fxdung said:
Which books say about SR in curvilinear cordinates?
Most GR books will discuss the tensor math you need in one form or other. Rindler's book "Relativity, special and general" comes to mind - not because it's such a fantastic book, but because it's a bit less advanced than some of the full-blown GR treatments. But this would be overkill for your problem, if you just want to understand the Sagnac effect. You don't need curvilinear coordinates or tensors for that.
The first thing to realize is that in order to measure velocities fairly, most particularly when you're measuring the speed of light, you need some concept of how to synchronize clocks. To take an example, suppose you're trying to find the speed of an airpline flying from Chicago to Los Angeles and back again, and you use daylight savings time to time the flights. You find the flight takes half an hour or so in one direction, and six hours in the other. So if you weren't careful, you might claim that the speed of the plane was much faster going from Chicago to LA than it was going back. Unfortunately, this would basically be wrong-headed. I hope this is obvious, but I suppose we could talk about it more if it isn't.
The particular issues with light are that it travels very fast, so a small synchronization error shows up as a large "velocity error" in the calculation. The other issue is that you might twig to the clock synchronization issue with the planes by carrying a clock onboard the plane and measuring your trip-time, but you can't pull that particular trick with light.
Thee question then becomes up - how do you synchronize clocks on a rotating platform. You might not think about this issue, and just assume that "surely, there's some way of doing this", and assume that's that's part of some hypothetical "rotating frame of reference" that does this for you. Unfortunately, this turns out to be a bad assumption. If you're careful to synchronize your clocks fairly using the Einstein convention, you find that all clocks on a rotating platform can not be synchronized. If you label 6 clocks around the periprhery, say A, B, C, D, E, F, you can syncnronize A with B, B with C, C with D, D with E, E with F, but when you do this, F is right next to A, as you've goon around the loop, but F is NOT synchyronized properly with A.
This is what the Sagnac effect is all about - it's really got very little to do with velocities, and a lot more to do with clock synchronization, something that people tend not to think about much if they are not familiar with SR - and sometimes even when they are.
