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A '''variable speed of light''' ('''VSL''') is a feature of a family of hypotheses stating that the [[speed of light in vacuum]], usually denoted by ''c'', may in some way not be constant, e.g. varying in space or time, or depending on frequency.
A variable speed of light occurs in some situations of classical physics as equivalent formulations of accepted theories, but also in various alternative theories of gravitation and cosmology, many of them non-mainstream.

Notable attempts to incorporate a variable speed of light into [[physics]] have been made by [[Albert Einstein|Einstein]] in 1911, by [[Robert Dicke]] in 1957, and by several researchers starting from the late 1980s.

The speed of light in vacuum instead is considered a [[Physical constant|constant]], and defined by the [[SI]] as 299792458&nbsp;m/s. Variability of the speed of light is therefore equivalent with a variability of the SI  [[meter]] and/or the SI [[second]].

VSL should not be confused with [[faster than light]] theories; nor should it be confused with the fact that the speed of light in a [[condensed matter physics|medium]] is slower than the speed of light in vacuum depending on the medium's  [[refractive index]].

==Historical proposals==

===Einstein's early proposal (1911)===
While Einstein first mentioned a variable speed of light in 1907,<ref>{{cite journal |first=Albert |last=Einstein |authorlink=Albert Einstein |journal=Jahrbuch der Radioaktivität und Elektronik |title=Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen |date=4 December 1907 |volume=4 |pages=411–462 |url=http://wikilivres.ca/wiki/Über_das_Relativitätsprinzip_und_die_aus_demselben_gezogenen_Folgerungen}}</ref> he reconsidered the idea more thoroughly in 1911.<ref>{{cite journal |author=A. Einstein |title=Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes |journal=Annalen der Physik |year=1911 |volume=35 |issue=10 |pages=898–906 |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1911_35_898-908.pdf |doi=10.1002/andp.19113401005 |bibcode=1911AnP...340..898E }}</ref> In analogy to the situation in a medium, where a shorter wavelength <math> \lambda </math>, by means of <math> c = \nu \lambda </math>, leads to a lower speed of light, Einstein assumed that clocks in a gravitational field run slower, whereby the corresponding frequencies <math> \nu </math> are influenced by the gravitational potential (eq.2, p.&nbsp;903):

: <math> \nu_1 = \nu_2 \left(1+\frac{GM}{r c^2}\right). </math>

Einstein commented (pages 906–907):
{{quote| "Aus dem soeben bewiesenen Satze, daß die Lichtgeschwindigkeit im Schwerefelde eine Funktion des Ortes ist, läßt sich leicht mittels des Huygensschen Prinzipes schließen, daß quer zum Schwerefeld sich fortpflanzende Lichtstrahlen eine Krümmung erfahren müssen."
("From the just proved assertion, that the speed of light in a gravity field is a function of position, it is easily deduced from [[Huygens–Fresnel principle|Huygens's principle]] that light rays propagating at right angles to the gravity field must experience curvature.") }}
In a subsequent paper in 1912,<ref>{{cite journal|author=A. Einstein|title = Lichtgeschwindigkeit und Statik des Gravitationsfeldes| journal = Annalen der Physik|year = 1912|volume = 38|issue = 7|pages = 355–369 |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1912_38_355-369.pdf|doi=10.1002/andp.19123430704|bibcode = 1912AnP...343..355E }}</ref> he concluded that:
{{quote| “Das Prinzip der Konstanz der Lichtgeschwindigkeit kann nur insofern aufrechterhalten werden, als man sich auf für Raum-Zeitliche-Gebiete mit konstantem Gravitationspotential beschränkt.“ 
(“The principle of the constancy of the speed of light can be kept only when one restricts oneself to space-time regions of constant gravitational potential.”)}}
However, Einstein deduced a light deflection at the [[Sun]] of “almost one arcsecond” which is just one-half of the [[Tests of general relativity|measured value]] later derived by his theory of [[general relativity]]. The measured value was provided by [[Arthur Eddington|Eddington]] in 1919. The accuracy of this value was improved by measurements done by Donald Bruns during the 2017 total eclipse of the sun <ref>http://stellarproducts.com/2017eclipse/BrunsEclipse.pdf</ref>.

===Einstein's updated proposals (1905–1915)===
Albert Einstein went through several versions of light speed theory between 1905 and 1915, eventually concluding that light speed is constant when gravity does not have to be considered<ref>{{cite book|last1=Einstein|first1=Albert|title=Relativity - The Special And The General Theory|date=1961|publisher=Bonanza|location=New York|isbn=978-0-517-029619|page=[https://archive.org/details/relativityspecia00eins_0/page/76 76]|edition=15th reprint from 1952|url=https://archive.org/details/relativityspecia00eins_0/page/76}}</ref> but that the speed of light cannot be constant in a gravitational field with variable strength. In the same book Einstein explained that he intended light speed to be a vector when it was described by coordinates in a reference frame.<ref>{{cite book|last1=Einstein|first1=Albert|title=Relativity - The Special And The General Theory|date=1961|publisher=Bonanza|location=New York|isbn=978-0-517-029619|page=[https://archive.org/details/relativityspecia00eins_0/page/145 145]|edition=15th reprint from 1952|url=https://archive.org/details/relativityspecia00eins_0/page/145}}</ref>

