View source for Variable speed of light

{{short description|Non-mainstream theory in physics}}

A '''variable speed of light''' ('''VSL''') is a feature of a family of hypotheses stating that the [[speed of light]] may in some way not be [[Physical constant|constant]], for example, that it varies in space or time, or depending on [[frequency]]. Accepted [[Classical physics|classical theories of physics]], and in particular [[general relativity]], predict a constant speed of light in any [[Principle of locality|local]] [[frame of reference]] and in some situations these predict apparent variations of the speed of light depending on frame of reference, but this article does not refer to this as a variable speed of light. Various alternative theories of [[gravitation]] and [[cosmology]], many of them non-mainstream, incorporate variations in the local speed of light.

Attempts to incorporate a variable speed of light into physics were made by [[Robert Dicke]] in 1957, and by several researchers starting from the late 1980s.

VSL should not be confused with [[faster than light]] theories, its dependence on a [[condensed matter physics|medium]]'s [[refractive index]] or its measurement in a remote observer's frame of reference in a [[gravitational potential]]. In this context, the "speed of light" refers to the limiting speed ''c'' of the theory rather than to the velocity of propagation of [[photon]]s.

== Historical proposals ==

=== Background ===
Einstein's [[equivalence principle]], on which [[general relativity]] is founded, requires that in any local, freely falling reference frame, the speed of light is always the same.<ref>{{Cite book|last=Will|first=Clifford M.|author-link=Clifford Martin Will |url=https://books.google.com/books?id=gf1uDwAAQBAJ|title=Theory and Experiment in Gravitational Physics|date=2018-09-30|publisher=Cambridge University Press|isbn=978-1-108-57749-6|pages=238|language=en}}</ref><ref>{{Cite book|last1=Misner |first1=Charles W. |title=Gravitation |title-link=Gravitation (book) |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John Archibald |author-link1=Charles W. Misner |author-link2=Kip Thorne |author-link3=John Archibald Wheeler |date=2017-10-03 |publisher=Princeton University Press |isbn=978-1-4008-8909-9 |pages=297 |language=en}}</ref> This leaves open the possibility, however, that an inertial observer inferring the apparent speed of light in a distant region might calculate a different value. Spatial variation of the speed of light in a gravitational potential as measured against a distant observer's time reference is implicitly present in general relativity.<ref>{{cite book|first=S. |last=Weinberg|author-link=Steven Weinberg |url=https://archive.org/details/gravitationcosmo00stev_0|title=Gravitation and Cosmology|publisher=Wiley|year=1972|location=London|page=[https://archive.org/details/gravitationcosmo00stev_0/page/222 222]|isbn=9780471925675|url-access=registration}}</ref> The apparent speed of light will change in a gravity field and, in particular, go to zero at an event horizon as viewed by a distant observer.<ref>{{cite book|last1=Bergmann|first1=Peter|author-link=Peter Bergmann |title=The Riddle of Gravitation|url=https://archive.org/details/riddlegravitatio00berg_292|url-access=limited|date=1992|publisher=Dover|location=New York|isbn=978-0-486-27378-5|page=[https://archive.org/details/riddlegravitatio00berg_292/page/n116 94]|edition=1st reprint from 1968}}</ref> In deriving the [[gravitational redshift]] due to a spherically symmetric massive body, a radial speed of light ''dr''/''dt'' can be defined in [[Schwarzschild coordinates]], with ''t'' being the time recorded on a stationary clock at infinity. The result is
: <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.<ref>{{cite book|last1=Tolman|first1=Richard|author-link=Richard Tolman |title=Relativity Cosmology and Thermodynamics|date=1958|publisher=Oxford|location=Oxford UK|page=212|edition=1st reprint from 1934}}</ref><ref>{{Cite book |last=Stavrov |first=Iva |title=Curvature of Space and Time, with an Introduction to Geometric Analysis |title-link=Curvature of Space and Time, with an Introduction to Geometric Analysis |date=2020 |publisher=American Mathematical Society |isbn=978-1-4704-6313-7 |location=Providence, Rhode Island |oclc=1202475208 |page=179}}</ref>

