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] ]
The '''Fizeau experiment'''<ref name="Becker">{{cite book |last1=Becker |first1=Richard |last2=Sauter |first2=Fritz |title=Electromagnetic Fields and Interactions |date=1 January 1982 |publisher=Courier Corporation |isbn=978-0-486-64290-1 |page=308 |url=https://books.google.com/books?id=5U7HVjHbphwC |access-date=9 March 2023 |language=en}}</ref><ref name="Rohrlich">{{cite book |last1=Rohrlich |first1=Fritz |title=From Paradox to Reality: Our Basic Concepts of the Physical World |date=25 August 1989 |publisher=Cambridge University Press |isbn=978-0-521-37605-1 |page=54 |url=https://books.google.com/books?id=3TqA1394OVcC |access-date=9 March 2023 |language=en}}</ref><ref name="https://www.google.com/books/edition/Introductory_Special_Relativity/zpjBEBbIjAIC">{{cite book |last1=Rosser |first1=W. G. V. |title=Introductory Special Relativity |date=6 January 1992 |publisher=CRC Press |isbn=978-0-85066-838-4 |page=113 |url=https://books.google.com/books?id=zpjBEBbIjAIC |access-date=9 March 2023 |language=en}}</ref> was carried out by ] in 1851 to measure the relative speeds of light in moving water. Fizeau used a special ] arrangement to measure the effect of movement of a medium upon the speed of light. The '''Fizeau experiment''' was carried out by ] in 1851 to measure the relative speeds of light in moving water. Fizeau used a special ] arrangement to measure the effect of movement of a medium upon the speed of light.


According to the theories prevailing at the time, light traveling through a moving medium would be dragged along by the medium, so that the measured speed of the light would be a simple sum of its speed ''through'' the medium plus the speed ''of'' the medium. Fizeau indeed detected a dragging effect, but the magnitude of the effect that he observed was far lower than expected. When he repeated the experiment with air in place of water he observed no effect. His results seemingly supported the ] of ], a situation that was disconcerting to most physicists. Over half a century passed before a satisfactory explanation of Fizeau's unexpected measurement was developed with the advent of ]'s theory of ]. Einstein later pointed out the importance of the experiment for special relativity, in which it corresponds to the relativistic ] when restricted to small velocities. According to the theories prevailing at the time, light traveling through a moving medium would be dragged along by the medium, so that the measured speed of the light would be a simple sum of its speed ''through'' the medium plus the speed ''of'' the medium. Fizeau indeed detected a dragging effect, but the magnitude of the effect that he observed was far lower than expected. When he repeated the experiment with air in place of water he observed no effect. His results seemingly supported the ] of ], a situation that was disconcerting to most physicists. Over half a century passed before a satisfactory explanation of Fizeau's unexpected measurement was developed with the advent of ]'s theory of ]. Einstein later pointed out the importance of the experiment for special relativity, in which it corresponds to the relativistic ] when restricted to small velocities.


Although it is referred to as ''the'' Fizeau experiment, Fizeau was an active experimenter who carried out a wide variety of different experiments involving measuring the speed of light in various situations. Although it is referred to as ''the'' Fizeau experiment, Fizeau was an active experimenter who carried out a wide variety of different experiments involving measuring the speed of light in various situations.

== Background ==
{{main| History of electromagnetism}}
As scientists in the 1700's worked on a theory of light and of electromagnetism, ], a medium that would support waves, was the focus of many experiments.<ref group=S name=Whittaker>{{Cite book |last=Whittaker |first=E. T. |title=A history of the theories of aether & electricity |date=1989 |publisher=Dover Publications |isbn=978-0-486-26126-3 |location=New York}}</ref>{{rp|98}} Two critical issues were the relation of aether to motion and its relation to matter. For example, ], the apparent motion of stars observed at different times of year, was proposed to be related to starlight propagated through an aether.<ref group=S name=Whittaker/>{{rp|108}} In 1846 Fresnel proposed that the portion aether that will move with an object relates to the object's index of refraction of light, which was take to be the ratio of the speed of light in the material to the speed of light in interstellar space.<ref group=S name=Whittaker/>{{rp|110}}
Having recently measured the speed of light in air and water, Fizeau set out to measure the speed of light in moving water.<ref name=Mermin group=S/>


== Experimental setup == == Experimental setup ==
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The simplified arrangement illustrated in Fig.&nbsp;2 would have required the use of monochromatic light, which would have enabled only dim fringes. Because of white light's short ], use of white light would have required matching up the optical paths to an impractical degree of precision, and the apparatus would have been extremely sensitive to vibration, motion shifts, and temperature effects. The simplified arrangement illustrated in Fig.&nbsp;2 would have required the use of monochromatic light, which would have enabled only dim fringes. Because of white light's short ], use of white light would have required matching up the optical paths to an impractical degree of precision, and the apparatus would have been extremely sensitive to vibration, motion shifts, and temperature effects.


On the other hand, Fizeau's actual apparatus, illustrated in Fig.&nbsp;3 and Fig.&nbsp;4, was set up as a ]. This guaranteed that the opposite beams would pass through equivalent paths, so that fringes readily formed even when using the sun as a light source. Fizeau's actual apparatus, illustrated in Fig.&nbsp;3 and Fig.&nbsp;4, was set up as a ]. This guaranteed that the opposite beams would pass through equivalent paths, so that fringes readily formed even when using the sun as a light source.
{{Quote {{Quote
|text=The double transit of the light was for the purpose of augmenting the distance traversed in the medium in motion, and further to compensate entirely any accidental difference of temperature or pressure between the two tubes, from which might result a displacement of the fringes, which would be mingled with the displacement which the motion alone would have produced; and thus have rendered the observation of it uncertain.<ref name=fiz1 group=P /> |text=The double transit of the light was for the purpose of augmenting the distance traversed in the medium in motion, and further to compensate entirely any accidental difference of temperature or pressure between the two tubes, from which might result a displacement of the fringes, which would be mingled with the displacement which the motion alone would have produced; and thus have rendered the observation of it uncertain.<ref name=fiz1 group=P />
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A light ray emanating from the source ''{{prime|S}}'' is reflected by a ] ''G'' and is ] into a parallel beam by lens ''L''. After passing the slits ''O''<sub>1</sub> and ''O''<sub>2</sub>, two rays of light travel through the tubes ''A''<sub>1</sub> and ''A''<sub>2</sub>, through which water is streaming back and forth as shown by the arrows. The rays reflect off a mirror ''m'' at the focus of lens ''{{prime|L}}'', so that one ray always propagates in the same direction as the water stream, and the other ray opposite to the direction of the water stream. After passing back and forth through the tubes, both rays unite at ''S'', where they produce interference fringes that can be visualized through the illustrated eyepiece. The ] can be analyzed to determine the speed of light traveling along each leg of the tube.<ref name=fiz1 group=P/><ref name=fiz2 group=P /><ref name=Mascart group=S/> A light ray emanating from the source ''{{prime|S}}'' is reflected by a ] ''G'' and is ] into a parallel beam by lens ''L''. After passing the slits ''O''<sub>1</sub> and ''O''<sub>2</sub>, two rays of light travel through the tubes ''A''<sub>1</sub> and ''A''<sub>2</sub>, through which water is streaming back and forth as shown by the arrows. The rays reflect off a mirror ''m'' at the focus of lens ''{{prime|L}}'', so that one ray always propagates in the same direction as the water stream, and the other ray opposite to the direction of the water stream. After passing back and forth through the tubes, both rays unite at ''S'', where they produce interference fringes that can be visualized through the illustrated eyepiece. The ] can be analyzed to determine the speed of light traveling along each leg of the tube.<ref name=fiz1 group=P/><ref name=fiz2 group=P /><ref name=Mascart group=S/>