Einstein was not explicit about whether the speed of light would change in a gravity field, or whether just the direction of the vector would change; this can be clarified from the text implicitly, however.<ref>{{cite book|last1=Einstein|first1=Albert|title=The Meaning of Relativity|date=1984|publisher=Barns & Noble|location=New York|isbn=978-1-56731-136-5|page=[https://archive.org/details/meaningofrelati00eins/page/93 93]|edition=5th reprint from 1953|url=https://archive.org/details/meaningofrelati00eins/page/93}}</ref> A calculation of alpha (''α'') follows equation 107 and makes an unambiguous use of variable scalar light velocity (L) both as the argument of a partial differential function (proving a variable) and as the denominator in a fraction (proving not a vector) both in the same integrated quantity. Division of a scalar by a vector is not defined, so there is no other way to interpret the velocity of light in this usage except as a variable scalar speed.

In this calculation ''L'' = ''c''/''c''<sub>0</sub> where ''c''<sub>0</sub> is light speed in flat space.

: <math> \alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\left(\frac{\partial L}{\partial x_1}\right) dx_3 </math>

Peter Bergmann did not agree with Einstein, but left the dispute out of his earlier book<ref>{{cite book|last1=Bergmann|first1=Peter|title=Introduction to the Theory of Relativity|date=1976|publisher=Dover|location=New York|isbn=978-0-486-63282-7|page=[https://archive.org/details/introductiontoth00berg/page/221 221]|edition=1st reprint from 1942|url=https://archive.org/details/introductiontoth00berg/page/221}}</ref> in 1942 to get Einstein’s endorsement. After Einstein died Bergmann wrote a new book<ref>{{cite book|last1=Bergmann|first1=Peter|title=The Riddle of Gravitation|date=1992|publisher=Dover|location=New York|isbn=978-0-486-27378-5|page=66|edition=1st reprint from 1968}}</ref> in 1968 claiming that vector light velocity could change direction but not speed. This has become a prevailing opinion in science, but not in agreement with Einstein’s unambiguous math. Bergman did acknowledge that the apparent speed of light would change in a gravity field and go to zero at an event horizon as viewed by a distant observer.<ref>{{cite book|last1=Bergmann|first1=Peter|title=The Riddle of Gravitation|date=1992|publisher=Dover|location=New York|isbn=978-0-486-27378-5|page=94|edition=1st reprint from 1968}}</ref>

Max Born agreed with Einstein and stated both speed and direction of light change in a gravity field<ref>{{cite book|last1=Born|first1=Max|title=Einstein's Theory of Relativity|date=1965|publisher=Dover|location=New York|isbn=978-0-486-60769-6|page=357|edition=1st reprint from 1923}}</ref> first published 1923 in English, 1920 in German. Born expressed the magnitude of light speed as the square root of (−''g''<sub>44</sub>/''g''<sub>11</sub>) gravity components in general relativity to leave no doubt about his intent.

Richard Tolman also agreed with Einstein and expressed the radial speed of light as ''dr''/''dt'' in a gravity field.<ref>{{cite book|last1=Tolman|first1=Richard|title=Relativity Cosmology and Thermodynamics|date=1958|publisher=Oxford|location=Oxford UK|page=212|edition=1st reprint from 1934}}</ref>

: <math> \frac{dr}{dt} = 1 - \frac{2m}{r}, </math>

where ''m'' is ''MG''/''c''<sup>2</sup> and where natural units are used such that ''c''<sub>0</sub> is equal to one.

===Dicke's proposal (1957)===
[[Robert Dicke]], in 1957, developed a related VSL theory of gravity.<ref name="Dicke">{{cite journal|author=R. Dicke|title=Gravitation without a Principle of Equivalence|doi=10.1103/RevModPhys.29.363|journal= Reviews of Modern Physics|year=1957|volume = 29|issue=3|pages=363–376|bibcode=1957RvMP...29..363D}}</ref> In contrast to Einstein, Dicke assumed not only the frequencies to vary, but also the wavelengths. Since <math> c = \nu \lambda </math>, this resulted in a relative change of ''c'' twice as much as considered by Einstein. Dicke assumed a refractive index <math> n= \frac{c}{c_0} = 1+\frac{2 GM}{r c^2} </math> (eqn.5) and proved it to be consistent with the observed value for light deflection. In a comment related to [[Mach's principle]], Dicke suggested that, while the right part of the term in eq. 5 is small, the left part, 1, could have   “its origin in the remainder of the matter in the universe”.

Given that in a universe with an increasing horizon more and more masses contribute to the above refractive index, Dicke considered a cosmology where ''c'' decreased in time, providing an alternative explanation to the [[Hubble's law|cosmological redshift]] <ref name="Dicke"/> (p.&nbsp;374). Dicke's theory is compatible with the [[International System of Units|SI]] definition of c since the time and length units [[second]] and [[meter]] can vary in concert, keeping the value of c fixed (p.&nbsp;366).

