Editing Michelson–Morley experiment (section)

== Light path analysis and consequences ==
=== Observer resting in the aether ===
[[File:Michelson-morley calculations.svg|thumb|300px|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 display="inline">c</math> in the aether. It passes through the half-silvered mirror at the origin at <math display="inline">T=0</math>.  The reflecting mirror is at that moment at distance <math display="inline">L</math> (the length of the interferometer arm) and is moving with velocity <math display="inline">v</math>. The beam hits the mirror at time <math display="inline">T_1</math> and thus travels the distance <math display="inline">cT_1</math>. At this time, the mirror has traveled the distance <math display="inline">vT_1</math>. Thus <math display="inline">cT_1 =L+vT_1</math> and consequently the travel time <math display="inline">T_1=L/(c-v)</math>. The same consideration applies to the backward journey, with the sign of <math display="inline">v</math> reversed, resulting in <math display="inline">cT_2 =L-vT_2</math> and <math display="inline">T_2 =L/(c+v)</math>. The total travel time <math display="inline">T_\ell=T_1+T_2</math> is:

:<math>T_\ell=\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 the increase in path length in the rest frame of the aether. This was corrected by [[Alfred Potier]] (1882) and [[Hendrik Lorentz]] (1886). The derivation in the transverse direction can be given as follows (analogous to the derivation of [[time dilation]] using a [[Time dilation#Simple inference of velocity time dilation|light clock]]): The beam is propagating at the speed of light <math display="inline">c</math> and hits the mirror at time <math display="inline">T_3</math>, traveling the distance <math display="inline">cT_3</math>. At the same time, the mirror has traveled the distance <math display="inline">vT_3</math> in the ''x'' direction. So in order to hit the mirror, the travel path of the beam is <math display="inline">L</math> in the ''y'' direction (assuming equal-length arms) and <math display="inline">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 display="inline"> \sqrt{L^2+\left(vT_3\right)^2}</math>. Thus <math display="inline"> cT_3 =\sqrt{L^2+\left(vT_3\right)^2}</math> and consequently the travel time <math display="inline"> T_3 =L/\sqrt{c^2-v^2}</math>, which is the same for the backward journey. The total travel time <math display="inline">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 <math>T_\ell</math> and <math>T_t</math> is given by<ref group=A>{{cite book |author=Albert Shadowitz  |title=Special relativity |url=https://archive.org/details/specialrelativit0000shad  |url-access=registration  |isbn=978-0-486-65743-1 |publisher=Courier Dover Publications |edition=Reprint of 1968 |year=1988|pages=[https://archive.org/details/specialrelativit0000shad/page/159 159–160]}}</ref>

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

To find the path difference, simply multiply by <math>c</math>;

<math>\Delta{\lambda}_1=2L\left(\frac{1}{1-\frac{v^2}{c^2}}-\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\right)</math>

The path difference is denoted by <math>\Delta \lambda</math> because the beams are out of phase by a some number of wavelengths (<math>\lambda</math>). To visualise this, consider taking the two beam paths along the longitudinal and transverse plane, and lying them straight (an animation of this is shown at minute 11:00, [[The Mechanical Universe|The Mechanical Universe, episode 41]]<ref name=":0">{{Cite web|title=The Mechanical Universe, Episode 41|website = [[YouTube]]|url=https://www.youtube.com/watch?v=Ip_jdcA8fcw&t=632s| archive-url=https://ghostarchive.org/varchive/youtube/20211118/Ip_jdcA8fcw| archive-date=2021-11-18 | url-status=live}}{{cbignore}}</ref>). One path will be longer than the other, this distance is <math>\Delta \lambda</math>. Alternatively, consider the rearrangement of the speed of light formula <math>c{\Delta}T = \Delta\lambda</math> .

If the relation <math>{v^2}/{c^2} << 1</math> is true (if the velocity of the aether is small relative to the speed of light), then the expression can be simplified using a first order binomial expansion;

<math>(1-x)^n \approx {1-nx}</math>

So, rewriting the above in terms of powers;

<math>\Delta{\lambda}_1 = 2L\left(\left({1-\frac{v^2}{c^2}}\right)^{-1}-\left(1-\frac{v^2}{c^2}\right)^{-1/2}\right)</math>

Applying binomial simplification;<ref name="Cengage Learning">{{cite book|last1=Serway|first1=Raymond|url=https://books.google.com/books?id=-g_y6CMkZ0IC|title=Physics for Scientists and Engineers, Volume 2|last2=Jewett|first2=John|publisher=Cengage Learning|year=2007|isbn=978-0-495-11244-0|edition=7th illustrated|page=1117}} [https://books.google.com/books?id=-g_y6CMkZ0IC&pg=PA1117 Extract of page 1117]</ref>

