Speed of light: Difference between revisions - Wikipedia
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{{Short description|Speed of electromagnetic waves in vacuum}}
{{Redirect|Lightspeed|other uses|Speed of light (disambiguation)|and|Lightspeed (disambiguation)}}
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{{infobox| title = Speed of light
{{Featured article}}
| image = [[File:Earth to Sun - en.png|300px|alt=The distance from the Sun to the Earth is shown as 150 million kilometers, an approximate average. Sizes to scale.]]
{{Use Oxford spelling|date=August 2022}}
| caption = [[Sunlight]] takes about 8 minutes 17 seconds to travel the average distance from the surface of the [[Sun]] to the [[Earth]]. <!-- (1 AU - 1 solar radius - 1 earth radius)/c = (149,597,871 - 695,500 - 6371)/299,792.458 = 496.66 s = 8 min 17 s -->
{{Use dmy dates|date=August 2022|cs1-dates=l}}
| header1 = Exact values
|labelstyle = font-weight:normal
{{Infobox
| label2 = [[Metre per second|metres per second]]
| data2title = {{val|299792458}}Speed of light
| image = [[File:Earth to Sun - en.png|frameless|upright=1.5|alt=The distance from the Sun to Earth is shown as 150 million kilometres, an approximate average. Sizes to scale.]]
| label3 = [[Planck length]] per [[Planck time]] <br>(i.e., [[Planck units]])
| caption = On average, [[sunlight]] takes 8{{nbsp}}minutes and 17{{nbsp}}seconds to travel from the [[Sun]] to [[Earth]].<!-- (1 AU − 1 solar radius − 1 earth radius)/c = (149597871 − 695500 − 6371)/299792.458 = 496.66 s = 8 min 17 s -->
| data3 = 1
| header1 = Exact value
| header4 = Approximate values <!--This section lists various values for c, to three significant digits. Please do not change to more exact values!-->
| labelstyle = font-weight:normal
| label5 = [[kilometre per second|kilometres per second]]
| label2 = [[metre per second|metres per second]]
| data5 = 300,000
| data2 = {{val|299792458}}
| label6 = [[kilometre per hour|kilometres per hour]]
| header4 = Approximate values (to three significant digits)<!-- This section lists various values for c, to three significant digits. Please do not change to more exact values! -->
| data6 = 1,080 million
| label7label5 = [[mile per second|mileskilometres per secondhour]]
| data7data5 = 186,000{{val|1080000000}}
| label8label6 = [[milemiles per hour|miles per hoursecond]]
| data8data6 = 671 million = {{val|186000}}
| label7 = [[miles per hour]]<ref>{{Cite book |title=Elementary and Intermediate Algebra: A Combined Course, Student Support Edition |edition=4th illustrated |first1=Ron |last1=Larson |first2=Robert P. |last2=Hostetler |publisher=Cengage Learning |year=2007 |isbn=978-0-618-75354-3 |page=197 |url=https://books.google.com/books?id=qe-YvKoeiasC&pg=PA179}}</ref>
| label9 = [[astronomical unit]]s per day
| data9data7 = 173{{val|671000000}}
| label8 = [[astronomical unit]]s per day
| header10 = Approximate light signal travel times
| data8 = 173{{#tag:ref|Exact value: {{nowrap|({{val|299792458}} × {{val|86400}}<!--60 × 60 × 24--> / {{val|149597870700}}) AU/day}}.|group="Note"}}
| label11 = '''Distance'''
| label9 = [[parsec]]s per year
| data11 = '''Time'''
| data9 = 0.307{{#tag:ref|Exact value: {{nowrap|({{val|999992651|end= π}} / {{val|10246429500}}) pc/y}}.|group="Note"}}
| label12 = one [[Foot (unit)|foot]]
| header10 = Approximate light signal travel times
| data12 = 1.0 [[Nanosecond|ns]]
| label11 = '''Distance'''
| label13 = one [[metre]]
| data11 = '''Time'''
| data13 = 3.3 ns
| label12 = one [[foot (unit)|foot]]
| label16 = from [[geostationary orbit]] to Earth
| data16data12 = 1191.0 [[millisecondNanosecond|msns]]
| label17label13 = the length of Earth's = one [[equatormetre]]
| data17data13 = 1343.3 msns
| label18label15 = from [[Moongeostationary orbit]] to Earth
| data18data15 = 1.3119 [[secondmillisecond|sms]]
| label19label16 = fromthe [[Sun]]length toof Earth (1's [[astronomical unit|AUequator]])
| data19data16 = 8.3134 [[minute|min]]ms
| label17 = from [[Moon]] to Earth
| label21 =
| data17 = 1.3 [[second|s]]
| data21 =
| label22label18 = from [[Proxima Centauri|nearest starSun]] to SunEarth (1.3 [[Parsecastronomical unit|pcAU]])
| data22data18 = 48.23 years[[minute|min]]
| label20 = one [[light-year]]
| label23 =
| data23data20 = 1.0 [[year]]
| label21 = one [[parsec]]
| label24 = from the nearest galaxy (the [[Canis Major Dwarf Galaxy]]) to Earth
| data24data21 = 25,0003.26 years