===Subsequent proposals===

Spatial variation of the speed of light{{elucidate|date=March 2017}} is implicitly present in general relativity, occurring in the coordinate space description.:<ref>{{cite book|author=C.M. Will |title=Theory and Experiment in Gravitational Physics|publisher=Cambridge University Press|year=1995|page=144}}</ref><ref>{{cite book|author=S. Weinberg |title=Gravitation and Cosmology|publisher=Wiley| year=1972|location=London |page=222}}</ref>

"note that the photon speed is ... <math> |u| = 1+2 \phi +O(v^3)</math>." (<math> \phi </math> denoting the gravitational potential&nbsp;&minus;''GM''/''r'')

Based on this observation, variable speed of light models, including Dicke's, have been developed which agree with all known tests of general relativity,<ref>{{cite journal|author=J. Broekaert |title =A Spatially-VSL Gravity Model with 1-PN Limit of GRT|journal=Foundations of Physics|year = 2008| volume = 38|issue =5| pages=409–435|arxiv=gr-qc/0405015|doi=10.1007/s10701-008-9210-8|bibcode = 2008FoPh...38..409B }}</ref> but some are distinct for higher-order tests.<ref>{{cite arXiv|author=K. Krogh |title=Gravitation Without Curved Space-time|year = 1999| eprint=astro-ph/9910325}}</ref>

Other models claim to shed light on the equivalence principle{{how|date=March 2017}}<ref>{{cite journal|author=M. Arminjon |title = Space Isotropy and Weak Equivalence Principle in a Scalar Theory of Gravity| journal = Brazilian Journal of Physics |arxiv=gr-qc/0412085|year = 2006|volume = 36|issue = 1B|pages = 177–189| doi = 10.1590/S0103-97332006000200010|bibcode = 2006BrJPh..36..177A }}</ref> or make a link to Dirac's [[Dirac large numbers hypothesis|Large Numbers Hypothesis]].<ref>{{cite journal|author=A. Unzicker |title=A look at the abandoned contributions to cosmology of Dirac, Sciama, and Dicke|journal=Annalen der Physik| year = 2009|volume=521|issue=1|pages=57–70| arxiv=0708.3518|doi=10.1002/andp.200810335|bibcode = 2009AnP...521...57U }}</ref>{{why|date=March 2017}}

Several hypotheses for varying speed of light, seemingly in contradiction to general relativity theory, have been published, including those of Giere and Tan (1986),<ref>{{cite journal|author=Giere, A. C.|author2=A. Tan|title=A Derivation of Hubble.|journal=Chinese Journal of Physics|volume=24|issue=3|year=1986|pages=217–219|url=https://www.airitilibrary.com/Publication/alDetailedMesh?docid=05779073-198610-201303280001-201303280001-217-219}}</ref> [[Louise Riofrio|Riofrio]] (2004),<ref>{{cite web|author=Riofrio, L.|title=''GM=tc<sup>3</sup>'' Space/Time Explanation of Supernova Data|website=Beyond Einstein Meeting, Stanford Linear Accelerator Center, May 12–14, 2004|url=http://www-conf.slac.stanford.edu/einstein/talks/aspauthor2004_3.pdf}}</ref><ref>{{cite journal|title=''GM=tc<sup>3</sup>'' Space/Time Explanation of Supernova Data|author=Riofrio, L.|journal=Observing Dark Energy, ASP Conference Series, Proceedings of a Meeting Held 18–20 March 2004 in Tucson, Arizona. Edited by Sidney C. Wolff and Tod R. Lauer|publisher=Astronomical Society of the Pacific|year=2005|location=San Francisco|bibcode=2005ASPC..339..181R|volume=339|pages=181}}</ref><ref name=Riofrio2012>{{cite journal|author=Riofrio, L.|title=Calculation of lunar orbit anomaly|year=2012|journal=Planetary Science|volume=1|issue=1|page=1|doi=10.1186/2191-2521-1-1|bibcode=2012PlSci...1....1R}}</ref> and Sanejouand (2005).<ref>{{cite arxiv|author=Sanejouand, Yves-Henri|title=A simple varying-speed-of-light hypothesis is enough for explaining high-redshift supernovae data|year=2005|eprint=astro-ph/0509582}}</ref><ref>{{cite journal|author=Sanejouand, Yves-Henri|title=Empirical evidences in favor of a varying-speed-of-light|year=2009|arxiv=0908.0249|doi=10.1209/0295-5075/88/59002|bibcode=2009arXiv0908.0249S}}</ref> In 2003 Magueijo gave a review of such hypotheses.<ref>{{cite journal|author=Magueijo, João|title=New varying speed of light theories|journal=Reports on Progress in Physics|volume=66|issue=11|year=2003|pages=2025–2068|doi=10.1088/0034-4885/66/11/R04|bibcode=2003RPPh...66.2025M|arxiv=astro-ph/0305457}} [https://arxiv.org/abs/astro-ph/0305457 arXiv preprint]</ref>

