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 $v/c$:

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

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

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

Generalization

$v^{\prime }=V+v/{\mathrm{\Gamma }}^2$