# I A clock’s double life?

#### pervect

Staff Emeritus
To all,
I noticed that no one denies that the printing of the front clock should be ahead of the printing of the rear clock, except that it is hard to define the proper frame for it.
I think there's an easier way to look at it. If the clocks all accelerate in a symmetrical manner in the station frame, we can say that they all print out exactly the same time.

Then it is a simple matter, if one is familiar with the relativity of simultaneity, to say that this directly implies that the clocks are NOT syncrhronized in the train frame.

I'll provide a reference to where Einstein mentioned this. Now, if you don't quite follow Einstein's argument, you would be far from the first. But at least you might have some inkling of where the rest of us are coming from, and what the issue is with your analysis.

The link is https://www.bartleby.com/173/9.html, and the reference is Einstein's book, "Relativity: the special and general theory", chapeter IX, 'The relativity of simultaneity'.

Einstein said:
 UP to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: 1​ Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative.
So there you have it. BECAUSE the events on two neary cars are simultaneous in the station frame (when we assume symmetry), it follows, from Einstein's argument, that they can't be simultaneous in the train's frame.

The "train's frame" is only defined in the limit as the two cars are very close to each other, which has caused some discussion. But after taking said limit, there's nothing at all ambiguous about the train frame - it's moving at some velocity v along the track, and because of the effect Einstein mentioned, the fact that the clocks are synchronized in the station frame implies they cannot be synhronized in the train frame.

#### PeterDonis

Mentor
I noticed that no one denies that the printing of the front clock should be ahead of the printing of the rear clock
"Ahead" is frame dependent. The invariant is what gets printed on each clock's receipt. The analysis in the SF that I already gave answers that question.

every single clock on the train has two different readings in the sequential frames it is acting in (as a front clock and as a rear clock)
I have no idea what you mean by this. The reading on a particular clock at a particular point in spacetime is an invariant. That particular point in spacetime will in general be labeled by different coordinates in different frames. But the invariant is the point itself and the reading of the clock at that point.

I think you would greatly benefit from forgetting all about coordinate-dependent quantities and focusing on invariants. Otherwise you will just continue to confuse yourself.

#### PeterDonis

Mentor
A single carriage is going round the track, equipped with clocks at the front and rear - the worldlines of these clocks are marked in blue.
Yes, and what you are basically drawing here is the start of a system of Fermi Normal Coordinates centered on the chosen carriage. The "simultaneous lines" are the surfaces of constant coordinate time, and the worldlines are the "grid lines" of the time coordinate.

All I was suggesting was that you can chain this process together along the train
As I understand the OP, the train is supposed to go all the way around the circle, so it's impossible to cover the whole train this way. But I think you could cover a portion of the train this way, yes.

Also, I'm not sure that the chaining process you describe, within the ange you can extend it, will be exactly the same as Fermi Normal Coordinates centered on the chosen carriage. The simultaneity conventions might not be the same when extended beyond the initial chosen carriage; I would have to look at the detailed math.

#### pervect

Staff Emeritus
A few more comments on the topic. If we assume that all the train cars are accelerating unfiormly (one way of doing this would be to assume they all start acclelerating with the same proper acceleration at the same time , "same time" being defined in the station frame, then in their own instantaneous frame, the nearby cars will get further apart as time goes on, in addition to the loss of Einstein synchronzation that has already been discussed.

It's somewhat similar to Bell's spaceship paradox, except to make life a little more complicated it's on a curved track rather than a straight one.

It's easiest to analyze the case of a straight track than a curved one.

In the case with a straight track, if the lead spaceship accelerates at 1 light year/year^2 (roughly one earth gravity) , when the rear spaceship accelerates at the same rate, it does not keep a constant distance behind the lead ship in the ship frame.

The rear ship has to accelerate harder at roughly 1.1 light years/year^2, to keep a constant distance away from the lead ship in the ship frame.

This also implies that if the two ships are to stop when they have the same velocities, they accelerte for different time periods, illustrating some physical consequences to the loss of synchronization we were talking about earlier. If the lead spaceship accelerates at 1 g for one month, the tailing spaceship accelerates at 1.1g for .9 moths. So obviously, their clocks do nor remain synchronized

#### Ibix

Building on my earlier post, I've constructed Minkowski diagrams for a train circling a track at constant speed. Unfortunately my arithmetic must have gone wrong somewhere because the train has a small gap between front and rear. I don't think this makes much difference to anything.

Again, the diagram is drawn in the rest frame of the circular track, the worldsheet of which is the translucent cylinder. Pale blue lines show the worldlines of clocks distributed evenly along the train (and you can see the small gap because the last line doesn't quite line up with the first), with red dots marking their ticks. One set of "simultaneous" red dots are joined by a green line - simultaneous, here, means in the sense that a clock is Einstein synchronised to its neighbours in the inertial frame in which they are instantaneously at rest. I've also added a white line, which is the worldline of an observer at rest in the track frame.

Note that if these clocks are equipped with printers and print at the green-connected times, then in the track frame they print one after the other. Thus the movement of the train means that when the last-to-print clock prints, it isn't in the same place as the first-to-print clock was when it printed, and the ends of the dot pattern thus printed overlap.

