What happens to speed and frequency of a light beam moving in transparent medium when observed from different inertial frame of reference?

Question

Suppose a transparent medium where speed of light is $c/n$, an inertial frame of reference $K$ which is stationary relatively to the medium and an inertial frame of reference $K'$ which is moving with speed $V$ according to $K$ along the $x$ axis. Also suppose that a light beam is travelling through the medium and its frequency measured from $K$ is $f$ and from $K'$ is $f'$ and the angle between the $x$ axis and the bean is $\theta$ and $\theta'$ respectively. If we know $n,f,\theta'$(or $\theta$ or both) and $V$ how can we calculate $f'$?

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From a book I have read the transformation was the following:

$$f'~=~\frac{f}{\gamma (1+\beta n\cos\theta')},$$ where $$\gamma~=~\frac{1}{\sqrt{1-\beta^2}},\qquad \text{and}\qquad \beta~=~v/c.$$

I can prove it with momentum-energy Lorentz transformations only if I suppose that light is moving with same speed ($c/n$) both in $K$ and $K'$ which (I think) is wrong according to speed transformation. Does the speed of light when moving through a transparent medium change when observed form a different inertial frame of reference or not?

Answer 1

You can transform the speed of the light just like you would transform the speed of a rocket ship. Why? Well, why not? The front of the light beam traces out a path through spacetime, and a Lorentz transformation transforms that path just like it transforms any other path through spacetime.

For the frequency...well, it's easier to think about the period of the light oscillation (the reciprocal of frequency). This is a length of time just like any other length of time, so the "relativistic time dilation formula" applies. If the light period in the lab frame is $T_{lab}$, and the velocity of light in the lab frame is $v_{lab}$, and then $\gamma_{lab}=(1-v_{lab}^2/c^2)^{-1/2}$, then the light period in the light beam's rest frame is $T_{proper}=T_{lab}/\gamma_{lab}$ and the light period in any other frame is $T_{otherframe}=\gamma_{otherframe}T_{proper}$.

(Funny to think of the "light's rest frame" but it certainly has one! In the light's rest frame, the transparent medium is moving very fast the opposite direction to the light; it's sort of like how a motor boat can be exactly stationary driving against the current of a very fast-flowing river.)