==Modern VSL theories as an alternative to cosmic inflation ==
{{Undue weight section|date=May 2019}}
The varying speed of light cosmology{{clarify|previous discussion was about models; what does a VSL cosmology mean?|date=March 2017}} has been proposed independently by [[Jean-Pierre Petit]] in 1988,<ref>
{{cite journal |title= An interpretation of cosmological model with variable light velocity |author= J.P. Petit |journal= Mod. Phys. Lett. A |volume= 3 |issue= 16 |year= 1988 |pages= 1527&ndash;1532 |doi= 10.1142/S0217732388001823 |bibcode= 1988MPLA....3.1527P |url= http://www.januscosmologicalmodel.com/pdf/1988-ModPhysLettA-1.pdf |citeseerx= 10.1.1.692.9603 }}</ref><ref>
{{Cite journal |title= Cosmological model with variable light velocity: the interpretation of red shifts |author= J.P. Petit |journal= Mod. Phys. Lett. A |volume= 3 |issue= 18 |year= 1988 |pages= 1733&ndash;1744 |doi= 10.1142/S0217732388002099 |bibcode= 1988MPLA....3.1733P |url= http://www.januscosmologicalmodel.com/pdf/1988-ModPhysLettA-2.pdf |citeseerx= 10.1.1.692.9067 }}</ref><ref>
{{cite journal |title= Gauge cosmological model with variable light velocity. Comparizon with QSO observational data |author1= J.P. Petit |author2= M. Viton |journal= Mod. Phys. Lett. A |volume= 4 |issue= 23 |year= 1989 |pages= 2201&ndash;2210 |doi= 10.1142/S0217732389002471 |bibcode= 1989MPLA....4.2201P |url= http://www.januscosmologicalmodel.com/pdf/1989-ModPhysLettA-3.pdf }}</ref><ref>
{{cite journal |title= Scale invariant cosmology |author1= P. Midy |author2= J.P. Petit |journal= Int. J. Mod. Phys. D |issue= 8 |year= 1989 |pages= 271&ndash;280 |url= http://www.bigravitytheory.com/pdf/scale_inv.pdf |access-date= 2014-12-24 |archive-url= https://web.archive.org/web/20140717164810/http://bigravitytheory.com/pdf/scale_inv.pdf |archive-date= 2014-07-17 |url-status= dead }}</ref> [[John Moffat (physicist)|John Moffat]] in 1992,<ref>
{{cite journal | title= Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology | author= J. Moffat | journal= Int. J. Mod. Phys. D |volume= 2 |issue= 3 |year=1993 | pages=351&ndash;366 | doi= 10.1142/S0218271893000246 |arxiv = gr-qc/9211020 |bibcode = 1993IJMPD...2..351M }}</ref> 
and the team of [[Andreas Albrecht (cosmologist)|Andreas Albrecht]] and [[João Magueijo]] in 1998<ref>
{{cite journal |author= J.D. Barrow |title= Cosmologies with varying light-speed |year= 1998 |doi= 10.1103/PhysRevD.59.043515 |journal= Physical Review D |volume= 59 |issue= 4 |pages= 043515 |arxiv=astro-ph/9811022|bibcode = 1999PhRvD..59d3515B }}</ref><ref>
{{cite journal |title= A time varying speed of light as a solution to cosmological puzzles |author1=A. Albrecht |author2=J. Magueijo |journal= Phys. Rev. |volume= D59 |issue=4 |pages= 043516 |year= 1999 |arxiv=astro-ph/9811018|bibcode = 1999PhRvD..59d3516A |doi = 10.1103/PhysRevD.59.043516 }}</ref><ref name="local">
{{cite journal |author= J. Magueijo |title= Covariant and locally Lorentz-invariant varying speed of light theories |journal=Phys. Rev. |volume= D62 |issue= 10 |year= 2000 |pages= 103521 |arxiv=gr-qc/0007036|bibcode = 2000PhRvD..62j3521M |doi = 10.1103/PhysRevD.62.103521 }}</ref><ref>
{{cite journal |author= J. Magueijo |title= Stars and black holes in varying speed of light theories |journal= Phys. Rev. |volume= D63 |issue= 4 |year= 2001 |pages= 043502 |arxiv=astro-ph/0010591|bibcode = 2001PhRvD..63d3502M |doi = 10.1103/PhysRevD.63.043502 }}</ref><ref>
{{cite journal |author= J. Magueijo |title= New varying speed of light theories |journal= Rep. Prog. Phys. |volume= 66 |year= 2003 |issue= 11 |pages= 2025–2068 |doi= 10.1088/0034-4885/66/11/R04 |arxiv = astro-ph/0305457 |bibcode = 2003RPPh...66.2025M }}</ref><ref>
{{cite book |title= Faster Than the Speed of Light: The Story of a Scientific Speculation |author= J. Magueijo |publisher= Perseus Books Group |location= Massachusetts |year= 2003 |isbn= 978-0-7382-0525-0 |url= https://archive.org/details/fasterthanspeedo00magu }}</ref> 
to explain the [[horizon problem]] of [[physical cosmology|cosmology]] and propose an alternative to [[cosmic inflation]].

