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

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 $f'$?

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?

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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 $T_{otherframe}=\gamma_{otherframe}T_{proper}$.