Fizeau Experiment - Derivation in Special Relativity

Derivation in Special Relativity

Einstein showed how Lorentz's equations could be derived as the logical outcome of a set of two simple starting postulates. In addition Einstein recognized that the stationary aether concept has no place in special relativity, and that the Lorentz transformation concerns the nature of space and time. The Fizeau experiment was one of the key experimental results that shaped Einstein's thinking about relativity. Robert S. Shankland reported some conversations with Einstein, in which Einstein emphasized the importance of the Fizeau experiment:

He continued to say the experimental results which had influenced him most were the observations of stellar aberration and Fizeau’s measurements on the speed of light in moving water. “They were enough,” he said.

Max von Laue (1907) demonstrated that the Fresnel drag coefficient can be easily explained as a natural consequence of the relativistic formula for addition of velocities, namely:

The speed of light in immobile water is c/n.

From the velocity composition law it follows that the speed of light observed in the laboratory, where water is flowing with speed v (in the same direction as light) is

Thus the difference in speed is (assuming v is small comparing to c, approximating to the first non-trivial correction)

 

This is accurate when v/c << 1, and agrees with the formula based upon Fizeau's measurements, which satisfied the condition v/c << 1.

Fizeau's experiment is hence supporting evidence for the collinear case of Einstein's velocity addition formula.