Editing Variable speed of light (section)

== 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]].