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!)