<math>\Delta{\lambda}_1 = 2L\left( (1+\frac{v^2}{c^2}) - (1+\frac{v^2}{2c^2})\right)={2L}\frac{v^2}{2c^2}</math>

Therefore;

<math>\Delta{\lambda}_1={L}\frac{v^2}{c^2}</math>

The derivation above shows that the presence of an aether wind would produce a difference in optical path lengths between the two arms of the interferometer. This path difference depends on the orientation of the interferometer relative to the aether wind. Specifically, the derivation assumes that the longitudinal arm is aligned parallel to the presumed direction of the aether wind. If instead the longitudinal arm is oriented perpendicular to the aether wind, the resulting path difference would have the opposite sign.

The magnitude of the path difference can vary continuously and may represent any fraction of the wavelength, depending on both the angle between the apparatus and the aether wind and the wind's speed.

To detect the existence of the aether, Michelson and Morley aimed to observe a "fringe shift" in the interference pattern. The underlying principle is straightforward: when the interferometer is rotated by 90°, the roles of the two arms are exchanged, altering the path difference due to the aether wind. The fringe shift is determined by calculating the difference in path differences between the two orientations, and then dividing that value by the wavelength.<ref name="Cengage Learning"/>

: <math>n=\frac{\Delta\lambda_1-\Delta\lambda_2}{\lambda}\approx\frac{2Lv^2}{\lambda c^2}.</math>

Note the difference between <math>\Delta \lambda</math>, which is some number of wavelengths, and <math>\lambda</math> which is a single wavelength. As can be seen by this relation, fringe shift n is a unitless quantity.

Since ''L''&nbsp;≈&nbsp;11 meters and λ&nbsp;≈&nbsp;500 [[nanometer]]s, the expected [[fringe shift]] was ''n''&nbsp;≈&nbsp;0.44. The negative result led Michelson to the conclusion that there is no measurable aether drift.<ref name="michel2" /> However, he never accepted this on a personal level, and the negative result haunted him for the rest of his life.<ref name=":0" />

=== 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 display="inline">c</math> against a river flowing with velocity <math display="inline">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 display="inline">c-v</math>. On his way back moving downstream, his velocity is increased to <math display="inline">c+v</math>. This gives the beam travel times <math display="inline">T_1</math> and  <math display="inline">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 display="inline">\sqrt{c^2-v^2}</math>, and gives the beam travel time <math display="inline">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 Francis FitzGerald|George FitzGerald]] (1889) in a letter to same journal that published the Michelson-Morley paper, as "almost the only hypothesis that can reconcile" the apparent contradictions. It was independently also proposed by [[Hendrik Lorentz]] (1892).<ref group=A name=lorentz95 /> According to this law all objects physically contract by <math display="inline">L/\gamma</math> along the line of motion (originally thought to be relative to the aether), <math display="inline">\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 display="inline">L</math> is inserted into the above formula for <math display="inline">T_\ell</math>, then the light propagation time in the longitudinal direction becomes equal to that in the transverse direction:

:<math>T_\ell=\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 display="inline">\gamma</math>. This can be achieved in many ways. If <math display="inline">L_1</math> is the moving longitudinal length and <math display="inline">L_2</math> the moving transverse length, <math display="inline">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}{\varphi}\left/\frac{L'_1}{\gamma\varphi}\right.=\gamma.</math>

<math display="inline">\varphi</math> can be arbitrarily chosen, so there are infinitely many combinations to explain the Michelson–Morley null result. For instance, if <math display="inline">\varphi=1</math> the relativistic value of length contraction of <math display="inline">L_1</math> occurs, but if <math display="inline">\varphi=1/\gamma</math> then no length contraction but an elongation of <math display="inline">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\varphi(x-vt),\ y'=\varphi y,\ z'=\varphi z,\ t'=\gamma\varphi\left(t-\frac{vx}{c^2}\right)</math>

It remained to define the value of <math display="inline">\varphi</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 display="inline">\varphi=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.}}

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 display="inline">\varphi</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 display="inline">\varphi=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 (relativity)|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|bibcode=1977PhRvL..39.1051B|issue=17}}</ref> Furthermore, Filippas and Fox (1964) conducted terrestrial [[particle accelerator]] tests specifically designed to address Fox's earlier "extinction" objection, the results being inconsistent with source dependence of the speed of light.<ref name=FilippasFox>{{cite journal|last1=Filippas |first1=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>