We know that light travels at the same speed in vacuum regardless of the observer.And light also travels at a speed of c/n in the medium.My question is since c/n is the limit speed in medium,then do we have the similar Lorentz transformation in the medium which just replace c with c/n? For example, suppose space is filled with medium.And the observer is in S-frame in the medium. A light is travelling in S'-frame which is also in the medium and is moving at the speed of u respect to S.Then what's the speed of light that the observer measures in S? c/n?
c/n is the phase velocity of the light in the medium. c/n is not the limit speed, c is. (Check out: Cherenkov radiation) The Lorentz transformations remain as usual. No. The speed measured by the observer will be the relativistic sum of those velocities (using the relativistic addition of velocity).
In fact, this was first observed by Fizeau in the 1850s. Of course, he didn't recognize it as relativistic velocity addition, but Einstein did, more than fifty years later! http://en.wikipedia.org/wiki/Fizeauâ€“Foucault_apparatus#Speed_of_light_in_moving_media
Nevertheless Fizeau's experiment was a major step toward to relativity because the experiments of Michelson and Morley can not be interpreted correctly without falsification of a complete aether drag.