== Result ==
== Fresnel drag coefficient ==
Fizeau's experiment showed a faster speed of light in water moving in the same direction and a slower speed when the water moved opposite the light. However the amount of difference in the speed of light was only a fraction of the water speed. Interpreted in terms of the aether theory, the water seemed to drag the aether and thus the light propagation, but only partially.<ref name="Rohrlich">{{cite book |last1=Rohrlich |first1=Fritz |title=From Paradox to Reality: Our Basic Concepts of the Physical World |date=25 August 1989 |publisher=Cambridge University Press |isbn=978-0-521-37605-1 |page=54 |url=https://books.google.com/books?id=3TqA1394OVcC |access-date=9 March 2023 |language=en}}</ref>{{rp|53}}

== Impact ==
{{main|Aether drag hypothesis}} {{main|Aether drag hypothesis}}
At the time of Fizeau's experiment, two different models of how aether related to moving bodies were discussed, Fresnel's ] and ]' ] hypothesis. Fresnel had ] (1818) proposed his model to explain ]. In 1845 Stokes showed that complete aether drag could also explain it. Since Fresnel had no model to explain partial drag, scientists favored Stokes explanation.<ref name=Stachel2005 group=S>{{cite book |last=Stachel |first=J. |title=The universe of general relativity|year=2005|publisher=Birkhäuser|location=Boston|isbn=0-8176-4380-X|pages=1–13 |chapter-url=https://books.google.com/books?id=-KlBhDwUKF8C&pg=PA1 |editor=Kox, A.J. |editor2=Eisenstaedt, J |access-date=17 April 2012 |chapter=Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies}}</ref>
Assume that water flows in the pipes with speed ''v''. According to the non-relativistic theory of the ], the speed of light should be increased or decreased when "dragged" along by the water through the aether frame, dependent upon the direction. According to Stokes' ] hypothesis, the overall speed of a beam of light should be a simple additive sum of its speed ''through'' the water plus the speed ''of'' the water.


According to the Stokes' hypothesis, the speed of light should be increased or decreased when "dragged" along by the water through the aether frame, dependent upon the direction.<ref group=S name=ferr />{{rp|33|q=Stokes' hypothesis corresponds to f=1 (total dragging), while Fresnel's partial dragging implies the value f=1-n<sup>-1</sup> = 0.43"}} The overall speed of a beam of light should be a simple additive sum of its speed ''through'' the water plus the speed ''of'' the water.
That is, if ''n'' is the ] of water, so that ''c/n'' is the speed of light in stationary water, then the predicted speed of light ''w'' in one arm would be That is, if ''n'' is the ] of water, so that ''c/n'' is the speed of light in stationary water, then the predicted speed of light ''w'' in one arm would be<ref name=Mermin group=S/>{{rp|40}}
: <math>w_+=\frac{c}{n}+v \ , </math> : <math>w_+=\frac{c}{n}+v \ , </math>
and the predicted speed in the other arm would be and the predicted speed in the other arm would be
: <math>w_-=\frac{c}{n} - v \ . </math> : <math>w_-=\frac{c}{n} - v \ , </math>
for water with velocity <math>v</math>.
Hence light traveling against the flow of water should be slower than light traveling with the flow of water. Hence light traveling against the flow of water should be slower than light traveling with the flow of water.
The ] between the two beams when the light is recombined at the observer depends upon the transit times over the two paths.<ref name=Wood group=S>{{cite book |title=Physical Optics |author=Robert Williams Wood |url=https://archive.org/details/bub_gb_Ohp5AAAAIAAJ |page= |year=1905 |publisher=The Macmillan Company}}</ref> The ] between the two beams when the light is recombined at the observer depends upon the transit times over the two paths.<ref name=Wood group=S>{{cite book |title=Physical Optics |author=Robert Williams Wood |url=https://archive.org/details/bub_gb_Ohp5AAAAIAAJ |page= |year=1905 |publisher=The Macmillan Company}}</ref>


Fizeau found that However Fizeau found that
: <math>w_+=\frac{c}{n}+ v\left(1-\frac{1}{n^2}\right) \ . </math> : <math>w_+=\frac{c}{n}+ v\left(1-\frac{1}{n^2}\right) \ . </math>
In other words, light appeared to be dragged by the water, but the magnitude of the dragging was much lower than expected. In other words, light appeared to be dragged by the water, but the magnitude of the dragging was much lower than expected.


The Fizeau experiment forced physicists to accept the empirical validity of an older theory of ] (1818) that had been invoked to explain ], namely, that a medium moving through the stationary aether drags light propagating through it with only a fraction of the medium's speed, with a dragging coefficient ''f'' given by The Fizeau experiment forced physicists to accept the empirical validity of an Fresnel's model, that a medium moving through the stationary aether drags light propagating through it with only a fraction of the medium's speed, with a dragging coefficient ''f'' related to the index of refraction:
: <math>f = 1-\frac{1}{n^2} \ . </math> : <math>f = 1-\frac{1}{n^2} \ . </math>


Although Fresnel's hypothesis was empirically successful in explaining Fizeau's results, many experts in the field, including Fizeau himself, found Fresnel's hypothesis partial aether-dragging unsatisfactory. Fresnel had found an empirical formula that worked but no mechanical model of the aether was used to derive it.<ref name=Stachel2005 group=S>{{cite book |last=Stachel |first=J. |title=The universe of general relativity|year=2005|publisher=Birkhäuser|location=Boston|isbn=0-8176-4380-X|pages=1–13 |chapter-url=https://books.google.com/books?id=-KlBhDwUKF8C&pg=PA1 |editor=Kox, A.J. |editor2=Eisenstaedt, J |access-date=17 April 2012 |chapter=Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies}}</ref>
In 1895, ] predicted the existence of an extra term due to ]:<ref name=paul group=S />{{rp|15–20}}
: <math> w_+ = \frac {c}{n} + v \left(1 - \frac{1}{n^2} - \frac{\lambda}{n} \! \cdot \! \frac{ \mathrm{d} n }{ \mathrm{d} \lambda } \right) \ . </math>
Since the medium is flowing towards or away from the observer, the light traveling through the medium is Doppler-shifted, and the refractive index used in the formula has to be that appropriate to the Doppler-shifted wavelength.<ref group=P name=jones1 /> Zeeman verified the existence of Lorentz' dispersion term in 1915.<ref name=zee2 group=P />

It turned out later that Fresnel's dragging coefficient is indeed in accordance with the relativistic velocity addition formula, see ''{{slink|#Derivation in special relativity}}''.