=== Dicke's proposal (1957) ===
[[Robert Dicke]], in 1957, developed a VSL theory of gravity, a theory in which (unlike general relativity) the speed of light measured locally by a free-falling observer could vary.<ref name="Dicke">{{cite journal |author=Dicke |first=Robert |year=1957 |title=Gravitation without a Principle of Equivalence |journal=Reviews of Modern Physics |volume=29 |issue=3 |pages=363–376 |bibcode=1957RvMP...29..363D |doi=10.1103/RevModPhys.29.363}}</ref> Dicke assumed that both frequencies and wavelengths could vary, which since <math> c = \nu \lambda </math> resulted in a relative change of ''c''. Dicke assumed a refractive index <math> n= \frac{c}{c_0} = 1+\frac{2 GM}{r c^2} </math> (eqn.&nbsp;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"/>{{rp|374}}

=== Subsequent proposals ===

Variable speed of light models, including Dicke's, have been developed which agree with all known tests of general relativity.<ref>{{cite journal |author=Broekaert |first=J. |year=2008 |title=A Spatially-VSL Gravity Model with 1-PN Limit of GRT |journal=Foundations of Physics |volume=38 |issue=5 |pages=409–435 |arxiv=gr-qc/0405015 |bibcode=2008FoPh...38..409B |doi=10.1007/s10701-008-9210-8 |s2cid=8955243}}</ref>

Other models make a link to [[Dirac large numbers hypothesis]].<ref>{{cite journal |author=Unzicker |first=A. |year=2009 |title=A look at the abandoned contributions to cosmology of Dirac, Sciama, and Dicke |journal=Annalen der Physik |volume=521 |issue=1 |pages=57–70 |arxiv=0708.3518 |bibcode=2009AnP...521...57U |doi=10.1002/andp.200810335 |s2cid=11248780}}</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 |first1=A. C. |last2=Tan |first2=A. |year=1986 |title=A Derivation of Hubble |url=https://www.airitilibrary.com/Publication/alDetailedMesh?docid=05779073-198610-201303280001-201303280001-217-219 |journal=Chinese Journal of Physics |volume=24 |issue=3 |pages=217–219}}</ref> and Sanejouand (2009).<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|journal=Europhysics Letters |volume=88 |pages=59002|s2cid=121784053 }}</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|s2cid=15716718}}</ref>

Cosmological models with varying speeds of light<ref>
{{cite journal |author=Barrow |first=J. D. |year=1998 |title=Cosmologies with varying light-speed |journal=Physical Review D |volume=59 |issue=4 |pages=043515 |arxiv=astro-ph/9811022 |bibcode=1999PhRvD..59d3515B |doi=10.1103/PhysRevD.59.043515 |s2cid=119374406}}</ref> have been proposed independently by Jean-Pierre Petit in 1988,<ref>
{{cite journal |author=Petit |first=Jean-Pierre |year=1988 |title=An interpretation of cosmological model with variable light velocity |url=http://www.januscosmologicalmodel.com/pdf/1988-ModPhysLettA-1.pdf |journal=Mod. Phys. Lett. A |volume=3 |issue=16 |pages=1527&ndash;1532 |bibcode=1988MPLA....3.1527P |citeseerx=10.1.1.692.9603 |doi=10.1142/S0217732388001823}}</ref> [[John Moffat (physicist)|John Moffat]] in 1992,<ref>
{{cite journal |author=Moffat |first=John |year=1993 |title=Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology |journal=International Journal of Modern Physics D |volume=2 |issue=3 |pages=351&ndash;366 |arxiv=gr-qc/9211020 |bibcode=1993IJMPD...2..351M |doi=10.1142/S0218271893000246 |s2cid=17978194}}</ref> and the team of [[Andreas Albrecht (cosmologist)|Andreas Albrecht]] and [[João Magueijo]] in 1998<ref>{{cite journal |last1=Albrecht |first1=A. |last2=Magueijo |first2=J. |year=1999 |title=A time varying speed of light as a solution to cosmological puzzles |journal=Physical Review |volume=D59 |issue=4 |pages=043516 |arxiv=astro-ph/9811018 |bibcode=1999PhRvD..59d3516A |doi=10.1103/PhysRevD.59.043516 |s2cid=56138144}}</ref> to explain the [[horizon problem]] of [[physical cosmology|cosmology]] and propose an alternative to [[cosmic inflation]].

