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<title>Aether drag hypothesis</title>

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Aether drag hypothesis

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In the 19th century, the theory of the <a href="Luminiferous_aether" title="Luminiferous aether">luminiferous aether</a> as the hypothetical <a href="Transmission_medium" title="Transmission medium">medium</a> for the propagation of light waves was widely discussed. The aether hypothesis arose because physicists of that era could not conceive of light waves propagating without a physical medium in which to do so. When experiments failed to detect the hypothesized luminiferous aether, physicists conceived explanations for the experiments' failure which preserved the hypothetical aether's existence.

The aether drag hypothesis proposed that the luminiferous aether is dragged by or entrained within moving matter. According to one version of this hypothesis, no relative motion exists between Earth and aether. According to another version, the Earth does move relative to the aether and the measured <a href="Speed_of_light" title="Speed of light">speed of light</a> should depend on the speed of this motion ("aether wind"), which should be measurable by instruments at rest on Earth's surface. In 1818, <a href="Augustin-Jean_Fresnel" title="Augustin-Jean Fresnel">Augustin-Jean Fresnel</a> proposed that the aether is partially entrained by matter. In 1845, <a href="Sir_George_Stokes%2C_1st_Baronet" title="Sir George Stokes, 1st Baronet">George Stokes</a> proposed that the aether is completely entrained within or in the vicinity of matter.

Although Fresnel's almost-stationary theory was apparently confirmed by the <a href="Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a> (1851), Stokes' theory was apparently confirmed by the <a href="Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a> (1881, 1887). <a href="Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a> resolved this contradictory situation in <a href="Lorentz_ether_theory" title="Lorentz ether theory">his own aether theory</a>, which banished any form of aether dragging. <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="Special_relativity" title="Special relativity">special theory of relativity</a> (1905) excludes aether as a mechanical medium.<a href="#cite_note-whit-1">[1]</a><a href="#cite_note-jann-2">[2]</a><a href="#cite_note-ferr-3">[3]</a>

In modern physics (which is based on the <a href="Theory_of_relativity" title="Theory of relativity">theory of relativity</a> and <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>), the aether as a "material substance" with a "state of motion" no longer plays any role. So questions concerning a possible "aether drag" are no longer considered meaningful by the scientific community. However, <a href="Frame-dragging" title="Frame-dragging">frame-dragging</a> as predicted by <a href="General_relativity" title="General relativity">general relativity</a>, in which rotating masses distort the <a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">spacetime metric</a>, causing a <a href="Precession" title="Precession">precession</a> of the orbit of nearby particles, does exist. But this effect is orders of magnitude weaker than any "aether drag" discussed in this article.

<div class="mw-heading mw-heading2"><h2 id="Partial_aether_dragging">Partial aether dragging</h2></div>

In 1810, <a href="Fran%C3%A7ois_Arago" title="François Arago">François Arago</a> realised that variations in the refractive index of a substance predicted by the corpuscular theory would provide a useful method for measuring the velocity of light. These predictions arose because the <a href="Refractive_index" title="Refractive index">refractive index</a> of a substance such as glass depends on the ratio of the velocities of light in air and in the glass. Arago attempted to measure the extent to which corpuscles of light would be refracted by a glass prism at the front of a telescope. He expected that there would be a range of different angles of refraction due to the variety of different velocities of the stars and the motion of the Earth at different times of the day and year. Contrary to this expectation, he found that there was no difference in refraction between stars, between times of day or between seasons. All Arago observed was ordinary <a href="Aberration_of_light" class="mw-redirect" title="Aberration of light">stellar aberration</a>.<a href="#cite_note-4">[4]</a>

In 1818, <a href="Augustin-Jean_Fresnel" title="Augustin-Jean Fresnel">Augustin-Jean Fresnel</a> examined Arago's results using a wave theory of light. He realised that even if light were transmitted as waves the refractive index of the glass-air interface should have varied as the glass moved through the aether to strike the incoming waves at different velocities when the Earth rotated, and the seasons changed. Fresnel proposed that the glass prism would carry some of the aether along with it so that "...the aether is in excess inside the prism".<a href="#cite_note-5">[5]</a> He realised that the velocity of propagation of waves depends on the density of the medium and so proposed that the velocity of light in the prism would need to be adjusted by an amount of 'drag'. The velocity of light <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{n}}</annotation>