[[File:Secular variation of constants.jpg|thumb|Secular variation of the constants of Physics.[34]]]
In Petit's VSL model, the variation of ''c'' accompanies the joint variations of all [[physical constant]]s combined to space and time [[Scale factor (Universe)|scale factors]] changes, so that all equations and measurements of these constants remain unchanged  through the evolution of the universe. This provides a universal gauge relationship and the [[secular variation]] of the parameters usually taken to be constants:<ref>{{cite journal|last1=PETIT|first1=Jean-Pierre|title=Twin Universe Cosmology|journal=Astrophys. And Sp. Science|date=1995|volume=226|issue=2|pages=273–307|doi=10.1007/bf00627375|bibcode=1995Ap&SS.226..273P|citeseerx=10.1.1.692.7762}}</ref><ref name=":0" />

<math>G=G_0 \sqrt[3]{1+\frac{1}{\tau^2}}\quad m=\frac{m_0}{\sqrt[3]{1+\frac{1}{\tau^2}}}\quad c=c_0 \sqrt[6]{1+\frac{1}{\tau^2}}</math>

<math>h=\frac{h_0}{\sqrt{1+\frac{1}{\tau^2}}}\quad e=\frac{e_0}{\sqrt[6]{1+\frac{1}{\tau^2}}}\quad \mu=\frac{\mu_0}{\sqrt[3]{1+\frac{1}{\tau^2}}}</math>

see figure "secular variation of constant".  The [[Einstein field equations]] remain invariant through convenient joint variations of ''c'' and ''G'' in [[Einstein's constant]]. According to this model, the cosmological horizon grows like R, the space scale, which ensures the homogeneity of the primeval universe, which fits the observational data. Subsequent developments of the model restricts the variation of constants to the higher [[energy density]] of the early universe, at the very beginning of the [[radiation-dominated era]] where spacetime is identified to space-entropy with a [[Metric (mathematics)|metric]] [[conformally flat]].<ref name=":0">{{cite conference |title= Twin matter against dark matter |author1= J.P. Petit |author2= P. Midy |author3= F. Landsheat |conference= Int. Conf. on Astr. & Cosm |booktitle= "Where's the matter?" (See sections '''14''' and '''15''' pp. 21&ndash;26) |year= 2001 |url= http://www.januscosmologicalmodel/pdf/http://www.januscosmologicalmodel.com/pdf/2001-where-is-the-matter.pdf }}</ref><ref>{{cite arXiv |title= Bigravity: a bimetric model of the Universe with variable constants, including VSL (variable speed of light) |author1=J.P Petit |author2=G. d'Agostini |year = 2007 |eprint=0803.1362 |class= physics.gen-ph}}</ref><ref>{{Cite journal |last1=Petit |first1=J.-P. |last2=d'Agostini |first2=G. |date=10 November 2014 |title=Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe |url=http://ayuba.fr/pdf/janus_vsl.pdf |journal=Modern Physics Letters A |volume=29 |issue=34 |doi=10.1142/S021773231450182X |pages=1450182|bibcode=2014MPLA...2950182P }}</ref>

Moffat and the team Albrecht–Magueijo present the idea that the speed of light had a very large value of about 10<sup>35</sup> km/s in the [[early universe]], thus their models show distant regions of the expanding universe interacting at the beginning of the universe{{clarify|connection with horizon problem needs to be explicit to help reader|date=March 2017}}.  There is no known way to solve the horizon problem with a variation of the [[fine-structure constant]]{{clarify|this seems like a complete non-sequitur; why is this parameter being discussed when the topic is VSL? unstated at best|date=March 2017}}, because its variation does not change the [[causal structure]] of [[spacetime]]. To do so would require{{why|date=March 2017}} modifying gravity by varying [[Newton's constant]] or redefining [[special relativity]].{{clarify|date=May 2016}}

Classically,{{clarify|date=May 2016}} varying speed of light cosmologies propose to circumvent this by varying the dimensionful quantity ''c'' by breaking the [[Lorentz invariance]] of [[Albert Einstein|Einstein]]'s [[theory of relativity|theories of general and special relativity]] in a particular way.<ref>{{cite journal |title= Dynamical Mechanism for Varying Light Velocity as a Solution to Cosmological Problems |author1=M. A. Clayton |author2=J. W. Moffat |journal= Phys. Lett. |volume= B460 |issue=3–4 |year= 1999 |pages= 263–270 |arxiv=astro-ph/9812481|bibcode = 1999PhLB..460..263C |doi = 10.1016/S0370-2693(99)00774-1 }}</ref><ref>
{{cite journal |title= Geometrodynamics of variable-speed-of-light cosmologies |author1=B.A. Bassett |author2=S. Liberati |author3=C. Molina-Paris |author4=M. Visser |journal= Phys. Rev. |volume= D62 |issue=10 |year= 2000 |pages= 103518 |arxiv=astro-ph/0001441|bibcode = 2000PhRvD..62j3518B |doi = 10.1103/PhysRevD.62.103518 }}</ref>  Magueijo has also proposed a formulation that preserves local Lorentz invariance.<ref name="local"/> The [[Louise Riofrio|Riofrio cosmological model]] predicts a precise relation between the speed of light (''in vacuo'') and the age of the universe; the model is consistent with data from the [[Lunar Laser Ranging experiment]].<ref name=Riofrio2012/>

==Various other VSL occurrences ==

===Virtual photons===
[[Virtual photon]]s in some calculations in [[quantum field theory]] may also travel at a different speed for short distances; however, this doesn't imply that anything can travel [[faster than light]]. While it has been claimed (see VSL criticism below) that no meaning can be ascribed to a dimensional quantity such as the speed of light varying in time (as opposed to a [[dimensionless number]] such as the [[fine structure constant]]), in some controversial theories in cosmology, the speed of light also varies by changing the postulates of [[special relativity]].{{citation needed|date=February 2013}}