== 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=Dirac |first=Paul A. M. |year=1938 |title=A New Basis for Cosmology |journal=[[Proceedings of the Royal Society A]] |volume=165 |issue=921 |pages=199–208 |bibcode=1938RSPSA.165..199D |doi=10.1098/rspa.1938.0053|s2cid=121069801 }}</ref> For example, Dirac proposed a change of only 5 parts in 10<sup>11</sup> per year of the [[Gravitational constant|Newtonian constant of gravitation]] ''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 |last1=Feynman |first1=Richard P. |title=The Feynman Lectures on Physics |title-link=The Feynman Lectures on Physics |last2=Leighton |first2=R. |last3=Sands |first3=M. |publisher=Addison Wesley Longman |year=2006 |isbn=0-8053-9045-6 |edition=definitive |volume=1 |language=en-us |chapter=7: The Theory of Gravitation |orig-year=1964}}</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 ''G'' varying in isolation. (See also [[strong equivalence principle]].)

=== Fine-structure constant ''α'' ===
{{See also|Fine-structure constant#Potential variation over time}}
One group, studying distant quasars, has claimed to detect a variation of the fine-structure constant<ref>{{cite journal |author=Webb |first1=J. K. |last2=Murphy |first2=M. T. |last3=Flambaum |first3=V. V. |last4=Dzuba |first4=V. A. |last5=Barrow |first5=J. D. |last6=Churchill |first6=C. W. |last7=Prochaska |first7=J. X. |last8=Wolfe |first8=A. M. |year=2001 |title=Further Evidence for Cosmological Evolution of the Fine Structure Constant |journal=Physical Review Letters |volume=87 |issue=9 |pages=091301 |arxiv=astro-ph/0012539 |bibcode=2001PhRvL..87i1301W |doi=10.1103/PhysRevLett.87.091301 |pmid=11531558 |s2cid=40461557}}</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=Chand |first1=H. |last2=Srianand |first2=R. |last3=Petitjean |first3=P. |last4=Aracil |first4=B. |year=2004 |title=Probing the cosmological variation of the fine-structure constant: results based on VLT-UVES sample |journal=Astron. Astrophys. |volume=417 |issue=3 |pages=853–871 |arxiv=astro-ph/0401094 |bibcode=2004A&A...417..853C |doi=10.1051/0004-6361:20035701 |s2cid=17863903}}</ref><ref>{{cite journal |author=Srianand |first1=R. |last2=Chand |first2=H. |last3=Petitjean |first3=P. |last4=Aracil |first4=B. |year=2004 |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=Physical Review Letters |volume=92 |issue=12 |pages=121302 |arxiv=astro-ph/0402177 |bibcode=2004PhRvL..92l1302S |doi=10.1103/PhysRevLett.92.121302 |pmid=15089663 |s2cid=29581666}}</ref><ref>{{cite journal |author=Levshakov |first1=S. A. |last2=Centurion |first2=M. |last3=Molaro |first3=P. |last4=D'Odorico |first4=S. |year=2005 |title=VLT/UVES constraints on the cosmological variability of the fine-structure constant |journal=Astron. Astrophys. |volume=434 |issue=3 |pages=827–838 |arxiv=astro-ph/0408188 |bibcode=2005A&A...434..827L |doi=10.1051/0004-6361:20041827 |s2cid=119351573}}</ref>

