Required math: algebra, vectors
Required physics: basics
The outcome of the Michelson-Morley experiment was that the speed of light appeared to be independent of the velocity of the apparatus relative to the postulated universal ether, which was the medium in which light was presumed to travel. If light really did travel in some substance, so that the wave nature of light was due to its propagation through it, then the speed of light should be fixed relative to this ether in the same way that the speed of other waves such as sound or water are fixed relative to their propagation medium, and if the apparatus is moving relative to the ether, then the velocity of light relative to the apparatus should vary.
Michelson’s explanation for the null result of his experiment was that the Earth dragged the ether along with it, so that the Earth remained at rest in the ether. This didn’t seem to convince many physicists of the time, since it would imply that every mass dragged its own little aura of ether along with it, which didn’t seem likely (of course, the final explanation – special relativity – didn’t seem very likely at first either).
One other explanation that was proposed at the time was a suggestion made independently by the Dutch physicist Hendrik Lorentz and the Irish physicist George Fitzgerald. This was that all objects contracted in the direction of their motion relative to the ether. As we saw when we analyzed the Michelson-Morley experiment, the round trip time for the light travelling parallel to the motion relative to the ether is
while the time for the round trip perpendicular to the direction of motion is
where is the distance from the source to the mirror in each case, and is the velocity of the Earth relative to the ether. The result of the experiment was that .
If is actually different in the two cases, then the equality of the two times could be explained. In particular if we write
the two times are equal. Thus lengths in the direction of motion are contracted by a factor of , where (although at the time, the restriction of to be less than 1 wasn’t formally imposed).
This is, of course, the same result as is obtained in special relativity, since there, objects do in fact appear contracted in the direction of motion of one observer with respect to another. There are two crucial differences, however. In relativity, the contraction is a result of the motion of one observer relative to any other observer, not with respect to some background ether. The second difference is the most fundamental; it is that the time as measured by the two observers is not the same.
The possibility that time wasn’t absolute did not occur to Lorentz or Fitzgerald, and as a result their proposal doesn’t work properly. To see this a bit more quantitatively, we can work out the transformations between two coordinate systems assuming only their contraction hypothesis. If our two observers are (at rest relative to the ether, and using Greek letters) and (moving at speed in the direction relative to the ether, and using Roman letters), then will say that any distance measured by as is actually shorter, so that , but that the two observers will agree on the times at which events occur, so that . We can write this as a transformation matrix:
The formal transformations are therefore
The term, of course, just represents the fact that any point on the axis in ‘s frame is moving with speed relative to . In particular, at , this transformation provides a uniform contraction of all distances in the direction.
This is fine as far as it goes and seems to explain the null result of the experiment, but there are a couple of problems. First, we would expect that the inverse transformation (from to ) should be obtained by plugging in in place of , but if we try that, we get
and it can be seen by direct multiplication that . For reference, the inverse matrix turns out to be
which doesn’t have any obvious meaning as a transformation matrix.
Another problem appears when we consider how this transformation affects light. If we fire two photons in opposite directions along the axis, clearly they travel with speed 1 (in the positive direction) and (in the opposite direction). Under transformation, since also sees the two photons take equal travel times, the two photons should transform the same way. In ‘s frame, the equations of the two world lines of the photons are
Under the transformation we get
since . At the two extremes, we get, first for
which is as it should be, since if , and are in the same frame.
For though, we get
For values of in between 0 and 1, the slope of versus is always greater than 1 (it has a maximum value of when ), and the slope of versus is always greater than (the slope increases monotonically from at to at ). Thus the photons have different speeds in ‘s frame, and will take different travel times to travel the same length, so this transformation still contradicts the results of the experiment.
The solution to the problem requires special relativity, in which the ether is abolished, the speed of light is independent of the observer’s frame, and the universality of time is abolished. On reflection, this solution is probably much more radical and non-intuitive than the simple contraction proposed by Lorentz and Fitzgerald, but it has passed many experimental tests since.