what you *mean* by "events X and Y are simultaneous" is that you have a pair of synchronized clocks, one clock is present at event X, one clock is present at event Y, and the time that the first clock shows at event X is the same as the time the second clock shows at event Y.

5. Jan 5, 2014

### WannabeNewton

But how is one to understand clock "synchronization" without a notion of simultaneity present? If I use some operational procedure that claims two clocks are "synchronized" if I do such and such then in what sense are they "synchronized"? Is "synchronization" in such a context to be interpreted solely as what the synchronization convention defines, with no availability of the simultaneity dependent "intuitive" definition of synchronized clocks that Jammer defines in the quote above?

Well this is the synchronization convention that the paper I linked defined, and is the same one that Poincare used historically when investigating simultaneity, and it was the definition in the paper that prompted my question.

6. Jan 5, 2014

### WannabeNewton

Yes I have no problems with this at all. My problem is with dichotomy between the use of the terminology "synchronization" as you have used it (and as I have understood it from a purely operational standpoint) with that of Jammer's: synchronized clocks are those whose hands are positioned the same at simultaneous events (see quote in post #1).

In other words, since Poincare's synchronization convention is trivially equivalent to the usual simultaneity convention but has no intuitive meaning a priori in the sense described in the preceding paragraph, why do we use the term "synchronization" to label this convention? The terminology confuses me because almost subsequently in various relativity books, "synchronization" is used in the intuitive sense to mean what Jammer defines as clock synchrony, which depends on simultaneity.

For example, a relativity book would say two uniformly accelerating observers who "synchronize" their clocks at the simultaneity slice $t = 0$ of one of the observers to read $t_A = t_B =0$ will find that their clocks desynchronize, that is the readings on the clocks disagree, at a later simultaneity slice of said observer.

Thanks for the replies guys! Like I said this is a purely terminological issue for me.

7. Jan 5, 2014

### pervect

Staff Emeritus
The important thing from my viewpoint is that Einstein's clock synchronization scheme is the unique clock synchronization in which equal masses, moving in opposite directions at the same velocity, come to rest when they collide.

This is assuming that velocity is operationally defined by dividing the distance an object travels by the time interval, where said travel time is just the difference between two synchronized clocks that are at rest at two points along the object's path. This is basically the standard defintion of velocity in a standard "frame" of reference.

I have used the concept "at rest", but I don't think it's terribly problematic, "no doppler shift" is a physical means of verifying the "at rest" condition.

To be complete about my assumptions, I assume that we have the concepts of "at rest", "straight lines", "equal masses", and "distance". Given that we know what all of these mean, then we have all we need _in principle_ to experimentally distinguish the isotropic Einstein clock synchronization from alternate non-isotropic non-Einstein clock synchronizations is to collide equal masses with equal velocities moving along a straight line in opposite directions.

For a given physical setup, one where the masses come to rest after the collision - if we change the clock synchronization, we change the velocity (as we have defined it) - but we don't change the actual physics, the physics doesn't care whether or not our clocks our synchronized. So when we change the clock synchronization, the two objects collide and come to rest, but their velocities are no longer equal when this happens.

I should add that it's obviously necessary for equal masses moving in opposite directions at the same velocity to stop in order for Newton's laws (as generally understood at the high school level) to work properly. So while clock synchronization is a "convention", it's required to use the right convention (Einstein's, or some equivalent method) if you wish to apply Newton's laws.

To try to put it another way, if your clocks aren't Einstein synchronized, you don't have a "valid frame of reference" in the Newtonian sense.

8. Jan 5, 2014

### ghwellsjr

You started off with two mutually at-rest inertial observers for which the definition of simultaneous and synchronous applies and now you're switching to accelerating observers for which it doesn't apply. Is this the reason for this thread?

In his 1905 paper (near the end of the first article), Einstein equated the definition of the terms “simultaneous” and “synchronous”:

Are you trying to find a difference between them?

9. Jan 5, 2014

### WannabeNewton

Sorry I should have been more clear. I was using that as an example of where a text defines two clocks at rest with respect to one another as "synchronous" if the readings on the clocks are the same at simultaneous events as determined relative to one of the observers.

I wasn't trying to find a difference between them but rather asking if there actually is a difference between the terms as used in standard language. Jammer's book was in fact distinguishing the two, while acknowledging that they are intimately related, which is why I got confused, especially when the book said that synchrony "presupposes" simultaneity.

But your comment cleared it up for me again. Thank you very much George, I appreciate it :)

10. Jan 5, 2014

### PAllen

You can invent arbitrary simultaneity conventions, especially in GR. All I am saying is that given a simultaneity convention, there is an implied clock synch; and given an operational clock synch, you also have a simultaneity convention.

Note, I agree with Pervect that for initial observers in SR, there is only one reasonable simultaneity convention. However, for all other cases (noninertial, GR) every convention involves tradeoffs.

