Special Relativity/Aether

Introduction
Many students confuse Relativity Theory with a theory about the propagation of light. According to modern Relativity Theory the constancy of the speed of light is a consequence of the geometry of spacetime rather than something specifically due to the properties of photons; but the statement "the speed of light is constant" often distracts the student into a consideration of light propagation. This confusion is amplified by the importance assigned to interferometry experiments, such as the Michelson-Morley experiment, in most textbooks on Relativity Theory.

The history of theories of the propagation of light is an interesting topic in physics and was indeed important in the early days of Relativity Theory. In the seventeenth century two competing theories of light propagation were developed. Christiaan Huygens published a wave theory of light which was based on Huygen's principle whereby every point in a wavelike disturbance can give rise to further disturbances that spread out spherically. In contrast Newton considered that the propagation of light was due to the passage of small particles or "corpuscles" from the source to the illuminated object. His theory is known as the corpuscular theory of light. Newton's theory was widely accepted until the nineteenth century.

In the early nineteenth century Thomas Young performed his Young's slits experiment and the interference pattern that occurred was explained in terms of diffraction due to the wave nature of light. The wave theory was accepted generally until the twentieth century when quantum theory confirmed that light had a corpuscular nature and that Huygen's principle could not be applied.

The idea of light as a disturbance of some medium, or aether, that permeates the universe was problematical from its inception (US spelling: "ether"). The first problem that arose was that the speed of light did not change with the velocity of the observer. If light were indeed a disturbance of some stationary medium then as the earth moves through the medium towards a light source the speed of light should appear to increase. It was found however that the speed of light did not change as expected. Each experiment on the velocity of light required corrections to existing theory and led to a variety of subsidiary theories such as the "aether drag hypothesis". Ultimately it was experiments that were designed to investigate the properties of the aether that provided the first experimental evidence for Relativity Theory.

The aether drag hypothesis
The aether drag hypothesis was an early attempt to explain the way experiments such as Arago's experiment showed that the speed of light is constant. The aether drag hypothesis is now considered to be incorrect.

According to the aether drag hypothesis light propagates in a special medium, the aether, that remains attached to things as they move. If this is the case then, no matter how fast the earth moves around the sun or rotates on its axis, light on the surface of the earth would travel at a constant velocity.



The primary reason the aether drag hypothesis is considered invalid is because of the occurrence of stellar aberration. In stellar aberration the position of a star when viewed with a telescope swings each side of a central position by about 20.5 seconds of arc every six months. This amount of swing is the amount expected when considering the speed of earth's travel in its orbit. In 1871, George Biddell Airy demonstrated that stellar aberration occurs even when a telescope is filled with water. It seems that if the aether drag hypothesis were true then stellar aberration would not occur because the light would be travelling in the aether which would be moving along with the telescope.



If you visualize a bucket on a train about to enter a tunnel and a drop of water drips from the tunnel entrance into the bucket at the very centre, the drop will not hit the centre at the bottom of the bucket. The bucket is the tube of a telescope, the drop is a photon and the train is the earth. If aether is dragged then the droplet would be travelling with the train when it is dropped and would hit the centre of bucket at the bottom.

The amount of stellar aberration, &alpha; is given by:


 * $$tan(\alpha) = v \delta t / c \delta t$$

So:


 * $$tan(\alpha) = v / c$$

The speed at which the earth goes round the sun, v = 30 km/s, and the speed of light is c = 300,000,000 m/s which gives &alpha; = 20.5 seconds of arc every six months. This amount of aberration is observed and this contradicts the aether drag hypothesis.

In 1818, Augustin Jean Fresnel introduced a modification to the aether drag hypothesis that only applies to the interface between media. This was accepted during much of the nineteenth century but has now been replaced by special theory of relativity (see below).

The aether drag hypothesis is historically important because it was one of the reasons why Newton's corpuscular theory of light was replaced by the wave theory and it is used in early explanations of light propagation without relativity theory. It originated as a result of early attempts to measure the speed of light.

In 1810, François Arago realised that variations in the refractive index of a substance predicted by the corpuscular theory would provide a useful method for measuring the velocity of light. These predictions arose because the refractive index of a substance such as glass depends on the ratio of the velocities of light in air and in the glass. Arago attempted to measure the extent to which corpuscles of light would be refracted by a glass prism at the front of a telescope. He expected that there would be a range of different angles of refraction due to the variety of different velocities of the stars and the motion of the earth at different times of the day and year. Contrary to this expectation he found that there was no difference in refraction between stars, between times of day or between seasons. All Arago observed was ordinary stellar aberration.

In 1818 Fresnel examined Arago's results using a wave theory of light. He realised that even if light were transmitted as waves the refractive index of the glass-air interface should have varied as the glass moved through the aether to strike the incoming waves at different velocities when the earth rotated and the seasons changed.

Fresnel proposed that the glass prism would carry some of the aether along with it so that "...the aether is in excess inside the prism". He realised that the velocity of propagation of waves depends on the density of the medium so proposed that the velocity of light in the prism would need to be adjusted by an amount of 'drag'.

The velocity of light $$ v_n$$ in the glass without any adjustment is given by:


 * $$ v_n = c / n $$

The drag adjustment $$ v_d$$ is given by:


 * $$ v_d = v (1 - \frac {\rho_e}{\rho_g}) $$

Where $$ \rho_e$$ is the aether density in the environment, $$\rho_g$$ is the aether density in the glass and $$v$$ is the velocity of the prism with respect to the aether.

The factor $$(1 - \frac {\rho_e}{\rho_g})$$ can be written as $$ (1 - \frac{1}{n^2})$$ because the refractive index, n, would be dependent on the density of the aether. This is known as the Fresnel drag coefficient.

The velocity of light in the glass is then given by:


 * $$ V = \frac {c}{n} + v (1 - \frac{1}{n^2}) $$

This correction was successful in explaining the null result of Arago's experiment. It introduces the concept of a largely stationary aether that is dragged by substances such as glass but not by air. Its success favoured the wave theory of light over the previous corpuscular theory.

The Fresnel drag coefficient was confirmed by an interferometer experiment performed by Fizeau. Water was passed at high speed along two glass tubes that formed the optical paths of the interferometer and it was found that the fringe shifts were as predicted by the drag coefficient.



The special theory of relativity predicts the result of the Fizeau experiment from the velocity addition theorem without any need for an aether.

If $$V$$ is the velocity of light relative to the Fizeau apparatus and $$U$$ is the velocity of light relative to the water and $$v$$ is the velocity of the water:


 * $$ U = \frac {c}{n} $$


 * $$ V = \frac {c/n + v}{1 + v/nc}$$

which, if v/c is small can be expanded using the binomial expansion to become:


 * $$ V = \frac {c}{n} + v (1 - \frac{1}{n^2}) $$

This is identical to Fresnel's equation.

It may appear as if Fresnel's analysis can be substituted for the relativistic approach, however, more recent work has shown that Fresnel's assumptions should lead to different amounts of aether drag for different frequencies of light and violate Snell's law (see Ferraro and Sforza (2005)).

The aether drag hypothesis was one of the arguments used in an attempt to explain the Michelson-Morley experiment before the widespread acceptance of the special theory of relativity.

The Fizeau experiment is consistent with relativity and approximately consistent with each individual body, such as prisms, lenses etc. dragging its own aether with it. This contradicts some modified versions of the aether drag hypothesis that argue that aether drag may happen on a global (or larger) scale and stellar aberration is merely transferred into the entrained "bubble" around the earth which then faithfully carries the modified angle of incidence directly to the observer.

References


 * Rafael Ferraro and Daniel M Sforza 2005. Arago (1810): the first experimental result against the ether Eur. J. Phys. 26 195-204

The Michelson-Morley experiment
The Michelson-Morley experiment, one of the most important and famous experiments in the history of physics, was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University, and is considered to be the first strong evidence against the theory of a luminiferous aether.

Physics theories of the late 19th century postulated that, just as water waves must have a medium to move across (water), and audible sound waves require a medium to move through (air), so also light waves require a medium, the "luminiferous aether". The speed of light being so great, designing an experiment to detect the presence and properties of this aether took considerable thought.

Measuring aether
A depiction of the concept of the “aether wind”. Each year, the Earth travels a tremendous distance in its orbit around the sun, at a speed of around 30 km/second, over 100,000 km per hour. It was reasoned that the Earth would at all times be moving through the aether and producing a detectable "aether wind". At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analysing the effective wind at various different times, it should be possible to separate out components due to motion of the Earth relative to the Solar System from any due to the overall motion of that system.

The effect of the aether wind on light waves would be like the effect of wind on sound waves. Sound waves travel at a constant speed relative to the medium that they are travelling through (this varies depending on the pressure, temperature etc (see sound), but is typically around 340 m/s). So, if the speed of sound in our conditions is 340 m/s, when there is a 10 m/s wind relative to the ground, into the wind it will appear that sound is travelling at 330 m/s (340 - 10). Downwind, it will appear that sound is travelling at 350 m/s (340 + 10). Measuring the speed of sound compared to the ground in different directions will therefore enable us to calculate the speed of the air relative to the ground.

If the speed of the sound cannot be directly measured, an alternative method is to measure the time that the sound takes to bounce off of a reflector and return to the origin. This is done parallel to the wind and perpendicular (since the direction of the wind is unknown before hand, just determine the time for several different directions). The cumulative round trip effects of the wind in the two orientations slightly favors the sound travelling at right angles to it. Similarly, the effect of an aether wind on a beam of light would be for the beam to take slightly longer to travel round-trip in the direction parallel to the “wind” than to travel the same round-trip distance at right angles to it.

“Slightly” is key, in that, over a distance such as a few meters, the difference in time for the two round trips would be only about a millionth of a millionth of a second. At this point the only truly accurate measurements of the speed of light were those carried out by Albert Abraham Michelson, which had resulted in measurements accurate to a few meters per second. While a stunning achievement in its own right, this was certainly not nearly enough accuracy to be able to detect the aether.

The experiments
Michelson, though, had already seen a solution to this problem. His design, later known as an interferometer, sent a single source of white light through a half-silvered mirror that was used to split it into two beams travelling at right angles to one another. After leaving the splitter, the beams travelled out to the ends of long arms where they were reflected back into the middle on small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive interference based on the length of the arms. Any slight change in the amount of time the beams spent in transit would then be observed as a shift in the positions of the interference fringes. If the aether were stationary relative to the sun, then the Earth's motion would produce a shift of about 0.04 fringes.

Michelson had made several measurements with an experimental device in 1881, in which he noticed that the expected shift of 0.04 was not seen, and a smaller shift of about 0.02 was. However his apparatus was a prototype, and had experimental errors far too large to say anything about the aether wind. For a measurement of the aether wind, a much more accurate and tightly controlled experiment would have to be carried out. The prototype was, however, successful in demonstrating that the basic method was feasible.

He then combined forces with Edward Morley and spent a considerable amount of time and money creating an improved version with more than enough accuracy to detect the drift. In their experiment the light was repeatedly reflected back and forth along the arms, increasing the path length to 11m. At this length the drift would be about .4 fringes. To make that easily detectable the apparatus was located in a closed room in the basement of a stone building, eliminating most thermal and vibrational effects. Vibrations were further reduced by building the apparatus on top of a huge block of marble, which was then floated in a pool of mercury. They calculated that effects of about 1/100th of a fringe would be detectable.

The mercury pool allowed the device to be turned, so that it could be rotated through the entire range of possible angles to the "aether wind". Even over a short period of time some sort of effect would be noticed simply by rotating the device, such that one arm rotated into the direction of the wind and the other away. Over longer periods day/night cycles or yearly cycles would also be easily measurable.

During each full rotation of the device, each arm would be parallel to the wind twice (facing into and away from the wind) and perpendicular to the wind twice. This effect would show readings in a sine wave formation with two peaks and two troughs. Additionally if the wind was only from the earth's orbit around the sun, the wind would fully change directions east/west during a 12 hour period. In this ideal conceptualization, the sine wave of day/night readings would be in opposite phase.

Because it was assumed that the motion of the solar system would cause an additional component to the wind, the yearly cycles would be detectable as an alteration of the magnitude of the wind. An example of this effect is a helicopter flying forward. While on the ground, a helicopter's blades would be measured as travelling around at 50 km/h at the tips. However, if the helicopter is travelling forward at 50 km/h, there are points at which the tips of the blades are travelling 0 km/h and 100 km/h with respect to the air they are travelling through. This increases the magnitude of the lift on one side and decreases it on the other just as it would increase and decrease the magnitude of an ether wind on a yearly basis.

The most famous failed experiment
Ironically, after all this thought and preparation, the experiment became what might be called the most famous failed experiment to date. Instead of providing insight into the properties of the aether, Michelson and Morley's 1887 article in the American Journal of Science reported the measurement to be as small as one-fortieth of the expected displacement but "since the displacement is proportional to the square of the velocity" they concluded that the measured velocity was approximately one-sixth of the expected velocity of the Earth's motion in orbit and "certainly less than one-fourth". Although this small "velocity" was measured, it was considered far too small to be used as evidence of aether, it was later said to be within the range of an experimental error that would allow the speed to actually be zero.

Although Michelson and Morley went on to different experiments after their first publication in 1887, both remained active in the field. Other versions of the experiment were carried out with increasing sophistication. Kennedy and Illingsworth both modified the mirrors to include a half-wave "step", eliminating the possibility of some sort of standing wave pattern within the apparatus. Illingsworth could detect changes on the order of 1/300th of a fringe, Kennedy up to 1/1500th. Miller later built a non-magnetic device to eliminate magnetostriction, while Michelson built one of non-expanding invar to eliminate any remaining thermal effects. Others from around the world increased accuracy, eliminated possible side effects, or both. All of these with the exception of Dayton Miller also returned what is considered a null result.

Morley was not convinced of his own results, and went on to conduct additional experiments with Dayton Miller. Miller worked on increasingly large experiments, culminating in one with a 32m (effective) arm length at an installation at the Mount Wilson observatory. To avoid the possibility of the aether wind being blocked by solid walls, he used a special shed with thin walls, mainly of canvas. He consistently measured a small positive effect that varied, as expected, with each rotation of the device, the sidereal day and on a yearly basis. The low magnitude of the results he attributed to aether entrainment (see below). His measurements amounted to only ~10 kps instead of the expected ~30 kps expected from the earth's orbital motion alone. He remained convinced this was due to partial entrainment, though he did not attempt a detailed explanation.

Though Kennedy later also carried out an experiment at Mount Wilson, finding 1/10 the drift measured by Miller, and no seasonal effects, Miller's findings were considered important at the time, and were discussed by Michelson, Hendrik Lorentz and others at a meeting reported in 1928 (ref below). There was general agreement that more experimentation was needed to check Miller's results. Lorentz recognised that the results, whatever their cause, did not quite tally with either his or Einstein's versions of special relativity. Einstein was not present at the meeting and felt the results could be dismissed as experimental error (see Shankland ref below).

In recent times versions of the MM experiment have become commonplace. Lasers and masers amplify light by repeatedly bouncing it back and forth inside a carefully tuned cavity, thereby inducing high-energy atoms in the cavity to give off more light. The result is an effective path length of kilometers. Better yet, the light emitted in one cavity can be used to start the same cascade in another set at right angles, thereby creating an interferometer of extreme accuracy.

The first such experiment was led by Charles H. Townes, one of the co-creators of the first maser. Their 1958 experiment put an upper limit on drift, including any possible experimental errors, of only 30 m/s. In 1974 a repeat with accurate lasers in the triangular Trimmer experiment reduced this to 0.025 m/s, and included tests of entrainment by placing one leg in glass. In 1979 the Brillet-Hall experiment put an upper limit of 30 m/s for any one direction, but reduced this to only 0.000001 m/s for a two-direction case (ie, still or partially entrained aether). A year long repeat known as Hils and Hall, published in 1990, reduced this to 2x10 -13.

Fallout
This result was rather astounding and not explainable by the then-current theory of wave propagation in a static aether. Several explanations were attempted, among them, that the experiment had a hidden flaw (apparently Michelson's initial belief), or that the Earth's gravitational field somehow "dragged" the aether around with it in such a way as locally to eliminate its effect. Miller would have argued that, in most if not all experiments other than his own, there was little possibility of detecting an aether wind since it was almost completely blocked out by the laboratory walls or by the apparatus itself. Be this as it may, the idea of a simple aether, what became known as the First Postulate, had been dealt a serious blow.

A number of experiments were carried out to investigate the concept of aether dragging, or entrainment. The most convincing was carried out by Hamar, who placed one arm of the interferometer between two huge lead blocks. If aether were dragged by mass, the blocks would, it was theorised, have been enough to cause a visible effect. Once again, no effect was seen.

Walter Ritz's Emission theory (or ballistic theory), was also consistent with the results of the experiment, not requiring aether, more intuitive and paradox-free. This became known as the Second Postulate. However it also led to several "obvious" optical effects that were not seen in astronomical photographs, notably in observations of binary stars in which the light from the two stars could be measured in an interferometer.

The Sagnac experiment placed the MM apparatus on a constantly rotating turntable. In doing so any ballistic theories such as Ritz's could be tested directly, as the light going one way around the device would have different length to travel than light going the other way (the eyepiece and mirrors would be moving toward/away from the light). In Ritz's theory there would be no shift, because the net velocity between the light source and detector was zero (they were both mounted on the turntable). However in this case an effect was seen, thereby eliminating any simple ballistic theory. This fringe-shift effect is used today in laser gyroscopes.

Another possible solution was found in the Lorentz-FitzGerald contraction hypothesis. In this theory all objects physically contract along the line of motion relative to the aether, so while the light may indeed transit slower on that arm, it also ends up travelling a shorter distance that exactly cancels out the drift.

In 1932 the Kennedy-Thorndike experiment modified the Michelson-Morley experiment by making the path lengths of the split beam unequal, with one arm being very long. In this version the two ends of the experiment were at different velocities due to the rotation of the earth, so the contraction would not "work out" to exactly cancel the result. Once again, no effect was seen.

Ernst Mach was among the first physicists to suggest that the experiment actually amounted to a disproof of the aether theory. The development of what became Einstein's special theory of relativity had the Fitzgerald-Lorentz contraction derived from the invariance postulate, and was also consistent with the apparently null results of most experiments (though not, as was recognised at the 1928 meeting, with Miller's observed seasonal effects). Today relativity is generally considered the "solution" to the MM null result.

The Trouton-Noble experiment is regarded as the electrostatic equivalent of the Michelson-Morley optical experiment, though whether or not it can ever be done with the necessary sensitivity is debatable. On the other hand, the 1908 Trouton-Rankine experiment that spelled the end of the Lorentz-FitzGerald contraction hypothesis achieved an incredible sensitivity.

References


 * W. Ritz, Recherches Critiques sur l'Electrodynamique Generale, Ann. Chim., Phys., 13, 145, (1908) - English Translation


 * W. de Sitter, Ein astronomischer Bewis für die Konstanz der Lichgeshwindigkeit, Physik. Zeitschr, 14, 429 (1913) - English Translation


 * The Michelson Morley and the Kennedy Thorndike tests of STR


 * The Trouton-Rankine Experiment and the Refutation of the FitzGerald Contraction


 * High Speed Ives-Stilwell Experiment Used to Disprove the Emission Theory

Mathematical analysis of the Michelson Morley Experiment
The Michelson interferometer splits light into rays that travel along two paths then recombines them. The recombined rays interfere with each other. If the path length changes in one of the arms the interference pattern will shift slightly, moving relative to the cross hairs in the telescope. The Michelson interferometer is arranged as an optical bench on a concrete block that floats on a large pool of mercury. This allows the whole apparatus to be rotated smoothly.

If the earth were moving through an aether at the same velocity as it orbits the sun (30 km/sec) then Michelson and Morley calculated that a rotation of the apparatus should cause a shift in the fringe pattern. The basis of this calculation is given below.