== Repetitions ==
]
] and ] (1886)<ref name=mich group=P /> repeated Fizeau's experiment with improved accuracy, addressing several concerns with Fizeau's original experiment: (1) Deformation of the optical components in Fizeau's apparatus could cause artifactual fringe displacement; (2) observations were rushed, since the pressurized flow of water lasted only a short time; (3) the ] profile of water flowing through Fizeau's small diameter tubes meant that only their central portions were available, resulting in faint fringes; (4) there were uncertainties in Fizeau's determination of flow rate across the diameter of the tubes. Michelson redesigned Fizeau's apparatus with larger diameter tubes and a large reservoir providing three minutes of steady water flow. His ] design provided automatic compensation of path length, so that white light fringes were visible at once as soon as the optical elements were aligned. Topologically, the light path was that of a ] with an even number of reflections in each light path.<ref name=Hariharan2007 group=S/> This offered extremely stable fringes that were, to first order, completely insensitive to any movement of its optical components. The stability was such that it was possible for him to insert a glass plate at '''''h''''' or even to hold a lighted match in the light path without displacing the center of the fringe system. Using this apparatus, Michelson and Morley were able to completely confirm Fizeau's results not just in water, but also in air.<ref name=mich group=P />

Other experiments were conducted by ] in 1914–1915. Using a scaled-up version of Michelson's apparatus connected directly to ]'s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å) to confirm Lorentz's modified coefficient.<ref name=zee1 group=P /><ref name=zee2 group=P />
In 1910, ] used a ''rotating'' device and overall confirmed Fresnel's dragging coefficient. However, he additionally found a "systematic bias" in the data, which later turned out to be the ].<ref name=and group=S />

Since then, many experiments have been conducted measuring such dragging coefficients in a diversity of materials of differing refractive index, often in combination with the Sagnac effect.<ref group=S name=sted /> For instance, in experiments using ]s together with rotating disks,<ref group=P name=macek /><ref group=P name=bilger1 /><ref group=P name=bilger2 /><ref group=P name=sanders /> or in ] experiments.<ref group=P name=klein /><ref group=P name=bonse /><ref group=P name=arif /> Also a transverse dragging effect was observed, i.e. when the medium is moving at right angles to the direction of the incident light.<ref group=P name=jones1 /><ref group=P name=jones2 />


== Hoek experiment == == Confirmation ==
=== Wilhelm Veltmann's colors of light ===
An indirect confirmation of Fresnel's dragging coefficient was provided by ] (1868).<ref group=P name=hoek /><ref group=S name=ferr />
In 1870 Wilhelm Veltmann demonstrated that Fresnel's formula worked for different frequencies (colors) of light. According the Fresnel's model this would imply different amounts of eather drag for different colors of light. The velocity with white light, a mixture of colors, would be unexplained.<ref name=Stachel2005 group=S/>
=== Hoek experiment ===
{{See also|Hammar experiment}}
An indirect confirmation of Fresnel's dragging coefficient was provided by ] (1868).<ref group=P name=hoek />
His apparatus was similar to Fizeau's, though in his version only one arm contained an area filled with resting water, while the other arm was in the air. As seen by an observer resting in the aether, Earth and hence the water is in motion. So the following travel times of two light rays traveling in opposite directions were calculated by Hoek (neglecting the transverse direction, see image): His apparatus was similar to Fizeau's, though in his version only one arm contained an area filled with resting water, while the other arm was in the air. As seen by an observer resting in the aether, Earth and hence the water is in motion. So the following travel times of two light rays traveling in opposite directions were calculated by Hoek (neglecting the transverse direction, see image):


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|} |}


The travel times are not the same, which should be indicated by an interference shift. However, if Fresnel's dragging coefficient is applied to the water in the aether frame, the travel time difference (to first order in ''v/c'') vanishes. Using different setups Hoek actually obtained a null result, confirming Fresnel's dragging coefficient. (For a similar experiment refuting the possibility of ''shielding'' the aether wind, see '']''). The travel times are not the same, which should be indicated by an interference shift. However, if Fresnel's dragging coefficient is applied to the water in the aether frame, the travel time difference (to first order in ''v/c'') vanishes. Upon turning the apparatus table 180 degrees, altering the direction of a hypothetical aether wind, Hoek obtained a null result, confirming Fresnel's dragging coefficient.<ref group=S name=ferr /><ref group=S name=Whittaker/>{{rp|111}}


In the particular version of the experiment shown here, Hoek used a prism ''P'' to disperse light from a slit into a spectrum which passed through a collimator ''C'' before entering the apparatus. With the apparatus oriented parallel to the hypothetical aether wind, Hoek expected the light in one circuit to be retarded 7/600&nbsp;mm with respect to the other. Where this retardation represented an integral number of wavelengths, he expected to see constructive interference; where this retardation represented a half-integral number of wavelengths, he expected to see destructive interference. In the absence of dragging, his expectation was for the observed spectrum to be continuous with the apparatus oriented transversely to the aether wind, and to be banded with the apparatus oriented parallel to the aether wind. His actual experimental results were completely negative.<ref group=P name=hoek /><ref group=S name=ferr /> In the particular version of the experiment shown here, Hoek used a prism ''P'' to disperse light from a slit into a spectrum which passed through a collimator ''C'' before entering the apparatus. With the apparatus oriented parallel to the hypothetical aether wind, Hoek expected the light in one circuit to be retarded 7/600&nbsp;mm with respect to the other. Where this retardation represented an integral number of wavelengths, he expected to see constructive interference; where this retardation represented a half-integral number of wavelengths, he expected to see destructive interference. In the absence of dragging, his expectation was for the observed spectrum to be continuous with the apparatus oriented transversely to the aether wind, and to be banded with the apparatus oriented parallel to the aether wind. His actual experimental results were completely negative.<ref group=P name=hoek /><ref group=S name=ferr />


=== Mascart's birefringence experiment ===
==Controversy==
Éleuthère Mascart (1872) demonstrated a result for polarized light traveling through a birefringent medium gives different velocities in accordance with Fresnel's empirical formula. However, the result in terms of Fresnel's physical model requires different aether drag in different direction in the medium.<ref name=Stachel2005 group=S/>
Although Fresnel's hypothesis was empirically successful in explaining Fizeau's results, many experts in the field, including Fizeau himself (1851), ] (1872), Ketteler (1873), Veltmann (1873), and Lorentz (1886) found Fresnel's mechanical reasoning for partial aether-dragging unpalatable for various reasons. For example, Veltmann (1870) Explains that Fresnel's hypothesis was proposed as a "so-called compensation" of aberration which will "exactly cancel out" the deflection of Arago experiment. He then goes on to demonstrate a method for using Stokes' fully dragged aether in lieu of Fresnel's hypothesis which would still be "necessary at the end of the development". At the end he returns to the principle of Fresnel emphasizing that it is a mathematical relationship that represents a "common principle" to a "class of explanations" of starlight aberration by clarifying:


=== Michelson and Morley confirmation ===
{{quote|The speed with which the movement of light takes part in the movement of the medium depends on the speed of propagation and must therefore be different for each color. (translation by Google) ''Die Geschwindigkeit, mit welcher die Lichtbewegung an der Bewegung des Mediums theilnimmt, hängt von der Fortpflanzungsgeschwindigkeit ab und müsste deshalb für jede Farbe eine andere sein.''}}
]
] and ] (1886)<ref name=mich group=P /> repeated Fizeau's experiment with improved accuracy,<ref name="Rosser1992" group=S>{{cite book |last1=Rosser |first1=W. G. V. |title=Introductory Special Relativity |date=6 January 1992 |publisher=CRC Press |isbn=978-0-85066-838-4 |page=113 |url=https://books.google.com/books?id=zpjBEBbIjAIC |access-date=9 March 2023 |language=en}}</ref>{{rp|113}} addressing several concerns with Fizeau's original experiment: (1) Deformation of the optical components in Fizeau's apparatus could cause artifactual fringe displacement; (2) observations were rushed, since the pressurized flow of water lasted only a short time; (3) the ] profile of water flowing through Fizeau's small diameter tubes meant that only their central portions were available, resulting in faint fringes; (4) there were uncertainties in Fizeau's determination of flow rate across the diameter of the tubes. Michelson redesigned Fizeau's apparatus with larger diameter tubes and a large reservoir providing three minutes of steady water flow. His ] design provided automatic compensation of path length, so that white light fringes were visible at once as soon as the optical elements were aligned. Topologically, the light path was that of a ] with an even number of reflections in each light path.<ref name=Hariharan2007 group=S/> This offered extremely stable fringes that were, to first order, completely insensitive to any movement of its optical components. The stability was such that it was possible for him to insert a glass plate at '''''h''''' or even to hold a lighted match in the light path without displacing the center of the fringe system. Using this apparatus, Michelson and Morley were able to completely confirm Fizeau's results not just in water, but also in air.<ref name=mich group=P />


=== Zeeman and Lorentz's improved formula ===
This line can be more directly translated as "the speed with which the movement of light to the movement of the medium depends on the propagation speed and therefore is needed a different one for each color." Thus confirming Fresnel's mathematical principle (but not his explanation) that rate at which a medium affects the speed of light is dependent upon the index of refraction which was already established to be a measure of alterations to light's speed dependent on frequency.
In 1895, ] predicted the existence of an extra term due to ]:<ref name=paul group=S />{{rp|15–20}}
: <math> w_+ = \frac {c}{n} + v \left(1 - \frac{1}{n^2} - \frac{\lambda}{n} \! \cdot \! \frac{ \mathrm{d} n }{ \mathrm{d} \lambda } \right) \ . </math>
Since the medium is flowing towards or away from the observer, the light traveling through the medium is Doppler-shifted, and the refractive index used in the formula has to be that appropriate to the Doppler-shifted wavelength.<ref group=P name=jones1 /> Zeeman verified the existence of Lorentz' dispersion term in 1915.<ref name=zee2 group=P /> Using a scaled-up version of Michelson's apparatus connected directly to ]'s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å) to confirm Lorentz's modified coefficient.<ref name=zee1 group=P /><ref name=zee2 group=P />


=== Later confirmations ===
However the historian Stachel in 2005 gives us a different interpretation that assumes the "one for each color" to mean ether instead of differing "rates" or "speeds".
In 1910, ] used a ''rotating'' device and overall confirmed Fresnel's dragging coefficient. However, he additionally found a "systematic bias" in the data, which later turned out to be the ].<ref name=and group=S />


Since then, many experiments have been conducted measuring such dragging coefficients in a diversity of materials of differing refractive index, often in combination with the Sagnac effect.<ref group=S name=sted /> For instance, in experiments using ]s together with rotating disks,<ref group=P name=macek /><ref group=P name=bilger1 /><ref group=P name=bilger2 /><ref group=P name=sanders /> or in ] experiments.<ref group=P name=klein /><ref group=P name=bonse /><ref group=P name=arif /> Also a transverse dragging effect was observed, i.e. when the medium is moving at right angles to the direction of the incident light.<ref group=P name=jones1 /><ref group=P name=jones2 />
{{quote|Veltmann (1870) demonstrates experimentally that Fresnel's formula must be applied using the appropriate (different) index of refraction for each color of light. This means that, however the ether moves, it must move differently for each frequency of light. But what happens when white light (or indeed any mixture of frequencies) passes through a transparent medium?<ref name=Stachel2005 group=S>{{cite book |last=Stachel |first=J. |title=The universe of general relativity|year=2005|publisher=Birkhäuser|location=Boston|isbn=0-8176-4380-X|pages=1–13 |chapter-url=https://books.google.com/books?id=-KlBhDwUKF8C&pg=PA1 |editor=Kox, A.J. |editor2=Eisenstaedt, J |access-date=17 April 2012 |chapter=Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies}}</ref>}}

Mascart (1872) demonstrated a result for polarized light traveling through a birefringent medium is insensitive to the motion of the earth. After establishing that Fresnel's theory represents an exact compensatory mechanism that cancels aberration effects, he discusses various other exact compensatory mechanisms in mechanical wave systems including the insensitivity to the doppler effect of co-moving experiments. He concludes " formula is not applicable to birefringent media." He finalized this report on his experiments in birefringent media with his finding that the experiment in anisotropic media produced a resulting quantity which was "four times lower than that which we would obtain by applying to the propagation of circularly polarized waves the formula demonstrated by Fresnel for the case of isotropic bodies."

Fizeau himself shows he was aware of the mechanical feasibility of Fresnel's hypothesis earlier in his report, but Fizeau's surprise and defied expectation of Stokes' complete drag was intimated at the conclusion to the report:
<blockquote>
<span style="line-height: 1.5em;">Lastly, if only one part of the æther is carried along, the velocity of light would be increased, but only by a fraction of the velocity of the body, and not, as in the first hypothesis, by the whole velocity. This consequence is not so obvious as the former, but Fresnel has shown that it may be supported by mechanical arguments of great probability.
The success of the experiment seems to me to render the adoption of Fresnel's hypothesis necessary, or at least the law which he found for the expression of the alteration of the velocity of light by the effect of motion of a body; for although that law being found true may be a very strong proof in favour of the hypothesis of which it is only a consequence, perhaps the conception of Fresnel may appear so extraordinary, and in some respects so difficult, to admit, that other proofs and a profound examination on the part of geometricians will still be necessary before adopting it as an expression of the real facts of the case.<ref name=fiz1 group=P /></span>
</blockquote>

Despite the dissatisfaction of most physicists{{citation needed|reason=appears to be synthesis based on Stachel's focus on partial-drag detractors. Perhaps more references from Buchwald and Schaffner (which Stachel mentions) are needed to remove article bias |date=September 2022}} with Fresnel's partial aether-dragging hypothesis, repetitions and improvements to Fizeau's experiment (]) by others confirmed his results to high accuracy.

