Clocks ticks keep track of the natural passage of time in whatever frame they are in. However, the starting point for counting those ticks may be arbitrary. Unless clocks at different locations are synchronized, they are useful for only local activities.
Clock setting in SRT is a critical but straightforward task in inertial frames. Einstein proposed one method in his 1905 paper. Send a light pulse from the master clock to a distant slave clock, where it initializes that clock and is reflected back to the master. The round trip time is measured, then the second clock is notified to adjust its setting by half the round trip time. This is called Einstein synchronization.
A second method involves bringing the master and slave clock together and synchronizing them, then taking them back to their intended locations. This is called ‘slow transport’ because the movement speed is very much less than c, so slow that time dilation is negligible.
As we watch from the train frame, the train conductor walks from the caboose to the locomotive, synchronizing clocks in the cars as he goes. He returns to the caboose, and all clocks he passes agree with his.
Einstein synchronization would work as well, had he a clear path through the cars for his light pulse.
But we’ve noted from the lab frame that the moving train exhibits Lorentz contraction, mass increase, and time dilation. It also exhibits relativity of simultaneity, with clocks at the locomotive reading earlier than clocks at the caboose. The conductor has synched his trains clocks, but from the lab frame we see a spread of times, with events happening first in the caboose, then flowing up to the locomotive.
Seen as the contracted train passes, the clocks in the caboose read later than the locomotive's.
The train is travelling at .866 c, and one time unit long in its own frame, here Lorentz contracted by the Lorentz factor of 2. The animation has clock rates slowed for clarity, but relative times are to scale for .866 c.
All well and good for straight line travel, but Ehrenfest is about going around in circles. How do we synchronize clocks going .866 c on the rim of a disk?
You can’t use Einstein synchronization. Sagnac demonstrated that the time of travel around the circumference of a rotating disk depends on the direction of the light beam, co- or contra-rotation, so the speed of light measured in a rotating frame is not constant, much to the consternation of relativists. The reason is obvious from the fixed (non-rotating) frame – the starting/ending point has moved while the light pulse is travelling around. Within the rotating frame, that point hasn’t moved, but that points out a difference between inertial and rotating frames. Unlike inertial frames, where any frame is as valid as another, in rotation there exists a ‘preferred’ frame – the inertial frame in which the center of rotation is fixed. Rotation rates, the frame’s equivalent of ‘speed,’ combine with simple addition, not with the Einsteinian ‘addition of velocities’ formula, and there is no lightspeed-like limitation on rotation rate.
The rotating frame is therefore not relativistic, though it may retain some relativistic traits like the time dilation and mass increase demonstrated in particle accelerators. Given that, can we use ‘slow transport’ to synch clocks?
We know that clocks at different radii run at different rates, so synchronization of clocks is radius dependent. We’ll deal with clocks on the disc rim moving at .866 c.
There are three scenarios mentioned earlier.
1. The circumference contracts, yielding a reduced value for pi.
2. The measuring rods contract, yielding an increased value of pi.
3. Lorentz contraction doesn’t occur in rotation, keeping pi at 3.14…
The Lorentz factor for .866c is 2, so in case 1, the rim circumference will contract to pi*r, in case 2, contracted measuring rods will yield 4 pi*r, and in case 3, we have the non-relativistic normal value of 2 pi*r.
In the earlier example above, we’ve seen how synchronized clocks in a contracted train frame appear evenly unsynched in the lab or fixed frame, and that this spread of time comes as part of contraction.
Let’s assume both contraction and valid slow transport, and have a rim-riding conductor travel around the circumference setting clocks.
He synchs his watch with a master clock on the rim, then as he travels in the co-rotation direction, we see his watch and the clocks he sets running progressively earlier than the master clock. Here is a significant difference between rotating and non-rotating frames. Where in an inertial frame the train will run off toward infinity and the spread of time will have no consequences on board the train, in the rotating disk, the conductor will eventually come upon the starting point, the locomotive will be nudging the caboose.
In accordance with our formula of v/c time unit change per contracted distance unit, the conductor will now find his watch either 2.7 or 10.9 time units earlier than the master clock, depending on the case 1 or case 2 scenario. If the conductor reverses course and returns the way he came, travelling contra-rotation, he will find all clocks he passes still in synch with his watch, and arriving back at the master clock, again still in synch. Should he continue contra-rotation for another circuit, he will find his watch unsynched the same 2.7 or 10.9 time units, only now unsynched later instead of earlier.
This conflicts with Einstein’s simultaneity relationship, where if clock A is synchronized with both clock B and clock C, then clocks B and C will be synchronized with each other. This means that either clocks cannot be synchronized in the rotating frame, or that the time spread doesn’t occur, and therefore we cannot have Lorentz contraction.
Of course, rim clocks can be synchronized on the disk. A light pulse from the center of the disk will reach all rim clocks simultaneously, and no reflected pulse is required, or indeed even useful, since the center and rim clocks run at different rates. Indeed, if the master clock is placed on the rim, a center clock may be synchronized to its dilated tick rate, and negatively (earlier) offset by the round trip light time, so that all rim clocks may continuously read and synch to the master by observing the center clock.
Because assuming either contraction/time spread scenario produces non-sensical results, we look at slow transport. To produce anything other than good synchronization around the rim would require the slowly transported clock to drift one way or the other from the continuously synched center clock, and then the only clock out of synch would be our transported clock. Again, a non-sensical outcome.
So now that we have our rim clocks running in synch, let’s put that in our tool kit and look at Einstein’s take on stellar aberration.