The Fizeau Experiment
In 1851 Hyppolyte Fizeau carried out a remarkable experiment to answer what might appear to be a simple question. The question was if (or how) is the velocity of a photon effected by a moving fluid.
As an example, consider a photon which moves accross a cart of water. We might recall that the speed of light in a medium is related to the index of refraction. The relation being:
\[ n = \frac{c}{v} \]
Where \(n\) is a measured index of refraction, \(c\) is the speed of light in a vacuum, and \(v\) is the speed of light in the medium.
Armed with this knowledge, we ask
What is the net speed of a photon moving in a fluid where its speed is \(v\) and where the fluid is moving at speed \(u\)?At first, there is the obvious possibility that we can simply add the two velocities so that the total veloctiy is simply \(v_{\text{tot}} = v + u \). In this case, known as Galilean velocity addition, light has no idea that it is moving in a fluid! However, there is a major problem that would arise in such a world: information could move faster than the speed of light! Clearly, that is not the case.
Another possibility is that regardless of what the fluid is doing, the absolute speed of light is always the same. This way, there is no way we can violate the law that nothing travels faster than the speed of light in a vacuum. Both of these two scenarios are illistrated below.
Galilean
Strongly Coupled
The Truth
What actually happens is neither of the above, but some in-between case. Relativity, which is based on the principle that nothing can travel faster than the speed of light forces us to add the velocities in a special way such that their addition never surpasses the speed of light in a vacuum. This is known as relativistic addition