| label22 = from the [[Proxima Centauri|nearest star]] to Sun ({{val|1.3|u={{abbr|pc|parsec}}}})
| label25 = across the [[Milky Way]]
| data25data22 = 100,0004.2 years
| label26label23 = from the [[AndromedaSagittarius Galaxy]]Dwarf (theSpheroidal Galaxy|nearest spiral galaxy)]] to Earth
| data26data23 = 2.5 million = {{val|70,000|u=years}}
| label24 = across the [[Milky Way]]
| data24 = {{val|87,400|u=years}}
| label25 = from the [[Andromeda Galaxy]] to Earth
| data25 = 2.5 million years
}}
{{Special relativity sidebar}}
The '''speed of light''' in [[vacuum]], commonly denoted {{mathmvar|'''''c'''''}}, is a universal [[physical constant]] importantthat inis manyexactly areasequal ofto [[physics]].{{convert|299792458|m/s|km/s Itsmi/s valuee6mph|abbr=off|sigfig=3|disp=x| (approximately }}).{{refn|group="Note"|It is exactlyexact '''299because,792 by a 1983 international agreement,458 a [[metreMetre#Speed perof secondlight definition|metresmetre]] peris seconddefined as the length of the path travelled by [[light]]''', in vacuum during a figuretime thatinterval isof exact{{frac|1|{{val|299792458}}}} because[[second]]. theThis lengthparticular value was chosen to provide a more accurate definition of the metre isthat definedstill fromagreed thisas constantmuch andas possible with the definition used before. See, for example, the [[Second#International_second|internationalNIST]] standardwebsite<ref forname="nist-definitions"/> timeor the explanation by [[Roger Penrose|Penrose]].<ref name="penrose">{{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=410–1 |[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 299,792,458{{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 [[History of the metre#International prototype metre|standard metre rule]] in Paris. |title-link=The Road to Reality: A Complete Guide to the Laws of the Universe }}</ref> ThisThe second is approximately 186,282.4 [[mile]]s perin secondturn, ordefined aboutto 671be millionthe [[mileslength perof hour]].time Accordingoccupied to [[special relativity]],by {{mathval|''c''9192631770|u=cycles}} isof the maximumradiation speedemitted atby which all energy, matter, anda [[physical information|informationcaesium]] in the-133 [[universeatom]] canin travel.a Ittransition isbetween the speed at whichtwo allspecified [[masslessenergy particle]]s and associated [[field (physics)level|fields]]energy (including [[electromagnetic radiationstates]].<ref suchname="nist-definitions">{{Cite asweb [[light]]) travel in vacuum|url=https://physics.nist.gov/cuu/Units/current.html It|title=Definitions is alsoof the [[speedSI ofbase gravity]]units (i|website=physics.enist.gov of|date=29 [[gravitationalMay wave]]s)2019 predicted|access-date=8 byFebruary current2022}}</ref>}} theories.According Suchto particlesthe and[[special wavesrelativity|special traveltheory atof relativity]], {{mathMvar|''c''}} regardless ofis the motionupper oflimit for the sourcespeed orat thewhich conventional [[inertialmatter]] frameor of reference[[energy]] of(and thethus observer.any In[[signal]] thecarrying [[theory of relativityinformation]],) {{math|''c''}}can interrelatestravel through [[spacetimeSpace#Relativity|space and time]],.<ref>{{cite book |title=Special Relativity and alsoHow appearsit inWorks the|author1=Moses famousFayngold equation|edition=illustrated of|publisher=John [[mass–energyWiley equivalence]]& Sons {{math|''E''year=2008 |isbn=978-3-527-40607-4 |page=497 |url=https://books.google.com/books?id=Q3egk8Ds6ogC}} [https://books.google.com/books?id=Q3egk8Ds6ogC {{math|''mc''<sup>2pg=PA497 Extract of page 497].</supref>}}.<ref name=LeClerq>{{Citecite book |title=Special Relativity last|author1=UzanAlbert Shadowitz |firstedition=J-Prevised |last2publisher=LeclercqCourier Corporation |first2year=B1988 |isbn=978-0-486-65743-1 |page=79 year|url=2008https://books.google.com/books?id=1axfJqUT6R0C}} [https://books.google.com/books?id=1axfJqUT6R0C&pg=PA79 Extract of page 79].</ref><ref>{{Cite journal |last1=Peres |first1=Asher title|author-link=TheAsher NaturalPeres Laws|last2=Terno of|first2=Daniel theR. Universe:|date=2004-01-06 Understanding|title=Quantum Fundamentalinformation Constantsand |relativity theory |url=httphttps://bookslink.googleaps.comorg/?iddoi/10.1103/RevModPhys.76.93 |journal=dSAWX8TNpScC&pgReviews of Modern Physics |language=PA43en |volume=76 |issue=1 |pages=43–493–123 |doi=10.1103/RevModPhys.76.93 publisher|arxiv=[[Springerquant-ph/0212023 (publisher)|Springer]]bibcode=2004RvMP...76...93P |s2cid=7481797 isbn|issn=00034-387-73454-6 6861}}</ref>
All forms of [[electromagnetic radiation]], including [[visible light]], travel at the speed of light. For many practical purposes, light and other electromagnetic waves will appear to propagate instantaneously, but for long distances and very sensitive measurements, their finite speed has noticeable effects. Much [[starlight]] viewed on [[Earth]] is from the distant past, allowing humans to study the history of the universe by viewing distant objects. When [[Data communication|communicating]] with distant [[space probe]]s, it can take minutes to hours for signals to travel. In [[computing]], the speed of light fixes the ultimate minimum [[Latency (engineering)|communication delay]]. The speed of light can be used in [[time of flight]] measurements to measure large distances to extremely high precision.
The speed at which light propagates through [[transparent material]]s, such as glass or air, is less than {{math|''c''}}. The ratio between {{math|''c''}} and the speed {{math|''v''}} at which light travels in a material is called the [[refractive index]] {{math|''n''}} of the material ({{math|''n''}} = {{math|''c''}} / {{math|''v''}}). For example, for [[visible light]] the refractive index of glass is typically around 1.5, meaning that light in glass travels at {{nowrap|{{math|''c''}} / 1.5 ≈ {{val|200000|u=km/s}}}}; the [[refractive index of air]] for visible light is 1.000293, so the speed of light in air is {{val|299705|u=km/s}} or about {{val|88|u=km/s}} slower than {{math|''c''}}.
[[Ole Rømer]] first [[Rømer's determination of the speed of light|demonstrated in 1676]] that light does not travel instantaneously by studying the apparent motion of [[Jupiter]]'s moon [[Io (moon)|Io]]. Progressively more accurate measurements of its speed came over the following centuries. In a [[A Dynamical Theory of the Electromagnetic Field|paper]] published in 1865, [[James Clerk Maxwell]] proposed that light was an [[Electromagnetic radiation|electromagnetic wave]] and, therefore, travelled at speed {{Mvar|c}}.<ref>{{Cite web |title=How is the speed of light measured? |url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html |url-status=dead |website=The Physics and Relativity FAQ |first=Philip |last=Gibbs |date=1997 |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 }}</ref> In 1905, [[Albert Einstein]] postulated that the speed of light {{Mvar|c}} with respect to any [[inertial frame of reference]] is a constant and is independent of the motion of the light source.<ref name="stachel">{{Cite book |title=Einstein from "B" to "Z" – Volume 9 of Einstein studies |first1=J. J. |last1=Stachel |publisher=Springer |year=2002 |isbn=978-0-8176-4143-6 |page=226 |url=https://books.google.com/books?id=OAsQ_hFjhrAC&pg=PA226}}</ref> He explored the consequences of that postulate by deriving the [[theory of relativity]] and, in doing so, showed that the parameter {{Mvar|c}} had relevance outside of the context of light and electromagnetism.
In most practical cases, light and other electromagnetic waves can be thought of as moving "instantaneously", but for long distances and very sensitive measurements their finite speed has noticeable effects. For example, in videos of an intense lightning storm on the Earth's surface taken from the International Space Station, the expansion of light wavefronts from individual flashes of lightning is clearly visible, and allows estimates of the speed of light to be made from frame-to-frame analysis of the position of the light wavefront. This is not surprising, as the time for light to propagate completely around the Earth is of the order of 140 milliseconds. This transit time is what causes the [[Schumann resonance]]. In communicating with distant [[space probes]], it can take minutes to hours for a message to get from Earth to the spacecraft, or vice versa. The light we see from stars left them many years ago, allowing us to study the history of the universe by looking at distant objects. The finite speed of light also limits the theoretical maximum speed of [[computer]]s, since information must be sent within the computer from chip to chip. Finally, the speed of light can be used with [[time of flight]] measurements to measure large distances to high precision.
[[Massless particle]]s and [[field (physics)|field]] perturbations, such as [[gravitational wave]]s, also travel at speed {{Mvar|c}} in vacuum. Such particles and waves travel at {{Mvar|c}} regardless of the motion of the source or the inertial reference frame of the [[observer (special relativity)|observer]]. Particles with nonzero [[rest mass]] can be accelerated to approach {{Mvar|c}} but can never reach it, regardless of the frame of reference in which their speed is measured. In the [[theory of relativity]], {{Mvar|c}} interrelates [[spacetime|space and time]] and appears in the famous [[mass–energy equivalence]], {{math|1=''E'' = ''mc''{{i sup|2}}}}.<ref>See, for example:
[[Ole Rømer]] first demonstrated in 1676 that light travelled at a finite speed (as opposed to instantaneously) by studying the apparent motion of [[Jupiter]]'s moon [[Io (moon)|Io]]. In 1865, [[James Clerk Maxwell]] proposed that light was an electromagnetic wave, and therefore travelled at the speed {{math|''c''}} appearing in his theory of electromagnetism.<ref>{{cite web|title=How is the speed of light measured?|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html}}</ref> In 1905, [[Albert Einstein]] postulated that the speed of light with respect to any inertial frame is independent of the motion of the light source,<ref name="stachel">{{cite book |title=Einstein from "B" to "Z" – Volume 9 of Einstein studies |first1=JJ |last1=Stachel |publisher=Springer |year=2002 |isbn=0-8176-4143-2 |page=226 |url=http://books.google.com/books?id=OAsQ_hFjhrAC&pg=PA226}}</ref> and explored the consequences of that postulate by deriving the [[special theory of relativity]] and showing that the parameter {{math|''c''}} had relevance outside of the context of light and electromagnetism. After centuries of increasingly precise measurements, in 1975 the speed of light was known to be {{val|299792458|u=m/s}} with a [[measurement uncertainty]] of 4 parts per billion. In 1983, the [[metre]] was redefined in the [[International System of Units]] (SI) as the distance travelled by light in vacuum in 1/299,792,458 of a [[second]]. As a result, the numerical value of {{math|''c''}} in metres per second is now fixed exactly by the definition of the metre.<ref name=BIPM_SI_units>{{SIbrochure|page=112}}</ref>
* {{Cite journal|last1=Feigenbaum|first1=Mitchell J.|author-link=Mitchell Feigenbaum|last2=Mermin|first2=N. David|author-link2=N. David Mermin|date=January 1988|title=E = mc<sup>2</sup>|url=http://aapt.scitation.org/doi/10.1119/1.15422|journal=[[American Journal of Physics]]|language=en|volume=56|issue=1|pages=18–21|doi=10.1119/1.15422|bibcode=1988AmJPh..56...18F|issn=0002-9505}}
* {{Cite book |last1=Uzan |first1=J-P |last2=Leclercq |first2=B |year=2008 |title=The Natural Laws of the Universe: Understanding Fundamental Constants |url=https://books.google.com/books?id=dSAWX8TNpScC&pg=PA43 |pages=43–44 |publisher=Springer |isbn=978-0-387-73454-5 }}</ref>
In some cases, objects or waves may appear to travel [[#Faster-than-light observations and experiments|faster than light]] (e.g., [[phase velocities]] of waves, [[Superluminal motion|the appearance of certain high-speed astronomical objects]], and particular [[quantum effects]]). The [[expansion of the universe]] is understood to exceed the speed of light beyond [[Hubble volume|a certain boundary]].
The speed at which light propagates through [[Transparency and translucency|transparent materials]], such as glass or air, is less than {{Mvar|c}}; similarly, the speed of [[electromagnetic waves]] in wire cables is slower than {{Mvar|c}}. The ratio between {{Mvar|c}} and the speed {{Mvar|v}} at which light travels in a material is called the [[refractive index]] {{mvar|n}} of the material ({{math|1={{Mvar|n}} = {{sfrac|{{Mvar|c}}|{{Mvar|v}}}}}}). For example, for visible light, the refractive index of glass is typically around 1.5, meaning that light in glass travels at {{nowrap|{{sfrac|{{Mvar|c}}|1.5}} ≈ {{cvt|200000|km/s|mi/s|comma=gaps|sigfig=3}}}}; the [[refractive index of air]] for visible light is about 1.0003, so the speed of light in air is about {{cvt|{{#expr:299792.458*(1-1/1.0003) round 0}}|km/s|mi/s|comma=gaps|sigfig=2}} slower than {{Mvar|c}}.
{{TOC limit}}
== Numerical value, notation, and units ==
The speed of light in vacuum is usually denoted by a lowercase {{math|''c''}}, for "constant" or the Latin {{lang|la|''[[:wikt:celeritas|celeritas]]''}} (meaning "swiftness"). (Capital '''[[C]]''' is the [[SI]] unit for [[coulomb]] of [[electric charge]].) Originallyswiftness, the symbol celerity''V'' was used for the speed of light, introduced by [[James Clerk Maxwell]] in 1865). 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 1894, [[Paul Drude]] redefined {{math|''c''}} with its modern meaning. [[Albert Einstein|Einstein]] used ''V'' in his [[Annus Mirabilis papers|original German-language papers]] on special relativity in 1905<!--, the 1923 English translation of them by Perrett and Jeffery using {{math|''c''}}-->, but in 1907 he switched to {{math|''c''}}, which by then had become the standard symbol for the speed of light.<ref name=Yc>
{{citeCite web
|last=Gibbs
|first=P.
|year=2004 |origyear=1997
|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
|accessdate=2009-11-16
|archive-url=https://web.archive.org/web/20100325220247/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/c.html
|archiveurl=http://www.webcitation.org/5lLMPPN4L
|archive-date=25 March 2010
|archivedate=2009-11-17
|url-status=dead
}}</ref><ref>
}}
{{cite journal
"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'."
|last=Mendelson |first=KS
</ref><ref>
{{Cite journal
|last=Mendelson |first=K. S.
|year=2006
|title=The story of ''c''
Line 79 ⟶ 96:
|volume=74 |issue=11 |pages=995–997
|doi=10.1119/1.2238887
|bibcode = 2006AmJPh..74..995M | issn=0002-9505}}</ref>
Sometimes {{math|''c''}} is used for the speed of waves in ''any'' material medium, and {{math|''c''}}<sub>0</sub> for the speed of light in vacuum.<ref name=handbook>See, for example:
* {{Cite book
|last=Lide |first=DRD. R.
|year=2004
|title=CRC Handbook of Chemistry and Physics
|url=httphttps://books.google.com/books?id=WDll8hA006AC&pg=PT76&dqq=speed+of+light+%22c0+OR+%22&pg=PT76
|pages=2–9
|publisher=[[CRC Press]]
|isbn=978-0-8493-0485-79
}}
* {{Cite book
|last=Harris |first=JWJ. W. |coauthorsyear=''et al.''2002
|year=2002
|title=Handbook of Physics
|url=httphttps://books.google.com/books?id=c60mCxGRMR8C&pg=PA499&dqq=speed+of+light+%22c0+OR+%22+date:2000-2009&pg=PA499
|page=499
|publisher=[[Springer (publisher)|Springer]]
|isbn=978-0-387-95269-17
|display-authors=etal}}
}}
* {{Cite book
|last=Whitaker |first=JCJ. C.
|year=2005
|title=The Electronics Handbook
|url=httphttps://books.google.com/books?id=FdSQSAC3_EwC&pg=PA235&dqq=speed+of+light+c0+handbook&pg=PA235
|page=235
|publisher=[[CRC Press]]
|isbn=978-0-8493-1889-04
}}
* {{Cite book
|last=Cohen |first=ERE. R. |coauthorsyear=''et al.''2007
|year=2007
|title=Quantities, Units and Symbols in Physical Chemistry
|url=httphttps://books.google.com/books?id=TElmhULQoeIC&pg=PA143&dqq=speed+of+light+c0+handbook&pg=PA143
|page=184
|edition=3rd3
|publisher=[[Royal Society of Chemistry]]
|isbn=978-0-85404-433-7
|display-authors=etal
}}</ref> This subscripted notation, which is endorsed in official SI literature,<ref name=BIPM_SI_units /> has the same form as other related constants: namely, ''μ''<sub>0</sub> for the [[vacuum permeability]] or magnetic constant, ''ε''<sub>0</sub> for the [[vacuum permittivity]] or electric constant, and ''Z''<sub>0</sub> for the [[impedance of free space]]. This article uses ''c'' exclusively for the speed of light in vacuum.
}}</ref> This subscripted notation, which is endorsed in official SI literature,<ref name=BIPM_SI_units>{{SIbrochure8th|page=112}}</ref> has the same form as related electromagnetic constants: namely, ''μ''<sub>0</sub> for the [[vacuum permeability]] or magnetic constant, ''ε''<sub>0</sub> for the [[vacuum permittivity]] or electric constant, and ''Z''<sub>0</sub> for the [[impedance of free space]]. This article uses {{math|''c''}} exclusively for the speed of light in vacuum.
=== Use in unit systems ===
In the [[International System of Units]] (SI), the metre is defined as the distance light travels in vacuum in 1/{{val|299792458}} of a second. This definition fixes the speed of light in vacuum at exactly {{val|299792458|u=m/s}}.<ref name=Boyes>
{{Further information|Metre#Speed of light definition}}
{{Cite book
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]]. By using the value of {{math|''c''}}, as well as an accurate measurement of the [[second]], one can thus establish a standard for the metre.<ref name="fixes">See, for example:
|last=Sydenham |first=PH
* {{Cite book
|last=Sydenham |first=P. H.
|year=2003
|chapter=Measurement of length
|chapterurlchapter-url=httphttps://books.google.com/books?id=sarHIbCVOUAC&pg=PA56
|editor=Boyes, W
|title=Instrumentation Reference Book
|edition=3rd3
|page=56
|publisher=[[Butterworth–Heinemann]]
|isbn=978-0-7506-7123-81
|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}}...
}}
}}</ref><ref name="Fundamental Physical Constants">
* {{citeCite 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=[[National Institute of Standards and Technology|NIST]]
|accessdateaccess-date=21 August 2009-08-21
}}
}}</ref><ref name=Jespersen>
* {{Cite book
|lastlast1=Jespersen |firstfirst1=J. |last2=Fitz-Randolph |first2=J. |last3=Robb |first3=J.
|year=1999
|title=From Sundials to Atomic Clocks: Understanding Time and Frequency
|url=httphttps://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
}}</ref> As a [[Physical constant#Dimensional and dimensionless physical constants|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
}}</ref>
: {{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 (unit)|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"}}<ref>{{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}}</ref><ref>{{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 Hopper|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}}</ref>
As a [[Physical constant#Dimensionful and dimensionless physical constants|dimensional physical constant]], the numerical value of ''c'' is different for different unit systems.{{#tag:ref|The speed of light in [[imperial units]] and [[United States customary units|US units]] is based on an inch of exactly {{val|2.54|u=cm}} and is exactly 186,282 miles, 698 yards, 2 feet, and {{sfrac|5|21|127}} inches per second.<ref>
{{cite web
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}}.<ref name="Lawrie">
|last=Savard |first=J
|title=From Gold Coins to Cadmium Light
|url=http://www.quadibloc.com/other/cnv03.htm
|work=[http://www.quadibloc.com/ John Savard's Home Page]
|accessdate=2009-11-14
|archiveurl=http://www.webcitation.org/5lHYVsp5E
|archivedate=2009-11-14
}}</ref>|group="Note"}}
In branches of physics in which ''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|''c'' {{=}} 1}}.<ref name=Lawrie>
{{Cite book
|last=Lawrie |first=IDI. D.
|year=2002
|chapter=Appendix C: Natural units
|chapterurlchapter-url=httphttps://books.google.com/books?id=9HZStxmfi3UC&pg=PA540
|title=A Unified Grand Tour of Theoretical Physics
|page=540
|edition=2nd2
|publisher=[[CRC Press]]
|isbn=978-0-7503-0604-1
}}</ref><ref name=Hsu"Hsu1">
{{Cite book
|last=Hsu |first=L.
|year=2006
|chapter=Appendix A: Systems of units and the development of relativity theories
|chapterurlchapter-url=httphttps://books.google.com/books?id=amLqckyrvUwC&pg=PA428
|title=A Broader View of Relativity: General Implications of Lorentz and Poincaré Invariance
|pages=427–8427–428
|edition=2nd2
|publisher=[[World Scientific]]
|isbn=978-981-256-651-15
}}</ref> 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.
== Fundamental role in physics ==
{{See also|Introduction to special relativity|Special relativity|One-way speed of light}}
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.<ref group="Note">However, the [[frequency]] of light can depend on the motion of the source relative to the observer, due to the [[Doppler effect]].</ref> This invariance of the speed of light was postulated by Einstein in 1905,<ref name="stachel" /> after being motivated by [[Maxwell's theory of electromagnetism]] and the lack of evidence for motion against the [[luminiferous aether]];.<ref>
{{citeCite 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=Germande
|bibcode=1905AnP...322..891E
|url=http://sedici.unlp.edu.ar/handle/10915/2786
}} English translation: {{cite web
|type=Submitted manuscript
|last=Perrett |first=W
|doi-access=free
|last2=Jeffery |first2=GB (tr.)
}} English translation:
|last3=Walker |first3=J (ed.)