In addition to Mascart's experiments which demonstrated an insensitivity to earth's motion and complaints about the partial aether-dragging hypothesis, another major problem arose with the ] (1887). Mascart's claims that optical experiments of refraction and reflection would be insensitive to the earth's motion were proven out by this later experiment. In Fresnel's theory, the aether is almost stationary and the Earth is moving through it, so the experiment should have given a partially reduced, but net positive, result. Only a complete aether drag by the medium of the air would result in a null. However, the result of this experiment was reported as null. Thus from the viewpoint of the aether models at that time, the experimental situation was contradictory: On one hand, the ], the Fizeau experiment and its repetition by Michelson and Morley in 1886 appeared to support only a small degree of aether-dragging. On the other hand, the Michelson–Morley experiment of 1887 appeared to prove that the aether is at rest with respect to Earth, apparently supporting the idea of complete aether-dragging (see '']'').<ref group=S name=jan /> So the success of Fresnel's hypothesis in explaining Fizeau's results helped lead to a theoretical crisis, which was only resolved by the introduction of relativistic theory.

<blockquote>
Is it fantastic to imagine that someone might have been led to develop some or all of these kinematical responses to the challenge presented by the situation in the optics of moving bodies around 1880, given that an optical principle of relative motion had been formulated by Mascart? Perhaps no more fantastic than what actually happened: Einstein's development around 1905 of a kinematical response to the challenge presented by the situation in the electrodynamics of moving bodies, given that an electrodynamic principle of relative motion had already been formulated by Poincaré.
<ref name=Stachel2005 group=S/>
</blockquote>


== Lorentz's interpretation == == Lorentz's interpretation ==
Line 110: Line 100:
In 1895, Lorentz more generally explained Fresnel's coefficient based on the concept of local time. However, Lorentz's theory had the same fundamental problem as Fresnel's: a stationary aether contradicted the ]. So in 1892 Lorentz proposed that moving bodies contract in the direction of motion (], since ] had already arrived in 1889 at this conclusion). The equations that he used to describe these effects were further developed by him until 1904. These are now called the ] in his honor, and are identical in form to the equations that Einstein was later to derive from first principles. Unlike Einstein's equations, however, Lorentz's transformations were strictly ''ad hoc'', their only justification being that they seemed to work.<ref group=S name=jan /><ref group=S name=mil />{{rp|27–30}} In 1895, Lorentz more generally explained Fresnel's coefficient based on the concept of local time. However, Lorentz's theory had the same fundamental problem as Fresnel's: a stationary aether contradicted the ]. So in 1892 Lorentz proposed that moving bodies contract in the direction of motion (], since ] had already arrived in 1889 at this conclusion). The equations that he used to describe these effects were further developed by him until 1904. These are now called the ] in his honor, and are identical in form to the equations that Einstein was later to derive from first principles. Unlike Einstein's equations, however, Lorentz's transformations were strictly ''ad hoc'', their only justification being that they seemed to work.<ref group=S name=jan /><ref group=S name=mil />{{rp|27–30}}


== Einstein's use of Fizeau's experiment ==
== Derivation in special relativity ==
{{Main article|Special relativity}} {{Main article|Special relativity}}
Einstein showed how Lorentz's equations could be derived as the logical outcome of a set of two simple starting postulates. In addition Einstein recognized that the stationary aether concept has no place in special relativity, and that the Lorentz transformation concerns the nature of space and time. Together with the ], the ], and the ], the Fizeau experiment was one of the key experimental results that shaped Einstein's thinking about relativity.<ref name=lah group=S /><ref name="norton" group=S>{{citation |last1=Norton, John D. |year=2004 |first1=John D. |journal=Archive for History of Exact Sciences |title= Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905 |pages= 45–105 |volume=59 |issue=1 |url=http://philsci-archive.pitt.edu/archive/00001743/ |doi=10.1007/s00407-004-0085-6 |bibcode=2004AHES...59...45N |s2cid=17459755 }}</ref> ] reported some conversations with Einstein, in which Einstein emphasized the importance of the Fizeau experiment:<ref name=shank group=S /> Einstein showed how Lorentz's equations could be derived as the logical outcome of a set of two simple starting postulates. In addition Einstein recognized that the stationary aether concept has no place in special relativity, and that the Lorentz transformation concerns the nature of space and time. Together with the ], the ], and the ], the Fizeau experiment was one of the key experimental results that shaped Einstein's thinking about relativity.<ref name=lah group=S /><ref name="norton" group=S>{{citation |last1=Norton, John D. |year=2004 |first1=John D. |journal=Archive for History of Exact Sciences |title= Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905 |pages= 45–105 |volume=59 |issue=1 |url=http://philsci-archive.pitt.edu/archive/00001743/ |doi=10.1007/s00407-004-0085-6 |bibcode=2004AHES...59...45N |s2cid=17459755 }}</ref> ] reported some conversations with Einstein, in which Einstein emphasized the importance of the Fizeau experiment:<ref name=shank group=S />
Line 116: Line 106:
{{Quote|He continued to say the experimental results which had influenced him most were the observations of ] and Fizeau's measurements on the speed of light in moving water. "They were enough," he said.}} {{Quote|He continued to say the experimental results which had influenced him most were the observations of ] and Fizeau's measurements on the speed of light in moving water. "They were enough," he said.}}


==Modern interpretation==
] (1907) demonstrated that the Fresnel drag coefficient can be easily explained as a natural consequence of the relativistic formula for ],<ref name=Mermin group=S>{{cite book |title=It's about time: understanding Einstein's relativity |author=N David Mermin |url=https://archive.org/details/itsabouttimeunde0000merm |url-access=registration |pages=''ff'' |isbn=0-691-12201-6 |publisher=Princeton University Press |year=2005}}</ref> namely:
] (1907) demonstrated<ref name=laue group=P />
: The speed of light in immobile water is ''c/n''.
that the Fresnel drag coefficient can be explained as a natural consequence of the relativistic formula for ].<ref name=Mermin group=S>{{cite book |title=It's about time: understanding Einstein's relativity |author=N David Mermin |url=https://archive.org/details/itsabouttimeunde0000merm |url-access=registration |pages=''ff'' |isbn=0-691-12201-6 |publisher=Princeton University Press |year=2005}}</ref>
: From the ] it follows that the speed of light observed in the laboratory, where water is flowing with speed ''v'' (in the same direction as light) is The speed of light in immobile water is ''c/n''. From the ] it follows that the speed of light observed in the laboratory, where water is flowing with speed ''v'' (in the same direction as light) is
:: <math>V_\mathrm{lab}=\frac{\frac{c}{n}+v}{1+\frac{\frac{c}{n}v}{c^2}}=\frac{\frac{c}{n}+v}{1+\frac{v}{cn}} \ .</math> <math display="block">V_\mathrm{lab}=\frac{\frac{c}{n}+v}{1+\frac{\frac{c}{n}v}{c^2}}=\frac{\frac{c}{n}+v}{1+\frac{v}{cn}} \ .</math>
: Thus the difference in speed is (assuming ''v'' is small comparing to ''c'', dropping higher order terms) Thus the difference in speed is (assuming ''v'' is small comparing to ''c'', dropping higher order terms)
:: <math>V_\mathrm{lab}-\frac{c}{n} = \frac{\frac{c}{n}+v}{1+\frac{v}{cn}}-\frac{c}{n}=\frac{\frac{c}{n}+v-\frac{c}{n}(1+\frac{v}{cn})}{1+\frac{v}{cn}} </math> <math> = \frac{v\left(1-\frac{1}{n^2}\right)}{1+\frac{v}{cn}}\approx v\left(1-\frac{1}{n^2}\right) \ .</math> <math display="block">V_\mathrm{lab}-\frac{c}{n} = \frac{\frac{c}{n}+v}{1+\frac{v}{cn}}-\frac{c}{n}=\frac{\frac{c}{n}+v-\frac{c}{n}(1+\frac{v}{cn})}{1+\frac{v}{cn}} </math> <math display="block"> = \frac{v\left(1-\frac{1}{n^2}\right)}{1+\frac{v}{cn}}\approx v\left(1-\frac{1}{n^2}\right) \ .</math>
: This is accurate when {{nowrap|''v''/''c'' ≪ 1}}, and agrees with the formula based upon Fizeau's measurements, which satisfied the condition {{nowrap|''v''/''c'' ≪ 1}}. This is accurate when {{nowrap|''v''/''c'' ≪ 1}}, and agrees with the formula based upon Fizeau's measurements, which satisfied the condition {{nowrap|''v''/''c'' ≪ 1}}.


