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[[File:Michelson morley experiment 1887.jpg|thumb|300px|Figure 1. Michelson and Morley's interferometric setup, mounted on a stone slab that floats in an annular trough of mercury.]]
The '''Michelson–Morley experiment''' was performed over the spring and summer of 1887 by [[Albert A. Michelson]] and [[Edward W. Morley]] at what is now [[Case Western Reserve University]] in [[Cleveland]], [[Ohio]], and published in November of the same year.<ref name=michel2/> It compared the speed of light in perpendicular directions, in an attempt to detect the [[relative motion]] of matter through the stationary [[luminiferous aether]] ("aether wind"). The result was negative, in that the expected difference between the speed of light in the direction of movement through the presumed aether, and the speed at right angles, was found not to exist; this result is generally considered to be the first strong evidence against the then-prevalent [[Aether theories|aether theory]], and initiated a line of research that eventually led to [[special relativity]], which rules out a stationary aether.<ref group=A name=staley/> The experiment has been referred to as "the moving-off point for the theoretical aspects of the Second Scientific Revolution".<ref group=A name=hoover/>

Michelson–Morley type experiments have been repeated many times with steadily increasing sensitivity. These include experiments from 1902 to 1905, and a series of experiments in the 1920s. More recent [[optical resonator]] experiments confirmed the absence of any aether wind at the 10<sup>−17</sup> level.<ref name=Eisele /><ref name=Herrmann2 /> Together with the [[Ives–Stilwell experiment|Ives–Stilwell]] and [[Kennedy–Thorndike experiment]]s, Michelson–Morley type experiments form one of the fundamental [[tests of special relativity]] theory.<ref name=rob group=A />

== Detecting the aether ==
[[Physics]] theories of the late 19th century assumed that just as surface water waves must have a supporting substance, i.e. a "medium", to move across (in this case water), and audible [[sound]] requires a medium to transmit its wave motions (such as air or water), so light must also require a medium, the "[[luminiferous aether]]", to transmit its wave motions. Because light can travel through a vacuum, it was assumed that even a vacuum must be filled with aether. Because the [[speed of light]] is so large, and because material bodies pass through the ''aether'' without obvious friction or drag, it was assumed to have a highly unusual combination of properties. Designing experiments to test the properties of the aether was a high priority of 19th century physics.<ref group=A name=Whittaker />{{rp|411ff}}

[[Earth]] orbits around the [[Sun]] at a speed of around {{Convert|30|km/s|4 = 2|abbr = on}}, or {{Convert| 108000|km/h|abbr = on}}. The Earth is in motion, so two main possibilities were considered: (1)&nbsp;The aether is stationary and only partially [[drag (physics)|dragged]] by Earth (proposed by [[Augustin-Jean Fresnel]] in 1818), or (2)&nbsp;the aether is completely dragged by Earth and thus shares its motion at Earth's surface (proposed by [[Sir George Stokes, 1st Baronet]] in 1844).<ref group=A name=Jan /> In addition, [[James Clerk Maxwell]] (1865) recognized the [[Electrodynamics|electromagnetic]] nature of light and developed what are now called [[Maxwell's equations]], but these equations were still interpreted as describing the motion of waves through an aether, whose state of motion was unknown. Eventually, Fresnel's idea of an (almost) stationary aether was preferred because it appeared to be confirmed by the [[Fizeau experiment]] (1851) and the [[aberration of light|aberration of star light]].<ref group=A name=Jan />

[[Image:aetherWind.svg|thumb|right|300px|Figure 2. A depiction of the concept of the "aether wind"]]
According to this hypothesis, Earth and the aether are in relative motion, implying that a so-called "aether wind" (Fig.&nbsp;2) should exist. Although it would be possible, in theory, for the Earth's motion to match that of the aether at one moment in time, it was not possible for the Earth to remain at rest with respect to the aether at all times, because of the variation in both the direction and the speed of the motion. 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 analyzing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the aether. The expected relative difference in the measured speed of light was quite small, given that the velocity of the Earth in its orbit around the Sun has a magnitude of about one hundredth of one percent of the speed of light.<ref group=A name=Whittaker />{{rp|417ff}}

During the mid-19th century, measurements of aether wind effects of first order, i.e. effects proportional to ''v''/''c'' (''v'' being Earth's velocity, ''c'' the speed of light) were thought to be possible, but no direct measurement of the speed of light was possible with the accuracy required. For instance, the [[Fizeau–Foucault apparatus]] could measure the speed of light to perhaps 5% accuracy, which was quite inadequate for measuring directly a first-order 0.01% change in the speed of light. A number of physicists therefore attempted to make measurements of indirect first-order effects not of the speed of light itself, but of variations in the speed of light (see [[Luminiferous aether#First order experiments|First order aether-drift experiments]]). The [[Hoek experiment]], for example, was intended to detect [[interferometry|interferometric]] [[fringe shift]]s due to speed differences of oppositely propagating light waves through water at rest. The results of such experiments were all negative.<ref group="A" name="laub" /> This could be explained by using [[Aether drag hypothesis#Partial aether dragging|Fresnel's dragging coefficient]], according to which the aether and thus light are partially dragged by moving matter. Partial aether-dragging would thwart attempts to measure any first order change in the speed of light. As pointed out by Maxwell (1878), only experimental arrangements capable of measuring second order effects would have any hope of detecting aether drift, i.e. effects proportional to ''v''<sup>2</sup>/''c''<sup>2</sup>.<ref group=A name=maxa /><ref group=A name=maxb /> Existing experimental setups, however, were not sensitive enough to measure effects of that size.

== 1881 and 1887 experiments ==

=== Michelson experiment (1881) ===
[[File:Michelson1881c.png|thumb|300px|Figure 3. Michelson's 1881 interferometer. Although ultimately it proved incapable of distinguishing between differing theories of aether-dragging, its construction provided important lessons for the design of Michelson and Morley's 1887 instrument.<ref group=note>Among other lessons was the need to control for vibration. Michelson (1881) wrote: "...&nbsp;owing to the extreme sensitiveness of the instrument to vibrations, the work could not be carried on during the day. Next, the experiment was tried at night. When the mirrors were placed half-way on the arms the fringes were visible, but their position could not be measured till after twelve o'clock, and then only at intervals. When the mirrors were moved out to the ends of the arms, the fringes were only occasionally visible. It thus appeared that the experiments could not be performed in Berlin, and the apparatus was accordingly removed to the Astrophysicalisches Observatorium in Potsdam&nbsp;... Here, the fringes under ordinary circumstances were sufficiently quiet to measure, but so extraordinarily sensitive was the instrument that the stamping of the pavement, about 100 meters from the observatory, made the fringes disappear entirely!"</ref>]]
{{Wikisource|The Relative Motion of the Earth and the Luminiferous Ether|The Relative Motion of the Earth and the Luminiferous Ether (1881)}} 
Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether flow. In 1877, while teaching at his alma mater, the [[United States Naval Academy]] in Annapolis, Michelson conducted his first known light speed experiments as a part of a classroom demonstration. In 1881, he left active U.S. Naval service while in Germany concluding his studies. In that year, Michelson used a prototype experimental device to make several more measurements.

The device he designed, later known as a [[Michelson interferometer]], sent [[yellow]] light from a [[sodium]] flame (for alignment), or [[white]] light (for the actual observations), through a [[beam splitter|half-silvered mirror]] that was used to split it into two beams traveling at right angles to one another. After leaving the splitter, the beams traveled out to the ends of long arms where they were reflected back into the middle by small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive [[Interference (wave propagation)|interference]] whose transverse displacement would depend on the relative time it takes light to transit the longitudinal ''vs.'' the transverse arms. If the Earth is traveling through an aether medium, a beam reflecting back and forth parallel to the flow of aether would take longer than a beam reflecting perpendicular to the aether because the time gained from traveling downwind is less than that lost traveling upwind. Michelson expected that the Earth's motion would produce a [[fringe shift]] equal to 0.04 fringes—that is, of the separation between areas of the same intensity. He did not observe the expected shift; the greatest average deviation that he measured (in the northwest direction) was only 0.018 fringes; most of his measurements were much less. His conclusion was that Fresnel's hypothesis of a stationary aether with partial aether dragging would have to be rejected, and thus he confirmed Stokes' hypothesis of complete aether dragging.<ref name=michel1/>

However, [[Alfred Potier]] (and later [[Hendrik Lorentz]]) pointed out to Michelson that he had made an error of calculation, and that the expected fringe shift should have been only 0.02 fringes.  Michelson's apparatus was subject to experimental errors far too large to say anything conclusive about the aether wind. Definitive measurement of the aether wind would require an experiment with greater accuracy and better controls than the original. Nevertheless, the prototype was successful in demonstrating that the basic method was feasible.<ref group=A name=Jan /><ref group=A name=AIMiller />

=== Michelson–Morley experiment (1887) ===
{{Wikisource|On the Relative Motion of the Earth and the Luminiferous Ether|On the Relative Motion of the Earth and the Luminiferous Ether (1887)}}
[[Image:On the Relative Motion of the Earth and the Luminiferous Ether - Fig 4.png|300px|thumb|right|Figure 5. This figure illustrates the folded light path used in the Michelson–Morley interferometer that enabled a path length of 11 m. ''a'' is the light source, an oil lamp. ''b'' is a beam splitter. ''c'' is a compensating plate so that both the reflected and transmitted beams travel through the same amount of glass (important since experiments were run with white light which has an extremely short [[coherence length]] requiring precise matching of optical path lengths for fringes to be visible; monochromatic sodium light was used only for initial alignment<ref name=michel1/><ref group=note>Michelson (1881) wrote: "...&nbsp;a sodium flame placed at ''a'' produced at once the interference bands. These could then be altered in width, position, or direction, by a slight movement of the plate ''b'', and when they were of convenient width and of maximum sharpness, the sodium flame was removed and the lamp again substituted. The screw ''m'' was then slowly turned till the bands reappeared. They were then of course colored, except the central band, which was nearly black."</ref>). ''d'', ''d' '' and ''e'' are mirrors. ''e' '' is a fine adjustment mirror. ''f'' is a telescope.]]

In 1885, Michelson began a collaboration with [[Edward Morley]], spending considerable time and money to confirm with higher accuracy [[Fizeau experiment|Fizeau's 1851 experiment]] on Fresnel's drag coefficient,<ref name=michel1a /> to improve on Michelson's 1881 experiment,<ref name=michel2 /> and to establish the wavelength of light as a [[Length measurement|standard of length]].<ref name=michel3 /><ref name=michel4 /> At this time Michelson was professor of physics at the Case School of Applied Science, and Morley was professor of chemistry at Western Reserve University (WRU), which shared a campus with the Case School on the eastern edge of Cleveland. Michelson suffered a [[nervous breakdown]] in September 1885, from which he recovered by October 1885. Morley ascribed this breakdown to the intense work of Michelson during the preparation of the experiments. In 1886, Michelson and Morley successfully confirmed Fresnel's drag coefficient – this result was also considered as a confirmation of the stationary aether concept.<ref group=A name=staley />

This result strengthened their hope of finding the aether wind. Michelson and Morley created an improved version of the Michelson experiment with more than enough accuracy to detect this hypothetical effect. The experiment was performed in several periods of concentrated observations between April and July 1887, in the basement of Adelbert Dormitory of WRU (later renamed Pierce Hall, demolished in 1962).<ref group=A name=Fickinger /><ref group=A name=hamerla />

As shown in Fig.&nbsp;5, the light was repeatedly reflected back and forth along the arms of the interferometer, increasing the path length to {{Convert|11|m|abbr = on}}. At this length, the drift would be about 0.4 fringes. To make that easily detectable, the apparatus was assembled in a closed room in the basement of the heavy stone dormitory, eliminating most thermal and vibrational effects. Vibrations were further reduced by building the apparatus on top of a large block of sandstone (Fig.&nbsp;1), about a foot thick and five feet square, which was then floated in a circular trough of mercury. They estimated that effects of about 0.01 fringe would be detectable.

[[File:MichelsonCoinAirLumiereBlanche.JPG|thumb|Figure 6. Fringe pattern produced with a Michelson interferometer using white light. As configured here, the central fringe is white rather than black.]]
Michelson and Morley and other early experimentalists using interferometric techniques in an attempt to measure the properties of the luminiferous aether, used (partially) monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Purely monochromatic light would result in a uniform fringe pattern. Lacking modern means of [[air conditioning|environmental temperature control]], experimentalists struggled with continual fringe drift even when the interferometer was set up in a basement. Because the fringes would occasionally disappear due to vibrations caused by passing horse traffic, distant thunderstorms and the like, an observer could easily "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, far outweighed the difficulties of aligning the apparatus due to its low [[coherence length]]. As [[Dayton Miller]] wrote, "White light fringes were chosen for the observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all readings."<ref group=A name=Miller1933/><ref group=note>If one uses a half-silvered mirror as the beam splitter, the reflected beam will undergo a different number of front-surface reflections than the transmitted beam. At each front-surface reflection, the light will undergo a phase inversion. Because the two beams undergo a different number of phase inversions, when the path lengths of the two beams match or differ by an integral number of wavelengths (e.g. 0, 1, 2&nbsp;...), there will be destructive interference and a weak signal at the detector. If the path lengths of the beams differ by a half-integral number of wavelengths (e.g., 0.5, 1.5, 2.5&nbsp;...), constructive interference will yield a strong signal. The results are opposite if a cube beam-splitter is used, because a cube beam-splitter makes no distinction between a front- and rear-surface reflection.</ref> Use of partially monochromatic light (yellow sodium light) during initial alignment enabled the researchers to locate the position of equal path length, more or less easily, before switching to white light.<ref group=note>Sodium light produces a fringe pattern that displays cycles of fuzziness and sharpness that repeat every several hundred fringes over a distance of approximately a millimeter. This pattern is due to the yellow sodium D line being actually a doublet, the individual lines of which have a limited [[coherence length]]. After aligning the interferometer to display the centermost portion of the sharpest set of fringes, the researcher would switch to white light.</ref>

The mercury trough allowed the device to turn with close to zero friction, so that once having given the sandstone block a single push it would slowly rotate through the entire range of possible angles to the "aether wind," while measurements were continuously observed by looking through the eyepiece. The hypothesis of aether drift implies that because one of the arms would inevitably turn into the direction of the wind at the same time that another arm was turning perpendicularly to the wind, an effect should be noticeable even over a period of minutes.

The expectation was that the effect would be graphable as a sine wave with two peaks and two troughs per rotation of the device. This result could have been expected because during each full rotation, each arm would be parallel to the wind twice (facing into and away from the wind giving identical readings) and perpendicular to the wind twice. Additionally, due to the Earth's rotation, the wind would be expected to show periodic changes in direction and magnitude during the course of a [[sidereal day]].

Because of the motion of the Earth around the Sun, the measured data were also expected to show annual variations.

=== Most famous "failed" experiment ===
[[File:Michelson Morley 1887 Figure 6.png|thumb|300px|Figure 7. Michelson and Morley's results. The upper solid line is the curve for their observations at noon, and the lower solid line is that for their evening observations. Note that the theoretical curves and the observed curves are not plotted at the same scale: the dotted curves, in fact, represent only ''one-eighth of the theoretical displacements.'']]
After all this thought and preparation, the experiment became what has been called the most famous failed experiment in history.<ref group=A name=blum /> Instead of providing insight into the properties of the aether, Michelson and Morley's article in the ''[[American Journal of Science]]'' reported the measurement to be as small as one-fortieth of the expected displacement (Fig.&nbsp;7), but "since the displacement is proportional to the square of the velocity" they concluded that the measured velocity was "probably less than one-sixth" of the expected velocity of the Earth's motion in orbit and "certainly less than one-fourth."<ref name=michel2 /> Although this small "velocity" was measured, it was considered far too small to be used as evidence of speed relative to the aether, and it was understood to be within the range of an experimental error that would allow the speed to actually be zero.<ref group=A name=staley /> For instance, Michelson wrote about the "decidedly negative result" in a letter to [[Lord Rayleigh]] in August 1887:<ref group=A name=shankland2 />

{{Quote|The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been 0.40 of a fringe – the maximum displacement was 0.02 and the average much less than 0.01 – and then not in the right place. As displacement is proportional to squares of the relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth’s velocity.|Albert Abraham Michelson, 1887}}

From the standpoint of the then current aether models, the experimental results were conflicting. The [[Fizeau experiment]] and its 1886 repetition by Michelson and Morley apparently confirmed the stationary aether with partial aether dragging, and refuted complete aether dragging. On the other hand, the much more precise Michelson–Morley experiment (1887) apparently confirmed complete aether dragging and refuted the stationary aether.<ref group=A name=Jan /> In addition, the Michelson–Morley null result was further substantiated by the null results of other second-order experiments of different kind, namely the [[Trouton–Noble experiment]] (1903) and the [[Experiments of Rayleigh and Brace]] (1902–1904). These problems and their solution led to the development of the [[Lorentz transformation]] and [[special relativity]].

After the "failed" experiment Michelson and Morley ceased their aether drift measurements and started to use their newly developed technique to establish the wavelength of light as a [[Length measurement|standard of length]].<ref name=michel3 /><ref name=michel4 />

== Light path analysis and consequences ==

=== Observer resting in the aether ===
{{multiple image
| align     = right
| direction = vertical
| width     = 300

| image1    = Michelson-morley calculations.svg
| alt1      = Graphical presentation of the expected differential phase shifts in the Michelson–Morley apparatus

| image2    = MichelsonMorleyAnimationDE.gif
| alt2      = Animated presentation of the expected differential phase shifts
| caption2  = Figure 4. Expected differential phase shift between light traveling the longitudinal versus the transverse arms of the Michelson–Morley apparatus
}}

The beam travel time in the longitudinal direction can be derived as follows:<ref group=A name=Feynman /> Light is sent from the source and propagates with the speed of light <math>c</math> in the aether. It passes through the half-silvered mirror at the origin at <math>T=0</math>.  The reflecting mirror is at that moment at distance <math>L</math> (the length of the interferometer arm) and is moving with velocity <math>v</math>. The beam hits the mirror at time <math>T_1</math> and thus travels the distance <math>cT_1</math>. At this time, the mirror has traveled the distance <math>vT_1</math>. Thus <math>cT_1 =L+vT_1</math> and consequently the travel time <math>T_1=L/(c-v)</math>. The same consideration applies to the backward journey, with the sign of <math>v</math> reversed, resulting in <math>cT_2 =L-vT_2</math> and <math>T_2 =L/(c+v)</math>. The total travel time <math>T_{l}=T_{1}+T_{2}</math> is:

:<math>T_l=\frac{L}{c-v}+\frac{L}{c+v} =\frac{2L}{c}\frac{1}{1-\frac{v^{2}}{c^2}} \approx\frac{2L}{c}\left(1+\frac{v^2}{c^2}\right)</math>

Michelson obtained this expression correctly in 1881, however, in transverse direction he obtained the incorrect expression

:<math>T_{t}=\frac{2L}{c},</math>

because he overlooked that the aether wind also affects the transverse beam travel time. This was corrected by [[Alfred Potier]] (1882) and Lorentz (1886). The derivation in the transverse direction can be given as follows (analoguous to the derivation of [[time dilation]] using a [[Time dilation#Simple inference of time dilation due to relative velocity|light clock]]): The beam is propagating at the speed of light <math>c</math> and hits the mirror at time <math>T_3</math>, traveling the distance <math>cT_3</math>. At the same time, the mirror has traveled the distance <math>vT_3</math> in the ''x'' direction. So in order to hit the mirror, the travel path of the beam is <math>L</math> in the ''y'' direction (assuming equal-length arms) and <math>vT_3</math> in the ''x'' direction. This inclined travel path follows from the transformation from the interferometer rest frame to the aether rest frame. Therefore, the [[Pythagorean theorem]] gives the actual beam travel distance of <math>\scriptstyle \sqrt{L^{2}+\left(vT_{3}\right)^{2}}</math>. Thus <math>\scriptstyle cT_{3} =\sqrt{L^{2}+\left(vT_{3}\right)^{2}}</math> and consequently the travel time <math>\scriptstyle T_{3} =L/\sqrt{c^{2}-v^{2}}</math>, which is the same for the backward journey. The total travel time <math>T_{t}=2T_{3}</math> is:

:<math>T_t=\frac{2L}{\sqrt{c^2-v^2}}=\frac{2L}{c}\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\approx\frac{2L}{c}\left(1+\frac{v^2}{2c^2}\right)</math>

The time difference between ''T<sub>l</sub>'' and ''T<sub>t</sub>'' before rotation is given by<ref group=A>{{cite book |author=Albert Shadowitz  |title=Special relativity |isbn=0-486-65743-4 |publisher=Courier Dover Publications |edition=Reprint of 1968 |year=1988|pages=159–160}}</ref>

:<math>T_l-T_t=\frac{2}{c}\left(\frac{L}{1-\frac{v^2}{c^2}}-\frac{L}{\sqrt{1-\frac{v^2}{c^2}}}\right).</math>

By multiplying with ''c'', the corresponding length difference before rotation is

:<math>\Delta_1=2\left(\frac{L}{1-\frac{v^2}{c^2}}-\frac{L}{\sqrt{1-\frac{v^2}{c^2}}}\right),</math>

and after rotation

:<math>\Delta_2=2\left(\frac{L}{\sqrt{1-\frac{v^2}{c^2}}}-\frac{L}{1-\frac{v^2}{c^2}}\right).</math>

Dividing <math>\Delta_{1}-\Delta_{2}</math> by the [[wavelength]] λ, the fringe shift ''n'' is found:

:<math>n=\frac{\Delta_1-\Delta_2}{\lambda}\approx\frac{2Lv^2}{\lambda c^2}.</math>

Since ''L''&nbsp;≈&nbsp;11 meters and λ≈500 [[nanometer]]s, the expected [[fringe shift]] was ''n'' ≈0.44. So the result would be a delay in one of the light beams that could be detected when the beams were recombined through interference. Any slight change in the spent time would then be observed as a shift in the positions of the interference fringes. The negative result led Michelson to the conclusion that there is no measurable aether drift.<ref name=michel2 />

=== Observer comoving with the interferometer ===
If the same situation is described from the view of an observer co-moving with the interferometer, then the effect of aether wind is similar to the effect experienced by a swimmer, who tries to move with velocity <math>c</math> against a river flowing with velocity <math>v</math>.<ref group=A name=teller />

In the longitudinal direction the swimmer first moves upstream, so his velocity is diminished due to the river flow to <math>c-v</math>. On his way back moving downstream, his velocity is increased to <math>c+v</math>. This gives the beam travel times <math>T_1</math> and  <math>T_2</math> as mentioned above.

In the transverse direction, the swimmer has to compensate for the river flow by moving at a certain angle against the flow direction, in order to sustain his exact transverse direction of motion and to reach the other side of the river at the correct location. This diminishes his speed to <math>\sqrt{c^{2}-v^{2}}</math>, and gives the beam travel time <math>T_3</math> as mentioned above.

=== Mirror reflection ===
The classical analysis predicted a relative phase shift between the longitudinal and transverse beams which in Michelson and Morley's apparatus should have been readily measurable. What is not often appreciated (since there was no means of measuring it), is that motion through the hypothetical aether should also have caused the two beams to diverge as they emerged from the interferometer by about 10<sup>−8</sup> radians.<ref group=A name=schum94 />

For an apparatus in motion, the classical analysis requires that the beam-splitting mirror be slightly offset from an exact 45° if the longitudinal and transverse beams are to emerge from the apparatus exactly superimposed. In the relativistic analysis, Lorentz-contraction of the beam splitter in the direction of motion causes it to become more perpendicular by precisely the amount necessary to compensate for the angle discrepancy of the two beams.<ref group=A name=schum94 />

=== Length contraction and Lorentz transformation ===
{{Further|History of special relativity|History of Lorentz transformations}}
A first step to explaining the Michelson and Morley experiment's null result was found in the [[Length contraction|FitzGerald–Lorentz contraction hypothesis]], now simply called length contraction or Lorentz contraction, first proposed by [[George FitzGerald]] (1889) and [[Hendrik Lorentz]] (1892).<ref group=A name=lorentz95 /> According to this law all objects physically contract by <math>L/\gamma</math> along the line of motion (originally thought to be relative to the aether), <math>\gamma=1/\sqrt{1-v^{2}/c^{2}}</math> being the [[Lorentz factor]]. This hypothesis was partly motivated by [[Oliver Heaviside]]'s discovery in 1888, that electrostatic fields are contracting in the line of motion. But since there was no reason at that time to assume that binding forces in matter are of electric origin, length contraction of matter in motion with respect to the aether was considered an [[Ad hoc hypothesis]].<ref group=A name=AIMiller />

If length contraction of <math>L</math> is inserted into the above formula for <math>T_{l}</math>, then the light propagation time in the longitudinal direction becomes equal to that in the transverse direction:

:<math>T_l=\frac{2L\sqrt{1-\frac{v^2}{c^2}}}{c}\frac{1}{1-\frac{v^2}{c^2}}=\frac{2L}{c} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=T_t</math>

However, length contraction is only a special case of the more general relation, according to which the transverse length is larger than the longitudinal length by the ratio <math>\gamma</math>. This can be achieved in many ways. If <math>L_{1}</math> is the moving longitudinal length and <math>L_{2}</math> the moving transverse length, <math>L'_{1}=L'_{2}</math> being the rest lengths, then it is given:<ref group=A name=lorentz04 />

:<math>\frac{L_2}{L_1}=\frac{L'_2}{\phi}\left/\frac{L'_1}{\gamma\phi}\right.=\gamma.</math>

<math>\phi</math> can be arbitrarily chosen, so there are infinitely many combinations to explain the Michelson–Morley null result. For instance, if <math>\phi=1</math> the relativistic value of length contraction of <math>L_1</math> occurs, but if <math>\phi=1/\gamma</math> then no length contraction but an elongation of <math>L_2</math> occurs. This hypothesis was later extended by [[Joseph Larmor]] (1897), Lorentz (1904) and [[Henri Poincaré]] (1905), who developed the complete [[Lorentz transformation]] including [[time dilation]] in order to explain the [[Trouton–Noble experiment]], the [[Experiments of Rayleigh and Brace]], and [[Kaufmann–Bucherer–Neumann experiments|Kaufmann's experiments]]. It has the form

:<math>x'=\gamma\phi(x-vt),\ y'=\phi y,\ z'=\phi z,\ t'=\gamma\phi\left(t-\frac{vx}{c^2}\right)</math>

It remained to define the value of <math>\phi</math>, which was shown by Lorentz (1904) to be unity.<ref group=A name=lorentz04 /> In general, Poincaré (1905)<ref group=A name=poincare05 /> demonstrated that only <math>\phi=1</math> allows this transformation to form a [[Lorentz group|group]], so it is the only choice compatible with the [[principle of relativity]], ''i.e.'' making the stationary aether undetectable. Given this, length contraction and time dilation obtain their exact relativistic values.

=== Special relativity ===
[[Albert Einstein]] formulated the theory of [[special relativity]] by 1905, deriving the Lorentz transformation and thus length contraction and time dilation from the relativity postulate and the constancy of the speed of light, thus removing the ''ad hoc'' character from the contraction hypothesis. Einstein emphasized the [[Kinematics|kinematic]] foundation of the theory and the modification of the notion of space and time, with the stationary aether no longer playing any role in his theory. He also pointed out the group character of the transformation. Einstein was motivated by [[Maxwell's theory of electromagnetism]] (in the form as it was given by Lorentz in 1895) and the lack of evidence for the [[luminiferous aether]].<ref group=A name=einstein />

This allows a more elegant and intuitive explanation of the Michelson-Morley null result. In a comoving frame the null result is self-evident, since the apparatus can be considered as at rest in accordance with the relativity principle, thus the beam travel times are the same. In a frame relative to which the apparatus is moving, the same reasoning applies as described above in "Length contraction and Lorentz transformation", except the word "aether" has to be replaced by "non-comoving inertial frame". Einstein wrote in 1916:<ref group=A name=einstein2 />

{{Quote|Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result — a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a "specially favoured" (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun.|Albert Einstein, 1916}}

The extent to which the null result of the Michelson–Morley experiment influenced Einstein is disputed. Alluding to some statements of Einstein, many historians argue that it played no significant role in his path to special relativity,<ref group=A name=stachel /><ref group=A name=Polanyi /> while other statements of Einstein probably suggest that he was influenced by it.<ref group=A name=dongen /> In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.<ref group=A name=stachel />

It was later shown by [[Howard Percy Robertson]] (1949) and others<ref name=rob group=A /><ref name=sexl group=A /> (see [[Test theories of special relativity|Robertson–Mansouri–Sexl test theory]]), that it is possible to derive the Lorentz transformation entirely from the combination of three experiments. First, the Michelson–Morley experiment showed that the speed of light is independent of the ''orientation'' of the apparatus, establishing the relationship between longitudinal (β) and transverse (δ) lengths. Then in 1932, Roy Kennedy and Edward Thorndike modified the Michelson–Morley experiment by making the path lengths of the split beam unequal, with one arm being very short.<ref name=KennedyThorndike/> The [[Kennedy–Thorndike experiment]] took place for many months as the Earth moved around the sun. Their negative result showed that the speed of light is independent of the ''velocity'' of the apparatus in different inertial frames. In addition it established that besides length changes, corresponding time changes must also occur, i.e. it established the relationship between longitudinal lengths (β) and time changes (α). So both experiments do not provide the individual values of these quantities. This uncertainty corresponds to the undefined factor <math>\phi</math> as described above. It was clear due to theoretical reasons (the [[Lorentz group|group character]] of the Lorentz transformation as required by the relativity principle) that the individual values of length contraction and time dilation must assume their exact relativistic form. But a direct measurement of one of these quantities was still desirable to confirm the theoretical results. This was achieved by the [[Ives–Stilwell experiment]] (1938), measuring α in accordance with time dilation. Combining this value for α with the Kennedy–Thorndike null result shows that β must assume the value of relativistic length contraction. Combining β with the Michelson–Morley null result shows that δ must be zero. Therefore, the Lorentz transformation with <math>\phi=1</math> is an unavoidable consequence of the combination of these three experiments.<ref name=rob group=A />

Special relativity is generally considered the solution to all negative aether drift (or [[isotropy]] of the speed of light) measurements, including the Michelson–Morley null result. Many high precision measurements have been conducted as [[tests of special relativity]] and [[modern searches for Lorentz violation]] in the [[photon]], [[electron]], [[nucleon]], or [[neutrino]] sector, all of them confirming relativity.

=== Incorrect alternatives ===
As mentioned above, Michelson initially believed that his experiment would confirm Stokes' theory, according to which the aether was fully dragged in the vicinity of the earth (see [[Aether drag hypothesis]]). However, complete aether drag contradicts the observed [[aberration of light]] and was contradicted by other experiments as well. In addition, Lorentz showed in 1886 that Stokes's attempt to explain aberration is contradictory.<ref group=A name=Jan /><ref group=A name=Whittaker />

Furthermore, the assumption that the aether is not carried in the vicinity, but only ''within'' matter, was very problematic as shown by the [[Hammar experiment]] (1935). Hammar directed one leg of his interferometer through a heavy metal pipe plugged with lead. If aether were dragged by mass, it was theorized that the mass of the sealed metal pipe would have been enough to cause a visible effect. Once again, no effect was seen, so aether-drag theories are considered to be disproven.

[[Walther Ritz]]'s [[emission theory]] (or ballistic theory), was also consistent with the results of the experiment, not requiring aether. The theory postulates that light has always the same velocity in respect to the source.<ref name=norton group=A /> However [[De Sitter double star experiment|de Sitter]] noted that emitter theory predicted several optical effects that were not seen in observations of binary stars in which the light from the two stars could be measured in a [[spectrometer]]. If emission theory were correct, the light from the stars should experience unusual fringe shifting due to the velocity of the stars being added to the speed of the light, but no such effect could be seen. It was later shown by [[J. G. Fox]] that the original de Sitter experiments were flawed due to [[Extinction theorem of Ewald and Oseen|extinction]],<ref name=fox65>{{Citation|last=Fox |first=J. G.|title=Evidence Against Emission Theories|journal=American Journal of Physics|volume=33|issue=1|year=1965|pages=1–17|doi=10.1119/1.1971219|postscript=.|bibcode = 1965AmJPh..33....1F }}</ref> but in 1977 Brecher observed X-rays from binary star systems with similar null results.<ref>{{Cite journal|last=Brecher |first=K.|title=Is the speed of light independent of the velocity of the source|journal=Physical Review Letters|volume=39|year=1977|pages=1051–1054|doi=10.1103/PhysRevLett.39.1051|postscript=.|bibcode=1977PhRvL..39.1051B|issue=17}}</ref> Also terrestrial tests using [[particle accelerator]]s have been made that were inconsistent with source dependence of the speed of light.<ref name=FilippasFox>{{cite journal|last=Filippas |first=T.A. |last2=Fox |first2=J.G.|title=Velocity of Gamma Rays from a Moving Source|journal=Physical Review|year=1964|volume=135|issue=4B|pages=B1071-1075|bibcode = 1964PhRv..135.1071F |doi = 10.1103/PhysRev.135.B1071 }}</ref> In addition, [[Emission theory]] might fail the [[Ives–Stilwell experiment]], but Fox questioned that as well.

== Subsequent experiments ==
[[File:Illingworth simulation.png|thumb|250px|Figure 8. Simulation of the Kennedy/Illingworth refinement of the Michelson–Morley experiment. (a) Michelson–Morley interference pattern in monochromatic mercury light, with a dark fringe precisely centered on the screen. (b) The fringes have been shifted to the left by 1/100 of the fringe spacing. It is extremely difficult to see any difference between this figure and the one above. (c) A small step in one mirror causes two views of the same fringes to be spaced 1/20 of the fringe spacing to the left and to the right of the step. (d) A telescope has been set to view only the central dark band around the mirror step. Note the symmetrical brightening about the center line. (e) The two sets of fringes have been shifted to the left by 1/100 of the fringe spacing. An abrupt discontinuity in luminosity is visible across the step.]]
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.<ref group=A name=Swenson1 /><ref group=A name=Swenson2 /> Morley was not convinced of his own results, and went on to conduct additional experiments with [[Dayton Miller]] from 1902 to 1904. Again, the result was negative within the margins of error.<ref name=morley1 /><ref name=morley2/>

Miller worked on increasingly larger interferometers, culminating in one with a {{Convert|32|m|adj = on|sp = us}} (effective) arm length that he tried at various sites, including on top of a mountain at the [[Mount Wilson Observatory]]. To avoid the possibility of the aether wind being blocked by solid walls, his mountaintop observations used a special shed with thin walls, mainly of canvas. From noisy, irregular data, he consistently extracted a small positive signal that varied with each rotation of the device, with the [[Sidereal time|sidereal day]], and on a yearly basis. His measurements in the 1920s amounted to approximately {{Convert|10|km/s|abbr = on}} instead of the nearly {{Convert|30|km/s|4 = 1|abbr = on}} expected from the Earth's orbital motion alone. He remained convinced this was due to [[aether drag hypothesis#Partial aether dragging|partial entrainment or aether dragging]], though he did not attempt a detailed explanation. He ignored critiques demonstrating the inconsistency of his results and the refutation by the [[Hammar experiment]].<ref group=A name=Thirring /><ref group=note name=Thirring>Thirring (1926) as well as Lorentz pointed out that Miller's results failed even the most basic criteria required to believe in their celestial origin, namely that the azimuth of supposed drift should exhibit daily variations consistent with the source rotating about the celestial pole. Instead, while Miller's observations showed daily variations, their oscillations in one set of experiments might center, say, around a northwest–southeast line.</ref> Miller's findings were considered important at the time, and were discussed by Michelson, [[Hendrik Lorentz|Lorentz]] and others at a meeting reported in 1928.<ref group=A name=michel1928 /> There was general agreement that more experimentation was needed to check Miller's results. Miller later built a non-magnetic device to eliminate [[magnetostriction]], while Michelson built one of non-expanding [[Invar]] to eliminate any remaining thermal effects. Other experimenters from around the world increased accuracy, eliminated possible side effects, or both. So far, no one has been able to replicate Miller's results, and modern experimental accuracies have ruled them out.<ref group=A name=shankland /> Roberts (2006) has pointed out that the primitive data reduction techniques used by Miller and other early experimenters, including Michelson and Morley, were capable of ''creating'' apparent periodic signals even when none existed in the actual data. After reanalyzing Miller's original data using modern techniques of quantitative error analysis, Roberts found Miller's apparent signals to be statistically insignificant.<ref name=Roberts2006 group=A>{{cite web |last=Roberts |first=T.J. |title=An Explanation of Dayton Miller's Anomalous "Ether Drift" Result |url=http://arxiv.org/abs/physics/0608238 |year=2006 |accessdate=7 May 2012}}</ref>

Using a special optical arrangement involving a 1/20 wave step in one mirror, Roy J. Kennedy (1926) and K.K. Illingworth (1927) (Fig.&nbsp;8) converted the task of detecting fringe shifts from the relatively insensitive one of estimating their lateral displacements to the considerably more sensitive task of adjusting the light intensity on both sides of a sharp boundary for equal luminance.<ref name=Kennedy/><ref name=Illingworth/> If they observed unequal illumination on either side of the step, such as in Fig.&nbsp;8e, they would add or remove calibrated weights from the interferometer until both sides of the step were once again evenly illuminated, as in Fig.&nbsp;8d. The number of weights added or removed provided a measure of the fringe shift. Different observers could detect changes as little as 1/300 to 1/1500 of a fringe. Kennedy also carried out an experiment at Mount Wilson, finding only about 1/10 the drift measured by Miller and no seasonal effects.<ref group=A name=michel1928 />

In 1930, [[Georg Joos]] conducted an experiment using an automated interferometer with {{Convert|21|m|ft|-long|sp = us|adj = mid}} arms forged from pressed quartz having very low thermal coefficient of expansion, that took continuous photographic strip recordings of the fringes through dozens of revolutions of the apparatus. Displacements of 1/1000 of a fringe could be measured on the photographic plates. No periodic fringe displacements were found, placing an upper limit to the aether wind of {{Convert|1.5|km/s|abbr = on}}.<ref name=joos />

In the table below, the expected values are related to the relative speed between Earth and Sun of {{Convert|30|km/s|4 = 1|abbr = on}}. With respect to the speed of the solar system around the galactic center of about {{Convert|220|km/s|abbr = on}}, or the speed of the solar system relative to the [[cosmic microwave background radiation#CMBR dipole anisotropy|CMB rest frame]] of about {{Convert|368|km/s|abbr = on}}, the null results of those experiments are even more obvious.

{| class="wikitable"
|-
!Name
!Location
!Year
!Arm length (meters)
!Fringe shift expected
!Fringe shift measured
!Ratio
!Upper Limit on V<sub>aether</sub>
!Experimental Resolution
!Null result
|-
|Michelson<ref name=michel1 />||[[Potsdam]]||1881||1.2||0.04||≤ 0.02||2||∼ 20&nbsp;km/s||0.02||<math>\approx</math> yes
|-
|Michelson and Morley<ref name=michel2 />||[[Cleveland]]||1887||11.0||0.4||< 0.02<br />or ≤ 0.01|| 40||∼ 4–8&nbsp;km/s||0.01||<math>\approx</math> yes
|-
|Morley and Miller<ref name=morley1 /><ref name=morley2 />||[[Cleveland]]||1902–1904||32.2||1.13||≤ 0.015||80||∼ 3.5&nbsp;km/s||0.015||yes
|-
|Miller<ref name=mill />||[[Mount Wilson (California)|Mt. Wilson]]||1921||32.0||1.12||≤ 0.08||15||∼ 8–10&nbsp;km/s||unclear||unclear
|-
|Miller<ref name=mill/>||[[Cleveland]]||1923–1924||32.0||1.12||≤ 0.03||40||∼ 5&nbsp;km/s||0.03||yes
|-
|Miller <small>(sunlight)</small><ref name=mill/>||[[Cleveland]]||1924||32.0||1.12||≤ 0.014||80||∼ 3&nbsp;km/s||0.014||yes
|-
|[[Rudolf Tomaschek|Tomaschek]] <small>(star light)</small><ref name=Tomaschek />||[[Heidelberg]]||1924||8.6||0.3||≤ 0.02||15||∼ 7&nbsp;km/s||0.02||yes
|-
|Miller<ref name=mill/><ref group=A name=Miller1933 />||[[Mount Wilson (California)|Mt. Wilson]]||1925–1926||32.0||1.12||≤ 0.088||13||∼ 8–10&nbsp;km/s||unclear||unclear
|-
|Kennedy<ref name=Kennedy />||[[Pasadena, California|Pasadena]]/[[Mount Wilson (California)|Mt. Wilson]]||1926||2.0||0.07||≤ 0.002||35||∼ 5&nbsp;km/s||0.002||yes
|-
|Illingworth<ref name=Illingworth />||[[Pasadena, California|Pasadena]]||1927||2.0||0.07||≤ 0.0004||175||∼ 2&nbsp;km/s||0.0004||yes
|-
|Piccard & Stahel<ref name=piccard1 />||with a [[Balloon]]||1926||2.8||0.13||≤ 0.006||20||∼ 7&nbsp;km/s||0.006||yes
|-
|Piccard & Stahel<ref name=piccard2 />||[[Brussels]]||1927||2.8||0.13||≤ 0.0002||185||∼ 2.5&nbsp;km/s||0.0007||yes
|-
|Piccard & Stahel<ref name=piccard3 />||[[Rigi]]||1927||2.8||0.13||≤ 0.0003||185||∼ 2.5&nbsp;km/s||0.0007||yes
|-
|Michelson ''et al.''<ref name=michel5 />||[[Mount Wilson (California)|Mt. Wilson]]||1929||25.9||0.9||≤ 0.01||90||∼ 3&nbsp;km/s||0.01||yes
|-
|[[Georg Joos|Joos]]<ref name=joos />||[[Jena]]||1930||21.0||0.75||≤ 0.002||375||∼ 1.5&nbsp;km/s||0.002||yes
|}

== Recent experiments ==

===Optical tests===
Optical tests of the isotropy of the speed of light became commonplace.<ref group=A >Relativity FAQ (2007): [http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html What is the experimental basis of Special Relativity?]</ref> New technologies, including the use of  [[laser]]s and [[maser]]s, have significantly improved measurement precision. (In the following table, only Essen (1955), Jaseja (1964), and Shamir/Fox (1969) are experiments of Michelson–Morley type, ''i.e.'' comparing two perpendicular beams. The other optical experiments employed different methods.)

{| class=wikitable
|-
! Author !! Year !! Description !! Upper bounds
|-
| [[Louis Essen]]<ref name=essen />|| 1955 || The frequency of a rotating microwave [[optical cavity|cavity resonator]] is compared with that of a [[quartz clock]] || ~3&nbsp;km/s
|-
| Cedarholm ''et al''.<ref name=cedarholm /><ref name=cedarholm2 />|| 1958 || Two [[ammonia]] masers were mounted on a rotating table, and their beams were directed in opposite directions. || ~30&nbsp;m/s
|-
| [[Ives–Stilwell experiment#Mössbauer rotor experiments|Mössbauer rotor experiments]] || 1960–63 || In a series of experiments by different researchers, the frequencies of [[gamma rays]] were observed using the [[Mössbauer effect]]. || ~3–4&nbsp;m/s
|-
| Jaseja ''et al''.<ref name=Jaseja />|| 1964 || The frequencies of two [[Helium–neon laser|He–Ne masers]], mounted on a rotating table, were compared. Unlike Cedarholm ''et al.'', the masers were placed perpendicular to each other. || ~30&nbsp;m/s
|-
| nowrap=nowrap| Shamir and Fox<ref name=shamir />|| 1969 || Both arms of the interferometer were contained in a transparent solid ([[Poly(methyl methacrylate)|plexiglass]]). The light source was a [[Helium–neon laser]]. || ~7&nbsp;km/s
|-
| Trimmer ''et al''.<ref name=trimmer /><ref name=trimmer2 />|| 1973 || They searched for anisotropies of the speed of light behaving as the first and third of the [[Legendre polynomials]]. They used a triangle interferometer, with one portion of the path in glass. (In comparison, the Michelson–Morley type experiments test the second Legendre polynomial)<ref name=sexl group=A />|| ~2.5&nbsp;cm/s
|}

[[File:MMX with optical resonators.svg|thumb|250px |Figure 9. Michelson–Morley experiment with cryogenic optical resonators of a form such as was used by Müller ''et al.'' (2003).<ref name=Muller2003/>]]

=== Recent optical resonator experiments ===
Over the last several years, there has been a resurgence in interest in performing precise Michelson–Morley type experiments using lasers, masers, cryogenic [[optical resonator]]s, etc. This is in large part due to predictions of quantum gravity that suggest that special relativity may be violated at scales accessible to experimental study. The first of these highly accurate experiments was conducted by Brillet & Hall (1979), in which they analyzed a laser frequency stabilized to a resonance of a rotating optical [[Fabry–Pérot interferometer|Fabry–Pérot]] cavity. They set a limit on the anisotropy of the speed of light resulting from the Earth's motions of Δ''c''/''c''&nbsp;≈&nbsp;10<sup>−15</sup>, where Δ''c'' is the difference between the speed of light in the ''x''- and ''y''-directions.<ref name=brillet />

As of 2009, optical and microwave resonator experiments have improved this limit to Δ''c''/''c''&nbsp;≈&nbsp;10<sup>−17</sup>. In some of them, the devices were rotated or remained stationary, and some were combined with the [[Kennedy–Thorndike experiment]]. In particular, Earth's direction and velocity (ca.&nbsp;{{Convert|368|km/s|abbr = on}}) relative to the [[Cosmic microwave background radiation#CMBR dipole anisotropy|CMB rest frame]] are ordinarily used as references in these searches for anisotropies.
{{Clear}}

{| class=wikitable
|-
! Author !! Year !! Description !! Δ''c/c''
|-
| Wolf ''et al.''<ref name=wolf1 />||2003 || The frequency of a stationary cryogenic microwave oscillator, consisting of sapphire crystal operating in a [[Whispering-gallery wave|whispering gallery mode]], is compared to a [[hydrogen maser]] whose frequency was compared to [[caesium]] and [[rubidium]] [[atomic fountain]] clocks. Changes during Earth's rotation have been searched for. Data between 2001–2002 was analyzed.||rowspan=4|<center><math>\lesssim10^{-15}</math></center>
|-
| Müller ''et al.''<ref name=Muller2003 />||2003 ||Two optical resonators constructed from crystalline sapphire, controlling the frequencies of two [[Nd:YAG laser]]s, are set at right angles within a helium cryostat. A frequency comparator measures the beat frequency of the combined outputs of the two resonators.
|-
| Wolf ''et al.''<ref name=wolf2 />||2004 || See Wolf ''et al.'' (2003). An active temperature control was implemented. Data between 2002–2003 was analyzed.
|-
| Wolf ''et al.''<ref name=wolf3 />||2004 || See Wolf ''et al.'' (2003). Data between 2002–2004 was analyzed.
|-
| Antonini ''et al.''<ref name=antonini />||2005|| Similar to Müller ''et al.'' (2003), though the apparatus itself was set into rotation. Data between 2002–2004 was analyzed.||rowspan=5|<center><math>\lesssim10^{-16}</math></center>
|-
| Stanwix ''et al.''<ref name=stanwix />||2005 || Similar to Wolf ''et al.'' (2003). The frequency of two cryogenic oscillators was compared. In addition, the apparatus was set into rotation. Data between 2004–2005 was analyzed.
|-
| Herrmann ''et al.''<ref name=Herrmann1 />||2005 || Similar to Müller ''et al.'' (2003). The frequencies of two optical [[Fabry–Pérot interferometer|Fabry–Pérot resonators]] cavities are compared – one cavity was continuously rotating while the other one was stationary oriented north–south. Data between 2004–2005 was analyzed.
|-
| Stanwix ''et al.''<ref name=stanwix2 />||2006 || See Stanwix ''et al.'' (2005). Data between 2004–2006 was analyzed.
|-
| Müller ''et al.''<ref name=Muller2007 />||2007 || See Herrmann ''et al.'' (2005) and Stanwix ''et al.'' (2006). Data of both groups collected between 2004–2006 are combined and further analyzed. Since the experiments are located at difference continents, at [[Berlin]] and [[Perth]] respectively, the effects of both the rotation of the devices themselves and the rotation of Earth could be studied.
|-
| Eisele ''et al.''<ref name=Eisele />||2009|| The frequencies of a pair of orthogonal oriented optical standing wave cavities are compared. The cavities were interrogated by a [[Nd:YAG laser]]. Data between 2007–2008 was analyzed. ||rowspan=2|<center><math>\lesssim10^{-17}</math></center>
|-
| style="white-space:nowrap;"| Herrmann ''et al.''<ref name=Herrmann2 />||2009 || Similar to Herrmann ''et al.'' (2005). The frequencies of a pair of rotating, orthogonal optical [[Fabry–Pérot interferometer|Fabry–Pérot resonators]] are compared. The frequencies of two [[Nd:YAG laser]]s are stabilized to resonances of these resonators.
|}

===Other tests of Lorentz invariance===
{{Further|Modern searches for Lorentz violation}}
[[File:Lithium-7-NMR spectrum of LiCl (1M) in D2O.gif|thumb|225px|Figure 10. <sup>7</sup>Li-NMR spectrum of LiCl (1M) in D<sub>2</sub>O. The sharp, unsplit NMR line of this isotope of lithium is evidence for the isotropy of mass and space.]]
Examples of other experiments not based on the Michelson–Morley principle, i.e. non-optical isotropy tests achieving an even higher level of precision, are [[Hughes–Drever experiment|Clock comparison or Hughes–Drever experiments]]. In Drever's 1961 experiment, <sup>7</sup>Li nuclei in the ground state, which has total angular momentum ''J''=3/2, were split into four equally spaced levels by a magnetic field. Each transition between a pair of adjacent levels should emit a photon of equal frequency, resulting in a single, sharp spectral line. However, since the nuclear wave functions for different ''M<sub>J</sub>'' have different orientations in space relative to the magnetic field, any orientation dependence, whether from an aether wind or from a dependence on the large-scale distribution of mass in space (see [[Mach's principle]]), would perturb the energy spacings between the four levels, resulting in an anomalous broadening or splitting of the line. No such broadening was observed. Modern repeats of this kind of experiment have provided some of the most accurate confirmations of the principle of [[Lorentz covariance|Lorentz invariance]].<ref group=A name=haugan/>

== See also ==
* [[Michelson–Morley Award]]
* [[Moving magnet and conductor problem]]
* [[The Light (Glass)|''The Light'' (Glass)]]
* [[LIGO]]

== References ==

=== Experiments ===
{{Reflist|30em|refs=
<ref name=antonini>{{cite journal |last1=Antonini |first1=P. |last2=Okhapkin |first2=M. |last3=Göklü |first3=E. |last4=Schiller |first4=S. |title=Test of constancy of speed of light with rotating cryogenic optical resonators |journal=Physical Review A |volume=71 |issue=5 |year=2005 |page=050101 |doi=10.1103/PhysRevA.71.050101|arxiv=gr-qc/0504109|bibcode = 2005PhRvA..71e0101A }}</ref>

<ref name=brillet>{{cite journal |last1=Brillet |first=A. |last2=Hall |first2=J. L. |title=Improved laser test of the isotropy of space|journal=Phys. Rev. Lett. |volume=42 |pages=549–552 |year=1979 |doi=10.1103/PhysRevLett.42.549|bibcode = 1979PhRvL..42..549B|issue=9 }}</ref>

<ref name=cedarholm>{{Cite journal |last1=Cedarholm |first=J. P. |last2=Bland |first2=G. F. |last3=Havens |first3=B. L. |last4=Townes |first4=C. H. |title=New Experimental Test of Special Relativity|journal=Physical Review Letters |volume=1 |issue=9 |pages=342–343 |year=1958 |doi=10.1103/PhysRevLett.1.342 |bibcode = 1958PhRvL...1..342C }}</ref>

<ref name=cedarholm2>{{Cite journal|last1=Cedarholm |first1=J. P. |last2=Townes |first2=C. H. |title=New Experimental Test of Special Relativity |journal=Nature |volume=184 |issue=4696 |pages=1350–1351 |year=1959 |doi=10.1038/1841350a0|bibcode = 1959Natur.184.1350C }}</ref>

<!--ref name=desit>{{Citation|last=Sitter |first=Willem de |title=[[s:On the constancy of the velocity of light|On the constancy of the velocity of light]] |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=16 |issue=1 |year=1913 |pages=395–396}}</ref-->

<ref name=Eisele>{{cite journal |last1=Eisele |first=Ch. |last2=Nevsky |first2=A. Yu. |last3=Schillerv |first3=S. |title=Laboratory Test of the Isotropy of Light Propagation at the 10<sup>−17</sup> level |journal=Physical Review Letters |volume=103 |issue=9|page=090401 |year=2009 |doi=10.1103/PhysRevLett.103.090401 |bibcode=2009PhRvL.103i0401E|pmid=19792767 |url=http://www.exphy.uni-duesseldorf.de/Publikationen/2009/Eisele%20et%20al%20Laboratory%20Test%20of%20the%20Isotropy%20of%20Light%20Propagation%20at%20the%2010-17%20Level%202009.pdf}}</ref>

<ref name=essen>{{Cite journal|last=Essen |first=L. |title=A New Æther-Drift Experiment |journal=Nature |volume=175 |issue=4462 |pages=793–794 |year=1955 |doi=10.1038/175793a0|bibcode = 1955Natur.175..793E }}</ref>

<ref name=Herrmann1>{{cite journal |last1=Herrmann |first=S. |last2=Senger |first2=A. |last3=Kovalchuk |first3=E. |last4=Müller |first4=H. |last5=Peters |first5=A. |title=Test of the Isotropy of the Speed of Light Using a Continuously Rotating Optical Resonator |journal=Phys. Rev. Lett. |volume=95 |issue=15 |year=2005 |page=150401 |doi=10.1103/PhysRevLett.95.150401 |arxiv=physics/0508097 |bibcode=2005PhRvL..95o0401H |pmid=16241700 }}</ref>

<ref name=Herrmann2>{{cite journal |last1=Herrmann |first1=S. |last2=Senger |first2=A. |last3=Möhle |first3=K. |last4=Nagel |first4=M. |last5=Kovalchuk |first5=E. V. |last6=Peters |first6=A. |title=Rotating optical cavity experiment testing Lorentz invariance at the 10<sup>−17</sup> level|journal=Physical Review D |volume=80 |issue=100|page=105011 |year=2009 |doi=10.1103/PhysRevD.80.105011 |arxiv=1002.1284 |bibcode = 2009PhRvD..80j5011H }}</ref>

<ref name=Illingworth>{{cite journal |last=Illingworth |first=K. K. |title=A Repetition of the Michelson–Morley Experiment Using Kennedy's Refinement|journal=Physical Review |volume=30 |issue=5 |year=1927 |pages=692–696 |doi=10.1103/PhysRev.30.692|bibcode = 1927PhRv...30..692I }}</ref>

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<ref name=Kennedy>{{cite journal|last=Kennedy |first=Roy J. |title=A Refinement of the Michelson–Morley Experiment |journal=Proceedings of the National Academy of Sciences |volume=12 |issue=11 |year=1926 |pages=621–629 |doi=10.1073/pnas.12.11.621 |bibcode = 1926PNAS...12..621K }}</ref>

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}}

=== Notes ===
{{Reflist|60em|group=note}}
<ref>E.W. silversmith "Special Relativity", Nature magazine, vol. 322 [AUG. 1986], P.590: the field exists, per the United States Air Force research, and it measured precisely as Michaelson and Morely predicted.</ref>

=== Bibliography (Series "A" references)===
{{Reflist|30em|group=A|refs=
<ref name=blum>{{cite book |first1=Edward K. |last1=Blum |first2=Sergey V. |last2=Lototsky |year=2006|title=Mathematics of physics and engineering |publisher=World Scientific  |isbn=981-256-621-X |page=98 |url=https://books.google.com/?id=nFRG2UizET0C}}, [https://books.google.com/books?id=nFRG2UizET0C&pg=PA98 Chapter 2, p. 98 ]</ref>

<ref name=dongen>{{Citation |last=Jeroen |first=van Dongen |year=2009 |journal=Archive for History of Exact Sciences |title=On the Role of the Michelson–Morley Experiment: Einstein in Chicago |pages=655–663 |issue=6 |volume=63 |doi=10.1007/s00407-009-0050-5 |arxiv=0908.1545}}</ref>

<ref name=einstein>{{cite journal|last=Einstein |first=A |authorlink=Albert Einstein |date=June 30, 1905 |title=Zur Elektrodynamik bewegter Körper |journal=Annalen der Physik |volume=17 |pages=890–921 |url=http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf |language=German |format=PDF |accessdate=2009-11-27 |doi=10.1002/andp.19053221004 |bibcode=1905AnP...322..891E |deadurl=yes |archiveurl=https://web.archive.org/web/20091229162203/http://www.pro-physik.de/Phy/pdfs/ger_890_921.pdf |archivedate=December 29, 2009 }} English translation: {{cite web
|url=http://www.fourmilab.ch/etexts/einstein/specrel/www/ |title=On the Electrodynamics of Moving Bodies |last=Perrett |first=W |translator-last=Jeffery |translator-first=GB  |editor-last=Walker |editor-first=J. | editorlink3 =John Walker (programmer) |publisher=[[Fourmilab]]
|accessdate=2009-11-27}}</ref>

<ref name=einstein2>{{Citation|last=Einstein |first=A. |year=1916 |title=[[s:Relativity: The Special and General Theory|Relativity: The Special and General Theory]]|publisher=H. Holt and Company|location=New York}}</ref>

<ref name=Feynman>{{Citation |last=Feynman |first=R.P. |year=1970 |title=The Feynman Lectures on Physics|chapter=The Michelson–Morley experiment (15-3) |volume=1 |location=Reading |publisher=Addison Wesley Longman |isbn=0-201-02115-3}}</ref>

<ref name=Fickinger>{{cite book |first=William |last=Fickinger |title=Physics at a Research University: Case Western Reserve, 1830–1990 |location=Cleveland |year=2005 |isbn=0977338606 |pages=18–22, 48 |quote=The Dormitory was located on a now largely unoccupied space between the Biology Building and the Adelbert Gymnasium, both of which still stand on the CWRU campus.}}</ref>

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<ref name=hoover>{{cite book |first=Earl R. |last=Hoover |title=Cradle of Greatness: National and World Achievements of Ohio's Western Reserve |publisher=Shaker Savings Association |location=Cleveland |year=1977 |oclc=2936545 }}</ref>

<ref name=Jan>{{Cite book|last=Janssen |first=Michel |last2=Stachel |first2=John |editor-first=John |editor-last=Stachel |title=Going Critical |publisher=Springer|isbn=1-4020-1308-6|year=2010|chapter=The Optics and Electrodynamics of Moving Bodies|chapter-url=http://www.mpiwg-berlin.mpg.de/Preprints/P265.PDF |chapter-format=pdf}}</ref>

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<ref name=lorentz95>{{Citation |last=Lorentz |first=Hendrik Antoon |year=1895 |title=[[s:Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies|Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies]] |location=Leiden |publisher=E.J. Brill}}</ref>

<ref name=lorentz04>{{Citation |last=Lorentz |first=Hendrik Antoon |year=1904 |title=[[s:Electromagnetic phenomena|Electromagnetic phenomena in a system moving with any velocity smaller than that of light]] |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=6 |pages=809–831}}</ref>

<ref name=maxa>{{cite EB9 |mode=cs2 |last=Maxwell |first=James Clerk |wstitle=Ether |volume=8 |pages=568–572}}</ref>

<ref name=maxb>{{Citation |last=Maxwell |first=James Clerk |year=1880
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<ref name=michel1928>{{cite journal |last=Michelson |first=A. A. |authorlink=  |year=1928 |title=Conference on the Michelson–Morley Experiment Held at Mount Wilson, February, 1927 |journal=Astrophysical Journal |volume=68 |issue= |pages=341–390 |doi=10.1086/143148 |url= |accessdate= |bibcode=1928ApJ....68..341M |display-authors=1 |last2=Lorentz |first2=H. A. |last3=Miller |first3=D. C. |last4=Kennedy |first4=R. J. |last5=Hedrick |first5=E. R. |last6=Epstein |first6=P. S.}}</ref>

<ref name=Miller1933>{{cite journal|last=Miller |first=Dayton C.|title=The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth|journal=Reviews of Modern Physics|volume=5|issue=3|year=1933|pages=203–242|doi=10.1103/RevModPhys.5.203|bibcode=1933RvMP....5..203M}}</ref>

<ref name=AIMiller>{{Cite book|last=Miller |first=A.I. |year=1981 |title= Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |location= Reading |publisher=Addison–Wesley |isbn=0-201-04679-2|page=24}}</ref>

<ref name=norton>{{Cite journal|last=Norton |first=John D.|year=2004|journal=Archive for History of Exact Sciences|title= Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905|pages= 45–105|volume=59|url=http://philsci-archive.pitt.edu/archive/00001743/|doi=10.1007/s00407-004-0085-6|bibcode=2004AHES...59...45N}}</ref>

<ref name=poincare05>{{Citation |last=Poincaré |first=Henri |year=1905 |title=[[s:On the Dynamics of the Electron (June)|On the Dynamics of the Electron]] |journal=Comptes Rendus |volume=140 |pages=1504–1508}} (Wikisource translation)</ref>

<ref name=Polanyi>[[Michael Polanyi]], ''Personal Knowledge: Towards a Post-Critical Philosophy'', {{ISBN|0-226-67288-3}}, footnote page&nbsp;10–11:  Einstein reports, via Dr N Balzas in response to Polanyi's query, that "The Michelson–Morley experiment had no role in the foundation of the theory." and "..the theory of relativity was not founded to explain its outcome at all."[https://books.google.com/books?id=0Rtu8kCpvz4C&lpg=PP1&pg=PT19#v=onepage&q=&f=false]</ref>

<ref name=rob>{{cite journal |last=Robertson |first=H. P. |year=1949 |title=Postulate versus Observation in the Special Theory of Relativity |journal=Reviews of Modern Physics |volume=21 |issue=3 |pages=378–382 |doi=10.1103/RevModPhys.21.378|bibcode = 1949RvMP...21..378R }}</ref>

<ref name=sexl>{{Cite journal |last=Mansouri |first=R. |last2=Sexl |first2=R.U. | year=1977 | title= A test theory of special relativity: III. Second-order tests| journal =Gen. Rel. Gravit. |volume=8 |issue=10 |pages=809–814 | doi=10.1007/BF00759585|bibcode = 1977GReGr...8..809M }}</ref>

<ref name=shankland>{{cite journal |last=Shankland |first=Robert S. |year=1955 |title=New Analysis of the Interferometer Observations of Dayton C. Miller |journal=Reviews of Modern Physics |volume=27 |issue=2 |pages=167–178 |doi=10.1103/RevModPhys.27.167|bibcode = 1955RvMP...27..167S |display-authors=1 |last2=McCuskey |first2=S. |last3=Leone |first3=F. |last4=Kuerti |first4=G. }}</ref>

<ref name=shankland2>{{cite journal |last=Shankland |first= R.S.|title=Michelson–Morley experiment|journal=American Journal of Physics |year=1964 |volume=31 |issue=1 |pages=16–35 |doi=10.1119/1.1970063|bibcode = 1964AmJPh..32...16S }}</ref>

<ref name=stachel>{{Citation |last=Stachel |first=John |authorlink=John Stachel |year=1982
|journal=Astronomische Nachrichten |title= Einstein and Michelson: the Context of Discovery and Context of Justification |pages=47–53 |issue=1 |volume=303 |bibcode=1982AN....303...47S |doi=10.1002/asna.2103030110}}</ref>

<ref name=staley>{{Citation|last=Staley |first=Richard |year=2009 |title=Einstein's generation. The origins of the relativity revolution|chapter=Albert Michelson, the Velocity of Light, and the Ether Drift|location=Chicago|publisher=University of Chicago Press |isbn=0-226-77057-5}}</ref>

<ref name=Swenson1>{{Cite journal | last=Swenson |first=Loyd S. |year=1970 |title=The Michelson–Morley–Miller Experiments before and after 1905 |journal=Journal for the History of Astronomy
|volume=1 |issue=2 |doi=10.1177/002182867000100108  |pages=56–78 |url=http://adsabs.harvard.edu//abs/1970JHA.....1...56S|bibcode = 1970JHA.....1...56S}}</ref>

<ref name=Swenson2>{{cite book |last=Swenson |first=Loyd S., Jr. |title=The Ethereal Aether: A History of the Michelson-Morley-Miller Aether-drift Experiments, 1880–1930 |url=https://books.google.com/books?id=kQTUAAAAQBAJ |year=2013 |publisher=University of Texas Press |isbn=978-0-292-75836-0  |origyear=1972}}</ref>

<ref name=teller>{{Citation |last=Teller |first=Edward |authorlink=Edward Teller |first2=Wendy |last2=Teller |first3=Wilson |last3=Talley|title=Conversations on the Dark Secrets of Physics|year=2002|publisher=Basic books|isbn=0786752378|pages=10–11|url=https://books.google.com/books?id=QClyAWecl60C&pg=PA10}}</ref>

<ref name=Thirring>{{Cite journal |last=Thirring |first=Hans |year=1926 |journal=Nature |title=Prof. Miller's Ether Drift Experiments|pages= 81–82 |volume=118 |issue=2959 |doi=10.1038/118081c0|bibcode = 1926Natur.118...81T }}</ref>

<ref name=Whittaker>{{Cite book |last=Whittaker |first=Edmund Taylor |year=1910 |title= A History of the theories of aether and electricity |edition=1.|place=Dublin|publisher=Longman, Green and Co. |url=https://archive.org/details/historyoftheorie00whitrich}}</ref>

<ref name=schum94>{{cite journal |last=Schumacher |first=Reinhard A. |title=Special Relativity and the Michelson-Morley Interferometer|journal=American Journal of Physics |year=1994 |volume=62 |pages=609–612 |doi=10.1119/1.17535|bibcode = 1994AmJPh..62..609S }}</ref>

}}

== External links ==
*{{Commons category-inline|Michelson-Morley experiment}}
*{{Wikibooks-inline|links=[[b:Special Relativity/Aether#Mathematical analysis of the Michelson Morley Experiment|Mathematical analysis of the Michelson Morley Experiment]]}}
* {{cite web |last=Roberts |first=T |last2=Schleif |first2=S |last3=Dlugosz |first3=JM (ed.) |year=2007 |title=What is the experimental basis of Special Relativity? |url=http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html |work=Usenet Physics FAQ |publisher=[[University of California, Riverside]]}}

{{Tests of special relativity}}

{{DEFAULTSORT:Michelson-Morley Experiment}}
[[Category:Aether theories]]
[[Category:Case Western Reserve University]]
[[Category:Tests of special relativity]]
[[Category:1887 in science]]