Last edited: Jan 5, 2014
11. Jan 5, 2014

### WannabeNewton

Thanks PAllen. Maybe if I explain in a bit more detail what I meant in my verbose OP, it will be clear to you what my confusion is with regards to the terminology. Here's the dictionary definition of synchronous: http://www.merriam-webster.com/dictionary/synchronous

Let's say I define the clock synchronization convention $t_B - t_A = t'_A - t_B$. Now when I see the phrase "clock A is synchronized with clock B" I think "if clock A's hand is positioned at say 12:00 AM then clock B's hand will also be positioned at 12:00 AM". However I can't actually prove this just by using the clock synchronization convention $t_B - t_A = t'_A - t_B$. In fact by defining operationally what it means for clocks A and B to be synchronous and then defining two events to be simultaneous in the vicinities of synchronized clocks A and B if clocks A and B display the same time at the respective events, I'm basically saying: "I'm going to give you this arbitrary operational procedure to adjust your clock and once you carry out this operational procedure, I'm going to say by definition that your clock is 'synchronous' with mine. Now that your clock is 'synchronized' with mine, if at some event in your vicinity your clock displayed 12:00 AM and at some event in my vicinity my clock displayed 12:00 AM then by definition these two events 'happened at the same time' i.e. simultaneously."

In other words, given an arbitrary operational procedure to "synchronize" a pair of clocks, one cannot prove just from this operational definition of clock synchrony that the pair of clocks is synchronized according to the dictionary definition. We just have this abstract operational definition of what it means for two clocks to be "synchronous". But we can give it concrete meaning by further adopting the convention that if two clocks are made "synchronous" according to this operational procedure then an event in the vicinity of one clock occurs simultaneously with an event in the vicinity of the other clock whenever the hands of both clocks are positioned identically; this is a convention we must accept once we arbitrarily define synchrony operationally, it is not something we can prove from the arbitrary operational definition of synchrony.

Would you agree with all that? This is why I was so confused by Jammer's discussion of the terminological aspects of the words "synchronization" and "simultaneity". He made it seem like one is a provable consequence of the other. This is basically why I opened the thread.

I agree as well that the $\epsilon = \frac{1}{2}$ simultaneity convention is the most "reasonable", if by "reasonable" you mean the most natural, since it only depends on the space-time metric and it makes the laws of physics look the simplest. Malament had a pretty convincing argument of this (in my opinion), if you're interested: http://plato.stanford.edu/entries/spacetime-convensimul/#4

12. Jan 5, 2014

### ghwellsjr

I would agree that the definition of both "synchronization" and "simultaneity" is non-provable and that they are merely restatements of Einstein's non-provable second postulate--that light propagates at c in any inertial frame.

It's only "reasonable" after Einstein came up with the idea that time (and space) can be relative. It certainly wasn't "reasonable" prior to Einstein. In fact, Einstein pointed out that his second postulate appeared to be irreconcilable with his first postulate, the Principle of Relativity. It really is more "reasonable" that light propagates at c only in one inertial frame. There is no way to prove how light propagates. That's why we need a postulate and the equivalent definition of "synchronization" and "simultaneity".

13. Jan 5, 2014

### PAllen

I agree with all of the above. I didn't like Jammer's discussion.

14. Feb 26, 2014

### JVNY

Here are a couple of thoughts that follow from this thread and others on similar topics. The dictionary definition that you link in post 11 includes coherence ("recurring . . . at exactly the same periods"), but I will separate coherence from synchronization.

(1) Two ideal clocks of the type you define in the original post can agree on simultaneity but not be coherent or synchronizable. An example discussed briefly above is clocks undergoing Born rigid acceleration. Two such clocks are at rest with respect to each other and agree on simultaneity. But they cannot be synchronized. Their tick rates are not coherent, so even if you set their hands to the same position simultaneously in their frame, the hands will subsequently diverge in their frame.

(2) Conversely, it appears that the ideal clocks can agree that they are coherent, but disagree on simultaneity. Consider clocks on the edge of a rotating rim that can only send signals to each other along the rim. These two clocks will agree that they are coherent, per post 39 here: https://www.physicsforums.com/showthread.php?t=738656&page=3. However, they will not agree on simultaneity ("light signals along the rim to and fro observers . . . will not give rise to a valid global time coordinate for the family of observers on the rim." See post 57 here: https://www.physicsforums.com/showthread.php?t=738656&page=4).

But intuitively the concepts of synchronization and simultaneity still seem to be linked. In the first example, one can make the clocks coherent and synchronizable by making the rear clock non-ideal, ticking faster than its proper time proportionate to its distance from the front clock.

In the second example, it seems intuitively hard to separate the identical periods of the two clocks as determined on the rim from simultaneity as determined on the rim (without reference to a different frame, such as the inertial frame of which the axis is a part). If the periods are the same, it seems intuitive that there should be a nonzero probability that the two clocks tick at the same time as determined on the rim. If one puts a nearly infinite number of clocks side by side on the rim in a row perpendicular to the direction of rotation, and starts the clocks ticking serially along the row each a fraction of a second apart from its neighbor, then it seems that there is a high probability that one of them ticks at the same time (as determined on the rim) as another clock located elsewhere on the rim.

15. Feb 26, 2014

### bcrowell

Staff Emeritus
We call it synchronization because in the low-velocity limit, it agrees with the methods of synchronization that had been used already for centuries. The Lorentz transformation has a term like $t'=\ldots+\gamma v c^{-2} x$, and this term is the relativistic effect that makes different observers disagree on the simultaneity of events. In the limit $v\ll c$, this term becomes negligible, synchronization becomes frame-independent, and Einstein synchronization gives the same result as other, traditional procedures.