Alternatively, the Fizeau result can be derived by applying ] to a moving medium.<ref name="Becker">{{cite book |last1=Becker |first1=Richard |last2=Sauter |first2=Fritz |title=Electromagnetic Fields and Interactions |date=1 January 1982 |publisher=Courier Corporation |isbn=978-0-486-64290-1 |page=308 |url=https://books.google.com/books?id=5U7HVjHbphwC |access-date=9 March 2023 |language=en}}</ref>
Fizeau's experiment is hence supporting evidence for the collinear case of Einstein's velocity addition formula.<ref name=laue group=P />


== See also == == See also ==

Latest revision as of 16:14, 7 January 2025

Experiment measuring the speed of light in moving water This article is about the experiment to measure the relative speed of light in a moving medium. For Fizeau's experiment to measure the absolute speed of light in air, see Fizeau's measurement of the speed of light in air.

Figure 1. Apparatus used in the Fizeau experiment

The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Fizeau used a special interferometer arrangement to measure the effect of movement of a medium upon the speed of light.

According to the theories prevailing at the time, light traveling through a moving medium would be dragged along by the medium, so that the measured speed of the light would be a simple sum of its speed through the medium plus the speed of the medium. Fizeau indeed detected a dragging effect, but the magnitude of the effect that he observed was far lower than expected. When he repeated the experiment with air in place of water he observed no effect. His results seemingly supported the partial aether-drag hypothesis of Augustin-Jean Fresnel, a situation that was disconcerting to most physicists. Over half a century passed before a satisfactory explanation of Fizeau's unexpected measurement was developed with the advent of Albert Einstein's theory of special relativity. Einstein later pointed out the importance of the experiment for special relativity, in which it corresponds to the relativistic velocity-addition formula when restricted to small velocities.

Although it is referred to as the Fizeau experiment, Fizeau was an active experimenter who carried out a wide variety of different experiments involving measuring the speed of light in various situations.

Background

Main article: History of electromagnetism

As scientists in the 1700's worked on a theory of light and of electromagnetism, luminiferous aether, a medium that would support waves, was the focus of many experiments. Two critical issues were the relation of aether to motion and its relation to matter. For example, astronomical aberration, the apparent motion of stars observed at different times of year, was proposed to be related to starlight propagated through an aether. In 1846 Fresnel proposed that the portion aether that will move with an object relates to the object's index of refraction of light, which was take to be the ratio of the speed of light in the material to the speed of light in interstellar space. Having recently measured the speed of light in air and water, Fizeau set out to measure the speed of light in moving water.

Experimental setup

Figure 2. Highly simplified representation of Fizeau's experiment.
Figure 3. Interferometer setup in the Fizeau Experiment (1851)

A highly simplified representation of Fizeau's 1851 experiment is presented in Fig. 2. Incoming light is split into two beams by a beam splitter (BS) and passed through two columns of water flowing in opposite directions. The two beams are then recombined to form an interference pattern that can be interpreted by an observer.

The simplified arrangement illustrated in Fig. 2 would have required the use of monochromatic light, which would have enabled only dim fringes. Because of white light's short coherence length, use of white light would have required matching up the optical paths to an impractical degree of precision, and the apparatus would have been extremely sensitive to vibration, motion shifts, and temperature effects.

Fizeau's actual apparatus, illustrated in Fig. 3 and Fig. 4, was set up as a common-path interferometer. This guaranteed that the opposite beams would pass through equivalent paths, so that fringes readily formed even when using the sun as a light source.

The double transit of the light was for the purpose of augmenting the distance traversed in the medium in motion, and further to compensate entirely any accidental difference of temperature or pressure between the two tubes, from which might result a displacement of the fringes, which would be mingled with the displacement which the motion alone would have produced; and thus have rendered the observation of it uncertain.

— Fizeau
Figure 4. Setup of the Fizeau Experiment (1851)

A light ray emanating from the source S′ is reflected by a beam splitter G and is collimated into a parallel beam by lens L. After passing the slits O1 and O2, two rays of light travel through the tubes A1 and A2, through which water is streaming back and forth as shown by the arrows. The rays reflect off a mirror m at the focus of lens L′, so that one ray always propagates in the same direction as the water stream, and the other ray opposite to the direction of the water stream. After passing back and forth through the tubes, both rays unite at S, where they produce interference fringes that can be visualized through the illustrated eyepiece. The interference pattern can be analyzed to determine the speed of light traveling along each leg of the tube.

Result

Fizeau's experiment showed a faster speed of light in water moving in the same direction and a slower speed when the water moved opposite the light. However the amount of difference in the speed of light was only a fraction of the water speed. Interpreted in terms of the aether theory, the water seemed to drag the aether and thus the light propagation, but only partially.

Impact

Main article: Aether drag hypothesis

At the time of Fizeau's experiment, two different models of how aether related to moving bodies were discussed, Fresnel's partial drag hypothesis and George Stokes' complete aether drag hypothesis. Fresnel had Augustin-Jean Fresnel (1818) proposed his model to explain an 1810 experiment by Arago. In 1845 Stokes showed that complete aether drag could also explain it. Since Fresnel had no model to explain partial drag, scientists favored Stokes explanation.

According to the Stokes' hypothesis, the speed of light should be increased or decreased when "dragged" along by the water through the aether frame, dependent upon the direction. The overall speed of a beam of light should be a simple additive sum of its speed through the water plus the speed of the water. That is, if n is the index of refraction of water, so that c/n is the speed of light in stationary water, then the predicted speed of light w in one arm would be

w + = c n + v   , {\displaystyle w_{+}={\frac {c}{n}}+v\ ,}

and the predicted speed in the other arm would be

w = c n v   , {\displaystyle w_{-}={\frac {c}{n}}-v\ ,}

for water with velocity v {\displaystyle v} . Hence light traveling against the flow of water should be slower than light traveling with the flow of water. The interference pattern between the two beams when the light is recombined at the observer depends upon the transit times over the two paths.

