Michelson-Morley over lightly
No grand revelations in this post. I'm just setting the record straight as to how I understand the Michelson-Morley experiment.
It is often casually said that Michelson and Morley established that the velocity of light is a constant.
This isn't quite correct. Their interferometer experiment tended to demonstrate that c was the maximum possible velocity in the ether, which, to be sure, was quite a shocking discovery.
Basically, the experiment checked a beam of reflected light crossing the presumed ether wind (relative to the moving earth and interferometer) against a beam traveling into or away from the wind. So they sought a velocity magnitude that did not equal c21/2, the magnitude for no ether flow. This difference would have been revealed by a difference in the interference pattern. That is, light crossing the ether wind would be reflected from a different part of the mirror than light going with the wind. This means that the interference pattern for a non-right angle of reflection will differ from the pattern for a right angle of reflection.
So they were testing for galilean velocity addition, which applies to a mechanical wave crossing a moving medium.
Another type of velocity addition is Doppler velocity addition.
So let us call v the constant of propagation in the medium, which doesn't change, and u the velocity of the observer or the source.
For galilean addition:
v + u = kv
so u = v(k-1)
For nonrelativistic doppler addition:
i. Observer moving toward source
(v + u)/v = kv
so u = v(kv-1)
also: f' = f(v+u)/v
indicating the change in frequency.
j. Source moving toward observer
v/(v-u) = kv
so u = v-(1/k)
also: f' = fv/(v-u)
In a mechanical system the elasticity of the medium emerges when the relative tensions differ, where T' = (f'/s)1/2, with s being the distance unit.
So the tension of the medium can be summarized by T'j - T'i
However, if the galilean vector c21/2 doesn't hold, then the medium effectively doesn't exist and one expects that v must be the top velocity in the "ether."
In that case one expects zero tension as deduced from the Doppler effect and we get relativistic Doppler addition, thus:
vi = vj
(v+u)(v-u) = v2
or, in the final analysis, u = 0.
When u =/= 0, we have the nonrelativistic doppler effect, of course.
In terms of proper time, relativistic velocity is
v = c2 - Tp1/2c
and obviously v cannot exceed c.