Editing Speed of light (section)

== Fundamental role in physics ==
{{See also|Special relativity}}

The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the [[inertial frame of reference]] of the observer. This invariance of the speed of light was postulated by Einstein in 1905,<ref name="stachel" /> after being motivated by [[Maxwell's theory of electromagnetism]] and the lack of evidence for motion against the [[luminiferous aether]].<ref>
{{Cite journal
 |last=Einstein |first=A.
 |year=1905
 |title=Zur Elektrodynamik bewegter Körper
 |journal=[[Annalen der Physik]]
 |volume=17 |issue=10
 |pages=890–921
 |doi=10.1002/andp.19053221004
 |language=de
 |bibcode=1905AnP...322..891E
 |url=http://sedici.unlp.edu.ar/handle/10915/2786
 |type=Submitted manuscript
 |doi-access=free
}} English translation: 
{{Cite web
 |last=Perrett |first=W.
 |translator-last=Jeffery |translator-first=G. B.
 |editor-last=Walker |editor-first=J
 |title=On the Electrodynamics of Moving Bodies
 |url=http://www.fourmilab.ch/etexts/einstein/specrel/www/
 |work=[[Fourmilab]]
 |access-date=27 November 2009
}}</ref> It has since been consistently confirmed by  experiments such as the [[Michelson–Morley experiment]] and [[Kennedy–Thorndike experiment]]. 

The [[special theory of relativity]] explores the consequences of this invariance of ''c'' with the assumption that the laws of physics are the same in all inertial frames of reference.<ref>
{{Cite book
 |last=d'Inverno
 |first=R.
 |year=1992
 |title=Introducing Einstein's Relativity
 |pages=[https://archive.org/details/introducingeinst0000dinv/page/19 19–20]
 |publisher=Oxford University Press
 |isbn=978-0-19-859686-8
 |url=https://archive.org/details/introducingeinst0000dinv/page/19
}}</ref><ref>
{{Cite book
 |last=Sriranjan |first=B.
 |year=2004
 |chapter=Postulates of the special theory of relativity and their consequences
 |chapter-url=https://books.google.com/books?id=FsRfMvyudlAC&pg=PA20
 |title=The Special Theory to Relativity
 |publisher=PHI Learning Pvt. Ltd.
 |isbn=978-81-203-1963-9
 |pages=20ff
}}</ref> One consequence is that ''c'' is the speed at which all massless particles and waves, including light, must travel in vacuum.<ref name=":0">{{Cite book|last1=Ellis|first1=George F. R.|url=https://www.worldcat.org/oclc/44694623|title=Flat and Curved Space-times|last2=Williams|first2=Ruth M.|date=2000|publisher=Oxford University Press|isbn=0-19-850657-0|edition=2|location=Oxford|oclc=44694623|author-link=George F. R. Ellis|author-link2=Ruth Margaret Williams |page=12}}</ref>
[[File:Lorentz factor.svg|thumb|upright|alt=γ starts at&nbsp;1 when&nbsp;v equals zero and stays nearly constant for small v's, then it sharply curves upwards and has a vertical asymptote, diverging to positive infinity as&nbsp;v approaches c. |The [[Lorentz factor]] ''γ'' as a function of velocity. It starts at{{nbsp}}1 and approaches infinity as ''v'' approaches&nbsp;''c''.]]
Special relativity has many counterintuitive and experimentally verified implications.<ref>
{{Cite web
 |last1        = Roberts
 |first1       = T.
 |last2        = Schleif
 |first2       = S.
 |editor-last  = Dlugosz
 |editor-first = J. M.
 |year         = 2007
 |title        = What is the experimental basis of Special Relativity?
 |url          = http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
 |work         = Usenet Physics FAQ
 |publisher    = [[University of California, Riverside]]
 |access-date  = 27 November 2009
 |archive-url  = https://web.archive.org/web/20091015153529/http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
 |archive-date = 15 October 2009
 |url-status=dead
}}</ref> These include the [[Mass–energy equivalence|equivalence of mass and energy]] {{nowrap|(''E'' {{=}} ''mc''{{i sup|2}})}}, [[length contraction]] (moving objects shorten), [[Terrell rotation]] (apparent rotation),<ref>
{{Cite journal
 |last=Terrell |first=J.
 |year=1959
 |title=Invisibility of the Lorentz Contraction
 |journal=[[Physical Review]]
 |volume=116
 |issue=4 |pages=1041–1045
 |doi=10.1103/PhysRev.116.1041
 |bibcode = 1959PhRv..116.1041T
}}</ref><ref>
{{Cite journal
 |last=Penrose |first=R. |authorlink=Roger Penrose
 |year=1959
 |title=The Apparent Shape of a Relativistically Moving Sphere
 |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
 |volume=55
 |issue=1 |pages=137–139
 |doi=10.1017/S0305004100033776
 |bibcode=1959PCPS...55..137P |s2cid=123023118
 }}</ref> and [[time dilation]] (moving clocks run more slowly). The factor&nbsp;''γ'' by which lengths contract and times dilate is known as the [[Lorentz factor]] and is given by {{nowrap|''γ'' {{=}} (1 − ''v''{{i sup|2}}/''c''{{i sup|2}})<sup>−1/2</sup>}}, where ''v'' is the speed of the object. The difference of ''γ'' from{{nbsp}}1 is negligible for speeds much slower than&nbsp;''c'', such as most everyday speeds{{snd}}in which case special relativity is closely approximated by [[Galilean relativity]]{{snd}}but it increases at relativistic speeds and diverges to infinity as ''v'' approaches ''c''. For example, a time dilation factor of ''γ''&nbsp;=&nbsp;2 occurs at a relative velocity of 86.6% of the speed of light (''v''&nbsp;=&nbsp;0.866&nbsp;''c''). Similarly, a time dilation factor of ''γ''&nbsp;=&nbsp;10 occurs at 99.5% the speed of light (''v''&nbsp;=&nbsp;0.995&nbsp;''c'').

