Light Propagation through Dielectric Media

Consider a transparent dielectric medium, such as air, water, or glass. Let $n$ be the refractive index of the medium. Thus, in the rest frame, $S$, of the medium, light propagates at the phase velocity $c/n$. Let us transform to a frame of reference, $S'$, that moves at velocity $\mp v\,{\bf e}_x$, and is in a standard configuration, with respect to frame $S$. Thus, the medium appears to flow at the velocity $\pm v\,{\bf e}_x$ in frame $S'$. According to Equation (3.122), the phase velocity of light propagating in the $+x$-direction in frame $S'$, in which the medium flows at velocity $\pm v\,{\bf e}_x$, is

$$u_\pm = \frac{c/n\pm v}{1\pm v/(c\,n)} = \frac{c}{n}\pm \frac{v\,(1-1/n^2)}{1\pm v/(c\,n)}.$$ (3.137)

In the limit in which the flow velocity of the medium is much smaller than the velocity of light in vacuum, $v/c\ll 1$, the previous equation reduces to

$$u_\pm \simeq \frac{c}{n} \pm v\left(1-\frac{1}{n^2}\right).$$ (3.138)

However, this is identical to the phase velocity in dielectric media measured by Fizeau. (See Section 3.1.4.) Thus, we can now appreciate that special relativity is capable of accounting for all of the experimental observations discussed in Section 3.1.