Consider the time taken $$t_1$$ for light to travel along Path 1 in the illustration:


 * $$t_1 = \frac{L_f}{c-v} + \frac{L_f}{c+v}$$

Rearranging terms:


 * $$\frac{L_f}{c-v} + \frac{L_f}{c+v} = \frac{2L_fc}{c^2-v^2}$$

further rearranging:

$$\frac{2L_fc}{c^2-v^2} = \frac{2L_f}{c}\frac{1}{1-v^2/c^2}$$

hence:

$$t_1 = \frac{2L_f}{c}\frac{1}{1-v^2/c^2}$$

Considering Path 2, the light traces out two right angled triangles so:


 * $$ct_2 = 2 \sqrt{L_m^2 + (vt_2/2)^2}$$

Rearranging:


 * $$t_2 = \frac{2L_m}{\sqrt{c^2-v^2}}$$

So:


 * $$t_2 =\frac{2L_m}{c} \frac{1}{\sqrt{1-(v/c)^2}}$$

It is now easy to calculate the difference ($$\Delta t$$ between the times spent by the light in Path 1 and Path 2:


 * $$\Delta t = \frac{2}{c} \left(\frac{L_m}{\sqrt{1-v^2/c^2}}-\frac{L_f}{1-v^2/c^2}\right)$$

If the apparatus is rotated by 90 degrees the new time difference is:


 * $$\Delta t^' = \frac{2}{c} \left(\frac{L_m}{1-v^2/c^2}-\frac{L_f}{\sqrt{1-v^2/c^2}}\right)$$

because $$L_m$$ and $$L_f$$ exchange roles.

The interference fringes due to the time difference between the paths will be different after rotation if $$\Delta t$$ and $$\Delta t^'$$ are different.


 * $$\Delta t^' - \Delta t = \frac{2}{c} \left(\frac{L_m+L_f}{1-v^2/c^2}-\frac{L_f+L_m}{\sqrt{1-v^2/c^2}}\right)$$

This difference between the two times can be calculated if the binomial expansions of $$\frac{1}{1-v^2/c^2}$$ and $$\frac{1}{\sqrt{1-v^2/c^2}}$$ are used:


 * $$\frac{1}{1-v^2/c^2}= 1 + \frac{v^2}{c^2} + \left(\frac{v^2}{c^2}\right)^2 + ....$$


 * $$\frac{1}{\sqrt{1-v^2/c^2}}= 1 + \frac{1}{2}\frac{v^2}{c^2} + \frac{3}{8}\left(\frac{v^2}{c^2}\right)^2 + ....$$

So:


 * $$\Delta t^' - \Delta t \approx \frac{L_f + L_m}{c}\frac{v^2}{c^2}$$

If the period of one vibration of the light is $$T$$ then the number of fringes ($$n$$), that will move past the cross hairs of the telescope when the apparatus is rotated will be:


 * $$n = \frac{\Delta t^' - \Delta t}{T}$$

Inserting the formula for $$\Delta t^' - \Delta t$$:


 * $$n \approx \frac{L_f + L_m}{cT}\frac{v^2}{c^2}$$

But $$cT$$ for a light wave is the wavelength of the light ie: $$cT = \lambda$$ so:


 * $$n \approx \frac{L_f + L_m}{\lambda}\frac{v^2}{c^2}$$

If the wavelength of the light is $$ 5 \times 10^{-7}$$ and the total path length is 20 metres then:


 * $$n = \left(\frac{20}{5 \times 10^{-7}}\right)10^{-8}$$

So the fringes will shift by 0.4 fringes (ie: 40%) when the apparatus is rotated.

However, no fringe shift is observed. The null result of the Michelson-Morley experiment is nowdays explained in terms of the constancy of the speed of light. The assumption that the light would have a velocity of $$c-v$$ and $$c+v$$ depending on the direction relative to the hypothetical "aether wind" is false, the light always travels at $$c$$ between two points in a vacuum and the speed of light is not affected by any "aether wind". This is because, in {special relativity} the Lorentz transforms induce a {length contraction}. Doing over the above calculations we obtain:


 * $$L_f=L_m{\sqrt{1-v^2/c^2}}$$

(taking into consideration the length contraction)

It is now easy to recalculate the difference $$\Delta t$$ between the times spent by the light in Path 1 and Path 2:


 * $$\Delta t = \frac{2}{c} \left(\frac{L_m}{\sqrt{1-v^2/c^2}}-\frac{L_f}{1-v^2/c^2}\right)=0$$ because $$L_f=L_m{\sqrt{1-v^2/c^2}}$$

If the apparatus is rotated by 90 degrees the new time difference is:


 * $$\Delta t^' = \frac{2}{c} \left(\frac{L_m}{1-v^2/c^2}-\frac{L_f}{\sqrt{1-v^2/c^2}}\right)=0$$

The interference fringes due to the time difference between the paths will be different after rotation if $$\Delta t$$ and $$\Delta t^'$$ are different.