The results of special relativity can be summarized by treating space and time as a unified structure known as [[spacetime]] (with&nbsp;''c'' relating the units of space and time), and requiring that physical theories satisfy a special [[Symmetry in physics|symmetry]] called [[Lorentz invariance]], whose mathematical formulation contains the parameter&nbsp;''c''.<ref>
{{Cite book
 |last=Hartle
 |first=J. B.
 |year=2003
 |title=Gravity: An Introduction to Einstein's General Relativity
 |pages=[https://archive.org/details/specialrelativit0000chan/page/52 52–59]
 |publisher=[[Addison-Wesley]]
 |isbn=978-981-02-2749-4
 |url=https://archive.org/details/specialrelativit0000chan/page/52
}}</ref> Lorentz invariance is an almost universal assumption for modern physical theories, such as [[quantum electrodynamics]], [[quantum chromodynamics]], the [[Standard Model]] of [[particle physics]], and [[general relativity]]. As such, the parameter&nbsp;''c'' is ubiquitous in modern physics, appearing in many contexts that are unrelated to light. For example, general relativity predicts that&nbsp;''c'' is also the [[speed of gravity]] and of [[gravitational waves]],<ref name="Hartle">
{{Cite book
 |last=Hartle
 |first=J. B.
 |year=2003
 |title=Gravity: An Introduction to Einstein's General Relativity
 |page=332
 |publisher=[[Addison-Wesley]]
 |isbn=978-981-02-2749-4
 |url=https://archive.org/details/specialrelativit0000chan
 |url-access=limited
}}</ref> and observations of gravitational waves have been consistent with this prediction.<ref>See, for example:
* {{Cite journal |last1=Abbott |first1=B. P. |display-authors=etal |year=2017 |title=Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A |journal=[[The Astrophysical Journal Letters]] |volume=848 |issue=2 |page=L13 |arxiv=1710.05834 |bibcode=2017ApJ...848L..13A |doi=10.3847/2041-8213/aa920c |doi-access=free}}
* {{Cite journal |last1=Cornish |first1=Neil |last2=Blas |first2=Diego |last3=Nardini |first3=Germano |date=18 October 2017 |title=Bounding the Speed of Gravity with Gravitational Wave Observations |url=https://link.aps.org/doi/10.1103/PhysRevLett.119.161102 |journal=[[Physical Review Letters]] |volume=119 |issue=16 |pages=161102 |doi=10.1103/PhysRevLett.119.161102 |pmid=29099221 |arxiv=1707.06101 |bibcode=2017PhRvL.119p1102C |s2cid=206300556}}
* {{Cite journal |last1=Liu |first1=Xiaoshu |last2=He |first2=Vincent F. |last3=Mikulski |first3=Timothy M. |last4=Palenova |first4=Daria |last5=Williams |first5=Claire E. |last6=Creighton |first6=Jolien |last7=Tasson |first7=Jay D. |date=7 July 2020 |title=Measuring the speed of gravitational waves from the first and second observing run of Advanced LIGO and Advanced Virgo |url=https://link.aps.org/doi/10.1103/PhysRevD.102.024028 |journal=[[Physical Review D]] |volume=102 |issue=2 |pages=024028 |doi=10.1103/PhysRevD.102.024028 |arxiv=2005.03121 |bibcode=2020PhRvD.102b4028L |s2cid=220514677}}</ref> In [[non-inertial frame]]s of reference (gravitationally curved spacetime or [[accelerated reference frame]]s), the ''local'' speed of light is constant and equal to&nbsp;''c'', but the speed of light can differ from&nbsp;''c'' when measured from a remote frame of reference, depending on how measurements are extrapolated to the region.<ref name="Gibbs1997" />