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</math><img src="./d5615ffa6233b0d09d5bafafb58a752c1e8de95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.346ex; height:2.009ex;" alt="{\displaystyle v_{n}}" loading="lazy"> in the glass without any adjustment is given by:

<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle v_{n}={\frac {c}{n}}}</annotation>

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</math><img src="./624a11490499a3aee41b64744b6d5e53f2384344.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.675ex; height:4.676ex;" alt="{\displaystyle v_{n}={\frac {c}{n}}}" loading="lazy"></dd></dl>

The drag adjustment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{d}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle v_{d}}</annotation>

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</math><img src="./eac4aa20f05fda74ee30f9f9842229d087f33133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.22ex; height:2.009ex;" alt="{\displaystyle v_{d}}" loading="lazy"> is given by:

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<annotation encoding="application/x-tex">{\displaystyle v_{d}=v\left(1-{\frac {\rho _{e}}{\rho _{g}}}\right)}</annotation>

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</math><img src="./2e4317cedfd10c0cd7405f7fd1d9d3f0d340671b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.316ex; height:6.176ex;" alt="{\displaystyle v_{d}=v\left(1-{\frac {\rho _{e}}{\rho _{g}}}\right)}" loading="lazy"></dd></dl>

Where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{e}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \rho _{e}}</annotation>

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</math><img src="./d972eda14d3f9d5d046cd8a1c8ba3dcd0f09a84d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.2ex; height:2.176ex;" alt="{\displaystyle \rho _{e}}" loading="lazy"> is the aether density in the environment, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{g}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \rho _{g}}</annotation>

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</math><img src="./453b6e0e4cc53968c3a7b6cbce65d4f8e2c4eaae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.223ex; height:2.343ex;" alt="{\displaystyle \rho _{g}}" loading="lazy"> is the aether density in the glass and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is the velocity of the prism with respect to the aether.

The factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\frac {\rho _{e}}{\rho _{g}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {\rho _{e}}{\rho _{g}}}\right)}</annotation>

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</math><img src="./1e67b26c541d184d7d8fe2e6584610ca79764d58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.483ex; height:6.176ex;" alt="{\displaystyle \left(1-{\frac {\rho _{e}}{\rho _{g}}}\right)}" loading="lazy"> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-{\frac {1}{n^{2}}}\right)}">

<mrow>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {1}{n^{2}}}\right)}</annotation>

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</math><img src="./6c17597ef5d9aae89421e286a53e277589386b78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.709ex; height:6.176ex;" alt="{\displaystyle \left(1-{\frac {1}{n^{2}}}\right)}" loading="lazy"> because the refractive index, n, would be dependent on the density of the aether. This is known as the Fresnel drag coefficient. The velocity of light in the glass is then given by:

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}\right)}</annotation>

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</math><img src="./b79d5539b932f1a25d078775ae6ef8985a626723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.181ex; height:6.176ex;" alt="{\displaystyle V={\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}\right)}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Problems_of_partial_aether_dragging">Problems of partial aether dragging</h3></div>

Fresnel's dragging coefficient was directly confirmed by the <a href="Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a> and its repetitions. In general, with the aid of this coefficient the negative result of all optical aether drift experiments sensitive enough to detect first order effects (such as the <a href="Luminiferous_aether#First_order_experiments" title="Luminiferous aether">experiments of Arago, Fizeau, Hoek, Airy, Mascart</a>) can be explained. The notion of an (almost) stationary aether is also consistent with <a href="Stellar_aberration" class="mw-redirect" title="Stellar aberration">stellar aberration</a>. However, this theory is considered to be refuted for the following reasons:<a href="#cite_note-whit-1">[1]</a><a href="#cite_note-jann-2">[2]</a><a href="#cite_note-ferr-3">[3]</a>

<ul><li>It was already known in the 19th century, that partial aether dragging requires the relative velocity of aether and matter to be different for light of different colours – which is evidently not the case.</li>

<li>Fresnel's theory of an (almost) stationary aether predicts positive results by experiments which are sensitive enough to detect second order effects. However, experiments such as the <a href="Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a> and the <a href="Trouton%E2%80%93Noble_experiment" title="Trouton–Noble experiment">Trouton–Noble experiment</a>, gave negative results within their margin of error and are therefore considered refutations of Fresnel's aether.</li>

