# Why do the Michelson-Morley (1887) & Michelson-Gale (1925) experiments both measure a ~0.02 fringe shift?

As I mentioned here, the 1887 Michelson-Morley experiment placed an upper bound on the fringe displacement of about 0.01 or 0.02. Why did the 1925 Michelson-Gale experiment also find a fringe shift of about 0.02?

Here's the distribution of the results of the Michelson-Gale experiment: The Michelson-Morley experiment was intended to measure the motion of the earth through the ether, and the Michelson-Gale experiment was intended to measure earth's rotation, but what exactly are the differences in the setup between these two experiments, even though they appear to produce the same fringe shifts?

## 2 Answers

First your very graph, and the paper you quote, show that the fringe shift in Michelson-Gale has a mean value of 0.23 with an average deviation from mean of 0.016. This is this accuracy which is of the same order of magnitude as the upper bound on the fringe shifts in Michelson-Morley. In passing, it shows how experimental technique had progressed: the same precision could be reached for an experiment with optical path miles long than for a tabletop experiment about 30 years before.

In the Michelson-Gale experiment, the light from the source is split with one ray going through a rectangular path in one direction and the other ray going through it in the opposite direction. The fringe shift between the two rays when they meet again is measured. This is very different from the Michelson-Morley experiments where each ray does a round trip from the source to a mirror back to the source.

As a result, the Michelson-Gale experiment probes a very different phenomenology. This is not a null result experiment, note, because Special Relativity (SR) predicts a non-zero fringe shift. Indeed the velocity of light is not isotropic in a rotating frame: the average speed of light for the ray going in one direction around the rectangular path is different from the average speed of light for the ray going in the opposite direction through the same path. The predicted fringe shift is

$$\Delta = \frac{4A}{\lambda c}\Omega\sin \theta,$$

where $\theta$ is the latitude, $\Omega$ the angular speed of the Earth, $\lambda$ the wavelength, and $A$ is the area of the rectangle. There are actually two rectangles in the Michelson-Gale experiment, a large one and a small one but this is just an experimental detail: the fringe shift for the smaller one is too small to be measured and the central interference fringe for that smaller rectangle is used as the reference point for the fringe shift for the bigger rectangle, for which the formula above holds.

As I mentioned here, the 1887 Michelson-Morley experiment placed an upper bound on the fringe displacement of about 0.01 or 0.02. Why did the 1925 Michelson-Gale experiment also find a fringe shift of about 0.02?

Well first off, the MGX measured a shift of 0.2, not 0.02.

Moreover, the MMX measured 0 +/- 0.02, the MGX measured 0.2 +/- 0.016

Those are not similar result sets! Nor should they be, they were measuring different things.