It is generally assumed that fundamental constants such as&nbsp;''c'' have the same value throughout spacetime, meaning that they do not depend on location and do not vary with time. However, it has been suggested in various theories that the [[Variable speed of light|speed of light may have changed over time]].<ref name=Ellis_Uzan>
{{Cite journal
 |last1=Ellis |first1=G. F. R. |last2=Uzan |first2=J.-P.
 |year=2005
 |title='c' is the speed of light, isn't it?
 |journal=[[American Journal of Physics]]
 |volume=73
 |issue=3 |pages=240–227
 |doi=10.1119/1.1819929
 |arxiv=gr-qc/0305099
 |quote=The possibility that the fundamental constants may vary during the evolution of the universe offers an exceptional window onto higher dimensional theories and is probably linked with the nature of the dark energy that makes the universe accelerate today.
 |bibcode = 2005AmJPh..73..240E |s2cid=119530637
}}</ref><ref name=Mota>
{{Cite thesis |type=PhD
 |last=Mota |first=D. F.
 |year=2006
 |title=Variations of the Fine Structure Constant in Space and Time
 |arxiv=astro-ph/0401631
 |bibcode=2004astro.ph..1631M
}}</ref> No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.<ref name=Uzan>
{{Cite journal
 |last=Uzan |first=J.-P.
 |year=2003
 |title=The fundamental constants and their variation: observational status and theoretical motivations
 |journal=[[Reviews of Modern Physics]]
 |volume=75
 |issue=2 |page=403
 |doi=10.1103/RevModPhys.75.403
 |arxiv=hep-ph/0205340
 |bibcode=2003RvMP...75..403U
 |s2cid=118684485
}}</ref><ref name=Camelia>
{{Cite journal
 |last=Amelino-Camelia |first=G.
 |year=2013
 |title=Quantum Gravity Phenomenology
 |arxiv=0806.0339
 |doi=10.12942/lrr-2013-5
 |pmid=28179844
 |pmc=5255913
 |volume=16
 |issue=1
 |pages=5
 |journal=Living Reviews in Relativity
 |doi-access=free
 |bibcode=2013LRR....16....5A
}}</ref>

It is generally assumed that the two-way speed of light is [[isotropy|isotropic]], meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear [[energy level]]s as a function of the orientation of the emitting [[atomic nucleus|nuclei]] in a magnetic field (see [[Hughes–Drever experiment]]), and of rotating [[optical resonator]]s (see [[Michelson–Morley experiment#Recent experiments|Resonator experiments]]) have put stringent limits on the possible two-way [[anisotropy]].<ref name=Herrmann>
{{Cite journal
 |last1=Herrmann |first1=S. |last2=Senger |first2=A. |last3=Möhle |first3=K. |last4=Nagel |first4=M. |last5=Kovalchuk |first5=E. V. |last6=Peters |first6=A.
 |title=Rotating optical cavity experiment testing Lorentz invariance at the 10<sup>−17</sup> level
 |journal=Physical Review D |volume=80 |issue=100 |pages=105011 |year=2009
 |doi=10.1103/PhysRevD.80.105011 |arxiv=1002.1284 |bibcode = 2009PhRvD..80j5011H |s2cid=118346408 }}</ref><ref name=Lang>{{Cite book
 |title=Astrophysical formulae
 |first=K. R. |last=Lang
 |url=https://books.google.com/books?id=OvTjLcQ4MCQC&pg=PA152
 |page=152
 |isbn=978-3-540-29692-8
 |publisher=Birkhäuser
 |edition=3
 |year=1999
}}</ref>