The above is not the scenario outlined in the OP. In the OP, all of the printers print simultaneously in the track frame (in contrast to earlier threads by the same poster). I've illustrated that circumstance below.

Note that this diagram is identical to the above, except that I deleted the green line and added an orange line that marks simultaneity in this different sense. If the clocks print at the time indicated by the orange line then there is no overlap and the dots printed on the track are uniformly separated (except for the one small gap, in this diagram). The orange line lies in a plane of simultaneity for the trackside observer, but note that it does not pass through a line of red dots. Thus observers on the train would not regard the printers at opposite ends of a carriage as printing simultaneously, assuming they adopt their instantaneous rest frame's simultaneity convention (life gets complex beyond the end of a short carriage because we haven't truly defined a generally applicable simultaneity convention for the on-train observers).

It has to be stressed that these two diagrams are (very nearly) identical, representing identical physical situations in terms of trains on the track. The only difference is the rule we use for defining "at the same time" in the sentence "the printers print at the same time". And the rule to use is a matter of choice by the experimenter.

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#### Nugatory

Mentor
Unfortunately my arithmetic must have gone wrong somewhere because the train has a small gap between front and rear.
I presume you used a computer in an interative loop? If so, I'd bet long odds that you're just seeing accumulated rounding errors.

#### Ibix

I presume you used a computer in an interative loop? If so, I'd bet long odds that you're just seeing accumulated rounding errors.
Turns out to be too many tools in my workflow. I simplified a bit - still don't know quite what I did wrong, but the numbers going in to the generation process were wrong. Corrected results are below:

No comments to add to my previous, except now you can see that the front of the train is coincident with the rear. Using the green "chain of local Einstein synchronisations" makes the pattern overlap due to the motion of the train while the printing happens. Using the orange "synchronised in rest frame" criterion for printing makes the pattern even, but adjacent clocks do not print simultaneously according to an observer in the carriage.

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#### Ibix

My last post with Minkowski diagrams, as no-one seems to be responding (no disrespect to Nugatory intended).

First of all, here's a (very long) train on a (very long) straight track. It is initially stationary and then accelerates to a constant speed. The acceleration profile is chosen so that the ends of the carriages are the same distance apart in the track frame (thus they are moving apart in their own frame).

The red line represents the front carriage and the blue line the back one. This is actually the standard Bell's spaceships configuration, just with eleven carriages instead of two ships. Clocks in the carriages tick every half-year of proper time, and these events are marked. The clocks are initially synchronised and remain synchronised with one another in the track frame, although they tick slower than coordinate time clocks - there are only ten ticks in six years. Acceleration terminates at $\tau=3$, which is about $t=3.25$

Next, here is a diagram of a train accelerating so that the distance between the carriages remains constant for observers on the train. It's fairly easy to see that this requires the train observers to be a family of Rindler observers with shared horizons.

As before, clock ticks are marked with crosses. They are initially synchronised in the track frame but, due to different velocity profiles, they de-synchronise during the acceleration. Because the requirement is that the final velocity be the same along the train, the carriages stop accelerating at different times - these are marked with red crosses. Note that, in the way I've chosen to set this up, the blue clock behaves identically to the blue clock in the first diagram. You can also see that the train is undergoing length contraction.

It's also interesting to see the above in the final rest frame of the train:

You can see that the carriages stop accelerating simultaneously in this frame, but their clocks are not synchronised. You can also see that the separation of the carriages after the acceleration is, in this frame, the same as it was before in the track frame, as promised.

Finally, why am I talking about this linear track? It's because the surface of a cylinder has the same geometry as a plane, albeit with a different topology. So you can simply wrap these diagrams into cylinders with the t-axis parallel to the cylinder axis to get cylindrical diagrams like the ones I posted before. Or one can imagine slitting the cylinders parallel to their axis and spreading them out to get flat diagrams.

Because the original problem specification was that the train exactly fitted around the track, in either of the first two diagrams we can simply cut off everything to the right of $x=2$ and paste it on to the left to get a Minkowski diagram of the "unwrapped" cylinder. So at the start of the experiment the red and blue lines are coincident, and anything that passes $x=2$ moves to $x=0$. Here, then, is the version keeping constant acceleration in the track frame:

The red and blue lines are initially coincident and remain coincident (this has been rendered as black by the colour-combining process I used). You can see that the clock ticks always remain simultaneous in the track frame, as they must from the symmetry of a situation where every clock does the same thing in this frame.

What about the case where the train is free to keep its natural length? That looks like this:

Here we can see that a gap opens up as the train length contracts and the clocks de-synchronise.

In both cases, if the clocks were re-synchronised after the acceleration, their "same time" ticks would look like the red events - non-simultaneous in the track frame. Note that the string of red events goes all the way around the cylinder and a little bit more, so if these were used to time printing then the end of the pattern would overlap the beginning.

So here's the takeaway message: the clock synchronisation you get is your choice. You can re-synchronise your clocks by some procedure after the acceleration, or you can accept whatever you get from synchronising pre-acceleration. The results of either approach will differ depending on the exact physical situation (which is also your choice!)

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"A clock’s double life?"

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