===Varying photon speed===

The [[photon]], the particle of light which mediates the [[electromagnetic force]] is believed to be massless. The so-called [[Proca action]] describes a theory of a massive photon.<ref>{{cite book |author=J. D. Jackson |title=Classical Electrodynamics |year=1998 |publisher=Wiley |edition=3rd}}</ref> Classically, it is possible to have a photon which is extremely light but nonetheless has a tiny mass, like the [[neutrino]]. These photons would propagate at less than the speed of light defined by [[special relativity]] and have three directions of [[Polarization (waves)|polarization]]. However, in [[quantum field theory]], the photon mass is not consistent with [[gauge invariance]] or [[renormalization|renormalizability]] and so is usually ignored. However, a quantum theory of the massive photon can be considered in the Wilsonian [[effective field theory]] approach to quantum field theory, where, depending on whether the photon mass is generated by a Higgs mechanism or is inserted in an ad hoc way in the Proca Lagrangian, the limits implied by various observations/experiments may be different. So, therefore, the speed of light is not constant.<ref name="adel">{{cite journal|author1=Eric Adelberger|author2=Gia Dvali|author3=Andrei Gruzinov|title=Photon Mass Bound Destroyed by Vortices|doi=10.1103/PhysRevLett.98.010402|year=2007|journal=Physical Review Letters|volume=98|issue=2|arxiv=hep-ph/0306245|bibcode = 2007PhRvL..98a0402A|pmid=17358459|pages=010402}}</ref>

===Varying ''c'' in quantum theory===
{{main|Propagator#Faster than light?|l1=Propagator (section "Faster than light?")}}
{{See also|Scharnhorst effect}}

In [[quantum field theory]] the [[Heisenberg uncertainty relations]] indicate that photons can travel at any speed for short periods. In the [[Feynman diagram]] interpretation of the theory, these are known as "[[virtual particle|virtual photons]]", and are distinguished by propagating off the [[mass shell]]. These photons may have any velocity, including velocities greater than the speed of light. To quote [[Richard Feynman]] "...there is also an amplitude for light to go faster (or slower) than the conventional speed of light. You found out in the last lecture that light doesn't go only in straight lines; now, you find out that it doesn't go only at the speed of light! It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, ''c''."<ref>{{cite book |author=R. Feynman |title=QED: the strange theory of light and matter |publisher=Princeton University Press |page=89 |year=1988}}</ref> These virtual photons, however, do not violate causality or special relativity, as they are not directly observable and information cannot be transmitted acausally in the theory. Feynman diagrams and virtual photons are usually interpreted not as a physical picture of what is taking place, but rather as a convenient calculation tool (which, in some cases, happen to involve faster-than-light velocity vectors).

==Relation to other constants and their variation==
{{further|Time-variation of fundamental constants}}

=== Gravitational constant ''G''===
{{See also | Dirac large numbers hypothesis}}
In 1937, [[Paul Dirac]] and others began investigating the consequences of natural constants changing with time.<ref>{{cite journal|author=P.A.M. Dirac|year=1938 |title=A New Basis for Cosmology |journal=[[Proceedings of the Royal Society A]] |volume=165 |issue=921 |pages=199–208 |doi=10.1098/rspa.1938.0053 |bibcode = 1938RSPSA.165..199D }}</ref> For example, Dirac proposed a change of only 5 parts in 10<sup>11</sup> per year of [[Newton's constant]] ''G'' to explain the relative weakness of the [[gravitational force]] compared to other [[fundamental forces]]. This has become known as the [[Dirac large numbers hypothesis]].

However, [[Richard Feynman]] showed<ref>{{cite book |author=R. P. Feynman |title=Lectures on Physics |year=1970 |publisher=Addison Wesley Longman |volume=1 |chapter=7}}</ref> that the [[gravitational constant]] most likely could not have changed this much in the past 4 billion years based on geological and solar system observations (although this may depend on assumptions about the constant not changing other constants). (See also [[strong equivalence principle]].)

===Fine structure constant ''α'' ===
{{See also| Fine-structure constant#Is_the_fine-structure_constant_actually_constant.3F}}
One group, studying distant quasars, has claimed to detect a variation of the fine structure constant <ref>{{cite journal |author=J.K. Webb, M.T. Murphy, V.V. Flambaum, V.A. Dzuba, J.D. Barrow, C.W. Churchill, J.X. Prochaska and A.M. Wolfe |title=Further Evidence for Cosmological Evolution of the Fine Structure Constant |journal=Phys. Rev. Lett. |volume=87 |year=2001 |pages=091301 |doi=10.1103/PhysRevLett.87.091301 |pmid=11531558 |issue=9 |bibcode=2001PhRvL..87i1301W |arxiv = astro-ph/0012539}}</ref> at the level in one part in 10<sup>5</sup>. Other authors dispute these results. Other groups studying quasars claim no detectable variation at much higher sensitivities.<ref>{{cite journal |author=H. Chand, R. Srianand, P. Petitjean and B. Aracil |title=Probing the cosmological variation of the fine-structure constant: results based on VLT-UVES sample |journal=Astron. Astrophys. |volume=417 |year=2004 |issue=3 |pages=853–871 |doi=10.1051/0004-6361:20035701 |bibcode=2004A&A...417..853C |arxiv = astro-ph/0401094}}</ref><ref>{{cite journal |author=R. Srianand, H. Chand, P. Petitjean and B. Aracil |title=Limits on the time variation of the electromagnetic ne-structure constant in the low energy limit from absorption lines in the spectra of distant quasars |journal=Phys. Rev. Lett. |volume=92 |year=2004 |pages=121302 |doi=10.1103/PhysRevLett.92.121302 |pmid=15089663 |issue=12 |bibcode=2004PhRvL..92l1302S |arxiv = astro-ph/0402177}}</ref><ref>{{cite journal |author=S. A. Levshakov, M. Centurion, P. Molaro and S. D’Odorico |title=VLT/UVES constraints on the cosmological variability of the fine-structure constant |journal=Astron. Astrophys. |arxiv=astro-ph/0408188|doi = 10.1051/0004-6361:20041827 |bibcode=2005A&A...434..827L |volume=434 |issue=3 |pages=827–838|year=2005 }}</ref>

