Addendum: Generalized Sagnac proof of sufficiency
A simple proof that the Generalized Sagnac effect carries light waves fully with the medium
Let me preface this by stating I am not a mathematician and don’t know the criteria for a formal proof. With that in mind, this is an informal, though logically sound, proof.
R. Wang’s experimentally derived Generalized Sagnac equation is as follows [1]:
Δφ = 4 π ν · Δl / c λ.
Δφ is the induced phase difference between light beams in radians
ν is the velocity of the medium (fiber optic cable)
Δl is the the projection of the segment length l on the moving direction
c is the speed of light
λ is the free space wavelength of light
Let’s first notice this equation consists of a ratio of two expressions: ν Δl and c λ. To understand this better let’s look at its limits. Let’s chose the case where the ratio is 1 and the phase difference is 4π radians between the beams. The bottom of the ratio, the speed of light and it’s wavelength is known so there is nothing to adjust there. The variables we can control are ν, the velocity of the medium and Δl the segment length. The simplest way to make ν Δl equal to c λ is to set ν to c and Δl to λ, so we do that.
Lastly we need to conceptualize what ν = c and Δl = λ means. This means the medium is moving as fast as the light and the length of the fiber optic cable is exactly the length needed for light to go through one wave cycle. Therefore, if we are pushing the light at the same velocity of our medium, over the distance of one wavelength, we would expect a shift of one full cycle 2π in each direction (+2π and -2π) resulting in a net difference of 4π between the light beams, which is our answer when ν Δl / c λ is 1.
With this in mind, we can be sure the equation covers the effects of the Michelson-Morley experiment (MMX). We would expect the moving medium of air in MMX to move light with velocity ν of Earth’s rotation thus removing any need for a length contraction hypothesis. QED.
References
[1] Wang R, Zheng Y, Yao A. Generalized sagnac effect. Physical review letters. 2004 Oct 1;93(14):143901.