PROBLEM $53$

In this problem we show that the relativistic velocity addition law can be used to explain the Fizeau experiment without invoking the existence of ether. The speed of light in stationary water is less than its speed $c$ in vacuum. Traditionally it is written as $\frac{c}{n},$ where $n$~~$\frac{4}{3}$ is the index of refraction of water. The water flows in the pipe with velocity $v.$ In the lower arm T$2$ of the interferometer $($as shown in Figure Problem $53\mathrm{.}1),$ one would expect that, from the nonrelativistic addition law, the speed of light in the moving water would be its speed in stationary water increased by the speed of the water in the pipe: $w=\frac{c}{n}+v.$ Show that the relativistic velocity addition law leads to$,$ up to higher order corrections,

$w=\frac{c}{n}+v(1-\frac{1}{n^2}).$

The result was observed by Fizeau in $1851,$ but for long time viewed as a confirmation of a rather elaborate contemporary ether$-$theoretical calculation based on the idea that the

water was partially successful in dragging ether along with it$.$ Einstein later said that it was of fundamental importance in his thinking.

$(25$ points$)$

$$lgure r rodiem j$.$: