Speed of light#Faster-than-light observations and experiments

The speed of light in vacuum is usually denoted by a lowercase {{math|c}}. The origin of the letter choice is unclear, with guesses including "c" for "constant" or the Latin {{lang|la|celeritas}} (meaning 'swiftness, celerity'). The "c" was used for "celerity" meaning a velocity in books by Leonhard Euler and others, but this velocity was not specifically for light; Isaac Asimov wrote a popular science article, "C for Celeritas", but did not explain the origin.{{Cite journal |last1=Pelosi |first1=G. |last2=Selleri |first2=S. |date=December 2010 |title="c" Utrum est ut Celeritas an Constantia? (Does "c" Stand for Speed or Constancy?) |url=https://ieeexplore.ieee.org/document/5723273 |journal=IEEE Antennas and Propagation Magazine |volume=52 |issue=6 |pages=207–219 |doi=10.1109/MAP.2010.5723273 |bibcode=2010IAPM...52..207P |issn=1045-9243}} In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used {{math|c}} for a different constant that was later shown to equal {{radic|2}} times the speed of light in vacuum. Historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1903, Max Abraham used {{math|c}} with its modern meaning in a widely read textbook on electromagnetism. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to {{math|c}}, which by then had become the standard symbol for the speed of light.

{{Cite web

|last=Gibbs

|first=P.

|year=2004

|orig-year=1997

|title=Why is c the symbol for the speed of light?

|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/c.html

|work=Usenet Physics FAQ

|publisher=University of California, Riverside

|access-date=16 November 2009

|archive-url=https://web.archive.org/web/20100325220247/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/c.html

|archive-date=25 March 2010

|url-status=dead

}}

"The origins of the letter c being used for the speed of light can be traced back to a paper of 1856 by Weber and Kohlrausch [...] Weber apparently meant c to stand for 'constant' in his force law, but there is evidence that physicists such as Lorentz and Einstein were accustomed to a common convention that c could be used as a variable for velocity. This usage can be traced back to the classic Latin texts in which c stood for 'celeritas', meaning 'speed'."

{{Cite journal

|last=Mendelson |first=K. S.

|year=2006

|title=The story of c

|journal=American Journal of Physics

|volume=74 |issue=11 |pages=995–997

|doi=10.1119/1.2238887

|bibcode = 2006AmJPh..74..995M | issn=0002-9505}}

Sometimes {{math|c}} is used for the speed of waves in any material medium, and {{math|c}}₀ for the speed of light in vacuum.See, for example:

• {{Cite book

|last=Lide |first=D. R.

|year=2004

|title=CRC Handbook of Chemistry and Physics

|url=https://books.google.com/books?id=WDll8hA006AC&q=speed+of+light+%22c0+OR+%22&pg=PT76

|pages=2–9

|publisher=CRC Press

|isbn=978-0-8493-0485-9

}}

• {{Cite book

|last=Harris |first=J. W. |year=2002

|title=Handbook of Physics

|url=https://books.google.com/books?id=c60mCxGRMR8C&q=speed+of+light+%22c0+OR+%22+date:2000-2009&pg=PA499

|page=499

|publisher=Springer

|isbn=978-0-387-95269-7

|display-authors=etal}}

• {{Cite book

|last=Whitaker |first=J. C.

|year=2005

|title=The Electronics Handbook

|url=https://books.google.com/books?id=FdSQSAC3_EwC&q=speed+of+light+c0+handbook&pg=PA235

|page=235

|publisher=CRC Press

|isbn=978-0-8493-1889-4

}}

• {{Cite book

|last=Cohen |first=E. R. |year=2007

|title=Quantities, Units and Symbols in Physical Chemistry

|url=https://books.google.com/books?id=TElmhULQoeIC&q=speed+of+light+c0+handbook&pg=PA143

|page=184

|edition=3

|publisher=Royal Society of Chemistry

|isbn=978-0-85404-433-7

|display-authors=etal

}} This subscripted notation, which is endorsed in official SI literature,{{SIbrochure8th|page=112}} has the same form as related electromagnetic constants: namely, μ₀ for the vacuum permeability or magnetic constant, ε₀ for the vacuum permittivity or electric constant, and Z₀ for the impedance of free space. This article uses {{math|c}} exclusively for the speed of light in vacuum.

{{Further information|Metre#Speed of light definition}}

Since 1983, the constant {{math|c}} has been defined in the International System of Units (SI) as exactly {{val|299792458|u=m/s}}; this relationship is used to define the metre as exactly the distance that light travels in vacuum in {{frac|1|{{val|299792458}}}} of a second. The second is, in turn, defined to be the length of time occupied by {{val|9192631770|u=cycles}} of the radiation emitted by a caesium-133 atom in a transition between two specified energy states.{{Cite web |url=https://physics.nist.gov/cuu/Units/current.html |title=Definitions of the SI base units |website=physics.nist.gov |date=29 May 2019 |access-date=8 February 2022}} By using the value of {{math|c}}, as well as an accurate measurement of the second, one can establish a standard for the metre.See, for example:

• {{Cite book

|last=Sydenham |first=P. H.

|year=2003

|chapter=Measurement of length

|chapter-url=https://books.google.com/books?id=sarHIbCVOUAC&pg=PA56

|editor=Boyes, W

|title=Instrumentation Reference Book

|edition=3

|page=56

|publisher=Butterworth–Heinemann

|isbn=978-0-7506-7123-1

|quote=...{{nbsp}}if the speed of light is defined as a fixed number then, in principle, the time standard will serve as the length standard{{nbsp}}...

}}

• {{Cite web

|title=CODATA value: Speed of Light in Vacuum

|url=http://physics.nist.gov/cgi-bin/cuu/Value?c

|work=The NIST reference on Constants, Units, and Uncertainty

|publisher=NIST

|access-date=21 August 2009

}}

• {{Cite book

|last1=Jespersen |first1=J. |last2=Fitz-Randolph |first2=J. |last3=Robb |first3=J.

|year=1999

|title=From Sundials to Atomic Clocks: Understanding Time and Frequency

|url=https://books.google.com/books?id=Z7chuo4ebUAC&pg=PA280

|page=280

|edition=Reprint of National Bureau of Standards 1977, 2nd

|publisher=Courier Dover

|isbn=978-0-486-40913-9

}}

The particular value chosen for the speed of light provided a more accurate definition of the metre that still agreed as much as possible with the definition used before 1983.{{Cite book |last=Penrose |first=R | author-link=Roger Penrose |year=2004 |title=The Road to Reality: A Complete Guide to the Laws of the Universe |pages=[https://archive.org/details/roadtoreality00penr_319/page/n438 410]–411 |publisher=Vintage Books |isbn=978-0-679-77631-4 |quote=...{{nbsp}}the most accurate standard for the metre is conveniently defined so that there are exactly {{val|299792458}} of them to the distance travelled by light in a standard second, giving a value for the metre that very accurately matches the now inadequately precise standard metre rule in Paris. |title-link=The Road to Reality: A Complete Guide to the Laws of the Universe }}

As a dimensional physical constant, the numerical value of {{math|c}} is different for different unit systems. For example, in imperial units, the speed of light is approximately {{val|186,282}} miles per second,{{#tag:ref|The speed of light in imperial is exactly

: {{val|186,282|u=miles}}, {{val|698|u=yards}}, {{val|2|u=feet}}, and {{sfrac|5|21|127}}{{nbsp}}inches per second.|group="Note"|name="imperial"}} or roughly 1 foot per nanosecond.{{#tag:ref|The exact value is {{sfrac|{{val|149,896,229}}|{{val|152,400,000}}}}{{nbsp}}{{sfrac|ft|ns}} ≈ 0.98{{nbsp}}{{sfrac|ft|ns}}.|group="Note"|name="nanosecond"}}{{Cite book|last=Mermin |first=N. David |url=https://www.worldcat.org/oclc/57283944 |title=It's About Time: Understanding Einstein's Relativity |date=2005 |publisher=Princeton University Press |isbn=0-691-12201-6 |location=Princeton |oclc=57283944 |author-link=N. David Mermin |page=22}}{{Cite web|url=https://americanhistory.si.edu/collections/search/object/nmah_692464 |title=Nanoseconds Associated with Grace Hopper |website=National Museum of American History |quote=Grace Murray Hopper (1906–1992), a mathematician who became a naval officer and computer scientist during World War II, started distributing these wire "nanoseconds" in the late 1960s in order to demonstrate how designing smaller components would produce faster computers. |access-date=1 March 2022}}

In branches of physics in which {{math|c}} appears often, such as in relativity, it is common to use systems of natural units of measurement or the geometrized unit system where {{nowrap|{{math|c}} {{=}} 1}}.

{{Cite book

|last=Lawrie |first=I. D.

|year=2002

|chapter=Appendix C: Natural units

|chapter-url=https://books.google.com/books?id=9HZStxmfi3UC&pg=PA540

|title=A Unified Grand Tour of Theoretical Physics

|page=540

|edition=2

|publisher=CRC Press

|isbn=978-0-7503-0604-1

}}

{{Cite book

|last=Hsu |first=L.

|year=2006

|chapter=Appendix A: Systems of units and the development of relativity theories

|chapter-url=https://books.google.com/books?id=amLqckyrvUwC&pg=PA428

|title=A Broader View of Relativity: General Implications of Lorentz and Poincaré Invariance

|pages=427–428

|edition=2

|publisher=World Scientific

|isbn=978-981-256-651-5

}} Using these units, {{math|c}} does not appear explicitly because multiplication or division by{{nbsp}}1 does not affect the result. Its unit of light-second per second is still relevant, even if omitted.

{{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, after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for motion against the luminiferous aether.

{{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

}} 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.

{{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

}}

{{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

}} One consequence is that c is the speed at which all massless particles and waves, including light, must travel in vacuum.{{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}}

File:Lorentz factor.svg γ as a function of velocity. It starts at{{nbsp}}1 and approaches infinity as v approaches c.]]

Special relativity has many counterintuitive and experimentally verified implications.

{{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

}} These include the equivalence of mass and energy {{nowrap|(E {{=}} mc{{i sup|2}})}}, length contraction (moving objects shorten), Terrell rotation (apparent rotation),

{{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

}}

{{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

}} and time dilation (moving clocks run more slowly). The factor 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}})^−1/2}}, where v is the speed of the object. The difference of γ from{{nbsp}}1 is negligible for speeds much slower than 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 = 2 occurs at a relative velocity of 86.6% of the speed of light (v = 0.866 c). Similarly, a time dilation factor of γ = 10 occurs at 99.5% the speed of light (v = 0.995 c).

The results of special relativity can be summarized by treating space and time as a unified structure known as spacetime (with c relating the units of space and time), and requiring that physical theories satisfy a special symmetry called Lorentz invariance, whose mathematical formulation contains the parameter c.

{{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

}} 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 c is ubiquitous in modern physics, appearing in many contexts that are unrelated to light. For example, general relativity predicts that c is also the speed of gravity and of gravitational waves,

{{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

}} and observations of gravitational waves have been consistent with this prediction.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}} In non-inertial frames of reference (gravitationally curved spacetime or accelerated reference frames), the local speed of light is constant and equal to c, but the speed of light can differ from c when measured from a remote frame of reference, depending on how measurements are extrapolated to the region.

It is generally assumed that fundamental constants such as 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 speed of light may have changed over time.

{{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

}}

{{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

}} No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.

{{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

}}

{{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

}}

It is generally assumed that the two-way speed of light is isotropic, meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear energy levels as a function of the orientation of the emitting nuclei in a magnetic field (see Hughes–Drever experiment), and of rotating optical resonators (see Resonator experiments) have put stringent limits on the possible two-way anisotropy.

{{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 10⁻¹⁷ 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 }}{{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

}}

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 c, and it would take an infinite amount of energy to accelerate an object with mass to the speed of light.{{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}}{{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.{{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}}

{{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 }}

This is experimentally established in many tests of relativistic energy and momentum.

{{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

}}

File:Relativity of Simultaneity.svg

More generally, it is impossible for signals or energy to travel faster than c. One argument for this is known as causality. If the spatial distance between two events A and B is greater than the time interval between them multiplied by c then there are frames of reference in which A precedes B, others in which B precedes A, and others in which they are simultaneous. As a result, if something were travelling faster than c relative to an inertial frame of reference, it would be travelling backwards in time relative to another frame, and causality would be violated.{{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}}{{rp|497}}

{{Cite journal

|last1=Liberati |first1=S. |last2=Sonego |first2=S. |last3=Visser |first3=M.

|year=2002

|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

|bibcode = 2002AnPhy.298..167L |s2cid=48166

}}

{{Cite book

|last1=Taylor

|first1=E. F.

|author-link1=Edwin F. Taylor

|last2=Wheeler

|first2=J. A.

|author-link2=John Archibald Wheeler

|year=1992

|title=Spacetime Physics

|pages=[https://archive.org/details/spacetimephysics00edwi_0/page/74 74–75]

|publisher=W. H. Freeman

|isbn=978-0-7167-2327-1

|url=https://archive.org/details/spacetimephysics00edwi_0/page/74

}} In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded, and would lead to paradoxes such as the tachyonic antitelephone.

{{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

|edition=Reprint |page=54

|publisher=BiblioLife

|isbn=978-1-103-17233-7

}}

In some theoretical treatments, the Scharnhorst effect allows signals to travel faster than c, by one part in 10³⁶.De Clark, S. G. (2016). The scharnhorst effect: Superluminality and causality in effective field theories. The University of Arizona. However other approaches to the same physical set up show no such effect.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}}
• {{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}} and it appears the special conditions in which this effect might occur would prevent one from using it to violate causality.

{{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.

{{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

|publisher=World Scientific

|series=Advanced Series on Theoretical Physical Science

|volume=8 |pages=543ff

|isbn=978-981-02-4721-8

}}

{{Cite book

|last = Zhang

|first = Y. Z.

|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

}}

{{See also|Faster-than-light|Superluminal motion}}

There are situations in which it may seem that matter, energy, or information-carrying signal travels at speeds greater than c, but they do not. For example, as is discussed in the propagation of light in a medium section below, many wave velocities can exceed c. The phase velocity of X-rays through most glasses can routinely exceed c,

{{Cite book

|last=Hecht |first=E.

|year=1987

|title=Optics

|page=62

|edition=2

|publisher=Addison-Wesley

|isbn=978-0-201-11609-0

}} but phase velocity does not determine the velocity at which waves convey information.

{{Cite book

|last=Quimby |first=R. S.

|title=Photonics and lasers: an introduction

|publisher=John Wiley and Sons

|year=2006

|page=9

|isbn=978-0-471-71974-8

|url=https://books.google.com/books?id=yWeDVfaVGxsC&pg=PA9

}}

If a laser beam is swept quickly across a distant object, the spot of light can move faster than c, although the initial movement of the spot is delayed because of the time it takes light to get to the distant object at the speed c. However, the only physical entities that are moving are the laser and its emitted light, which travels at the speed c from the laser to the various positions of the spot. Similarly, a shadow projected onto a distant object can be made to move faster than c, after a delay in time.

{{Cite news

|last=Wertheim |first=M.

|title=The Shadow Goes

|url=https://www.nytimes.com/2007/06/20/opinion/20wertheim.html?_r=1&scp=1&sq=%27the%20shadow%20goes%27&st=cse&oref=slogin

|work=The New York Times

|access-date=21 August 2009

|date=20 June 2007

}} In neither case does any matter, energy, or information travel faster than light.

{{Cite web

|last=Gibbs

|first=P.

|year=1997

|title=Is Faster-Than-Light Travel or Communication Possible?

|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html

|publisher=University of California, Riverside

|work=Usenet Physics FAQ

|access-date=20 August 2008

|archive-date=10 March 2010

|archive-url=https://web.archive.org/web/20100310205556/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html

|url-status=dead

}}

The rate of change in the distance between two objects in a frame of reference with respect to which both are moving (their closing speed) may have a value in excess of c. However, this does not represent the speed of any single object as measured in a single inertial frame.

Certain quantum effects appear to be transmitted instantaneously and therefore faster than c, as in the EPR paradox. An example involves the quantum states of two particles that can be entangled. Until either of the particles is observed, they exist in a superposition of two quantum states. If the particles are separated and one particle's quantum state is observed, the other particle's quantum state is determined instantaneously. However, it is impossible to control which quantum state the first particle will take on when it is observed, so information cannot be transmitted in this manner.See, for example:

• {{Cite book

|last=Sakurai |first=J. J. |author-link=J. J. Sakurai

|year=1994

|editor-last=Tuan |editor-first=S. F.

|title=Modern Quantum Mechanics |edition=Revised |pages=[https://archive.org/details/modernquantummec00saku_488/page/n243 231]–232

|publisher=Addison-Wesley

|isbn=978-0-201-53929-5

}}

• {{Cite book|last=Peres|first=Asher|title=Quantum Theory: Concepts and Methods|title-link=Quantum Theory: Concepts and Methods|publisher=Kluwer|year=1993|isbn=0-7923-2549-4|pages=170|oclc=28854083|author-link=Asher Peres}}
• {{Cite book

|first=Carlton M.

|last=Caves

|author-link=Carlton Caves

|chapter=Quantum Information Science: Emerging No More

|title=OSA Century of Optics

|pages=320–326

|arxiv=1302.1864

|publisher=Optica

|year=2015

|isbn=978-1-943-58004-0

|quote=[I]t was natural to dream that quantum correlations could be used for faster-than-light communication, but this speculation was quickly shot down, and the shooting established the principle that quantum states cannot be copied.

}}

Another quantum effect that predicts the occurrence of faster-than-light speeds is called the Hartman effect: under certain conditions the time needed for a virtual particle to tunnel through a barrier is constant, regardless of the thickness of the barrier.

{{Cite book

|editor-last=Muga |editor-first=J. G. |editor-last2=Mayato |editor-first2=R. S. |editor-last3=Egusquiza |editor-first3=I. L.

|year=2007

|title=Time in Quantum Mechanics

|url=https://books.google.com/books?id=InKru6zHQWgC&pg=PA48

|page=48

|publisher=Springer

|isbn=978-3-540-73472-7

}}

{{Cite book

|last1=Hernández-Figueroa |first1=H. E. |last2=Zamboni-Rached |first2=M. |last3=Recami |first3=E.

|year=2007

|title=Localized Waves

|url=https://books.google.com/books?id=xxbXgL967PwC&pg=PA26

|page=26

|publisher=Wiley Interscience

|isbn=978-0-470-10885-7

}} This could result in a virtual particle crossing a large gap faster than light. However, no information can be sent using this effect.

{{Cite journal

|last=Wynne

|first=K.

|year=2002

|title=Causality and the nature of information

|journal=Optics Communications

|volume=209

|issue=1–3

|pages=84–100

|doi=10.1016/S0030-4018(02)01638-3

|bibcode=2002OptCo.209...85W

|url=http://bcp.phys.strath.ac.uk/the_group/r/uf/2002-OC-causality.pdf

|archive-url=https://web.archive.org/web/20090325093856/http://bcp.phys.strath.ac.uk/the_group/r/uf/2002-OC-causality.pdf

|archive-date=2009-03-25

}}

So-called superluminal motion is seen in certain astronomical objects,

{{Cite journal

|last=Rees |first=M. |author-link=Martin Rees

|year=1966

|title=The Appearance of Relativistically Expanding Radio Sources

|journal=Nature

|volume=211

|issue=5048 |page=468

|doi=10.1038/211468a0

|bibcode = 1966Natur.211..468R |s2cid=41065207

}} such as the relativistic jets of radio galaxies and quasars. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection effect caused by objects moving near the speed of light and approaching Earth at a small angle to the line of sight: since the light which was emitted when the jet was farther away took longer to reach the Earth, the time between two successive observations corresponds to a longer time between the instants at which the light rays were emitted.

{{Cite web

|last=Chase |first=I. P.

|title=Apparent Superluminal Velocity of Galaxies

|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/Superluminal/superluminal.html

|publisher=University of California, Riverside

|work=Usenet Physics FAQ

|access-date=26 November 2009

}}

A 2011 experiment where neutrinos were observed to travel faster than light turned out to be due to experimental error.{{Cite journal |title=Embattled neutrino project leaders step down |journal=Nature News |first=Eugenie Samuel |last=Reich |date=2 April 2012 |access-date=11 February 2022 |doi=10.1038/nature.2012.10371 |s2cid=211730430 |url=http://www.nature.com/news/embattled-neutrino-project-leaders-step-down-1.10371}}{{Cite journal |author=OPERA Collaboration |author-link=OPERA experiment |title=Measurement of the neutrino velocity with the OPERA detector in the CNGS beam |date=12 July 2012 |arxiv=1109.4897 |doi=10.1007/JHEP10(2012)093 |volume=2012 |issue=10 |page=93 |journal=Journal of High Energy Physics |bibcode=2012JHEP...10..093A |s2cid=17652398 }}

In models of the expanding universe, the farther galaxies are from each other, the faster they drift apart. For example, galaxies far away from Earth are inferred to be moving away from the Earth with speeds proportional to their distances. Beyond a boundary called the Hubble sphere, the rate at which their distance from Earth increases becomes greater than the speed of light.

{{Cite book

|last= Harrison |first=E. R.

|year=2003

|title=Masks of the Universe

|url=https://books.google.com/books?id=tSowGCP0kMIC&pg=PA206

|page=206

|publisher=Cambridge University Press

|isbn=978-0-521-77351-5

}}

These recession rates, defined as the increase in proper distance per cosmological time, are not velocities in a relativistic sense. Faster-than-light cosmological recession speeds are only a coordinate artifact.

In classical physics, light is described as a type of electromagnetic wave. The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate in vacuum is related to the distributed capacitance and inductance of vacuum, otherwise respectively known as the electric constant ε₀ and the magnetic constant μ₀, by the equation{{Cite book

|last1=Panofsky |first1=W. K. H.

|last2=Phillips |first2=M.

|year=1962

|title=Classical Electricity and Magnetism

|url=https://archive.org/details/classicalelectri00pano_563 |url-access=limited |publisher=Addison-Wesley

|page=[https://archive.org/details/classicalelectri00pano_563/page/n192 182]

|isbn=978-0-201-05702-7

}}

: $c\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash varepsilon\_0\; \backslash mu\_0\}\}.$

In modern quantum physics, the electromagnetic field is described by the theory of quantum electrodynamics (QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, called photons. In QED, photons are massless particles and thus, according to special relativity, they travel at the speed of light in vacuum.

Extensions of QED in which the photon has a mass have been considered. In such a theory, its speed would depend on its frequency, and the invariant speed c of special relativity would then be the upper limit of the speed of light in vacuum.

{{Cite web

|last=Gibbs

|first=P.

|year=1997

|orig-year=1996

|title=Is The Speed of Light Constant?

|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

|editor-last=Carlip

|editor-first=S.

|work=Usenet Physics FAQ

|publisher=University of California, Riverside

|access-date=26 November 2009

|archive-url=https://web.archive.org/web/20100402090332/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

|archive-date=2 April 2010

|url-status=dead

}} No variation of the speed of light with frequency has been observed in rigorous testing, putting stringent limits on the mass of the photon.See, for example:

• {{Cite journal

|last=Schaefer |first=B. E.

|year=1999

|title=Severe limits on variations of the speed of light with frequency

|journal=Physical Review Letters

|volume=82

|issue=25 |pages=4964–4966

|doi=10.1103/PhysRevLett.82.4964

|arxiv=astro-ph/9810479

|bibcode=1999PhRvL..82.4964S

|s2cid=119339066

}}

• {{Cite journal

|last1=Ellis |first1=J.

|last2=Mavromatos |first2=N. E. |author-link2=N. E. Mavromatos

|last3=Nanopoulos |first3=D. V.

|last4=Sakharov |first4=A. S.

|year=2003

|title=Quantum-Gravity Analysis of Gamma-Ray Bursts using Wavelets

|journal=Astronomy & Astrophysics

|volume=402

|issue=2 |pages=409–424

|doi=10.1051/0004-6361:20030263

|arxiv=astro-ph/0210124 |bibcode=2003A&A...402..409E

|s2cid=15388873

}}

• {{Cite journal

|last=Füllekrug |first=M.

|year=2004

|title=Probing the Speed of Light with Radio Waves at Extremely Low Frequencies

|journal=Physical Review Letters

|volume=93

|issue=4 |page=043901

|doi=10.1103/PhysRevLett.93.043901

|bibcode=2004PhRvL..93d3901F

|pmid=15323762

}}

• {{Cite journal |last1=Bartlett |first1=D. J. |last2=Desmond |first2=H. |last3=Ferreira |first3=P. G. |last4=Jasche |first4=J. |date=17 November 2021 |title=Constraints on quantum gravity and the photon mass from gamma ray bursts |url=https://link.aps.org/doi/10.1103/PhysRevD.104.103516 |journal=Physical Review D |language=en |volume=104 |issue=10 |pages=103516 |arxiv=2109.07850 |bibcode=2021PhRvD.104j3516B |doi=10.1103/PhysRevD.104.103516 |s2cid=237532210 |issn=2470-0010}} The limit obtained depends on the model used: if the massive photon is described by Proca theory,

{{Cite journal

|last1=Adelberger |first1=E.

|last2=Dvali |first2=G.

|last3=Gruzinov |first3=A.

|year=2007

|title=Photon Mass Bound Destroyed by Vortices

|journal=Physical Review Letters

|volume=98

|issue=1 |page=010402

|doi=10.1103/PhysRevLett.98.010402

|arxiv=hep-ph/0306245 |pmid=17358459 |bibcode=2007PhRvL..98a0402A

|s2cid=31249827

}} the experimental upper bound for its mass is about 10⁻⁵⁷ grams;

{{Cite book

|last=Sidharth |first=B. G.

|year=2008

|title=The Thermodynamic Universe

|url=https://books.google.com/books?id=OUfHR36wSfAC&pg=PA134

|page=134

|publisher=World Scientific

|isbn=978-981-281-234-6

}} if photon mass is generated by a Higgs mechanism, the experimental upper limit is less sharp, {{nowrap|m ≤ {{val|e=-14|ul=eV/c2}}}}  (roughly 2 × 10⁻⁴⁷ g).

Another reason for the speed of light to vary with its frequency would be the failure of special relativity to apply to arbitrarily small scales, as predicted by some proposed theories of quantum gravity. In 2009, the observation of gamma-ray burst GRB 090510 found no evidence for a dependence of photon speed on energy, supporting tight constraints in specific models of spacetime quantization on how this speed is affected by photon energy for energies approaching the Planck scale.

{{Cite journal

|last=Amelino-Camelia |first=G.

|year=2009

|title=Astrophysics: Burst of support for relativity

|journal=Nature

|volume=462 |pages=291–292

|doi=10.1038/462291a

|pmid=19924200

|issue=7271

|bibcode = 2009Natur.462..291A |s2cid=205051022

|doi-access=free

}}

{{See also|Refractive index}}

In a medium, light usually does not propagate at a speed equal to c; further, different types of light wave will travel at different speeds. The speed at which the individual crests and troughs of a plane wave (a wave filling the whole space, with only one frequency) propagate is called the phase velocity v_p. A physical signal with a finite extent (a pulse of light) travels at a different speed. The overall envelope of the pulse travels at the group velocity v_g, and its earliest part travels at the front velocity v_f.{{Cite book|author=Milonni|first=Peter W.|url=https://books.google.com/books?id=kE8OUCvt7ecC&pg=PA26|title=Fast light, slow light and left-handed light|publisher=CRC Press|year=2004|isbn=978-0-7503-0926-4|pages=25 ff|authorlink1=Peter W. Milonni}}

File:frontgroupphase.gif

The phase velocity is important in determining how a light wave travels through a material or from one material to another. It is often represented in terms of a refractive index. The refractive index of a material is defined as the ratio of c to the phase velocity v_p in the material: larger indices of refraction indicate lower speeds. The refractive index of a material may depend on the light's frequency, intensity, polarization, or direction of propagation; in many cases, though, it can be treated as a material-dependent constant. The refractive index of air is approximately 1.0003.

{{Cite book

|last=de Podesta |first=M.

|year=2002

|title=Understanding the Properties of Matter

|url=https://books.google.com/books?id=h8BNvnR050cC&pg=PA131

|page=131

|publisher=CRC Press

|isbn=978-0-415-25788-6

}} Denser media, such as water,

{{Cite web

|title=Optical constants of H₂O, D₂O (Water, heavy water, ice)

|url=https://refractiveindex.info/?shelf=main&book=H2O&page=Hale

|publisher=Mikhail Polyanskiy

|work=refractiveindex.info

|access-date=7 November 2017

}} glass,

{{Cite web

|title=Optical constants of Soda lime glass

|url=https://refractiveindex.info/?shelf=glass&book=soda-lime&page=Rubin-clear

|publisher=Mikhail Polyanskiy

|work=refractiveindex.info

|access-date=7 November 2017

}} and diamond,

{{Cite web

|title=Optical constants of C (Carbon, diamond, graphite)

|url=https://refractiveindex.info/?shelf=main&book=C&page=Phillip

|publisher=Mikhail Polyanskiy

|work=refractiveindex.info

|access-date =7 November 2017

}} have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light.

In exotic materials like Bose–Einstein condensates near absolute zero, the effective speed of light may be only a few metres per second. However, this represents absorption and re-radiation delay between atoms, as do all slower-than-c speeds in material substances. As an extreme example of light "slowing" in matter, two independent teams of physicists claimed to bring light to a "complete standstill" by passing it through a Bose–Einstein condensate of the element rubidium. The popular description of light being "stopped" in these experiments refers only to light being stored in the excited states of atoms, then re-emitted at an arbitrarily later time, as stimulated by a second laser pulse. During the time it had "stopped", it had ceased to be light. This type of behaviour is generally microscopically true of all transparent media which "slow" the speed of light.{{Cite web |last=Cromie |first=William J. |url=http://www.news.harvard.edu/gazette/2001/01.24/01-stoplight.html |title=Researchers now able to stop, restart light |website=Harvard University Gazette |date=24 January 2001 |access-date=8 November 2011 |url-status=dead |archive-url=https://web.archive.org/web/20111028041346/http://www.news.harvard.edu/gazette/2001/01.24/01-stoplight.html |archive-date=28 October 2011 }}

In transparent materials, the refractive index generally is greater than 1, meaning that the phase velocity is less than c. In other materials, it is possible for the refractive index to become smaller than{{nbsp}}1 for some frequencies; in some exotic materials it is even possible for the index of refraction to become negative.

{{Cite book

|title=Fast light, slow light and left-handed light

|last=Milonni |first=P. W.

|author-link1=Peter W. Milonni

|url=https://books.google.com/books?id=kE8OUCvt7ecC&pg=PA25

|page=25

|isbn=978-0-7503-0926-4

|year=2004

|publisher=CRC Press

}} The requirement that causality is not violated implies that the real and imaginary parts of the dielectric constant of any material, corresponding respectively to the index of refraction and to the attenuation coefficient, are linked by the Kramers–Kronig relations.

{{Cite journal

|last=Toll |first=J. S.

|year=1956

|title=Causality and the Dispersion Relation: Logical Foundations

|journal=Physical Review

|volume=104

|issue=6 |pages=1760–1770

|doi=10.1103/PhysRev.104.1760

|bibcode = 1956PhRv..104.1760T

}}{{Cite book|last=Wolf|first=Emil|url=https://www.worldcat.org/oclc/261134839|title=Selected Works of Emil Wolf: with commentary|date=2001|publisher=World Scientific|isbn=978-981-281-187-5|location=River Edge, N.J.|pages=577–584|chapter=Analyticity, Causality and Dispersion Relations|oclc=261134839|author-link=Emil Wolf}} In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly.{{Cite journal |last1=Libbrecht |first1=K. G. |last2=Libbrecht |first2=M. W. |date=December 2006 |title=Interferometric measurement of the resonant absorption and refractive index in rubidium gas |url=https://authors.library.caltech.edu/12639/1/LIBajp06.pdf |journal=American Journal of Physics |language=en |volume=74 |issue=12 |pages=1055–1060 |doi=10.1119/1.2335476 |bibcode=2006AmJPh..74.1055L |issn=0002-9505}}

A pulse with different group and phase velocities (which occurs if the phase velocity is not the same for all the frequencies of the pulse) smears out over time, a process known as dispersion. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called slow light.See, for example:

• {{Cite journal

|last1=Hau |first1=L. V. |author-link1=Lene Hau

|last2=Harris |first2=S. E. |author-link2=Stephen E. Harris

|last3=Dutton |first3=Z. |author-link3=Zachary Dutton

|last4=Behroozi |first4=C. H.

|year=1999

|title=Light speed reduction to 17 metres per second in an ultracold atomic gas

|journal=Nature

|volume=397

|issue=6720 |pages=594–598

|doi=10.1038/17561

|bibcode = 1999Natur.397..594V |s2cid=4423307

|url=http://www.seas.harvard.edu/haulab/publications/pdf/Slow_Light_1999.pdf

}}

• {{Cite journal

|last1=Liu |first1=C. |last2=Dutton |first2=Z. |author-link2=Zachary Dutton |last3=Behroozi |first3=C. H. |last4=Hau |first4=L. V. |author-link4=Lene Hau

|year=2001

|title=Observation of coherent optical information storage in an atomic medium using halted light pulses

|journal=Nature

|volume=409 |issue=6819 |pages=490–493

|doi=10.1038/35054017

|pmid=11206540

|bibcode = 2001Natur.409..490L |s2cid=1894748 |url=http://www.seas.harvard.edu/haulab/publications/pdf/Stopped_Light_2001.pdf

}}

• {{Cite journal

|last1=Bajcsy |first1=M. |last2=Zibrov |first2=A. S. |last3=Lukin |first3=M. D.

|year=2003

|title=Stationary pulses of light in an atomic medium

|journal=Nature

|volume=426 |issue=6967 |pages=638–641

|doi=10.1038/nature02176

|pmid=14668857

|arxiv = quant-ph/0311092 |bibcode = 2003Natur.426..638B |s2cid=4320280

}}

• {{Cite web

|last=Dumé

|first=B.

|year=2003

|title=Switching light on and off

|url=http://physicsworld.com/cws/article/news/18724

|work=Physics World

|publisher=Institute of Physics

|access-date=8 December 2008

|archive-date=5 December 2008

|archive-url=https://web.archive.org/web/20081205051203/http://physicsworld.com/cws/article/news/18724

|url-status=dead

}}

The opposite, group velocities exceeding c, was proposed theoretically in 1993 and achieved experimentally in 2000.See, for example:

• {{Cite journal

|first=R. Y.

|last=Chiao |author-link=Raymond Chiao

|title=Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations

|journal=Physical Review A

|volume=48

|year=1993

|issue=1 |pages=R34–R37 |doi=10.1103/PhysRevA.48.R34

|pmid=9909684 |bibcode=1993PhRvA..48...34C

}}

• {{Cite journal

|first1=L. J. |last1=Wang

|first2=A. |last2=Kuzmich

|first3=A. |last3=Dogariu

|title=Gain-assisted superluminal light propagation

|journal=Nature

|volume=406

|pages=277–279

|year=2000

|issue=6793

|url=https://www.nature.com/articles/35018520

|doi=10.1038/35018520 |pmid=10917523

|bibcode=2000Natur.406..277W

|s2cid=4358601

}}

• {{Cite news

|last=Whitehouse |first=D.

|date=19 July 2000

|title=Beam Smashes Light Barrier

|url=http://news.bbc.co.uk/2/hi/science/nature/841690.stm

|work=BBC News

|access-date=9 February 2022

}}

• {{Cite web

|first=Greg |last=Gbur |author-link=Greg Gbur

|title=Light breaking its own speed limit: how 'superluminal' shenanigans work

|url=https://skullsinthestars.com/2008/02/26/light-breaking-its-own-speed-limit-how-superluminal-shenanigans-work/

|date=26 February 2008

|access-date=9 February 2022

}} It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.

None of these options allow information to be transmitted faster than c. It is impossible to transmit information with a light pulse any faster than the speed of the earliest part of the pulse (the front velocity). It can be shown that this is (under certain assumptions) always equal to c. {{Clear}}

It is possible for a particle to travel through a medium faster than the phase velocity of light in that medium (but still slower than c). When a charged particle does that in a dielectric material, the electromagnetic equivalent of a shock wave, known as Cherenkov radiation, is emitted.{{Cite journal |last=Cherenkov |first=Pavel A. |author-link=Pavel Alekseyevich Cherenkov |year=1934 |title=Видимое свечение чистых жидкостей под действием γ-радиации |trans-title=Visible emission of pure liquids by action of γ radiation |journal=Doklady Akademii Nauk SSSR |volume=2 |page=451}} Reprinted: {{Cite journal |last=Cherenkov |first=P. A. |date=1967 |title=Видимое свечение чистых жидкостей под действием γ-радиации |trans-title=Visible emission of pure liquids by action of γ radiation |journal=Usp. Fiz. Nauk |volume=93 |issue=10 |page=385 |doi=10.3367/ufnr.0093.196710n.0385}}, and in {{Cite book |title=Pavel Alekseyevich Čerenkov: Chelovek i Otkrytie |trans-title=Pavel Alekseyevich Čerenkov: Man and Discovery |editor1=A. N. Gorbunov |editor2=E. P. Čerenkova |location=Moscow |publisher=Nauka |date=1999 |pages=149–153}}

The speed of light is of relevance to telecommunications: the one-way and round-trip delay time are greater than zero. This applies from small to astronomical scales. On the other hand, some techniques depend on the finite speed of light, for example in distance measurements.

In computers, the speed of light imposes a limit on how quickly data can be sent between processors. If a processor operates at 1{{nbsp}}gigahertz, a signal can travel only a maximum of about {{convert|30|cm|ft|0}} in a single clock cycle – in practice, this distance is even shorter since the printed circuit board refracts and slows down signals. Processors must therefore be placed close to each other, as well as memory chips, to minimize communication latencies, and care must be exercised when routing wires between them to ensure signal integrity. If clock frequencies continue to increase, the speed of light may eventually become a limiting factor for the internal design of single chips.

{{Cite book

|last=Parhami |first=B.

|year=1999

|title=Introduction to parallel processing: algorithms and architectures

|url=https://books.google.com/books?id=ekBsZkIYfUgC

|page=5

|publisher=Plenum Press

|isbn=978-0-306-45970-2

}}

{{Cite conference

|url=https://books.google.com/books?id=sona_r6dPyQC&q=%22speed+of+light%22+processor+limit&pg=PA26

|title=Software Transactional Memories: An Approach for Multicore Programming

|first1=D. |last1=Imbs

|first2=Michel |last2=Raynal

|year=2009

|conference=10th International Conference, PaCT 2009, Novosibirsk, Russia, 31 August – 4 September 2009

|editor=Malyshkin, V.

|publisher=Springer

|isbn=978-3-642-03274-5

|page=26

}}

File:Light world trip.ogg

Given that the equatorial circumference of the Earth is about {{val|40075|u=km}} and that c is about {{val|300000|u=km/s}}, the theoretical shortest time for a piece of information to travel half the globe along the surface is about 67 milliseconds. When light is traveling in optical fibre (a transparent material) the actual transit time is longer, in part because the speed of light is slower by about 35% in optical fibre with an refractive index n around 1.52.

{{Cite book

| last = Midwinter |first=J. E.

| year = 1991

| title = Optical Fibers for Transmission

| edition = 2

| publisher = Krieger

| isbn = 978-0-89464-595-2

}} Straight lines are rare in global communications and the travel time increases when signals pass through electronic switches or signal regenerators.

{{Cite web

|date=June 2007

|title=Theoretical vs real-world speed limit of Ping

|url=http://royal.pingdom.com/2007/06/01/theoretical-vs-real-world-speed-limit-of-ping/

|website=Pingdom

|access-date=5 May 2010

|archive-date=2 September 2010

|archive-url=https://web.archive.org/web/20100902224536/http://royal.pingdom.com/2007/06/01/theoretical-vs-real-world-speed-limit-of-ping/

|url-status=dead

}}

Although this distance is largely irrelevant for most applications, latency becomes important in fields such as high-frequency trading, where traders seek to gain minute advantages by delivering their trades to exchanges fractions of a second ahead of other traders. For example, traders have been switching to microwave communications between trading hubs, because of the advantage which radio waves travelling at near to the speed of light through air have over comparatively slower fibre optic signals.{{Cite journal |last1=Buchanan |first1=Mark |date=11 February 2015 |title=Physics in finance: Trading at the speed of light |journal=Nature |volume=518 |issue=7538 |pages=161–163 |bibcode=2015Natur.518..161B |doi=10.1038/518161a |pmid=25673397 |doi-access=free}}{{Cite news |date=10 May 2013 |title=Time is money when it comes to microwaves |newspaper=Financial Times |url=http://www.ft.com/cms/s/2/2bf37898-b775-11e2-841e-00144feabdc0.html |archive-url=https://ghostarchive.org/archive/20221210211258/https://www.ft.com/content/2bf37898-b775-11e2-841e-00144feabdc0 |archive-date=10 December 2022 |url-access=subscription |access-date=25 April 2014 |url-status=live }}

File:Earth and Moon speed of light by James O'Donoghue.gif

Similarly, communications between the Earth and spacecraft are not instantaneous. There is a brief delay from the source to the receiver, which becomes more noticeable as distances increase. This delay was significant for communications between ground control and Apollo 8 when it became the first crewed spacecraft to orbit the Moon: for every question, the ground control station had to wait at least three seconds for the answer to arrive.

{{Cite web

|url = https://history.nasa.gov/ap08fj/15day4_orbits789.htm

|title = Day 4: Lunar Orbits 7, 8 and 9

|work = The Apollo 8 Flight Journal

|publisher = NASA

|access-date = 16 December 2010

|url-status = dead

|archive-url = https://web.archive.org/web/20110104032114/http://history.nasa.gov/ap08fj/15day4_orbits789.htm

|archive-date = 4 January 2011

}}

The communications delay between Earth and Mars can vary between five and twenty minutes depending upon the relative positions of the two planets. As a consequence of this, if a robot on the surface of Mars were to encounter a problem, its human controllers would not be aware of it until approximately {{nowrap|4–24 minutes}} later. It would then take a further {{nowrap|4–24 minutes}} for commands to travel from Earth to Mars.{{Cite web |last=Ormston |first=Thomas |date=2012-05-08 |title=Time delay between Mars and Earth – Mars Express |url=https://blogs.esa.int/mex/2012/08/05/time-delay-between-mars-and-earth/ |access-date=2024-07-16 |website=MARS EXPRESS ESA’s mission to the Red Planet |language=en-US}}{{Cite journal |last1=Parisi |first1=Megan |last2=Panontin |first2=Tina |last3=Wu |first3=Shu-Chieh |last4=Mctigue |first4=Kaitlin |last5=Vera |first5=Alonso |date=2023 |title=Effects of Communication Delay on Human Spaceflight Missions |url=https://openaccess.cms-conferences.org/publications/book/978-1-958651-74-2/article/978-1-958651-74-2_6 |journal=Human-Centered Aerospace Systems and Sustainability Applications |publisher=AHFE Open Acces |volume=98 |doi=10.54941/ahfe1003920 |isbn=978-1-958651-74-2}}

Receiving light and other signals from distant astronomical sources takes much longer. For example, it takes 13 billion (13{{e|9}}) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra-Deep Field images.

{{Cite press release

|date=5 January 2010

|title=Hubble Reaches the "Undiscovered Country" of Primeval Galaxies

|url=https://www.nasa.gov/mission_pages/hubble/science/undiscovered-country.html

|publisher=Space Telescope Science Institute

}}

{{Cite web

|title=The Hubble Ultra Deep Field Lithograph

|url=http://www.nasa.gov/pdf/283957main_Hubble_Deep_Field_Lithograph.pdf

|publisher=NASA

|access-date=4 February 2010

}} Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when the universe was less than a billion years old. The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars, of galaxies, and of the universe itself.{{Cite book|last=Mack|first=Katie|url=https://www.worldcat.org/oclc/1180972461|title=The End of Everything (Astrophysically Speaking)|date=2021|publisher=Penguin Books|isbn=978-0-141-98958-7|location=London|pages=18–19|oclc=1180972461|author-link=Katie Mack (astrophysicist)}}

Astronomical distances are sometimes expressed in light-years, especially in popular science publications and media.{{Cite web

|title=The IAU and astronomical units

|url=http://www.iau.org/public/measuring/

|publisher=International Astronomical Union

|access-date=11 October 2010

|archive-date=5 June 2013

|archive-url=https://web.archive.org/web/20130605024231/http://www.iau.org/public/measuring/

|url-status=dead

}} A light-year is the distance light travels in one Julian year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs. In round figures, a light year is nearly 10 trillion kilometres or nearly 6 trillion miles. Proxima Centauri, the closest star to Earth after the Sun, is around 4.2 light-years away.Further discussion can be found at

{{Cite web

|year=2000

|title=StarChild Question of the Month for March 2000

|url=http://starchild.gsfc.nasa.gov/docs/StarChild/questions/question19.html

|work=StarChild

|publisher=NASA

|access-date=22 August 2009

}}

{{Main|Distance measurement}}

Radar systems measure the distance to a target by the time it takes a radio-wave pulse to return to the radar antenna after being reflected by the target: the distance to the target is half the round-trip transit time multiplied by the speed of light. A Global Positioning System (GPS) receiver measures its distance to GPS satellites based on how long it takes for a radio signal to arrive from each satellite, and from these distances calculates the receiver's position. Because light travels about {{val|300000|u=kilometres}} ({{val|186000|u=miles}}) in one second, these measurements of small fractions of a second must be very precise. The Lunar Laser Ranging experiment, radar astronomy and the Deep Space Network determine distances to the Moon,

{{Cite journal

|last=Dickey |first=J. O.

|title=Lunar Laser Ranging: A Continuing Legacy of the Apollo Program

|journal=Science | volume=265 | issue=5171

|pages=482–490 |date=July 1994

|doi=10.1126/science.265.5171.482

|bibcode=1994Sci...265..482D | pmid=17781305|s2cid=10157934

|display-authors=etal|url=https://trs.jpl.nasa.gov/bitstream/2014/32452/1/94-0193.pdf}} planets{{Cite journal

|last=Standish |first=E. M.

|title=The JPL planetary ephemerides

|journal=Celestial Mechanics |volume=26 |date=February 1982

|issue=2 |pages=181–186 |doi=10.1007/BF01230883

|bibcode=1982CeMec..26..181S |s2cid=121966516

}} and spacecraft,

{{Cite journal

|last1=Berner |first1=J. B.

|last2=Bryant |first2=S. H.

|last3=Kinman |first3=P. W.

|title=Range Measurement as Practiced in the Deep Space Network

|journal=Proceedings of the IEEE |date=November 2007 |volume=95 |issue=11 |pages=2202–2214

|doi=10.1109/JPROC.2007.905128 |s2cid=12149700

|url=https://trs.jpl.nasa.gov/bitstream/2014/40972/1/07-0166.pdf}}

respectively, by measuring round-trip transit times.

There are different ways to determine the value of c. One way is to measure the actual speed at which light waves propagate, which can be done in various astronomical and Earth-based setups. It is also possible to determine c from other physical laws where it appears, for example, by determining the values of the electromagnetic constants ε₀ and μ₀ and using their relation to c. Historically, the most accurate results have been obtained by separately determining the frequency and wavelength of a light beam, with their product equalling c. This is described in more detail in the "Interferometry" section below.

In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of {{frac|1|{{val|299792458}}}} of a second", fixing the value of the speed of light at {{val|299792458|u=m/s}} by definition, as described below. Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value of c.

File:Io eclipse speed of light measurement.svg

Outer space is a convenient setting for measuring the speed of light because of its large scale and nearly perfect vacuum. Typically, one measures the time needed for light to traverse some reference distance in the Solar System, such as the radius of the Earth's orbit. Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units.

Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676.

{{Cite journal

|last=Cohen |first=I. B. |author-link=I. Bernard Cohen

|year=1940

|title=Roemer and the first determination of the velocity of light (1676)

|journal=Isis

|volume=31 |issue=2 |pages=327–379

|doi=10.1086/347594

|ref=cohen-1940

|hdl=2027/uc1.b4375710

|s2cid=145428377

|url=https://babel.hathitrust.org/cgi/imgsrv/download/pdf?id=uc1.b4375710;orient=0;size=100;seq=5;attachment=0

|hdl-access=free

}}

{{Cite journal

|year=1676

|title=Demonstration tovchant le mouvement de la lumiere trouvé par M. Rŏmer de l'Académie Royale des Sciences

|trans-title=Demonstration to the movement of light found by Mr. Römer of the Royal Academy of Sciences

|language=fr

|url=http://www-obs.univ-lyon1.fr/labo/fc/ama09/pages_jdsc/pages/jdsc_1676_lumiere.pdf

|journal=Journal des sçavans

|pages=233–236

|ref=roemer-1676

}}
Translated in

{{Cite journal

|doi=10.1098/rstl.1677.0024

|year=1677

|title=A demonstration concerning the motion of light, communicated from Paris, in the Journal des Sçavans, and here made English

|journal=Philosophical Transactions of the Royal Society

|volume=12 |issue=136 |pages=893–895

|ref=roemer-1676-EnglishTrans

|bibcode=1677RSPT...12..893.|doi-access=free

}}
Reproduced in

{{Cite book

|editor1-last=Hutton |editor1-first=C.

|editor2-last=Shaw |editor2-first=G.

|editor3-last=Pearson |editor3-first=R.

|year=1809

|title=The Philosophical Transactions of the Royal Society of London, from Their Commencement in 1665, in the Year 1800: Abridged

|chapter=On the Motion of Light by M. Romer

|chapter-url=https://archive.org/stream/philosophicaltra02royarich#page/397/mode/1up

|location=London |publisher=C. & R. Baldwin

|volume=II. From 1673 to 1682 |pages=397–398

}}

The account published in {{lang|fr|Journal des sçavans}} was based on a report that Rømer read to the French Academy of Sciences in November 1676 (Cohen, 1940, p. 346). When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it. The difference is small, but the cumulative time becomes significant when measured over months. The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the diameter of the Earth's orbit around the Sun. The observed change in the moon's orbital period is caused by the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect for Jupiter's innermost major moon Io and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.

File:SoL Aberration.svg

Another method is to use the aberration of light, discovered and explained by James Bradley in the 18th century.

{{Cite journal

|last=Bradley |first=J.

|year=1729

|title=Account of a new discovered Motion of the Fix'd Stars

|url=http://gallica.bnf.fr/ark:/12148/bpt6k55840n.image.f375.langEN

|journal=Philosophical Transactions

|volume=35 |pages=637–660

}} This effect results from the vector addition of the velocity of light arriving from a distant source (such as a star) and the velocity of its observer (see diagram on the right). A moving observer thus sees the light coming from a slightly different direction and consequently sees the source at a position shifted from its original position. Since the direction of the Earth's velocity changes continuously as the Earth orbits the Sun, this effect causes the apparent position of stars to move around. From the angular difference in the position of stars (maximally 20.5 arcseconds)

{{Cite book

|last=Duffett-Smith

|first=P.

|year=1988

|title=Practical Astronomy with your Calculator

|url=https://archive.org/details/practicalastrono0000duff

|url-access=registration

|page=[https://archive.org/details/practicalastrono0000duff/page/62 62]

|publisher=Cambridge University Press

|isbn=978-0-521-35699-2

}} [https://archive.org/details/practicalastrono0000duff/page/62 Extract of page 62]. it is possible to express the speed of light in terms of the Earth's velocity around the Sun, which with the known length of a year can be converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled {{val|10,210}} times faster than the Earth in its orbit (the modern figure is {{val|10,066}} times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.

{{main|Astronomical unit}}

Historically the speed of light was used together with timing measurements to determine a value for the astronomical unit (AU).{{Cite journal |last=Standish |first=E. M. |date=June 2004 |title=The Astronomical Unit now |url=https://www.cambridge.org/core/product/identifier/S1743921305001365/type/journal_article |journal=Proceedings of the International Astronomical Union |language=en |volume=2004 |issue=IAUC196 |pages=163–179 |doi=10.1017/S1743921305001365 |issn=1743-9213}} It was redefined in 2012 as exactly {{val|149597870700|u=m}}.{{Cite journal|journal=The International System of Units|title=Supplement 2014: Updates to the 8th edition (2006) of the SI Brochure|url=http://www.bipm.org/utils/common/pdf/si_supplement_2014.pdf|year=2014|publisher= International Bureau of Weights and Measures|page=14}}{{Cite web|title=Resolution B2 on the re-definition of the astronomical unit of length|url=https://www.iau.org/static/resolutions/IAU2012_English.pdf|year=2012|publisher=International Astronomical Union}} This redefinition is analogous to that of the metre and likewise has the effect of fixing the speed of light to an exact value in astronomical units per second (via the exact speed of light in metres per second).{{Cite journal|last=Brumfiel|first=Geoff|date=14 September 2012|title=The astronomical unit gets fixed|url=https://www.nature.com/articles/nature.2012.11416|journal=Nature|language=en|doi=10.1038/nature.2012.11416|s2cid=123424704|issn=1476-4687}}

File:Michelson speed of light measurement 1930.jpg

File:Fizeau-int.svg:{{image key|list type=ordered

|Light source

|Beam-splitting semi-transparent mirror

|Toothed wheel-breaker of the light beam

|Remote mirror

|Telescopic tube}}|alt=A light ray passes horizontally through a half-mirror and a rotating cog wheel, is reflected back by a mirror, passes through the cog wheel, and is reflected by the half-mirror into a monocular.]]

A method of measuring the speed of light is to measure the time needed for light to travel to a mirror at a known distance and back. This is the working principle behind experiments by Hippolyte Fizeau and Léon Foucault.

The setup as used by Fizeau consists of a beam of light directed at a mirror {{convert|8|km|mi|0}} away. On the way from the source to the mirror, the beam passes through a rotating cogwheel. At a certain rate of rotation, the beam passes through one gap on the way out and another on the way back, but at slightly higher or lower rates, the beam strikes a tooth and does not pass through the wheel. Knowing the distance between the wheel and the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light can be calculated.

{{Cite web

|last=Gibbs

|first=P.

|year=1997

|title=How is the speed of light measured?

|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html

|work=Usenet Physics FAQ

|publisher=University of California, Riverside

|access-date=13 January 2010

|url-status=dead

|archive-url=https://web.archive.org/web/20150821181850/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html

|archive-date=21 August 2015

}}

The method of Foucault replaces the cogwheel with a rotating mirror. Because the mirror keeps rotating while the light travels to the distant mirror and back, the light is reflected from the rotating mirror at a different angle on its way out than it is on its way back. From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated.

{{Cite web

|last=Fowler |first=M.

|title=The Speed of Light

|url=http://galileoandeinstein.physics.virginia.edu/lectures/spedlite.html

|publisher=University of Virginia

|access-date=21 April 2010

}} Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion by François Arago.{{Cite book|last1=Hughes|first1=Stephan|url=https://books.google.com/books?id=iZk5OOf7fVYC|title=Catchers of the Light: The Forgotten Lives of the Men and Women Who First Photographed the Heavens|date=2012|publisher=ArtDeCiel Publishing|isbn=978-1-62050-961-6|pages=210}}

Today, using oscilloscopes with time resolutions of less than one nanosecond, the speed of light can be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror. This method is less precise (with errors of the order of 1%) than other modern techniques, but it is sometimes used as a laboratory experiment in college physics classes.See, for example:

• {{Cite journal

|last1=Cooke |first1=J.

|last2=Martin |first2=M.

|last3=McCartney |first3=H.

|last4=Wilf |first4=B.

|year=1968

|title=Direct determination of the speed of light as a general physics laboratory experiment

|journal=American Journal of Physics

|volume=36 |issue=9 |page=847

|doi=10.1119/1.1975166

|bibcode = 1968AmJPh..36..847C

}}

• {{Cite journal

|last1=Aoki |first1=K. |last2=Mitsui |first2=T.

|year=2008

|title=A small tabletop experiment for a direct measurement of the speed of light

|journal=American Journal of Physics

|volume=76 |issue=9 |pages=812–815

|doi=10.1119/1.2919743

|arxiv=0705.3996

|bibcode = 2008AmJPh..76..812A |s2cid=117454437

}}

• {{Cite journal

|last1=James |first1=M. B. |last2=Ormond |first2=R. B. |last3=Stasch |first3=A. J.

|year=1999

|title=Speed of light measurement for the myriad

|journal=American Journal of Physics

|volume=67 |issue=8 |pages=681–714

|doi=10.1119/1.19352

|bibcode = 1999AmJPh..67..681J

}}

An option for deriving c that does not directly depend on a measurement of the propagation of electromagnetic waves is to use the relation between c and the vacuum permittivity ε₀ and vacuum permeability μ₀ established by Maxwell's theory: c² = 1/(ε₀μ₀). The vacuum permittivity may be determined by measuring the capacitance and dimensions of a capacitor, whereas the value of the vacuum permeability was historically fixed at exactly {{val|4|end=π|e=-7|u=H.m-1}} through the definition of the ampere. Rosa and Dorsey used this method in 1907 to find a value of {{val|299710|22|u=km/s}}. Their method depended upon having a standard unit of electrical resistance, the "international ohm", and so its accuracy was limited by how this standard was defined.{{Cite journal |last1=Rosa |first1=E. B. |author-link=Edward Bennett Rosa |last2=Dorsey |first2=N. E. |author-link2=Noah Ernest Dorsey |year=1907 |title=A new determination of the ratio of the electromagnetic to the electrostatic unit of electricity |journal=Bulletin of the Bureau of Standards |volume=3 |issue=6 |page=433 |doi=10.6028/bulletin.070 |doi-access=free}}

File:Waves in Box.svg in a cavity|alt=A box with three waves in it; there are one and a half wavelength of the top wave, one of the middle one, and a half of the bottom one.]]

Another way to measure the speed of light is to independently measure the frequency f and wavelength λ of an electromagnetic wave in vacuum. The value of c can then be found by using the relation c = fλ. One option is to measure the resonance frequency of a cavity resonator. If the dimensions of the resonance cavity are also known, these can be used to determine the wavelength of the wave. In 1946, Louis Essen and A.C. Gordon-Smith established the frequency for a variety of normal modes of microwaves of a microwave cavity of precisely known dimensions. The dimensions were established to an accuracy of about ±0.8 μm using gauges calibrated by interferometry. As the wavelength of the modes was known from the geometry of the cavity and from electromagnetic theory, knowledge of the associated frequencies enabled a calculation of the speed of light.

{{Cite journal

|last1=Essen |first1=L.

|last2=Gordon-Smith |first2=A. C.

|year=1948

|title=The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator

|journal=Proceedings of the Royal Society of London A

|volume=194 |issue=1038 |pages=348–361

|doi=10.1098/rspa.1948.0085

|bibcode=1948RSPSA.194..348E

|jstor=98293

|doi-access=free

}}

{{Cite journal

|last=Essen |first=L.

|year=1947

|title=Velocity of Electromagnetic Waves

|journal=Nature

|volume=159 |issue=4044 |pages=611–612

|doi=10.1038/159611a0

|bibcode=1947Natur.159..611E

|s2cid=4101717

}}

The Essen–Gordon-Smith result, {{val|299792|9|u=km/s}}, was substantially more precise than those found by optical techniques. By 1950, repeated measurements by Essen established a result of {{val|299792.5|3.0|u=km/s}}.

{{Cite journal

|last=Essen |first=L.

|year=1950

|title=The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator

|journal=Proceedings of the Royal Society of London A

|volume=204 |issue=1077 |pages=260–277

|doi=10.1098/rspa.1950.0172

|bibcode=1950RSPSA.204..260E

|jstor=98433

|s2cid=121261770

}}

A household demonstration of this technique is possible, using a microwave oven and food such as marshmallows or margarine: if the turntable is removed so that the food does not move, it will cook the fastest at the antinodes (the points at which the wave amplitude is the greatest), where it will begin to melt. The distance between two such spots is half the wavelength of the microwaves; by measuring this distance and multiplying the wavelength by the microwave frequency (usually displayed on the back of the oven, typically 2450 MHz), the value of c can be calculated, "often with less than 5% error".

{{Cite journal

|last = Stauffer | first = R. H.

|date=April 1997

|title = Finding the Speed of Light with Marshmallows

|journal = The Physics Teacher

|volume = 35

|issue = 4

|page = 231

|url = https://www.physics.umd.edu/icpe/newsletters/n34/marshmal.htm

|access-date = 15 February 2010

|bibcode = 1997PhTea..35..231S |doi = 10.1119/1.2344657

}}

{{Cite web

|url =http://www.bbc.co.uk/norfolk/features/ba_festival/bafestival_speedoflight_experiment_feature.shtml

|title = BBC Look East at the speed of light

|work = BBC Norfolk website

|access-date = 15 February 2010

}}

File:Interferometer sol.svg; Right: destructive interference.|alt=Schematic of the working of a Michelson interferometer.]]

Interferometry is another method to find the wavelength of electromagnetic radiation for determining the speed of light.

{{Cite book

|last=Vaughan |first=J. M.

|year=1989

|title=The Fabry-Perot interferometer

|url=https://books.google.com/books?id=mMLuISueDKYC

|pages=47, 384–391

|publisher=CRC Press

|isbn=978-0-85274-138-2

}} A coherent beam of light (e.g. from a laser), with a known frequency (f), is split to follow two paths and then recombined. By adjusting the path length while observing the interference pattern and carefully measuring the change in path length, the wavelength of the light (λ) can be determined. The speed of light is then calculated using the equation c = λf.

Before the advent of laser technology, coherent radio sources were used for interferometry measurements of the speed of light.

{{Cite journal

|doi=10.1098/rspa.1958.0172

|title=A New Determination of the Free-Space Velocity of Electromagnetic Waves

|first=K. D.

|last=Froome

|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences

|volume=247

|year=1958

|pages=109–122

|issue=1248

|bibcode = 1958RSPSA.247..109F

|jstor=100591 |s2cid=121444888

}} Interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~{{cvt|4|mm|in}}) of the radiowaves. The precision can be improved by using light with a shorter wavelength, but then it becomes difficult to directly measure the frequency of the light.

One way around this problem is to start with a low frequency signal of which the frequency can be precisely measured, and from this signal progressively synthesize higher frequency signals whose frequency can then be linked to the original signal. A laser can then be locked to the frequency, and its wavelength can be determined using interferometry.

{{Cite book

|title = A Century of Excellence in Measurements, Standards, and Technology

|editor-last = Lide

|editor-first = D. R.

|contribution = Speed of Light from Direct Frequency and Wavelength Measurements

|last = Sullivan

|first = D. B.

|year = 2001

|pages = 191–193

|publisher = CRC Press

|isbn = 978-0-8493-1247-2

|url = http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf

|url-status = dead

|archive-url = https://web.archive.org/web/20090813061018/http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf

|archive-date = 13 August 2009

}} This technique was due to a group at the National Bureau of Standards (which later became the National Institute of Standards and Technology). They used it in 1972 to measure the speed of light in vacuum with a fractional uncertainty of {{val|3.5|e=-9}}.

{{Cite journal

|last1=Evenson |first1=K. M. |year=1972

|title=Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser

|journal=Physical Review Letters

|volume=29

|issue=19 |pages=1346–1349

|doi=10.1103/PhysRevLett.29.1346

|bibcode=1972PhRvL..29.1346E

|s2cid=120300510 |display-authors=etal

}}

Until the early modern period, it was not known whether light travelled instantaneously or at a very fast finite speed. The first extant recorded examination of this subject was in ancient Greece. The ancient Greeks, Arabic scholars, and classical European scientists long debated this until Rømer provided the first calculation of the speed of light. Einstein's theory of special relativity postulates that the speed of light is constant regardless of one's frame of reference. Since then, scientists have provided increasingly accurate measurements.

Empedocles (c. 490–430 BCE) was the first to propose a theory of light

{{Cite book

|title=Light-Matter Interaction: Physics and Engineering at the Nanoscale |edition=illustrated

|first1=John

|last1=Weiner

|first2=Frederico

|last2=Nunes

|publisher=OUP Oxford

|year=2013

|isbn=978-0-19-856766-0

|page=1

|url=https://books.google.com/books?id=ctpG-kmmK8kC

}} [https://books.google.com/books?id=ctpG-kmmK8kC&pg=PA1 Extract of page 1]. and claimed that light has a finite speed.

{{Cite book

|last=Sarton |first=G. |author-link=George Sarton

|year=1993

|title=Ancient science through the golden age of Greece

|url=https://books.google.com/books?id=VcoGIKlHuZcC&pg=PA248

|page=248

|publisher=Courier Dover

|isbn=978-0-486-27495-9

}} He maintained that light was something in motion, and therefore must take some time to travel. Aristotle argued, to the contrary, that "light is due to the presence of something, but it is not a movement".

{{Cite journal

|last1=MacKay |first1=R. H. |last2=Oldford |first2=R. W.

|year=2000

|title=Scientific Method, Statistical Method and the Speed of Light

|url=http://sas.uwaterloo.ca/~rwoldfor/papers/sci-method/paperrev/

|journal=Statistical Science

|volume=15 |issue=3 |pages=254–278

|doi=10.1214/ss/1009212817

|doi-access=free

}} (click on "Historical background" in the table of contents) Euclid and Ptolemy advanced Empedocles' emission theory of vision, where light is emitted from the eye, thus enabling sight. Based on that theory, Heron of Alexandria argued that the speed of light must be infinite because distant objects such as stars appear immediately upon opening the eyes.{{Cite book |title=Electronic Microwave Imaging with Planar Multistatic Arrays |first1=Sherif Sayed |last1=Ahmed |publisher=Logos Verlag Berlin |year=2014 |isbn=978-3-8325-3621-3 |page=1 |url=https://books.google.com/books?id=ob79AgAAQBAJ}} [https://books.google.com/books?id=ob79AgAAQBAJ&pg=PA1 Extract of page 1]

Early Islamic philosophers initially agreed with the Aristotelian view that light had no speed of travel. In 1021, Alhazen (Ibn al-Haytham) published the Book of Optics, in which he presented a series of arguments dismissing the emission theory of vision in favour of the now accepted intromission theory, in which light moves from an object into the eye.

{{Cite journal

| last1 = Gross | first1 = C. G.

| title = The Fire That Comes from the Eye

| journal = Neuroscientist

| volume = 5

| pages = 58–64

| year = 1999

| doi = 10.1177/107385849900500108

| s2cid = 84148912

}} This led Alhazen to propose that light must have a finite speed,

{{Cite journal

|last=Hamarneh |first=S.

|year=1972

|title=Review: Hakim Mohammed Said, Ibn al-Haitham

|journal=Isis

|volume=63 |issue=1 |page=119

|doi=10.1086/350861

}}

{{Cite book

|last=Lester |first=P. M.

|year=2005

|title=Visual Communication: Images With Messages

|pages=10–11

|publisher=Thomson Wadsworth

|isbn=978-0-534-63720-0

}} and that the speed of light is variable, decreasing in denser bodies.

{{Cite web

|first1=J. J.

|last1=O'Connor

|author-link1=John J. O'Connor (mathematician)

|first2=E. F.

|last2=Robertson

|author-link2=Edmund F. Robertson

|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Haytham.html

|title=Abu Ali al-Hasan ibn al-Haytham

|work=MacTutor History of Mathematics archive

|publisher=University of St Andrews

|access-date=12 January 2010

}} He argued that light is substantial matter, the propagation of which requires time, even if this is hidden from the senses.

{{Cite conference

|last = Lauginie

|first = P.

|year = 2004

|title = Measuring Speed of Light: Why? Speed of what?

|url = http://sci-ed.org/documents/Lauginie-M.pdf

|conference = Fifth International Conference for History of Science in Science Education

|location = Keszthely, Hungary

|pages = 75–84

|access-date = 12 August 2017

|archive-url = https://web.archive.org/web/20150704043700/http://sci-ed.org/documents/Lauginie-M.pdf

|archive-date = 4 July 2015

|url-status=dead

}} Also in the 11th century, Abū Rayhān al-Bīrūnī agreed that light has a finite speed, and observed that the speed of light is much faster than the speed of sound.

{{Cite web

|first1=J. J.

|last1=O'Connor

|first2=E. F.

|last2=Robertson

|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Biruni.html

|title=Abu han Muhammad ibn Ahmad al-Biruni

|work=MacTutor History of Mathematics archive

|publisher=University of St Andrews

|access-date=12 January 2010

}}

In the 13th century, Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle.

{{Cite book

|last=Lindberg |first=D. C.

|year=1996

|title=Roger Bacon and the origins of Perspectiva in the Middle Ages: a critical edition and English translation of Bacon's Perspectiva, with introduction and notes

|url=https://books.google.com/books?id=jSPHMKbjYkQC&pg=PA143

|page=143

|isbn=978-0-19-823992-5

|publisher=Oxford University Press

}}

{{Cite book

|last=Lindberg |first=D. C.

|year=1974

|chapter=Late Thirteenth-Century Synthesis in Optics

|editor=Edward Grant

|title=A source book in medieval science

|chapter-url=https://books.google.com/books?id=fAPN_3w4hAUC&q=roger-bacon%20speed-of-light&pg=RA1-PA395

|page=396

|publisher=Harvard University Press

|isbn=978-0-674-82360-0

}} In the 1270s, Witelo considered the possibility of light travelling at infinite speed in vacuum, but slowing down in denser bodies.

{{Cite journal

|last=Marshall |first=P.

|year=1981

|title=Nicole Oresme on the Nature, Reflection, and Speed of Light

|journal=Isis

|volume=72 |issue=3 |pages=357–374 [367–374]

|doi=10.1086/352787

|s2cid=144035661

}}

In the early 17th century, Johannes Kepler believed that the speed of light was infinite since empty space presents no obstacle to it. René Descartes argued that if the speed of light were to be finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. Although this argument fails when aberration of light is taken into account, the latter was not recognized until the following century.{{Cite journal |last=Sakellariadis |first=Spyros |date=1982 |title=Descartes' Experimental Proof of the Infinite Velocity of Light and Huygens' Rejoinder |url=https://www.jstor.org/stable/41133639 |journal=Archive for History of Exact Sciences |volume=26 |issue=1 |pages=1–12 |doi=10.1007/BF00348308 |jstor=41133639 |s2cid=118187860 |issn=0003-9519}} Since such misalignment had not been observed, Descartes concluded the speed of light was infinite. Descartes speculated that if the speed of light were found to be finite, his whole system of philosophy might be demolished. Despite this, in his derivation of Snell's law, Descartes assumed that some kind of motion associated with light was faster in denser media.{{Cite book |last=Cajori |first=Florian |url=https://books.google.com/books?id=XNtUAAAAYAAJ |title=A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories |date=1922 |publisher=Macmillan |pages=76 |language=en}}{{Cite journal |last=Smith |first=A. Mark |date=1987 |title=Descartes's Theory of Light and Refraction: A Discourse on Method |url=https://www.jstor.org/stable/1006537 |journal=Transactions of the American Philosophical Society |volume=77 |issue=3 |pages=i–92 |doi=10.2307/1006537 |jstor=1006537 |issn=0065-9746}} Pierre de Fermat derived Snell's law using the opposing assumption, the denser the medium the slower light travelled. Fermat also argued in support of a finite speed of light.{{Cite book|author-link=Carl Benjamin Boyer |first=Carl Benjamin |last=Boyer |title=The Rainbow: From Myth to Mathematics |year=1959 |pages=205–206 |publisher=Thomas Yoseloff |oclc=763848561}}

In 1629, Isaac Beeckman proposed an experiment in which a person observes the flash of a cannon reflecting off a mirror about one mile (1.6 km) away. In 1638, Galileo Galilei proposed an experiment, with an apparent claim to having performed it some years earlier, to measure the speed of light by observing the delay between uncovering a lantern and its perception some distance away. He was unable to distinguish whether light travel was instantaneous or not, but concluded that if it were not, it must nevertheless be extraordinarily rapid.

{{Cite book

|last=Galilei

|first=G.

|year=1954

|orig-year=1638

|title=Dialogues Concerning Two New Sciences

|url=https://www.questia.com/read/88951396/dialogues-concerning-two-new-sciences

|page=43

|others=Crew, H.; de Salvio A. (trans.)

|publisher=Dover Publications

|isbn=978-0-486-60099-4

|ref=Reference-Galileo-1954

|access-date=29 January 2019

|archive-date=30 January 2019

|archive-url=https://web.archive.org/web/20190130053744/https://www.questia.com/read/88951396/dialogues-concerning-two-new-sciences

|url-status=dead

}}

{{Cite journal

|last=Boyer |first=C. B.

|year=1941

|title=Early Estimates of the Velocity of Light

|journal=Isis

|volume=33 |issue=1 |page=24

|doi=10.1086/358523

|s2cid=145400212

|ref=boyer-1941

}} According to Galileo, the lanterns he used were "at a short distance, less than a mile". Assuming the distance was not too much shorter than a mile, and that "about a thirtieth of a second is the minimum time interval distinguishable by the unaided eye", Boyer notes that Galileo's experiment could at best be said to have established a lower limit of about 60 miles per second for the velocity of light. In 1667, the Accademia del Cimento of Florence reported that it had performed Galileo's experiment, with the lanterns separated by about one mile, but no delay was observed.{{Cite journal|last1=Foschi|first1=Renato|last2=Leone|first2=Matteo|date=August 2009|title=Galileo, Measurement of the Velocity of Light, and the Reaction Times|url=http://journals.sagepub.com/doi/10.1068/p6263|journal=Perception|language=en|volume=38|issue=8|pages=1251–1259|doi=10.1068/p6263|pmid=19817156|hdl=2318/132957 |s2cid=11747908|issn=0301-0066|hdl-access=free}} The actual delay in this experiment would have been about 11 microseconds.

File:Illustration from 1676 article on Ole Rømer's measurement of the speed of light.png

The first quantitative estimate of the speed of light was made in 1676 by Ole Rømer. From the observation that the periods of Jupiter's innermost moon Io appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth's orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth's orbit to obtain an estimate of speed of light of {{val|220000|u=km/s}}, which is 27% lower than the actual value.

{{Cite book

|last=Huygens |first=C.

|year=1690

|title=Traitée de la Lumière |language=fr

|url=https://archive.org/details/bub_gb_kVxsaYdZaaoC

|pages=[https://archive.org/details/bub_gb_kVxsaYdZaaoC/page/n19 8]–9

|publisher=Pierre van der Aa

}}

In his 1704 book Opticks, Isaac Newton reported Rømer's calculations of the finite speed of light and gave a value of "seven or eight minutes" for the time taken for light to travel from the Sun to the Earth (the modern value is 8 minutes 19 seconds).

{{Cite book

|last=Newton |first=I.

|year=1704

|contribution=Prop. XI

|title=Optiks

|url=http://gallica.bnf.fr/ark:/12148/bpt6k3362k.image.f235.vignettesnaviguer

}} The text of Prop. XI is identical between the first (1704) and second (1719) editions. Newton queried whether Rømer's eclipse shadows were coloured. Hearing that they were not, he concluded the different colours travelled at the same speed. In 1729, James Bradley discovered stellar aberration. From this effect he determined that light must travel 10,210 times faster than the Earth in its orbit (the modern figure is 10,066 times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.

{{See also|History of electromagnetic theory|History of special relativity}}

In the 19th century Hippolyte Fizeau developed a method to determine the speed of light based on time-of-flight measurements on Earth and reported a value of {{val|315000|u=km/s}}.{{Cite journal|last=Guarnieri|first=M.|year=2015|title=Two Millennia of Light: The Long Path to Maxwell's Waves|journal=IEEE Industrial Electronics Magazine|volume=9|issue=2|pages=54–56, 60|doi=10.1109/MIE.2015.2421754|s2cid=20759821}} His method was improved upon by Léon Foucault who obtained a value of {{val|298000|u=km/s}} in 1862. In the year 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch measured the ratio of the electromagnetic and electrostatic units of charge, 1/{{radic|ε₀μ₀}}, by discharging a Leyden jar, and found that its numerical value was very close to the speed of light as measured directly by Fizeau. The following year Gustav Kirchhoff calculated that an electric signal in a resistanceless wire travels along the wire at this speed.

{{Cite journal

|last1=Kirchhoff |first1=G.

|title=Über die Bewegung der Elektricität

|url=http://gallica.bnf.fr/ark:/12148/bpt6k15187j/f549.item.r=

|journal=Annalen der Physik

|volume=178

|issue=12

|year=1857

|pages=529–244

|doi=10.1002/andp.18571781203

|bibcode=1857AnP...178..529K

}}

In the early 1860s, Maxwell showed that, according to the theory of electromagnetism he was working on, electromagnetic waves propagate in empty spaceSee, for example:

• {{Cite book

|title=College physics: reasoning and relationships

|first1=Nicholas J.

|last1=Giordano

|publisher=Cengage Learning

|year=2009

|isbn=978-0-534-42471-8

|page=787

|url=https://books.google.com/books?id=BwistUlpZ7cC

}} [https://books.google.com/books?id=BwistUlpZ7cC&pg=PA787 Extract of page 787]

• {{Cite book

|title=The riddle of gravitation

|first1=Peter Gabriel

|last1=Bergmann

|publisher=Courier Dover Publications

|year=1992

|isbn=978-0-486-27378-5

|page=17

|url=https://books.google.com/books?id=WYxkrwMidp0C

}} [https://books.google.com/books?id=WYxkrwMidp0C&pg=PA17 Extract of page 17]

• {{Cite book

|title=The equations: icons of knowledge

|first1=Sander

|last1=Bais

|publisher=Harvard University Press

|year=2005

|isbn=978-0-674-01967-6

|page=[https://archive.org/details/equationsiconsof0000bais/page/40 40]

|url=https://archive.org/details/equationsiconsof0000bais|url-access=registration

}} [https://archive.org/details/equationsiconsof0000bais/page/40 Extract of page 40]

at a speed equal to the above Weber/Kohlrausch ratio, and drawing attention to the numerical proximity of this value to the speed of light as measured by Fizeau, he proposed that light is in fact an electromagnetic wave.

{{Cite web

|last1=O'Connor

|first1=J. J.

|last2=Robertson

|first2=E. F.

|date=November 1997

|title=James Clerk Maxwell

|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Maxwell.html

|publisher=School of Mathematics and Statistics, University of St Andrews

|access-date=13 October 2010

|url-status=dead

|archive-url=https://web.archive.org/web/20110128034939/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Maxwell.html

|archive-date=28 January 2011

}} Maxwell backed up his claim with his own experiment published in the 1868 Philosophical Transactions which determined the ratio of the electrostatic and electromagnetic units of electricity.Campbell, Lewis; Garnett, William; Rautio, James C. "The Life of James Clerk Maxwell", p. 544, {{ISBN|978-1-77375-139-9}}.

{{main|Luminiferous aether}}

The wave properties of light were well known since Thomas Young. In the 19th century, physicists believed light was propagating in a medium called aether (or ether). But for electric force, it looks more like the gravitational force in Newton's law. A transmitting medium was not required. After Maxwell theory unified light and electric and magnetic waves, it was favored that both light and electric magnetic waves propagate in the same aether medium (or called the luminiferous aether).{{Cite journal |last=Watson |first=E. C. |date=1957-09-01 |title=On the Relations between Light and Electricity |url=https://doi.org/10.1119/1.1934460 |journal=American Journal of Physics |volume=25 |issue=6 |pages=335–343 |doi=10.1119/1.1934460 |bibcode=1957AmJPh..25..335W |issn=0002-9505}}

File:Einstein en Lorentz.jpg

It was thought at the time that empty space was filled with a background medium called the luminiferous aether in which the electromagnetic field existed. Some physicists thought that this aether acted as a preferred frame of reference for the propagation of light and therefore it should be possible to measure the motion of the Earth with respect to this medium, by measuring the isotropy of the speed of light. Beginning in the 1880s several experiments were performed to try to detect this motion, the most famous of which is the experiment performed by Albert A. Michelson and Edward W. Morley in 1887.{{Cite book |last1=Consoli |first1=Maurizio |last2=Pluchino |first2=Alessandro |date=2018 |title=Michelson-Morley Experiments: An Enigma for Physics & The History of Science |url=https://books.google.com/books?id=VdWEDwAAQBAJ&pg=PA118 |publisher=World Scientific |pages=118–119 |isbn=978-9-813-27818-9 |access-date=4 May 2020}}

{{Cite journal

|last1=Michelson |first1=A. A. |last2=Morley |first2=E. W.

|year=1887

|title=On the Relative Motion of the Earth and the Luminiferous Ether

|journal=American Journal of Science

|volume=34 |issue=203 |pages=333–345

|doi=10.1366/0003702874447824

|s2cid=98374065

|url=https://zenodo.org/record/1450078

}} The detected motion was found to always be nil (within observational error). Modern experiments indicate that the two-way speed of light is isotropic (the same in every direction) to within 6 nanometres per second.

{{Cite book

| last = French | first = A. P.

| year = 1983

| title = Special relativity

| pages = 51–57

| publisher = Van Nostrand Reinhold

| isbn = 978-0-442-30782-0

}}

Because of this experiment Hendrik Lorentz proposed that the motion of the apparatus through the aether may cause the apparatus to contract along its length in the direction of motion, and he further assumed that the time variable for moving systems must also be changed accordingly ("local time"), which led to the formulation of the Lorentz transformation. Based on Lorentz's aether theory, Henri Poincaré (1900) showed that this local time (to first order in v/c) is indicated by clocks moving in the aether, which are synchronized under the assumption of constant light speed. In 1904, he speculated that the speed of light could be a limiting velocity in dynamics, provided that the assumptions of Lorentz's theory are all confirmed. In 1905, Poincaré brought Lorentz's aether theory into full observational agreement with the principle of relativity.

{{Cite book

|last=Darrigol

|first=O.

|year=2000

|title=Electrodynamics from Ampére to Einstein

|publisher=Clarendon Press

|isbn=978-0-19-850594-5

|url-access=registration

|url=https://archive.org/details/electrodynamicsf0000darr

}}

{{Cite book

|last=Galison |first=P.

|author-link=Peter Galison

|year=2003

|title= Einstein's Clocks, Poincaré's Maps: Empires of Time

|publisher=W. W. Norton

|isbn=978-0-393-32604-8

}}

In 1905 Einstein postulated from the outset that the speed of light in vacuum, measured by a non-accelerating observer, is independent of the motion of the source or observer. Using this and the principle of relativity as a basis he derived the special theory of relativity, in which the speed of light in vacuum c featured as a fundamental constant, also appearing in contexts unrelated to light. This made the concept of the stationary aether (to which Lorentz and Poincaré still adhered) useless and revolutionized the concepts of space and time.

{{Cite book

|last=Miller

|first=A. I.

|year=1981

|title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911)

|publisher=Addison–Wesley

|isbn=978-0-201-04679-3

|url-access=registration

|url=https://archive.org/details/alberteinsteinss0000mill

}}

{{Cite book

|last=Pais |first=A.

|author-link=Abraham Pais

|year=1982

|title= Subtle is the Lord: The Science and the Life of Albert Einstein

|publisher=Oxford University Press

|isbn=978-0-19-520438-4

}}

{{See also|History of the metre}}

In the second half of the 20th century, much progress was made in increasing the accuracy of measurements of the speed of light, first by cavity resonance techniques and later by laser interferometer techniques. These were aided by new, more precise, definitions of the metre and second. In 1950, Louis Essen determined the speed as {{val|299792.5|3.0|u=km/s}}, using cavity resonance. This value was adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. In 1960, the metre was redefined in terms of the wavelength of a particular spectral line of krypton-86, and, in 1967, the second was redefined in terms of the hyperfine transition frequency of the ground state of caesium-133.

{{Cite web

|year=1967

|title=Resolution 1 of the 15th CGPM

|url=https://www.bipm.org/en/CGPM/db/13/1/

|publisher=BIPM

|access-date=14 March 2021

|archive-date=11 April 2021

|archive-url=https://web.archive.org/web/20210411132806/https://www.bipm.org/en/CGPM/db/13/1/

|url-status=dead

}}

In 1972, using the laser interferometer method and the new definitions, a group at the US National Bureau of Standards in Boulder, Colorado determined the speed of light in vacuum to be c = {{val|299792456.2|1.1|u=m/s}}. This was 100 times less uncertain than the previously accepted value. The remaining uncertainty was mainly related to the definition of the metre.

{{Cite web

|year=1967

|title=Resolution 6 of the 15th CGPM

|url=http://www.bipm.org/en/CGPM/db/11/6/

|publisher=BIPM

|access-date=13 October 2010

}} As similar experiments found comparable results for c, the 15th General Conference on Weights and Measures in 1975 recommended using the value {{val|299792458|u=m/s}} for the speed of light.