For over three decades since the discovery of the [[Oklo]] [[natural nuclear fission reactor]] in 1972, even more stringent constraints, placed by the study of certain [[isotope|isotopic]] abundances determined to be the products of a (estimated) 2 billion year-old fission reaction, seemed to indicate no variation was present.<ref>{{cite journal |author=A. I. Shlyakhter |title=Direct test of the constancy of fundamental nuclear constants |journal=Nature |volume=264 |year=1976 |issue=5584 |pages=340 |doi=10.1038/264340a0 |bibcode = 1976Natur.264..340S }}</ref><ref>{{cite journal |author1=T. Damour  |author2=F. Dyson  |lastauthoramp=yes |title=The Oklo bound on the time variation of the fine-structure constant revisited |journal=Nucl. Phys. |volume=B480 |year=1996 |issue=1–2 |pages=37–54 |doi=10.1016/S0550-3213(96)00467-1 |arxiv = hep-ph/9606486 |bibcode = 1996NuPhB.480...37D}}</ref> However, Lamoreaux and Torgerson of the [[Los Alamos National Laboratory]] conducted a new analysis of the data from Oklo in 2004, and concluded that ''α'' has changed in the past 2&nbsp;billion years by 4.5 parts in {{val|e=8}}. They claimed that this finding was "probably accurate to within 20%." Accuracy is dependent on estimates of impurities and temperature in the natural reactor. These conclusions have yet to be verified by other researchers.<ref>
{{Cite journal
 |author1=S.K. Lamoreaux |author2=J.R. Torgerson | year = 2004
 | title = Neutron Moderation in the Oklo Natural Reactor and the Time Variation of Alpha
 | journal = [[Physical Review D]]
 | volume = 69 |pages=121701
 | doi= 10.1103/PhysRevD.69.121701
 | arxiv = nucl-th/0309048
|bibcode = 2004PhRvD..69l1701L
 | issue = 12 }}</ref><ref>
{{Cite web
 | author=E.S. Reich
 | date=30 June 2004
 | title=Speed of Light May Have Changed Recently
 | url=https://www.newscientist.com/article/dn6092-speed-of-light-may-have-changed-recently.html
 | publisher=[[New Scientist]]

 | accessdate=30 January 2009
 }}</ref><ref>
{{Cite web
 | date=12 May 2005
 | title=Scientists Discover One Of The Constants Of The Universe Might Not Be Constant
 | url=https://www.sciencedaily.com/releases/2005/05/050512120842.htm
 | publisher=[[ScienceDaily]]
 | accessdate=30 January 2009
}}</ref>

[[Paul Davies]] and collaborators have suggested that it is in principle possible to disentangle which of the dimensionful constants (the [[elementary charge]], [[Planck's constant]], and the [[speed of light]]) of which the fine-structure constant is composed is responsible for the variation.<ref>{{cite journal |title = Cosmology: Black holes constrain varying constants |author1=P.C.W. Davies |author2=Tamara M. Davis |author3=Charles H. Lineweaver |year=2002 |journal=Nature |volume=418 |pages=602&ndash;603 |doi = 10.1038/418602a |pmid = 12167848 |issue = 6898|bibcode = 2002Natur.418..602D }}</ref> However, this has been disputed by others and is not generally accepted.<ref>{{cite arXiv |eprint=hep-th/0208093|last1= Duff|first1= M. J.|title= Comment on time-variation of fundamental constants|year= 2002}}</ref><ref>{{cite journal |title=Black holes may not constrain varying constants |author1=S. Carlip  |author2=S. Vaidya  |lastauthoramp=yes |year=2003 |journal=Nature |volume=421 |pages=498 |doi=10.1038/421498a |pmid=12556883 |issue=6922 |arxiv=hep-th/0209249|bibcode = 2003Natur.421..498C }}</ref>

