Note: It is highlighted from the outset that most mainstream discussions of the Michelson interferometer normally affirm that the null result is empirical evidence that a space-time wave medium cannot exist. However, the LaFreniere webpage on this topic explains the null results using a re-interpretation of Lorentz transform, i.e. the same transforms which Einstein used as the basis of special relativity. In contrast, the previous discussion of Ivanov Waves suggested a possible alternative set of transforms, which might also explain the null results, such that this section of discussions will consider these different perspectives.
Before pursuing some alternative interpretations, we will initially introduce the general idea of the interferometer in terms of the diagram below, where the light from the source is split into two paths, i.e. vertical and horizontal. In a stationary inertial frame, where [v=0], these two different paths are of equal length [L], such that the split light beams will return in phase. Of course, if [v>0], then the diagram suggests that the two paths would be different in length, such that the split beams would not be in-phase. However, the result of the experiment, regardless of rotational orientation, never produced any phase shift, which was then interpreted by the theory of special relativity to allow any two inertial reference frames to simply be reversed.
In the current context, we shall adopt the assumption that the reference frame is moving with velocity [v] through a wave media, i.e. the aether. Therefore, if the propagation velocity of light [c] is always relative to the wave media, then any velocity [v] of the equipment must affect the propagation times relative to the equipment in motion with velocity [v]. However, the propagation times in question will be different depending on the horizontal or transverse paths taken through an interferometer, which it was assumed would cause wave interference at the final destination. For ease of comparison, the mirrors on the horizontal and transverse paths are positioned at the same distance [L=LH=LT], such that [L] can be used as a common term. An analysis of the Michelson-Morley experiment can then proceed on the basis of the propagation time of a light-beam with velocity [c]. The light-beam is first transmitted from a source [S], where on route it passes through a half-silvered mirror at an origin, defined as [t=0], and continues on until it is reflected by a horizontal mirror positioned at distance [L], which is moving with velocity [v].
Note: Based on the wave media assumption, any observer of this experiment has to be comoving with the interferometer in a frame of reference with velocity [v] with respect to the wave media.
In the comoving frame, the beam will hit the mirror at time [t1] having propagated a distance [ct1], while the mirror has moved a distance [vt1], such that:
Where the roundtrip propagation time [tH] can be calculated, but now adopting the shorthand [β=v/c].
In the vertical direction, the beam is emitted from the same source having velocity [c] towards the half-silvered mirror, but now hits the vertical mirror at time [t3] having propagated the distance [ct3], while the mirror has moved a distance [vt3] along the [x] direction. However, in order to hit the mirror, the propagation path of the beam is [L] in the [y] direction plus [vt3] in the [x] direction. As such, the original transverse propagation path [L] is redefined using the Pythagorean theorem based on the ratio of [c] and [v].
However, in order to have some equivalence with the form in , we really want to know the propagation time [t3] in terms of the distance [L], which is achieved by simply rearranging .
In the transverse case, the forward and backward paths are the same, such that the total propagation time on the transverse path is simply double the result in .
So, based on the times in  and , we can calculate the propagation time difference between the two paths taken.
Based on , the initial path difference can be defined by the propagation velocity [c] and time difference, i.e. [LD=ctD], such that we can quantify this path difference based on  divided by [c].
This path difference after a 90 ° rotation simply reverses the order of the terms shown in the brackets, i.e.
By dividing [LD1−LD2] by the wavelength [λ] of light used, the fractional wavelength shift [n] can be calculated as a function of the velocity [v].
The equipment used in the Michelson-Morley experiment had an arm length [L=11m] and used light with a wavelength [λ=500nm], while the velocity [v] was initially approximated to the Earth’s velocity around the Sun, i.e. [v=30km/s]. On this basis, a fractional wavelength shift [n=0.44] can be calculated, which it was assumed would produce an interference shift that could be detected. However, the null result of the experiment led to the conclusion that there is no measurable aether drift and therefore no aether.
Note: In Michelson's day, the velocity of the Earth was assumed to be ~30km/s in the context of its rotation around the Sun. Today, it is recognised that the velocity [v] of the Earth through some potential ether might have to be interpreted on a much larger scale than the local solar system. By analysing the data collected by astronomical observation, the local group of galaxies, which includes the Milky Way, is moving at ~627 km/s relative to the reference frame of the CMB. However, the solar system within this larger system, which includes the Earth-frame, is only moving at ~360km/s with respect to the CMB frame. On this basis, the larger value of [v=360km/s] would suggest an even greater interference shift, i.e. by a factor greater than 10. As such, any WSE model has to proposed some other causal mechanism to explain the null results of the Michelson experiment.
So, based on  and the note above, the Michelson interferometer should have detected a fringe shift as it was rotated to reflect different orientations of its velocity [v] through the wave media generally assumed in 1887. Of course, the failure to detect any shift was clearly problematic to the idea of the wave media, as there was no obvious causal mechanism to explain the null results at that time. However, in the historical context surrounding the development of the Lorentz transforms, the idea of some form of length contraction was proposed as a possible explanation of the null results, which required all objects, i.e. including the interferometer, to physically contract by [g] along the axis of motion, where [g] is simply another form of the Lorentz factor [g] defined by the Lorentz transforms.
If length contraction of [L’=gL] is inserted into only the formula for [TH], i.e. along the axis of motion, then the light propagation time in  will equal the formula for [TT] given in , as shown in .
However, while this solution may appear to resolve the mathematical anomaly, it was never clear that this solution provided a causal explanation as to how length contraction physically took place within a localised moving reference frame. This said, the subsequent publishing of special relativity in 1905, based on the Lorentz transformation, also required the idea of time dilation to counter the effects of length contraction, if the postulate of the constancy and invariance of velocity [c] was to be maintained. As a consequence of the acceptance of special relativity, it appeared logical that the existence of the aether no longer had any role to played in the kinematic description of 4D spacetime.