The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed V relative to the lab frame is$\mathit{v}\mathbf{=}\frac{\mathbf{c}}{\mathbf{}\mathbf{n}}\mathbf{+}\mathit{K}\mathit{V}$

where$\mathit{n}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{333}$ is the index of refraction of water. Fizeau called K the dragging coefficient and obtained an experimental value of K = 0.44. What value of K do you calculate from relativistic transformations?

We call the lab and the water tank reference frames S and S’, respectively. The relative speed between S and S’ is . The speed of light relative to S’ (the tank water frame) is $v\text{'}=\frac{c}{n}$, where n is the refractive index of water. The speed of light relative to S (the lab frame) is v . The speeds of light in S and S’ are related by the Lorentz velocity transformation [Eq. ()], from which the dragging coefficient can be obtained.

From Eq. (),

$v=\frac{v\text{'}+V}{1+{\displaystyle \frac{v\text{'}V}{c^2}}}=\left(\frac{c}{n}+V\right){\left(1+\frac{V}{nc}\right)}^{-1}$

where we used$v\text{'}=\frac{c}{n}$ . The Taylor expansion of${\left(1+\frac{V}{nc}\right)}^{-1}$ in powers of$\frac{V}{nc}$ gives us the series

$$\begin{gathered} v=\left(\frac{c}{n}+V\right)\left[1-\frac{V}{nc}+{\left(\frac{V}{nc}\right)}^2+....\right] \\ v=\frac{c}{n}+\left(1-\frac{1}{n^2}\right)V-\frac{1-{\displaystyle \frac{1}{n^2}}}{nc}{V}^2+.............. \end{gathered}$$

We see that Fizeau's experimental formula is the linear approximation of the relativistic-based calculated series. Moreover, we find $K=1-\frac{1}{n^2}$. Using n = 1.333 (for water), we find

$K=1-\frac{1}{n^2}=1-\frac{1}{1.{333}^2}=0.4372$

In Fizeau's experiment the speed of the water tank is much smaller than c , so the higher order terms in the relativistic series are negligibly small and Fizeau's formula is a good approximation to the relativistic result; Fizeau's experimental value for K is within 1% of the relativistic calculated value.