{{Cite web

|year=1975

|title=Resolution 2 of the 15th CGPM

|url=http://www.bipm.org/en/CGPM/db/15/2/

|publisher=BIPM

|access-date=9 September 2009

}}

In 1983 the 17th meeting of the General Conference on Weights and Measures (CGPM) found that wavelengths from frequency measurements and a given value for the speed of light are more reproducible than the previous standard. They kept the 1967 definition of second, so the caesium hyperfine frequency would now determine both the second and the metre. To do this, they redefined the metre as "the length of the path traveled by light in vacuum during a time interval of 1/{{val|299792458}} of a second".

{{Cite web

|year=1983

|title=Resolution 1 of the 17th CGPM

|url=https://www.bipm.org/en/committees/cg/cgpm/17-1983/resolution-1

|publisher=BIPM

|access-date=23 August 2009

}}

As a result of this definition, the value of the speed of light in vacuum is exactly {{val|299792458|u=m/s}}

{{Cite book

|last1=Taylor |first1=E. F. |author-link1=Edwin F. Taylor |last2=Wheeler |first2=J. A. |author-link2=John Archibald Wheeler

|year=1992

|title=Spacetime Physics: Introduction to Special Relativity

|url=https://books.google.com/books?id=PDA8YcvMc_QC&pg=PA59

|edition=2

|publisher=Macmillan

|isbn=978-0-7167-2327-1

|page=59

}}

{{Cite web

|last=Penzes |first=W. B.

|year=2009

|title=Time Line for the Definition of the Meter

|url=https://www.nist.gov/pml/div683/upload/museum-timeline.pdf

|publisher=NIST

|access-date=11 January 2010

}} and has become a defined constant in the SI system of units. Improved experimental techniques that, prior to 1983, would have measured the speed of light no longer affect the known value of the speed of light in SI units, but instead allow a more precise realization of the metre by more accurately measuring the wavelength of krypton-86 and other light sources.

{{Cite book

|last=Adams |first=S.

|year=1997

|title=Relativity: An Introduction to Space–Time Physics

|url=https://books.google.com/books?id=1RV0AysEN4oC&pg=PA140

|page=140

|publisher=CRC Press

|isbn=978-0-7484-0621-0

|quote=One peculiar consequence of this system of definitions is that any future refinement in our ability to measure c will not change the speed of light (which is a defined number), but will change the length of the meter!

}}

{{Cite book

|last=Rindler |first=W.

|year=2006

|title=Relativity: Special, General, and Cosmological

|url=https://books.google.com/books?id=MuuaG5HXOGEC&pg=PT41

|page=41

|edition=2

|publisher=Oxford University Press

|isbn=978-0-19-856731-8

|quote=Note that [...] improvements in experimental accuracy will modify the meter relative to atomic wavelengths, but not the value of the speed of light!

}}

In 2011, the CGPM stated its intention to redefine all seven SI base units using what it calls "the explicit-constant formulation", where each "unit is defined indirectly by specifying explicitly an exact value for a well-recognized fundamental constant", as was done for the speed of light. It proposed a new, but completely equivalent, wording of the metre's definition: "The metre, symbol m, is the unit of length; its magnitude is set by fixing the numerical value of the speed of light in vacuum to be equal to exactly {{val|299792458}} when it is expressed in the SI unit {{nowrap|m s⁻¹}}."{{Cite web |url=http://www.bipm.org/en/si/new_si/explicit_constant.html |title=The "explicit-constant" formulation |archive-url=https://web.archive.org/web/20140811195806/http://www.bipm.org/en/si/new_si/explicit_constant.html |archive-date=11 August 2014 |website=BIPM |date=2011}} This was one of the changes that was incorporated in the 2019 revision of the SI, also termed the New SI.See, for example:

• {{Cite web |last=Conover |first=Emily |author-link=Emily Conover |date=2 November 2016 |title=Units of measure are getting a fundamental upgrade |url=https://www.sciencenews.org/article/units-measure-are-getting-fundamental-upgrade |access-date=6 February 2022 |website=Science News |language=en-US}}
• {{Cite journal |last1=Knotts |first1=Sandra |last2=Mohr |first2=Peter J. |last3=Phillips |first3=William D. |date=January 2017 |title=An Introduction to the New SI |url=http://scitation.aip.org/content/aapt/journal/tpt/55/1/10.1119/1.4972491 |journal=The Physics Teacher |language=en |volume=55 |issue=1 |pages=16–21 |doi=10.1119/1.4972491 |bibcode=2017PhTea..55...16K |s2cid=117581000 |issn=0031-921X}}
• {{Cite journal |date=11 May 2018 |title=SI Redefinition |url=https://www.nist.gov/si-redefinition |access-date=6 February 2022 |journal=National Institute of Standards and Technology |language=en}}

{{Portal|Physics|Astronomy|Outer space}}

{{cols |colwidth=20em}}

• Light-second
• Speed of electricity
• Speed of gravity
• Speed of sound
• Velocity factor
• Warp factor (fictional)

{{colend}}

{{reflist|group="Note"|30em}}

{{reflist}}

{{refbegin}}

• {{Cite journal

|first=O.

|last=Rømer

|author-link=Ole Rømer

|year=1676

|title=Démonstration touchant le mouvement de la lumière trouvé par M. Römer de l'Academie Royale des Sciences

|url=http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Roemer_1676.pdf

|journal=Journal des sçavans

|pages=223–236

|language=fr

|access-date=10 March 2020

|archive-date=8 September 2022

|archive-url=https://web.archive.org/web/20220908221513/http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Roemer_1676.pdf

|url-status=dead

}}

• Translated as {{Cite journal

|year = 1677

|title = A Demonstration concerning the Motion of Light

|url = http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html

|journal = Philosophical Transactions of the Royal Society

|issue = 136

|pages = 893–894

|archive-url = https://web.archive.org/web/20070729214326/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html

|archive-date = 29 July 2007

|doi = 10.1098/rstl.1677.0024

|volume = 12

|bibcode = 1677RSPT...12..893.

|doi-access = free

}}

• {{Cite journal

|first=E. |last=Halley |author-link=Edmond Halley

|year=1694

|title=Monsieur Cassini, his New and Exact Tables for the Eclipses of the First Satellite of Jupiter, reduced to the Julian Stile and Meridian of London

|journal=Philosophical Transactions of the Royal Society

|volume=18 |issue=214 |pages=237–256

|doi=10.1098/rstl.1694.0048 |doi-access=free

|bibcode=1694RSPT...18..237C

}}

• {{Cite journal |first=H. L. |last=Fizeau |author-link=Hippolyte Fizeau |year=1849 |title=Sur une expérience relative à la vitesse de propagation de la lumière |url=https://www.academie-sciences.fr/pdf/dossiers/Fizeau/Fizeau_pdf/CR1849_p90.pdf |journal=Comptes rendus de l'Académie des sciences |volume=29 |pages=90–92, 132 |language=fr }}
• {{Cite journal

|first=J. L.

|last=Foucault

|author-link=Léon Foucault

|year=1862

|title=Détermination expérimentale de la vitesse de la lumière: parallaxe du Soleil

|url=https://books.google.com/books?id=yYIIAAAAMAAJ&pg=PA216

|journal=Comptes rendus de l'Académie des sciences

|volume=55

|pages=501–503, 792–796

|language=fr

}}

• {{Cite journal

|first=A. A.

|last=Michelson

|author-link=Albert A. Michelson

|year=1878

|title=Experimental Determination of the Velocity of Light

|url=http://www.gutenberg.org/ebooks/11753

|journal=Proceedings of the American Association for the Advancement of Science

|volume=27

|pages=71–77

}}

• {{Cite journal

|first1=A. A. |last1=Michelson

|first2=F. G. |last2=Pease |author2-link=Francis G. Pease

|first3=F. |last3=Pearson

|title=Measurement of the Velocity of Light in a Partial Vacuum

|journal=Astrophysical Journal

|volume=82 |pages=26–61 |year=1935

|issue=2091

|doi=10.1086/143655 |pmid=17816642

|bibcode=1935ApJ....82...26M

|s2cid=123010613

}}

• {{Cite journal

|first=S. |last=Newcomb |author-link=Simon Newcomb

|year=1886

|title=The Velocity of Light

|journal=Nature

|volume=34

|issue=863 |pages=29–32

|doi=10.1038/034029c0 |doi-access=free

|bibcode = 1886Natur..34...29.

}}

• {{Cite journal

|first=J. |last=Perrotin |author-link=Henri Joseph Anastase Perrotin

|year=1900

|title=Sur la vitesse de la lumière

|journal=Comptes rendus de l'Académie des sciences

|volume=131 |pages=731–734

|language=fr

}}

{{refend}}

{{refbegin}}

• {{Cite book

|first=L. |last=Brillouin |author-link=Léon Brillouin

|year=1960

|title=Wave propagation and group velocity

|publisher=Academic Press

}}

• {{Cite book

|first=J. D. |last=Jackson |author-link=John David Jackson (physicist)

|year=1975

|title=Classical Electrodynamics

|edition=2

|publisher=John Wiley & Sons

|isbn=978-0-471-30932-1

}}

• {{Cite book

|first=G.

|last=Keiser

|year=2000

|title=Optical Fiber Communications

|page=[https://archive.org/details/opticalfibercomm00gerd/page/32 32]

|edition=3

|publisher=McGraw-Hill

|isbn=978-0-07-232101-2

|url=https://archive.org/details/opticalfibercomm00gerd/page/32

}}

• {{Cite book

|last=Ng |first=Y. J.

|year=2004

|chapter=Quantum Foam and Quantum Gravity Phenomenology

|chapter-url=https://books.google.com/books?id=RntpN7OesBsC

|editor=Amelino-Camelia, G |editor2=Kowalski-Glikman, J

|title=Planck Scale Effects in Astrophysics and Cosmology

|pages=321ff

|publisher=Springer

|isbn=978-3-540-25263-4

}}

• {{Cite book

|last1=Helmcke |first1=J. |last2=Riehle |first2=F.

|year=2001

|chapter=Physics behind the definition of the meter

|chapter-url=https://books.google.com/books?id=WE22Fez60EcC&pg=PA453

|editor=Quinn, T. J. |editor2=Leschiutta, S. |editor3=Tavella, P.

|title=Recent advances in metrology and fundamental constants

|page=453

|publisher=IOS Press

|isbn=978-1-58603-167-1

}}

• {{Cite arXiv

|last=Duff |first=M. J. |author-link=Michael James Duff

|year=2004

|title=Comment on time-variation of fundamental constants

|eprint=hep-th/0208093

}}

{{refend}}

{{wikiquote}}

• [https://books.google.com/books?id=UigDAAAAMBAJ&dq=1930+plane+%22Popular&pg=PA17 "Test Light Speed in Mile Long Vacuum Tube"]. Popular Science Monthly, September 1930, pp. 17–18.
• [http://www.bipm.org/en/si/si_brochure/chapter2/2-1/metre.html Definition of the metre] (International Bureau of Weights and Measures, BIPM)
• [http://physics.nist.gov/cgi-bin/cuu/Value?c Speed of light in vacuum] (National Institute of Standards and Technology, NIST)
• [https://web.archive.org/web/20040530221051/http://www.itl.nist.gov/div898/bayesian/datagall/michelso.htm Data Gallery: Michelson Speed of Light (Univariate Location Estimation)] (download data gathered by Albert A. Michelson)
• [https://www.gregegan.net/APPLETS/20/20.html Subluminal] (Java applet by Greg Egan demonstrating group velocity information limits)
• [http://www.ertin.com/sloan_on_speed_of_light.html Light discussion on adding velocities]
• [http://sixtysymbols.com/videos/light.htm Speed of Light] (Sixty Symbols, University of Nottingham Department of Physics [video])
• [https://www.bbc.co.uk/programmes/p0038x9h Speed of Light], BBC Radio{{nbsp}}4 discussion (In Our Time, 30 November 2006)
• [http://www.live-counter.com/speed-of-light/ Speed of Light] (Live-Counter – Illustrations)

• [https://www.sciencealert.com/simple-animations-by-a-nasa-scientist-prove-the-speed-of-light-is-torturously-slow Speed of Light – animated demonstrations]
• "[https://books.google.com/books?id=p4o9AQAAIAAJ The Velocity of Light]", Albert A. Nicholson, Scientific American, 28 September 1878, p. 193

{{Extreme motion}}

{{Gravitational waves}}

{{Records}}

{{Relativity}}

{{Special relativity}}

{{Authority control}}

Category:Fundamental constants

Category:Physical quantities

Category:Light

Category:Special relativity

Light