=== Upper limit on speeds ===
An object with [[rest mass]] ''m'' and speed ''v'' relative to a laboratory has [[kinetic energy]] {{nowrap|''(γ-1)mc''{{i sup|2}}}} with respect to that lab, where ''γ'' is the Lorentz factor defined above. The ''γ'' factor approaches infinity as ''v'' approaches&nbsp;''c'', and it would take an infinite amount of energy to accelerate an object with mass to the speed of light.<ref>{{Cite book |last1=Kleppner |first1=Daniel |title=An introduction to mechanics |last2=Kolenkow |first2=Robert J. |date=2014 |publisher=Cambridge university press |isbn=978-0-521-19811-0 |edition=2nd |location=Cambridge}}</ref>{{rp|loc=13.3}} The speed of light is the upper limit for the speeds of objects with positive rest mass. Analysis of individual photons confirm that information cannot travel faster than the speed of light.<ref>{{Cite web |last=Voss |first=David |date=2011-06-16 |title=Single photons obey the speed limits |url=https://physics.aps.org/articles/v4/s88 |access-date=2025-04-17 |website=Physics |pages=s88 |language=en |doi=10.1103/PhysRevLett.106.243602}}</ref><ref>
{{Cite journal |title=Optical Precursor of a Single Photon |author1=Shanchao Zhang |author2=J. F. Chen |author3=Chang Liu |author4=M. M. T. Loy |author5=G. K. L. Wong |author6=Shengwang Du |journal=[[Physical Review Letters]] |volume=106 |issue=243602 |pages=243602 |date=16 June 2011 |doi=10.1103/physrevlett.106.243602|pmid=21770570 |bibcode=2011PhRvL.106x3602Z |url=http://repository.ust.hk/ir/bitstream/1783.1-7246/1/PhysRevLett.106.243602.pdf }}
</ref> This is experimentally established in many [[tests of relativistic energy and momentum]].<ref>
{{Cite web
 |last=Fowler |first=M.
 |date=March 2008
 |title=Notes on Special Relativity
 |url=http://galileo.phys.virginia.edu/classes/252/SpecRelNotes.pdf
 |page=56
 |publisher=University of Virginia
 |access-date=7 May 2010
}}</ref>

[[File:Relativity of Simultaneity.svg|thumb|Event&nbsp;A precedes&nbsp;B in the red frame, is simultaneous with&nbsp;B in the green frame, and follows&nbsp;B in the blue frame.|alt=Three pairs of coordinate axes are depicted with the same origin&nbsp;A; in the green frame, the x axis is horizontal and the ct axis is vertical; in the red frame, the x′ axis is slightly skewed upwards, and the ct′ axis slightly skewed rightwards, relative to the green axes; in the blue frame, the x′′ axis is somewhat skewed downwards, and the ct′′ axis somewhat skewed leftwards, relative to the green axes. A point&nbsp;B on the green x axis, to the left of&nbsp;A, has zero ct, positive ct′, and negative ct′′.]]
More generally, it is impossible for signals or energy to travel faster than&nbsp;''c''. One argument for this is known as [[causality (physics)|causality]].  If the spatial distance between two events&nbsp;A and&nbsp;B is greater than the time interval between them multiplied by&nbsp;''c'' then there are frames of reference in which&nbsp;A precedes&nbsp;B, others in which&nbsp;B precedes&nbsp;A, and others in which they are simultaneous. As a result, if something were travelling faster than&nbsp;''c'' relative to an inertial frame of reference, it would be travelling backwards in time relative to another frame, and causality would be violated.<ref>{{Cite book |last=Fayngold |first=Moses |url=https://www.worldcat.org/title/180478876 |title=Special relativity and how it works |date=2008 |publisher=Wiley-VCH |isbn=978-3-527-40607-4 |series=Physics textbook |location=Weinheim |oclc=180478876}}</ref>{{rp|497}}<ref>
{{Cite journal
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 |title=Faster-than-''c'' signals, special relativity, and causality
 |journal=[[Annals of Physics]]
 |volume=298
 |issue=1 |pages=167–185
 |doi=10.1006/aphy.2002.6233
 |arxiv=gr-qc/0107091
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}}</ref><ref name="Taylor_p74">
{{Cite book
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 |url=https://archive.org/details/spacetimephysics00edwi_0/page/74
}}</ref> In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded,<ref name=Zhang/> and would lead to [[paradox]]es such as the [[tachyonic antitelephone]].<ref>
{{Cite book
 |last=Tolman |first=R. C.
 |year=2009 |orig-year=1917
 |chapter=Velocities greater than that of light
 |title=The Theory of the Relativity of Motion
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 |isbn=978-1-103-17233-7
}}</ref>