{{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
|accessdate=2009-11-27
}}</ref> itIt has since been consistently confirmed by many experiments.<ref group="Note">See [[Michelson–Morley experiment]] and [[Kennedy–Thorndike experiment]], for example.</ref> 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.<ref However,name=Hsu2/><ref byname=Zhang/>
By adopting [[Einstein synchronization]] for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition.<ref name=HsuHsu2>
{{Cite book
|lastlast1=Hsu |firstfirst1=J.-P. |last2=Zhang |first2=YZY. Z.
|year=2001
|title=Lorentz and Poincaré Invariance
|url=httphttps://books.google.com/books?id=jryk42J8oQIC&pg=RA1-PA541#v=onepage&q=
|publisher=[[World Scientific]]
|series=Advanced Series on Theoretical Physical Science
|volume=8 |pages=543''ff''543ff
|isbn=978-981-02-4721-48
}}</ref><ref name=Zhang>
{{Cite book
|last=Zhang |first =YZ Zhang
|first = Y. Z.
|year=1997
|year = 1997
|title = Special Relativity and Its Experimental Foundations
|url = https://archive.org/details/specialrelativit0000chan/page/172
|url=http://www.worldscibooks.com/physics/3180.html
|publisher = [[World Scientific]]
|series = Advanced Series on Theoretical Physical Science
|volume=4 |pages =172–3 4
|pages = [https://archive.org/details/specialrelativit0000chan/page/172 172–173]
|isbn=981-02-2749-3
|isbn = 978-981-02-2749-4
|access-date = 23 July 2009
}}</ref> The [[special theory of relativity]] explores the consequences of this invariance of ''c'' with the assumption that the laws of physics are the same in all inertial frames of reference.<ref>
{{Cite book
|last=d'Inverno
|first=R.
|year=1992
|title=Introducing Einstein's Relativity
|pages=[https://archive.org/details/introducingeinst0000dinv/page/19 19–20]
|pages=19–20
|publisher=[[Oxford University Press]]
|isbn=978-0-19-859686-38
|url=https://archive.org/details/introducingeinst0000dinv/page/19
}}</ref><ref>
{{Cite book
|last=Sriranjan |first=B.
|year=2004
|chapter=Postulates of the special theory of relativity and their consequences
|chapterurlchapter-url=httphttps://books.google.com/books?id=FsRfMvyudlAC&pg=PA20#v=onepage&q=&f=false
|title=The Special Theory to Relativity
|publisher=[[PHI Learning]] Pvt. Ltd.
|isbn=978-81-203-1963-X9
|pages=20 ''ff''20ff
}}</ref> One consequence is that ''c'' is the speed at which all [[massless particle]]s and waves, including light, must travel in vacuum.<ref name=":0">{{Cite book|last1=Ellis|first1=George F. R.|url=https://www.worldcat.org/oclc/44694623|title=Flat and Curved Space-times|last2=Williams|first2=Ruth M.|date=2000|publisher=Oxford University Press|isbn=0-19-850657-0|edition=2|location=Oxford|oclc=44694623|author-link=George F. R. Ellis|author-link2=Ruth Margaret Williams |page=12}}</ref>{{#tag:ref|Because [[neutrino]]s have a small but non-zero mass, they travel through empty space [[Measurements of neutrino speed|very slightly more slowly than light]]. However, because they pass through matter much more easily than light does, there are in theory occasions when the neutrino signal from an astronomical event might reach Earth before an optical signal can, like [[supernova]]e.<ref>{{Cite journal |last1=Antonioli |first1=Pietro |last2=Fienberg |first2=Richard Tresch |last3=Fleurot |first3=Fabrice |last4=Fukuda |first4=Yoshiyuki |last5=Fulgione |first5=Walter |last6=Habig |first6=Alec |last7=Heise |first7=Jaret |last8=McDonald |first8=Arthur B. |last9=Mills |first9=Corrinne |last10=Namba |first10=Toshio |last11=Robinson |first11=Leif J. |date=2 September 2004 |title=SNEWS: the SuperNova Early Warning System |url=https://iopscience.iop.org/article/10.1088/1367-2630/6/1/114 |journal=[[New Journal of Physics]] |volume=6 |pages=114 |doi=10.1088/1367-2630/6/1/114 |arxiv=astro-ph/0406214 |bibcode=2004NJPh....6..114A |s2cid=119431247 |issn=1367-2630}}</ref>|group="Note"}}
}}</ref> One consequence is that ''c'' is the speed at which all [[massless particle]]s and waves, including light, must travel in vacuum.
[[File:Lorentz factor.svg|thumb|upright|alt=γ starts at 1 when v equals zero and stays nearly constant for small v's, then it sharply curves upwards and has a vertical asymptote, diverging to positive infinity as v approaches c. |The [[Lorentz factor]] ''γ'' as a function of velocity. It starts at{{nbsp}}1 and approaches infinity as ''v'' approaches ''c''.]]