<li>In the <a href="Hammar_experiment" title="Hammar experiment">Hammar experiment</a>, conducted by <a href="Gustaf_Wilhelm_Hammar" title="Gustaf Wilhelm Hammar">Gustaf Wilhelm Hammar</a> in 1935, a <a href="Common-path_interferometer" title="Common-path interferometer">common-path interferometer</a> was used. Massive lead blocks were installed on both sides of only one leg of the interferometer. This arrangement should cause different amounts of aether drag and therefore produce a positive result. However, the result was again negative.<a href="#cite_note-6">[6]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Complete_aether_dragging">Complete aether dragging</h2></div>

For <a href="Sir_George_Stokes%2C_1st_Baronet" title="Sir George Stokes, 1st Baronet">George Stokes</a> (1845) the model of an aether which is totally unaffected or only partially affected by moving matter was unnatural and unconvincing, so he assumed that the aether is completely dragged within and in the vicinity of matter, partially dragged at larger distances, and stays at rest in free space.<a href="#cite_note-stokes1845-7">[7]</a><a href="#cite_note-stokes1846-8">[8]</a><a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> Also <a href="Heinrich_Rudolf_Hertz" class="mw-redirect" title="Heinrich Rudolf Hertz">Heinrich Rudolf Hertz</a> (1890) incorporated a complete aether drag model within his elaboration of Maxwell's theory of electromagnetism, to bring it into accord with the Galilean <a href="Principle_of_relativity" title="Principle of relativity">principle of relativity</a>. That is, if it is assumed that the aether is at rest within matter in one reference frame, the <a href="Galilean_transformation" title="Galilean transformation">Galilean transformation</a> gives the result that matter and (entrained) aether travel with the same speed in another frame of reference.<a href="#cite_note-whit-1">[1]</a>

<div class="mw-heading mw-heading3"><h3 id="Problems_of_complete_aether_dragging">Problems of complete aether dragging</h3></div>

Complete aether dragging can explain the negative outcome of all aether drift experiments (like the Michelson–Morley experiment). However, this theory is considered to be wrong for the following reasons:<a href="#cite_note-whit-1">[1]</a><a href="#cite_note-joos-11">[11]</a>

<ul><li>The <a href="Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a> (1851) indicated only a partial entrainment of light.</li>

<li>The <a href="Sagnac_effect" title="Sagnac effect">Sagnac effect</a> shows that two rays of light, emanated from the same light source in different directions on a rotating platform, require different times to come back to the light source. However, if the aether is completely dragged by the platform this effect should not occur at all.</li>

<li><a href="Oliver_Lodge" title="Oliver Lodge">Oliver Lodge</a> conducted experiments in the 1890s, seeking evidence that the propagation of light is influenced by being in the proximity of large rotating masses, and found no such influence.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a></li></ul>

<ul><li>It is inconsistent with the phenomenon of <a href="Stellar_aberration" class="mw-redirect" title="Stellar aberration">stellar aberration</a>. In stellar aberration the position of a star when viewed with a telescope swings each side of a central position by about 20.5 seconds of arc every six months. This amount of swing is the amount expected when considering the speed of Earth's travel in its orbit. In 1871 <a href="George_Biddell_Airy" title="George Biddell Airy">Airy</a> demonstrated that stellar aberration occurs even when a telescope is filled with water. It seems that if the aether drag hypothesis were true then stellar aberration would not occur because the light would be travelling in the aether which would be moving along with the telescope. Consider a bucket on a train about to enter a tunnel, and a drop of water drips from the tunnel entrance into the bucket at the very center. The drop will not hit the center at the bottom of the bucket. The bucket is analogous to the tube of a telescope, the drop is a photon and the train is the Earth. If aether is dragged then the droplet would be traveling with the train when it is dropped and would hit the center of bucket at the bottom. The amount of stellar aberration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">, is given by:</li></ul>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \tan(\alpha )={\frac {v\delta t}{c\delta t}}.}</annotation>

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</math><img src="./180081b30ff93b61e3fc3f5bf5ec0473f152fd10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.254ex; height:5.509ex;" alt="{\displaystyle \tan(\alpha )={\frac {v\delta t}{c\delta t}}.}" loading="lazy"> So: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\alpha )={\frac {v}{c}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(\alpha )={\frac {v}{c}}}</annotation>