The [[natural nuclear reactor]] of [[Oklo]] has been used to check whether the atomic [[fine-structure constant]] ''α'' might have changed over the past 2 billion years. That is because ''α'' influences the rate of various nuclear reactions. For example, {{SimpleNuclide|link=yes|samarium|149}} captures a neutron to become {{SimpleNuclide|samarium|150}}, and since the rate of neutron capture depends on the value of ''α'', the ratio of the two [[samarium]] isotopes in samples from Oklo can be used to calculate the value of ''α'' from 2 billion years ago. Several studies have analysed the relative concentrations of radioactive isotopes left behind at Oklo, and most have concluded that nuclear reactions then were much the same as they are today, which implies ''α'' was the same too.<ref>{{cite journal |last=Petrov |first=Yu. V. |author2=Nazarov, A. I. |author3=Onegin, M. S. |author4=Sakhnovsky, E. G.  |year=2006 |title=Natural nuclear reactor at Oklo and variation of fundamental constants: Computation of neutronics of a fresh core |journal=Physical Review C |volume=74 |issue=6 |pages=064610 |doi=10.1103/PHYSREVC.74.064610 |arxiv = hep-ph/0506186 |bibcode = 2006PhRvC..74f4610P |s2cid=118272311 }}</ref><ref>{{Cite journal|last1=Davis|first1=Edward D.|last2=Hamdan|first2=Leila|date=2015|title=Reappraisal of the limit on the variation in ''α'' implied by the Oklo natural fission reactors|journal=Physical Review C|volume=92|issue=1|pages=014319|bibcode=2015PhRvC..92a4319D|doi=10.1103/physrevc.92.014319|arxiv=1503.06011|s2cid=119227720}}</ref>

[[Paul Davies]] and collaborators have suggested that it is in principle possible to disentangle which of the dimensionful constants (the [[elementary charge]], the [[Planck constant]], and the speed of light) of which the fine-structure constant is composed is responsible for the variation.<ref>{{cite journal |last1=Davies |first1=P. C. W. |last2=Davis |first2=Tamara M. |last3=Lineweaver |first3=Charles H. |year=2002 |title=Cosmology: Black holes constrain varying constants |journal=Nature |volume=418 |issue=6898 |pages=602&ndash;603 |bibcode=2002Natur.418..602D |doi=10.1038/418602a |pmid=12167848 |s2cid=1400235}}</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 |last1=Carlip |first1=S. |last2=Vaidya |first2=S. |name-list-style=amp |year=2003 |title=Black holes may not constrain varying constants |journal=Nature |volume=421 |issue=6922 |pages=498 |arxiv=hep-th/0209249 |bibcode=2003Natur.421..498C |doi=10.1038/421498a |pmid=12556883 |s2cid=209814835}}</ref>

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

=== Dimensionless and dimensionful quantities ===
To clarify 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 the Planck 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. Physicists can ''choose'' their units so that the [[physical constants]] ''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 name="uzan">{{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|s2cid=118684485}}</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, as predicted by the [[Buckingham π theorem]]. Short of their [[anthropometric]] unit dependence, there is no speed of light, gravitational constant, nor the [[Planck 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]], could in principle have meaningful variance and their possible variation continues to be studied.<ref name="uzan"/>

=== General critique of varying ''c'' cosmologies ===
From a very general point of view, [[George Francis Rayner Ellis|G. F. R. Ellis]] and [[Jean-Philippe Uzan]] 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 |author=Ellis |first=George F. R. |date=April 2007 |title=Note on Varying Speed of Light Cosmologies |journal=General Relativity and Gravitation |volume=39 |issue=4 |pages=511–520 |arxiv=astro-ph/0703751 |bibcode=2007GReGr..39..511E |doi=10.1007/s10714-007-0396-4 |s2cid=119393303}}</ref><ref>{{Cite journal |last1=Ellis |first1=George F. R. |last2=Uzan |first2=Jean-Philippe |date=March 2005 |title=c is the speed of light, isn't it? |url=http://aapt.scitation.org/doi/10.1119/1.1819929 |journal=American Journal of Physics |language=en |volume=73 |issue=3 |pages=240–247 |arxiv=gr-qc/0305099 |bibcode=2005AmJPh..73..240E |doi=10.1119/1.1819929 |s2cid=119530637 |issn=0002-9505}}</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]]; and (5) must be done consistently with respect to all other physical theories. VSL cosmologies remain out of mainstream physics.

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

== External links ==
* [http://www.phys.unsw.edu.au/einsteinlight/jw/module6_constant.htm Is the speed of light constant? "Varying constants"]

[[Category:Hypotheses]]
[[Category:Electromagnetic radiation]]
[[Category:Light]]
[[Category:Physical cosmological concepts]]
[[Category:Special relativity]]
[[Category:Fringe physics]]

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