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 $v/c$:

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

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

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

Generalization

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