</semantics>

</math><img src="./e494a9215d3eb5a6c821f6de7afd719407a2a1e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.719ex; height:4.676ex;" alt="{\displaystyle \tan(\alpha )={\frac {v}{c}}}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>The speed at which the Earth goes round the Sun, v = 30 km/s, and the speed of light is c = 299,792,458 m/s which gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Stokes'_responses_to_those_problems">Stokes' responses to those problems</h3></div>

Stokes already in 1845 introduced some additional assumptions in order to bring his theory into accord with experimental results. To explain aberration, he assumed that his incompressible aether is irrotational as well, which would give, in connection with his specific model of aether drag, the correct law of aberration.<a href="#cite_note-stokes1845-7">[7]</a> To reproduce Fresnel's dragging coefficient (and therefore to explain the Fizeau experiment) he argued that the aether is completely dragged within a medium – i.e. the aether gets condensed when it enters the medium and rarefied when it leaves it again, which modifies the speed of the aether as well as that of light and leads to the same expression as Fresnel's.<a href="#cite_note-stokes1846-8">[8]</a>

Even though Stokes' aberration theory was considered viable for some time, it had to be given up because Lorentz argued in 1886, that when the aether is incompressible as in Stokes' theory, and if the aether has the same normal component of velocity as the Earth, it would not have the same tangential component of velocity, so all conditions posed by Stokes cannot be fulfilled at the same time.<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading3"><h3 id="Gravitational_aether_drag">Gravitational aether drag</h3></div>

Another version of Stokes' model was proposed by <a href="Theodor_des_Coudres" title="Theodor des Coudres">Theodor des Coudres</a> and <a href="Wilhelm_Wien" title="Wilhelm Wien">Wilhelm Wien</a> (1900). They assumed that aether dragging is proportional to the gravitational mass. That is, the aether is completely dragged by the Earth, and only partially dragged by smaller objects on Earth.<a href="#cite_note-15">[15]</a> And to save Stokes's explanation of aberration, <a href="Max_Planck" title="Max Planck">Max Planck</a> (1899) argued in a letter to Lorentz, that the aether might not be incompressible, but condensed by gravitation in the vicinity of Earth, and this would give the conditions needed for the theory of Stokes ("Stokes-Planck theory"). When compared with the experiments above, this model can explain the positive results of the experiments of Fizeau and Sagnac, because the small mass of those instruments can only partially (or not at all) drag the aether, and for the same reason it explains the negative result of Lodge's experiments. It is also compatible with Hammar's and Michelson–Morley experiment, as the aether is completely dragged by the large mass of Earth.

However, this theory was directly refuted by the <a href="Michelson%E2%80%93Gale%E2%80%93Pearson_experiment" title="Michelson–Gale–Pearson experiment">Michelson–Gale–Pearson experiment</a> (1925). The great difference of this experiment against the usual Sagnac experiments is the fact that the rotation of Earth itself was measured. If the aether is completely dragged by the Earth's gravitational field, a negative result has to be expected - but the result was positive.<a href="#cite_note-joos-11">[11]</a>

And from a theoretical side it was noted by <a href="Hendrik_Antoon_Lorentz" class="mw-redirect" title="Hendrik Antoon Lorentz">Hendrik Antoon Lorentz</a>, that the Stokes-Planck hypothesis requires that the speed of light is not affected by a density increase of 50,000 times of the aether. So Lorentz and Planck himself rejected this hypothesis as improbable.<a href="#cite_note-whit-1">[1]</a><a href="#cite_note-16">[16]</a>

<div class="mw-heading mw-heading2"><h2 id="Lorentz_and_Einstein">Lorentz and Einstein</h2></div>

Since Lorentz was forced to abandon Stokes' hypothesis, he chose Fresnel's model as a starting point. He was able to reproduce Fresnel's dragging coefficient in 1892, though in Lorentz's theory it represents a modification of the propagation of light waves, not the result of any aether entrainment. Therefore, <a href="Lorentz_ether_theory" title="Lorentz ether theory">Lorentz's aether</a> is fully stationary or immobile. However, this leads to the same problem that already afflicted Fresnel's model: it stood in contradiction with the Michelson–Morley experiment. Therefore, <a href="George_Francis_FitzGerald" title="George Francis FitzGerald">George Francis FitzGerald</a> (1889) and Lorentz (1892) introduced <a href="Length_contraction" title="Length contraction">length contraction</a>, that is, all bodies contract in the line of motion by the factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1-v^{2}/c^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {1-v^{2}/c^{2}}}}</annotation>

