Experiments by Fizeau and Airy

 

Fizeau’s experiment

 

Fizeau’s experiment that was designed to determine the convection coefficient provides an interesting topic of discussion. The apparent speed of the light going through moving water (moving with velocity v with respect to the apparatus) in the direction of the water flow was found by Fizeau to be:

 

.

 

Since the speed of light (apparently) changes in the moving water, and since the water is moving relative to the observer, the relative refractive index as measured for a specific frequency of light must also change to some value n’, which can be expressed as a function of n, v, and c. Also, since the water ‘carries’ the light 1‑1/n fraction of the time[1] thus giving the velocity due to this ‘carrying’ as v(1‑1/n), we can construct a modified formula as

 

.

 

Equating this equation with that found by Fizeau gives:

 

     or      .

 

Solving for n’ gives:

.                  (1)

 

Consider the water as being in a system S’, and the apparatus in a system S. A direct derivation from theoretical considerations can be made as follows. The fraction of time that the light is interacting with the water molecules is  as measured in S. As measured in S’, this time would have to be converted to S’ time by multiplying by the time correction factor t/t’, which is assumed at this point to be the basic conversion factor that is equal to . This gives

.

Solving for n’, we get

.                              (2)

 

This is the relative refractive index that an observer who is stationary with respect to the equipment that the water is flowing through would measure, and not the refractive index that an observer moving with the water would measure. That would presumably still be n. For a refractive index of around 1.3, equation (1) divided by equation (2) is greater than  and less than 1 for small values of v/c (less than about 0.5), approaching 1 as v approaches 0. The two equations for n’ are essentially the same for practical experimental conditions, differing by a factor of less than about 1x10^-7 for water velocities less than 30 m/s.

 

The moving water in S’ has a relative velocity of v with respect to S. The relative speed of light outside of the water in S’ for the initial time factor used above is c - v as viewed from S. The apparent speed of light in the water would be the denominator in equation (2). The denominator is , which leads to the equivalent formula , which gives  when .

 

Thus    provides essentially the same result as obtained by Fizeau for light traveling in the direction of the moving water (+) and against the moving water (–). The first component can be thought of as being due to the light traveling on its own, and the second part as being due to the light traveling with the water molecules.

 

If the time factor is assumed to be γ, the equation is a bit more complicated, but leads to the same general conclusion. With the water flowing with a velocity of less than 30 m/s, the experimental results are not sufficiently distant from the point of convergence of possible time factors to distinguish between them. Note: results from research subsequent to writing this suggest that γ is the more likely value for the time factor.

 

Airy’s experiment

 

Another experiment using water was done by Airy who filled a telescope with water and found that the aberration angle remained the same as when the telescope was filled with air. Light appears to travel slower through water than through air, so that if everything else were equal, the aberration angle would be expected to change slightly. Although a time gradient does not ‘carry’ light along with it, a substance such as water might be expected to do so to some extent (see footnote 1). While the light is interacting (1-1/n fraction of the time) the water molecules essentially carry it along. When not interacting, the light continues on at its normal speed. This gives an overall effect of the light slowing down in a refractive medium. The combination of the normal progress of the light and its movement with the water molecules (which are moving with the earth but at rest with respect to the observer) while it is interacting with them assures that the light will essentially follow the same path as it would without the water.

An interesting question here is the possibility that after a photon interacts with the water molecule and that energy is sent on its way from that molecule that the motion of the molecule is imparted to the ‘re-emitted' photon. This assumes that the velocity of the molecule is not in the same direction as the velocity of the photon. This sort of interaction is considered in the second experiment for detecting the ‘photon shift' in the section Examples - The Photon Shift .