The constancy of the speed of light imposes certain characteristics on objects moving inertially through another frame, as seen from that other frame:
- Time dilation
- Mass increase
- Lorentz-Fitzgerald contraction
- Relativity of simultaneity (RoS)
Particle accelerators have demonstrated time dilation and mass increase, but contraction and relativity of simultaneity, which by their nature aren’t easy to show, are necessary as complements to the first two effects in inertial frames.
Because a rotating frame is not inertial, and because the speed of light is neither constant nor a limit in a rotating frame, the contraction and RoS may not be required. RoS is the easiest to determine, so we’ll start with it.
Here’s Einstein’s train again, Lorentz-Fitzgerald contracted and with time-dilated clocks, seen as it passes the observer:
The clocks are synchronized in the train frame, but show RoS to the observer watching it go by, with the locomotive clock earlier than the caboose clock.
We now put a train on a circular track and make it long enough for the locomotive to come up behind the caboose. Once we get it running at relativistic speed, conditions on board are equivalent to those at the rim of a rotating disk.
The crew was having a lottery drawing in the caboose at noon, and the deadline for time-stamping the tickets was 11:50. If we assume relativity of simultaneity, we see the problem immediately.
The locomotive engineer can see the winning number drawn in the caboose and still have time to mark his ticket. Knowing the future and/or affecting the past aren’t allowed, so relativity of simultaneity can’t apply to a rotating frame.
We therefore have ordinary, everyday, plain vanilla simultaneity on the train. That requires synchronized clocks, but how can we synchronize them? The Sagnac experiment showed that we can’t use Einstein synchronization.
We can’t synch all the clocks across the face of a rotating disk because no two points in a rotating frame have the same acceleration or velocity. However, all the points at any given radius do have the same speed, and thus have the same time dilation. A light pulse from the disk center will reach all points at a given radius, such as the rim, simultaneously in the non-rotating frame of the disk center. Thus all points at that radius will be initialized together, and will run at their dilated rate in synch.
The slow transport method might also work here, as without RoS, there is little to suggest that the conductor walking the length of the train wouldn’t remain in synch with the caboose as he set clocks in the cars.
It is interesting to note that a rim-riding observer would see all clocks both interior to him and exterior in the non-rotating frame running at faster rates, quite the opposite of the inertial case where other clocks are seen to run more slowly.
The case against Lorentz-Fitzgerald contraction is more difficult, but with RoS, we’ve already knocked out one of the props it requires.