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</math><img src="./9fc73c0a2afca1aea4203edbca1c77cc9cb28985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.732ex; height:4.843ex;" alt="{\displaystyle {\sqrt {1-v^{2}/c^{2}}}}" loading="lazy">. In addition, in Lorentz's theory the <a href="Galilean_transformation" title="Galilean transformation">Galilean transformation</a> was replaced by the <a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a>.<a href="#cite_note-17">[17]</a>

However, the accumulation of hypotheses to rescue the stationary aether concept was considered to be very artificial. So, it was <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> (1905), who recognized that it is only required to assume the <a href="Principle_of_relativity" title="Principle of relativity">principle of relativity</a> and the constancy of the speed of light in all <a href="Inertial_frame_of_reference" title="Inertial frame of reference">inertial frames of reference</a>, in order to develop the theory of <a href="Special_relativity" title="Special relativity">special relativity</a> and to derive the complete Lorentz transformation. All this was done without using the stationary aether concept.<a href="#cite_note-18">[18]</a>

As shown by <a href="Max_von_Laue" title="Max von Laue">Max von Laue</a> (1907), special relativity predicts the result of the Fizeau experiment from the <a href="Velocity_addition" class="mw-redirect" title="Velocity addition">velocity addition</a> theorem without any need for an aether. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is the velocity of light relative to the Fizeau apparatus and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is the velocity of light relative to the water and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U={\frac {c}{n}}}</annotation>

</semantics>

</math><img src="./b63c575e43a3ea45c218852031649cbccb372477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.112ex; height:4.676ex;" alt="{\displaystyle U={\frac {c}{n}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\frac {c/n+v}{1+v/nc}}}</annotation>

</semantics>

</math><img src="./433d7a3f3a6c849f54492e682e47dbc02d61b9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.416ex; height:6.509ex;" alt="{\displaystyle V={\frac {c/n+v}{1+v/nc}}}" loading="lazy"></dd></dl>

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

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\approx {\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}\right)}</annotation>

</semantics>

</math><img src="./8e9605df2568effe0423e7b7223d0e4af01a3c2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.181ex; height:6.176ex;" alt="{\displaystyle V\approx {\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}\right)}" loading="lazy"></dd></dl>

This is identical to <a href="Fresnel_equation" class="mw-redirect" title="Fresnel equation">Fresnel equation</a>.<a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading2"><h2 id="Allais_aether_hypothesis">Allais aether hypothesis</h2></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Allais_effect" title="Allais effect">Allais effect</a></div>

<a href="Maurice_Allais" title="Maurice Allais">Maurice Allais</a> proposed in 1959 an aether hypothesis involving a wind velocity of about 8 km/s, much lower than the standard value of 30 km/s supported by scientists of the nineteenth century, and compatible with the Michelson–Morley and the disputed<a href="#cite_note-20">[20]</a> <a href="Dayton_Miller" title="Dayton Miller">Dayton Miller</a> experiments,<a href="#cite_note-Miller-21">[21]</a> as well as his own experiments.<a href="#cite_note-Allais_1959_A-22">[22]</a><a href="#cite_note-Allais_1959_B-23">[23]</a> The controversial <a href="Allais_effect" title="Allais effect">Allais effect</a> is not predicted by general relativity and his experimental results have been disputed.<a href="#cite_note-24">[24]</a>: <span title="Page / location: 211

Quotation: "The data showed, however, that there were no significant variations in the pendulum’s period during that eclipse."" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">211 <a href="#cite_note-Miller_reanalyzed-25">[25]</a><a href="#cite_note-26">[26]</a>

<ul><li><a href="History_of_special_relativity" title="History of special relativity">History of special relativity</a></li>

<li><a href="Tests_of_special_relativity" title="Tests of special relativity">Tests of special relativity</a></li>

<li><a href="Tests_of_general_relativity" title="Tests of general relativity">Tests of general relativity</a></li>

<li><a href="Frame-dragging" title="Frame-dragging">Frame-dragging</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWhittaker,_Edmund_Taylor1910" class="citation cs2"><a href="E._T._Whittaker" title="E. T. Whittaker">Whittaker, Edmund Taylor</a> (1910), <a href="A_History_of_the_Theories_of_Aether_and_Electricity" title="A History of the Theories of Aether and Electricity">A History of the Theories of Aether and Electricity</a> (1. ed.), Dublin: Longman, Green and Co.</cite>

