Light in moving media

Requisites
Relativistic addition of velocities

Refractive index

Problem
Light moves through a slowly moving medium with refractive index n. That medium moves with a speed v in parallel with the direction of light. What speed will be measured for that light by a rest observer?

Solution
If we have a medium with refractive index n, the speed of light relative to that medium is c/n.

Using relativistic addition of velocities, we get for the rest observer:

$$V=\frac{c/n+v}{1+v/nc}$$

But as $$v \ll c$$, we can expand that expression in terms of $$v/c$$:

$$V=c \frac{1/n+(v/c)}{1+(v/c)/n}$$ $$ \begin{align} V & =c \frac{1/n}{1} +c \frac{1 \cdot(1)-1/n \cdot(1/n)}{1} v/c + ... \\ & \approx \frac{c}{n} +c \left (1 - \frac{1}{n^2} \right ) v/c \\ & = \frac{c}{n} +v \left (1 - \frac{1}{n^2} \right ) \\ \end{align} $$

The factor $$\left (1 - \frac{1}{n^2} \right )$$ was known as the Fresnel drag coefficient. It is easily measured with interference experiments.

Generalization
$$v'=V+v/\Gamma^2$$