== Criticisms of various VSL concepts ==
{{synthesis|date=May 2016}}

=== Dimensionless and dimensionful quantities ===
It has to be clarified what a variation in a [[dimensionful]] quantity actually means, since any such quantity can be changed merely by changing one's choice of units. [[John D. Barrow|John Barrow]] wrote:
:"[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different.  The pure number we call the [[fine structure constant]] and denote by α is a combination of the [[electron charge]], ''e'', the [[speed of light]], ''c'', and [[Planck's constant]], ''h''.  At first we might be tempted to think that a world in which the speed of light was slower would be a different world.  But this would be a mistake.  If ''c'', ''h'', and ''e'' were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of  α remained the same, this new world would be ''observationally indistinguishable'' from our world.  The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature.  If all masses were doubled in value [including the [[Planck mass]] ''m<sub>P</sub>''] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."<ref>[[John D. Barrow]], ''The Constants of Nature; From Alpha to Omega &ndash; The Numbers that Encode the Deepest Secrets of the Universe,'' Pantheon Books, New York, 2002, {{ISBN|0-375-42221-8}}.</ref>
Any equation of [[physical law]] can be expressed in a form in which all dimensional quantities are normalized against like-dimensioned quantities (called ''[[nondimensionalization]]''), resulting in only [[dimensionless number|dimensionless quantities]] remaining. In fact, physicists can ''choose'' their units so that the [[physical constants]] [[speed of light|''c'']], [[gravitational constant|''G'']], [[Planck constant|''ħ''&nbsp;=&nbsp;''h''/(2π)]], [[vacuum permittivity|4π''ε''<sub>0</sub>]], and [[Boltzmann constant|''k''<sub>B</sub>]] take the value [[one]], resulting in every physical quantity being normalized against its corresponding [[Planck unit]]. For that, it has been claimed that specifying the evolution of a dimensional quantity is meaningless and does not make sense.<ref>{{Cite journal|arxiv=hep-ph/0205340|last1=Uzan|first1=Jean-Philippe|title=The fundamental constants and their variation: Observational status and theoretical motivations|journal=Reviews of Modern Physics|volume=75|issue=2|pages=403–455|year=2003|doi=10.1103/RevModPhys.75.403|bibcode=2003RvMP...75..403U}}</ref>  When Planck units are used and such equations of physical law are expressed in this nondimensionalized form, '''no''' dimensional physical constants such as ''c'', ''G'', ''ħ'', ''ε''<sub>0</sub>, nor ''k''<sub>B</sub> remain, only dimensionless quantities.  Short of their [[anthropometric]] unit dependence, there simply is no [[speed of light]], [[gravitational constant]], nor [[Planck's constant]], remaining in mathematical expressions of physical reality to be subject to such hypothetical variation.{{citation needed|date=February 2013}}  For example, in the case of a hypothetically varying gravitational constant, ''G'', the relevant dimensionless quantities that potentially vary ultimately become the ratios of the [[Planck mass]] to the masses of the [[fundamental particles]]. Some key dimensionless quantities (thought to be constant) that are related to the speed of light (among other dimensional quantities such as ''ħ'', ''e'', ''ε''<sub>0</sub>), notably the [[fine-structure constant]] or the [[proton-to-electron mass ratio]], does have meaningful variance and their possible variation continues to be studied.<ref>''ibid''</ref>

=== Relation to relativity and definition of  ''c'' ===

Specifically regarding VSL, if the [[SI]] [[meter]] definition was reverted to its pre-1960 definition as a length on a [[prototype]] bar (making it possible for the measure of ''c'' to change), then a conceivable change in ''c'' (the reciprocal of the amount of time taken for light to travel this prototype length) could be more fundamentally interpreted as a change in the dimensionless ratio of the meter prototype to the [[Planck length]] or as the dimensionless ratio of the SI [[second]] to the [[Planck time]] or a change in both.  If the number of atoms making up the meter prototype remains unchanged (as it should for a stable prototype), then a perceived change in the value of ''c'' would be the consequence of the more fundamental change in the dimensionless ratio of the Planck length to the sizes of atoms or to the [[Bohr radius]] or, alternatively, as the dimensionless ratio of the Planck time to the [[frequency|period]] of a particular [[caesium]]-133 [[atomic clock|radiation]] or both.

=== General critique of varying ''c'' cosmologies===

From a very general point of view, [[George Francis Rayner Ellis|G. Ellis]] expressed concerns that a varying ''c'' would require a rewrite of much of modern physics to replace the current system which depends on a constant c.<ref name="Ellis">{{cite journal | doi = 10.1007/s10714-007-0396-4 | title = Note on Varying Speed of Light Cosmologies | author = George F R Ellis | journal = General Relativity and Gravitation |date=April 2007 | issue = 4 | pages = 511–520 | volume = 39 | arxiv=astro-ph/0703751 |bibcode = 2007GReGr..39..511E }}</ref> Ellis claimed that any varying ''c'' theory (1) must redefine distance measurements (2) must provide an alternative expression for the metric tensor in [[general relativity]] (3) might contradict Lorentz invariance (4)  must modify [[Maxwell's equations]] (5) must be done consistently with respect to all other physical theories. Whether these concerns apply to the proposals of Einstein (1911) and Dicke (1957) is a matter of debate,<ref>{{cite arXiv|author=A. Unzicker |title=The VSL Discussion: What Does Variable Speed of Light Mean and Should we be Allowed to Think About? |eprint = 0708.2927 |year=2007|class=physics.gen-ph }}</ref> though VSL cosmologies remain out of mainstream physics.

==References==
{{reflist|30em}}

== External links ==
*http://www.phys.unsw.edu.au/einsteinlight/jw/module6_constant.htm

[[Category:Electromagnetic radiation]]
[[Category:Units of velocity]]
[[Category:Special relativity]]

[[de:Physikalische Konstante#Konstanz der Naturkonstanten]]