22. Invariance
In this section the flight time of a photon is calculated in a frame of reference moving at an absolute velocity. This includes the effects derived here from classical electromagnetism. It is seen that the absolute velocity cannot be determined from such a measurement. Thus if Michelson and Morley had known of these purely electromagnetic effects, they would not have been surprised by the null result of their experiment.
Imagine that a beam of light is timed for a trip between two points on an object which is moving at an absolute velocity v.
Fig.10: Light travelling a distance between two points.
From the law of cosines we have
(1)
where q is the angle of the path of distance d away from the direction of the absolute velocity. Solving for the time of the trip t yields
(2)
If there is a round trip the cosine term cancels out and thus
(3)
Note that it is dependent on q, so if times of different paths were compared, the absolute velocity could be determined. However, we still have to consider the change in path length caused by the length contraction of Eq.(14-2) which is
(4)
where y is the length in the direction of absolute velocity and x is in the direction transverse to the velocity. Since
(5ab)
we obtain
(6)
where q0 is the angle before contraction. This is related to the angle after contraction by:
(7)
If we solve for cosq0 , then substitute it into Eq.(6) and the result into Eq.(3), we obtain
(8)
which is independent of direction and thus a comparison of times would not reveal the absolute velocity1.
If two different distances were used to compare times then
(9)
where t is the difference in the time of the two trips. This difference could be measured by a phase shift thus,
(10)
where f is the frequency of the light. It appears that by comparing two different velocities the absolute motion could be determined. However, we still have to consider the change in frequency of Eq.(14-4) or Eq.(21-12) which cancels out the velocity dependent factor such that
(11)
where f0 is the frequency of the light as seen by an observer at rest. The phase is then independent of the absolute velocity2
1. This was verified by Michelson & Morley25 and more precisely by Brillet & Hall26. The Silvertooth and Whitney19 experiment measured the interference between the counter-travelling waves in one of the arms of the Michelson-Morley experiment as the distances were simultaneously altered. As a phase difference would be a consequence of an absolute frame, the results would be significant. They obtained a measurement that was diurnally cyclic and of similar magnitude to the back-ground radiation shift measured by Smoot, Gorenstein and Muller18.
While these results indicate that the speed of light is independent of the source, they do not indicate that the speed of light is independent of the observer. The null result of the Michelson-Morley experiment is merely a coincidence based on the above described contraction of physical length.
2. This was verified by Kennedy and Thorndike21
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