Monday, February 12, 2007

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

That is

(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.

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