Does the speed of light in different mediums affect the Lorentz transformation? We know that the Lorentz transformation is derived using the speed of light in a vacuum. But if we were to use it in water, would it change since the speed of light in water should also remain constant for every observer.
 A: No. The speed of light is always c. And only that is constant to every observer. You use only c in Lorentz transformation. In some medium light would appear to slow down because it would jiggle electrons of atoms in that medium which would generate electromagnetic waves themselves. The combined result is a wave traveling slower in a medium. You can see it classically (waves) or quantum-mechanically (photons), but the result is the same in this case. The resultant wave moves slower and is not constant to every observer. There are experiments where light slows down to snail pace and we certainly do not observe any weird time dilation effects in this case.
I don't know if it appropriate to link a youtube video here, but the video below sheds some "light" on to the issue using only simple words (no complex equations).
http://www.youtube.com/watch?v=CiHN0ZWE5bk
A: The medium (the water or whatever else) has a preferred frame, that is its frame of rest. In that medium, the speed of light is not the same for all observers and it doesn't have to be because there is a special restframe now. In vacuum there is no such preferred restframe, that being the root of the principle of relativity.
A: I doubt that the speed of light in water, is constant for every observer.
If a given velocity is constant when observed across any reference frame, it also means that velocity is an absolute speed limit. No one travels faster than it, and no actual body is observed to travel faster than it.
There are instances of electrons travelling faster than light in a given medium. See Cherenkov radiation. I imagine something similar would also apply to ordinary water.
If it can be proven however that some velocity is an absolute speed limit in a given medium (eg water), I think it should be possible to restructure the Lorentz transformation around that, so velocities change nonlinearly relative to that speed limit (as they do with the usual Lorentz transformations in free space) These transformations will only apply to events and observations within the given medium though.