</li>

<li id="cite_note-jann-2">^ <a href="#cite_ref-jann_2-0">a</a> <a href="#cite_ref-jann_2-1">b</a> <cite id="CITEREFJannsen,_MichelStachel,_John2008" class="citation cs2">Jannsen, Michel; Stachel, John (2008), <a rel="nofollow" class="external text" href="http://www.mpiwg-berlin.mpg.de/Preprints/P265.PDF">The Optics and Electrodynamics of Moving Bodies</a> (PDF)</cite>

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<li id="cite_note-ferr-3">^ <a href="#cite_ref-ferr_3-0">a</a> <a href="#cite_ref-ferr_3-1">b</a> <cite id="CITEREFRafael_FerraroDaniel_M_Sforza2005" class="citation cs2">Rafael Ferraro; Daniel M Sforza (2005), "Arago (1810): the first experimental result against the ether", Eur. J. Phys., 26 (1): 195–204, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0412055">physics/0412055</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005EJPh...26..195F">2005EJPh...26..195F</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0143-0807%2F26%2F1%2F020">10.1088/0143-0807/26/1/020</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119528074">119528074</a></cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFArago,_A.1810–1853" class="citation cs2">Arago, A. (1810–1853), "Mémoire sur la vitesse de la lumière, lu à la prémière classe de l'Institut, le 10 décembre 1810", Comptes Rendus de l'Académie des Sciences, 36: 38–49</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> Fresnel, A. (1818), "Lettre de M. Fresnel à M. Arago sur l'influence du mouvement terrestre dans quelques phénomènes d'optique", Annales de Chimie et de Physique, 9: 57–66 (Sep. 1818), 286–7 (Nov. 1818); reprinted in H. de Senarmont, E. Verdet, and L. Fresnel (eds.), Oeuvres complètes d'Augustin Fresnel, vol. 2 (1868), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g6tzUG7JmoQC&pg=PA627">pp. 627–36</a>; translated as <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9KQ3BQAAQBAJ&pg=PA125">"Letter from Augustin Fresnel to François Arago, on the influence of the movement of the earth on some phenomena of optics"</a> in K.F. Schaffner, Nineteenth-Century Aether Theories, Pergamon, 1972 (<a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FC2013-0-02335-3">10.1016/C2013-0-02335-3</a>), pp. 125–35; also translated (with several errors) by R.R. Traill as "Letter from Augustin Fresnel to François Arago concerning the influence of terrestrial movement on several optical phenomena", General Science Journal, 23 January 2006 (<a rel="nofollow" class="external text" href="https://www.gsjournal.net/Science-Journals/Historical%20Papers-Mechanics%20/%20Electrodynamics/Download/2496">PDF, 8 pp.</a>).

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFLorentz,_Hendrik_Antoon1904" class="citation cs2">Lorentz, Hendrik Antoon (1904), <a class="external text external" href="https://en.wikisource.org/wiki/Electromagnetic_phenomena">"Electromagnetic phenomena in a system moving with any velocity smaller than that of light" </a>, Proceedings of the Royal Netherlands Academy of Arts and Sciences, 6: 809–831</cite>

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<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFEinstein,_Albert1905" class="citation cs2">Einstein, Albert (1905), <a rel="nofollow" class="external text" href="http://www.fourmilab.ch/etexts/einstein/specrel/">"On the Electrodynamics of Moving Bodies"</a>, Annalen der Physik, 322 (10): 891–921, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1905AnP...322..891E">1905AnP...322..891E</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053221004">10.1002/andp.19053221004</a></cite>.

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<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFLaue,_Max_von1907" class="citation cs2 cs1-prop-foreign-lang-source">Laue, Max von (1907), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1424115">"Die Mitführung des Lichtes durch bewegte Körper nach dem Relativitätsprinzip"</a> [<a href="https://en.wikisource.org/wiki/Translation:The_Entrainment_of_Light_by_Moving_Bodies_in_Accordance_with_the_Principle_of_Relativity" class="extiw external" title="s:Translation:The Entrainment of Light by Moving Bodies in Accordance with the Principle of Relativity">The Entrainment of Light by Moving Bodies in Accordance with the Principle of Relativity</a>], Annalen der Physik (in German), 23 (10): 989–990, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1907AnP...328..989L">1907AnP...328..989L</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19073281015">10.1002/andp.19073281015</a></cite>