 * $$\Delta t^' - \Delta t =0$$

Note: if the rest length $$L0_f \ne L_m$$ then $$L_f \ne L_m{\sqrt{1-v^2/c^2}}$$ and then $$\Delta t^' \ne 0$$ and $$ \Delta t \ne 0$$ and, more importantly, $$\Delta t^' - \Delta t \ne 0$$. This is why Michelson took great pains in equalizing the arms of the interferometer.

Wave propagation in moving medium
To date, it is pointed out that the medium of light in Michelson-Morley experiment is the air. And the velocity of medium is zero. Therefore,


 * $$t_1 = \frac{L}{c-v} + \frac{L_f}{c+v}=\frac{2L}{c}$$
 * $$t_2=\frac{2L}{c}$$
 * $$\Delta t=0$$

After apparatus rotated 90°, there is no interference movement.

Coherence length
The coherence length of light rays from a source that has wavelengths that differ by $$\Delta \lambda $$ is:


 * $$x = \frac{\lambda^2}{2 \pi \Delta \lambda}$$

If path lengths differ by more than this amount then interference fringes will not be observed. White light has a wide range of wavelengths and interferometers using white light must have paths that are equal to within a small fraction of a millimetre for interference to occur. This means that the ideal light source for a Michelson Interferometer should be monochromatic and the arms should be as near as possible equal in length.

The calculation of the coherence length is based on the fact that interference fringes become unclear when light rays are about 60 degrees (about 1 radian or one sixth of a wavelength ($$\approx 1/2\pi$$)) out of phase. This means that when two beams are:


 * $$\frac{\lambda}{2 \pi}$$

metres out of step they will no longer give a well defined interference pattern. Suppose a light beam contains two wavelengths of light, $$\lambda$$ and $$\lambda + \Delta \lambda$$, then in:


 * $$\frac{\lambda}{2 \pi \Delta \lambda}$$

cycles they will be $$\frac{\lambda}{2 \pi}$$ out of phase.

The distance required for the two different wavelengths of light to be this much out of phase is the coherence length. Coherence length = number of cycles x length of each cycle so:

coherence length = $$\frac{\lambda^2}{2 \pi \Delta \lambda}$$.

Lorentz-Fitzgerald Contraction Hypothesis
After the first Michelson-Morley experiments in 1881 there were several attempts to explain the null result. The most obvious point of attack is to propose that the path that is parallel to the direction of motion is contracted by $$\sqrt{1-v^2/c^2}$$ in which case $$\Delta t$$ and $$\Delta t^'$$ would be identical and no fringe shift would occur. This possibility was proposed in 1892 by Fitzgerald. Lorentz produced an "electron theory of matter" that would account for such a contraction.

Students sometimes make the mistake of assuming that the Lorentz-Fitzgerald contraction is equivalent to the Lorentz transformations. However, in the absence of any treatment of the time dilation effect the Lorentz-Fitzgerald explanation would result in a fringe shift if the apparatus is moved between two different velocities. The rotation of the earth allows this effect to be tested as the earth orbits the sun. Kennedy and Thorndike (1932) performed the Michelson-Morley experiment with a highly sensitive apparatus that could detect any effect due to the rotation of the earth; they found no effect. They concluded that both time dilation and Lorentz-Fitzgerald Contraction take place, thus confirming relativity theory.

If only the Lorentz-Fitzgerald contraction applied then the fringe shifts due to changes in velocity would be: $$n = (v_1^2 - v_2^2)/c^2 \times (L_f-L_m)/\lambda$$. Notice how the sensitivity of the experiment is dependent on the difference in path length $$L_f-L_m$$ and hence a long coherence length is required.

Recent Michelson-Morley experiments
Optical tests of the isotropy of the speed of light have become commonplace. New technologies, including the use of lasers and masers, have significantly improved measurement precision.

More recent experiments still, using other types of experiment such as optical resonators (Eisele et al. ), have shown that the speed of light is constant to within $$10^{-8}$$ m/s.