However Fizeau found that

w + = c n + v ( 1 1 n 2 )   . {\displaystyle w_{+}={\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}\right)\ .}

In other words, light appeared to be dragged by the water, but the magnitude of the dragging was much lower than expected.

The Fizeau experiment forced physicists to accept the empirical validity of an Fresnel's model, that a medium moving through the stationary aether drags light propagating through it with only a fraction of the medium's speed, with a dragging coefficient f related to the index of refraction:

f = 1 1 n 2   . {\displaystyle f=1-{\frac {1}{n^{2}}}\ .}

Although Fresnel's hypothesis was empirically successful in explaining Fizeau's results, many experts in the field, including Fizeau himself, found Fresnel's hypothesis partial aether-dragging unsatisfactory. Fresnel had found an empirical formula that worked but no mechanical model of the aether was used to derive it.

Confirmation

Wilhelm Veltmann's colors of light

In 1870 Wilhelm Veltmann demonstrated that Fresnel's formula worked for different frequencies (colors) of light. According the Fresnel's model this would imply different amounts of eather drag for different colors of light. The velocity with white light, a mixture of colors, would be unexplained.

Hoek experiment

See also: Hammar experiment

An indirect confirmation of Fresnel's dragging coefficient was provided by Martin Hoek (1868). His apparatus was similar to Fizeau's, though in his version only one arm contained an area filled with resting water, while the other arm was in the air. As seen by an observer resting in the aether, Earth and hence the water is in motion. So the following travel times of two light rays traveling in opposite directions were calculated by Hoek (neglecting the transverse direction, see image):

t 1 = A B c + v + D E c n v   , {\displaystyle t_{1}={\frac {AB}{c+v}}+{\frac {DE}{{\frac {c}{n}}-v}}\ ,}

t 2 = A B c v + D E c n + v   . {\displaystyle t_{2}={\frac {AB}{c-v}}+{\frac {DE}{{\frac {c}{n}}+v}}\ .}

Figure 6. Hoek expected the observed spectrum to be continuous with the apparatus oriented transversely to the aether wind, and to be banded with the apparatus oriented parallel to the wind. In the actual experiment, he observed no banding regardless of the instrument's orientation.

The travel times are not the same, which should be indicated by an interference shift. However, if Fresnel's dragging coefficient is applied to the water in the aether frame, the travel time difference (to first order in v/c) vanishes. Upon turning the apparatus table 180 degrees, altering the direction of a hypothetical aether wind, Hoek obtained a null result, confirming Fresnel's dragging coefficient.

In the particular version of the experiment shown here, Hoek used a prism P to disperse light from a slit into a spectrum which passed through a collimator C before entering the apparatus. With the apparatus oriented parallel to the hypothetical aether wind, Hoek expected the light in one circuit to be retarded 7/600 mm with respect to the other. Where this retardation represented an integral number of wavelengths, he expected to see constructive interference; where this retardation represented a half-integral number of wavelengths, he expected to see destructive interference. In the absence of dragging, his expectation was for the observed spectrum to be continuous with the apparatus oriented transversely to the aether wind, and to be banded with the apparatus oriented parallel to the aether wind. His actual experimental results were completely negative.

Mascart's birefringence experiment

Éleuthère Mascart (1872) demonstrated a result for polarized light traveling through a birefringent medium gives different velocities in accordance with Fresnel's empirical formula. However, the result in terms of Fresnel's physical model requires different aether drag in different direction in the medium.

Michelson and Morley confirmation

Figure 5. Improved Fizeau type experiment by Michelson and Morley in 1886. Collimated light from source a falls on beam splitter b where it divides: one part follows the path b c d e f b g and the other the path b f e d c b g.

Albert A. Michelson and Edward W. Morley (1886) repeated Fizeau's experiment with improved accuracy, addressing several concerns with Fizeau's original experiment: (1) Deformation of the optical components in Fizeau's apparatus could cause artifactual fringe displacement; (2) observations were rushed, since the pressurized flow of water lasted only a short time; (3) the laminar flow profile of water flowing through Fizeau's small diameter tubes meant that only their central portions were available, resulting in faint fringes; (4) there were uncertainties in Fizeau's determination of flow rate across the diameter of the tubes. Michelson redesigned Fizeau's apparatus with larger diameter tubes and a large reservoir providing three minutes of steady water flow. His common-path interferometer design provided automatic compensation of path length, so that white light fringes were visible at once as soon as the optical elements were aligned. Topologically, the light path was that of a Sagnac interferometer with an even number of reflections in each light path. This offered extremely stable fringes that were, to first order, completely insensitive to any movement of its optical components. The stability was such that it was possible for him to insert a glass plate at h or even to hold a lighted match in the light path without displacing the center of the fringe system. Using this apparatus, Michelson and Morley were able to completely confirm Fizeau's results not just in water, but also in air.

Zeeman and Lorentz's improved formula

In 1895, Hendrik Lorentz predicted the existence of an extra term due to dispersion:

w + = c n + v ( 1 1 n 2 λ n d n d λ )   . {\displaystyle w_{+}={\frac {c}{n}}+v\left(1-{\frac {1}{n^{2}}}-{\frac {\lambda }{n}}\!\cdot \!{\frac {\mathrm {d} n}{\mathrm {d} \lambda }}\right)\ .}

Since the medium is flowing towards or away from the observer, the light traveling through the medium is Doppler-shifted, and the refractive index used in the formula has to be that appropriate to the Doppler-shifted wavelength. Zeeman verified the existence of Lorentz' dispersion term in 1915. Using a scaled-up version of Michelson's apparatus connected directly to Amsterdam's main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å) to confirm Lorentz's modified coefficient.

Later confirmations

In 1910, Franz Harress used a rotating device and overall confirmed Fresnel's dragging coefficient. However, he additionally found a "systematic bias" in the data, which later turned out to be the Sagnac effect.

Since then, many experiments have been conducted measuring such dragging coefficients in a diversity of materials of differing refractive index, often in combination with the Sagnac effect. For instance, in experiments using ring lasers together with rotating disks, or in neutron interferometric experiments. Also a transverse dragging effect was observed, i.e. when the medium is moving at right angles to the direction of the incident light.

Lorentz's interpretation

Main articles: Lorentz ether theory and History of Lorentz transformations

In 1892, Hendrik Lorentz proposed a modification of Fresnel's model, in which the aether is completely stationary. He succeeded in deriving Fresnel's dragging coefficient as the result of an interaction between the moving water with an undragged aether. He also discovered that the transition from one to another reference frame could be simplified by using an auxiliary time variable which he called local time:

t = t v x c 2   . {\displaystyle t^{\prime }=t-{\frac {vx}{c^{2}}}\ .}

In 1895, Lorentz more generally explained Fresnel's coefficient based on the concept of local time. However, Lorentz's theory had the same fundamental problem as Fresnel's: a stationary aether contradicted the Michelson–Morley experiment. So in 1892 Lorentz proposed that moving bodies contract in the direction of motion (FitzGerald-Lorentz contraction hypothesis, since George FitzGerald had already arrived in 1889 at this conclusion). The equations that he used to describe these effects were further developed by him until 1904. These are now called the Lorentz transformations in his honor, and are identical in form to the equations that Einstein was later to derive from first principles. Unlike Einstein's equations, however, Lorentz's transformations were strictly ad hoc, their only justification being that they seemed to work.