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<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFShanklandMcCuskeyLeoneKuerti1955" class="citation journal cs1">Shankland, R. S.; McCuskey, S. W.; Leone, F. C.; Kuerti, G. (1955-04-01). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.27.167">"New Analysis of the Interferometer Observations of Dayton C. Miller"</a>. Reviews of Modern Physics. 27 (2): 167–178. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.27.167">10.1103/RevModPhys.27.167</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>.</cite>

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<cite id="CITEREFMiller1933" class="citation journal cs1">Miller, Dayton C. (July 1933). "The Ether-Drift experiment and the determination of the absolute motion of the Earth". Reviews of Modern Physics. 5 (3): 203–254. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1933RvMP....5..203M">1933RvMP....5..203M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.5.203">10.1103/RevModPhys.5.203</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4119615">4119615</a>.</cite>

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<li id="cite_note-Allais_1959_A-22"><a href="#cite_ref-Allais_1959_A_22-0">^</a> <cite id="CITEREFAllais1959" class="citation journal cs1">Allais, M. (September 1959). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150720003631/http://home.t01.itscom.net/allais/blackprior/allais/lawgravit/lawgrav-one.pdf">"Should the Laws of Gravitation Be reconsidered? Part I – Abnormalities in the Motion of a Paraconical Pendulum on an Anisotropic Support"</a> (PDF). Aero/Space Engineering: 46–52. Archived from <a rel="nofollow" class="external text" href="http://home.t01.itscom.net/allais/blackprior/allais/lawgravit/lawgrav-one.pdf">the original</a> (PDF) on 2015-07-20. Retrieved 2017-03-30.</cite>

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<li id="cite_note-Allais_1959_B-23"><a href="#cite_ref-Allais_1959_B_23-0">^</a> <cite id="CITEREFAllais1959" class="citation journal cs1">Allais, M. (October 1959). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160622213903/http://home.t01.itscom.net/allais/blackprior/allais/lawgravit/lawgrav-two.pdf">"Should the Laws of Gravitation Be reconsidered? Part II – Experiments in Connection with the Abnormalities Noted in the Motion of the Paraconical Pendulum With an Anisotropic Support"</a> (PDF). Aero/Space Engineering: 51–55. Archived from <a rel="nofollow" class="external text" href="http://home.t01.itscom.net/allais/blackprior/allais/lawgravit/lawgrav-two.pdf">the original</a> (PDF) on 2016-06-22. Retrieved 2017-03-30.</cite>

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<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFGillies1997" class="citation journal cs1">Gillies, George T (1997-02-01). "The Newtonian gravitational constant: recent measurements and related studies". Reports on Progress in Physics. 60 (2): 151–225. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0034-4885%2F60%2F2%2F001">10.1088/0034-4885/60/2/001</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-4885">0034-4885</a>.</cite>

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<li id="cite_note-Miller_reanalyzed-25"><a href="#cite_ref-Miller_reanalyzed_25-0">^</a> <cite id="CITEREFDeloly" class="citation web cs1">Deloly, Jean-Bernard. <a rel="nofollow" class="external text" href="http://www.fondationmauriceallais.org/the-physicist/the-re-examination-of-millers-interferometric-observations-and-of-esclangons-observations/?lang=en">"The re-examination of Miller's interferometric observations and of Esclangon's observations"</a>. Maurice Allais Foundation.</cite>

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<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFSavrov1989" class="citation journal cs1">Savrov, L. A. (September 1989). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/BF02508025">"Paraconic pendulum: Proposal to detect the gravity screening effect during the full solar eclipse"</a>. Il Nuovo Cimento C. 12 (5): 681–683. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02508025">10.1007/BF02508025</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0390-5551">0390-5551</a>.</cite>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a class="external text external" href="https://en.wikibooks.org/wiki/Special_Relativity">Wikibooks: Special Relativity</a></li>

<li>Resnick, Robert, Basic Concepts in Relativity and Early Quantum Theory, 1972, John Wiley and Sons Inc.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Mathpages: <a rel="nofollow" class="external text" href="https://archive.today/20130128094248/http://www.mathpages.com/home/kmath561/kmath561.htm">Stokes’ Mistake</a></li></ul></div><div><div class="zim-footer">

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