In some theoretical treatments, the [[Scharnhorst effect]] allows signals to travel faster than&nbsp;''c'', by one part in 10<sup>36</sup>.<ref>De Clark, S. G. (2016). The scharnhorst effect: Superluminality and causality in effective field theories. The University of Arizona.</ref> However other approaches to the same physical set up show no such effect.<ref>See, for example:
* {{Cite journal|last=Ben-Menahem|first=Shahar|date=November 1990|title=Causality between conducting plates|url=https://linkinghub.elsevier.com/retrieve/pii/037026939091167A|journal=Physics Letters B|language=en|volume=250|issue=1–2|pages=133–138|doi=10.1016/0370-2693(90)91167-A|bibcode=1990PhLB..250..133B|osti=1449261}}
* {{Cite journal |last=Fearn |first=H. |date=10 November 2006 |title=Dispersion relations and causality: does relativistic causality require that n (ω) → 1 as ω → ∞ ? |url=http://www.tandfonline.com/doi/abs/10.1080/09500340600952085 |journal=Journal of Modern Optics |language=en |volume=53 |issue=16–17 |pages=2569–2581 |doi=10.1080/09500340600952085 |bibcode=2006JMOp...53.2569F |s2cid=119892992 |issn=0950-0340|url-access=subscription }}
* {{Cite journal |last=Fearn |first=H. |date=May 2007 |title=Can light signals travel faster than c in nontrivial vacua in flat space-time? Relativistic causality II |url=http://link.springer.com/10.1134/S1054660X07050155 |journal=Laser Physics |language=en |volume=17 |issue=5 |pages=695–699 |doi=10.1134/S1054660X07050155 |arxiv=0706.0553 |bibcode=2007LaPhy..17..695F |s2cid=61962 |issn=1054-660X}}</ref> and it appears the special conditions in which this effect might occur would prevent one from using it to violate causality.

=== One-way speed of light ===
{{main|One-way speed of light}}
It is only possible to verify experimentally that the two-way speed of light (for example, from a source to a mirror and back again) is frame-independent, because it is impossible to measure the [[one-way speed of light]] (for example, from a source to a distant detector) without some convention as to how clocks at the source and at the detector should be synchronized. By adopting [[Einstein synchronization]] for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition.<ref name=Hsu2>
{{Cite book
 |last1=Hsu |first1=J.-P. |last2=Zhang |first2=Y. Z.
 |year=2001
 |title=Lorentz and Poincaré Invariance
 |url=https://books.google.com/books?id=jryk42J8oQIC&pg=RA1-PA541
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 |series=Advanced Series on Theoretical Physical Science
 |volume=8 |pages=543ff
 |isbn=978-981-02-4721-8
}}</ref><ref name=Zhang>
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 |year        = 1997
 |title       = Special Relativity and Its Experimental Foundations
 |url         = https://archive.org/details/specialrelativit0000chan/page/172
 |publisher   = [[World Scientific]]
 |series      = Advanced Series on Theoretical Physical Science
 |volume      = 4
 |pages       = [https://archive.org/details/specialrelativit0000chan/page/172 172–173]
 |isbn        = 978-981-02-2749-4
 |access-date = 23 July 2009
}}</ref>