Einstein's use of Fizeau's experiment

Main article: Special relativity

Einstein showed how Lorentz's equations could be derived as the logical outcome of a set of two simple starting postulates. In addition Einstein recognized that the stationary aether concept has no place in special relativity, and that the Lorentz transformation concerns the nature of space and time. Together with the moving magnet and conductor problem, the negative aether drift experiments, and the aberration of light, the Fizeau experiment was one of the key experimental results that shaped Einstein's thinking about relativity. Robert S. Shankland reported some conversations with Einstein, in which Einstein emphasized the importance of the Fizeau experiment:

He continued to say the experimental results which had influenced him most were the observations of stellar aberration and Fizeau's measurements on the speed of light in moving water. "They were enough," he said.

Modern interpretation

Max von Laue (1907) demonstrated that the Fresnel drag coefficient can be explained as a natural consequence of the relativistic formula for addition of velocities. The speed of light in immobile water is c/n. From the velocity composition law it follows that the speed of light observed in the laboratory, where water is flowing with speed v (in the same direction as light) is V l a b = c n + v 1 + c n v c 2 = c n + v 1 + v c n   . {\displaystyle V_{\mathrm {lab} }={\frac {{\frac {c}{n}}+v}{1+{\frac {{\frac {c}{n}}v}{c^{2}}}}}={\frac {{\frac {c}{n}}+v}{1+{\frac {v}{cn}}}}\ .} Thus the difference in speed is (assuming v is small comparing to c, dropping higher order terms) V l a b c n = c n + v 1 + v c n c n = c n + v c n ( 1 + v c n ) 1 + v c n {\displaystyle V_{\mathrm {lab} }-{\frac {c}{n}}={\frac {{\frac {c}{n}}+v}{1+{\frac {v}{cn}}}}-{\frac {c}{n}}={\frac {{\frac {c}{n}}+v-{\frac {c}{n}}(1+{\frac {v}{cn}})}{1+{\frac {v}{cn}}}}} = v ( 1 1 n 2 ) 1 + v c n v ( 1 1 n 2 )   . {\displaystyle ={\frac {v\left(1-{\frac {1}{n^{2}}}\right)}{1+{\frac {v}{cn}}}}\approx v\left(1-{\frac {1}{n^{2}}}\right)\ .} This is accurate when v/c ≪ 1, and agrees with the formula based upon Fizeau's measurements, which satisfied the condition v/c ≪ 1.

Alternatively, the Fizeau result can be derived by applying Maxwell's equations to a moving medium.

See also

References

  1. Rohrlich, Fritz (25 August 1989). From Paradox to Reality: Our Basic Concepts of the Physical World. Cambridge University Press. p. 54. ISBN 978-0-521-37605-1. Retrieved 9 March 2023.
  2. Becker, Richard; Sauter, Fritz (1 January 1982). Electromagnetic Fields and Interactions. Courier Corporation. p. 308. ISBN 978-0-486-64290-1. Retrieved 9 March 2023.
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  1. ^ Whittaker, E. T. (1989). A history of the theories of aether & electricity. New York: Dover Publications. ISBN 978-0-486-26126-3.
  2. ^ N David Mermin (2005). It's about time: understanding Einstein's relativity. Princeton University Press. pp. 39ff. ISBN 0-691-12201-6.
  3. Mascart, Éleuthère Élie Nicolas (1889). Traité d'optique. Paris: Gauthier-Villars. p. 101. Retrieved 9 August 2015.
  4. ^ Stachel, J. (2005). "Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies". In Kox, A.J.; Eisenstaedt, J (eds.). The universe of general relativity. Boston: Birkhäuser. pp. 1–13. ISBN 0-8176-4380-X. Retrieved 17 April 2012.
  5. ^ Rafael Ferraro (2007). "Hoek's experiment". Einstein's Space-Time: An Introduction to Special and General Relativity. Springer. pp. 33–35. ISBN 978-0-387-69946-2.
  6. Robert Williams Wood (1905). Physical Optics. The Macmillan Company. p. 514.
  7. Rosser, W. G. V. (6 January 1992). Introductory Special Relativity. CRC Press. p. 113. ISBN 978-0-85066-838-4. Retrieved 9 March 2023.
  8. Hariharan, P. (2007). Basics of Interferometry, 2nd edition. Elsevier. p. 19. ISBN 978-0-12-373589-8.
  9. Pauli, Wolfgang (1981) . Theory of Relativity. New York: Dover. ISBN 0-486-64152-X.
  10. Anderson, R.; Bilger, H.R.; Stedman, G.E. (1994). "Sagnac effect: A century of Earth-rotated interferometers". Am. J. Phys. 62 (11): 975–985. Bibcode:1994AmJPh..62..975A. doi:10.1119/1.17656.
  11. Stedman, G. E. (1997). "Ring-laser tests of fundamental physics and geophysics". Reports on Progress in Physics. 60 (6): 615–688. Bibcode:1997RPPh...60..615S. doi:10.1088/0034-4885/60/6/001. S2CID 1968825.; see pp. 631–634, and references therein.
  12. ^ Janssen, Michel; Stachel, John (2010), "The Optics and Electrodynamics of Moving Bodies" (PDF), in John Stachel (ed.), Going Critical, Springer, ISBN 978-1-4020-1308-9
  13. ^ Miller, A.I. (1981). Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911). Reading: Addison–Wesley. ISBN 0-201-04679-2.
  14. Lahaye, Thierry; Labastie, Pierre; Mathevet, Renaud (2012). "Fizeau's "aether-drag" experiment in the undergraduate laboratory". American Journal of Physics. 80 (6): 497. arXiv:1201.0501. Bibcode:2012AmJPh..80..497L. doi:10.1119/1.3690117. S2CID 118401543.
  15. Norton, John D., John D. (2004), "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905", Archive for History of Exact Sciences, 59 (1): 45–105, Bibcode:2004AHES...59...45N, doi:10.1007/s00407-004-0085-6, S2CID 17459755
  16. Shankland, R. S. (1963). "Conversations with Albert Einstein". American Journal of Physics. 31 (1): 47–57. Bibcode:1963AmJPh..31...47S. doi:10.1119/1.1969236.
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    English: Fizeau, H. (1851). "The Hypotheses Relating to the Luminous Aether, and an Experiment which Appears to Demonstrate that the Motion of Bodies Alters the Velocity with which Light Propagates itself in their Interior" . Philosophical Magazine. 2: 568–573.
  2. Fizeau, H. (1859). "Sur les hypothèses relatives à l'éther lumineux". Ann. Chim. Phys. 57: 385–404.
    English: Fizeau, H. (1860). "On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light" . Philosophical Magazine. 19: 245–260.
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Tests of special relativity
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