[[File:Lorentz factor.svg|thumb|left|upright|alt=γ starts at 1 when v equals zero and stays nearly constant for small v's, then it sharply curves upwards and has a vertical asymptote, diverging to positive infinity as v approaches c. |The Lorentz factor ''γ'' as a function of velocity. It starts at 1 and approaches infinity as ''v'' approaches ''c''.]]
Special relativity has many counterintuitive and experimentally verified implications.<ref>
{{citeCite web
|last1 = Roberts
|last=Roberts |first=T
|first1 = T.
|last2=Schleif |first2=S
|last2 = Schleif
|last3=Dlugosz |first3=JM (ed.)
|first2 = S.
|year=2007
|editor-last = Dlugosz
|title=What is the experimental basis of Special Relativity?
|editor-first = J. M.
|url=http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
|year = 2007
|work=Usenet Physics FAQ
|title = What is the experimental basis of Special Relativity?
|publisher=[[University of California, Riverside]]
|url = http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
|accessdate=2009-11-27
|work = Usenet Physics FAQ
}}</ref> These include the [[Mass–energy equivalence|equivalence of mass and energy]] {{nowrap|(''E'' {{=}} ''mc''<sup>2</sup>)}}, [[length contraction]] (moving objects shorten),{{#tag:ref|Whereas moving objects are ''measured'' to be shorter along the line of relative motion, they are also ''seen'' as being rotated. This effect, known as [[Terrell rotation]], is due to the different times that light from different parts of the object takes to reach the observer.<ref>
|publisher = [[University of California, Riverside]]
{{cite journal
|access-date = 27 November 2009
|last=Terrell |first=J
|archive-url = https://web.archive.org/web/20091015153529/http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
|archive-date = 15 October 2009
|url-status=dead
}}</ref> These include the [[Mass–energy equivalence|equivalence of mass and energy]] {{nowrap|(''E'' {{=}} ''mc''{{i sup|2}})}}, [[length contraction]] (moving objects shorten),{{#tag:ref|Whereas moving objects are ''measured'' to be shorter along the line of relative motion, they are also ''seen'' as being rotated. This effect, known as [[Terrell rotation]], is due to the different times that light from different parts of the object takes to reach the observer.<ref>
{{Cite journal
|last=Terrell |first=J.
|year=1959
|title=Invisibility of the Lorentz Contraction
|journal=[[Physical Review]]
|volume=116
|issue=4 |pages=1041–51041–1045
|doi=10.1103/PhysRev.116.1041
|bibcode = 1959PhRv..116.1041T
}}</ref><ref>
{{citeCite journal
|last=Penrose |first=R.
|year=1959
|title=The Apparent Shape of a Relativistically Moving Sphere
|journal=[[Proceedings of the Cambridge Philosophical Society]]
|volume=55
|issue=011 |pages=137–9137–139
|doi=10.1017/S0305004100033776
|bibcode=1959PCPS...55..137P |s2cid=123023118
|bibcode = 1959PCPS...55..137P }}</ref>|group="Note"}} 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''<sup>2</sup>/''c''<sup>2</sup>)<sup>−1/2</sup>}}, where ''v'' is the speed of the object. The difference of ''γ'' from 1 is negligible for speeds much slower than ''c'', such as most everyday speeds—in which case special relativity is closely approximated by [[Galilean relativity]]—but it increases at relativistic speeds and diverges to infinity as ''v'' approaches ''c''.
}}</ref>|group="Note"}} 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}})<sup>−1/2</sup>}}, where ''v'' is the speed of the object. The difference of ''γ'' from{{nbsp}}1 is negligible for speeds much slower than ''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 in physics|symmetry]] called [[Lorentz invariance]], whose mathematical formulation contains the parameter ''c''.<ref>
{{Cite book
|last=Hartle
|first=JBJ. B.
|year=2003
|title=Gravity: An Introduction to Einstein's General Relativity
|pages=[https://archive.org/details/specialrelativit0000chan/page/52 52–59]
|pages=52–9
|publisher=[[Addison-Wesley]]
|isbn=978-981-02-2749-34
|url=https://archive.org/details/specialrelativit0000chan/page/52
}}</ref> Lorentz invariance is an almost universal assumption for modern physical theories, such as [[quantum electrodynamics]], [[quantum chromodynamics]], the [[Standard Model]] of [[particle physics]], and [[general relativity]]. As such, the parameter ''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]].<ref name="Hartle">
}}</ref> Lorentz invariance is an almost universal assumption for modern physical theories, such as [[quantum electrodynamics]], [[quantum chromodynamics]], the [[Standard Model]] of [[particle physics]], and [[general relativity]]. As such, the parameter ''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]],<ref name="Hartle">
{{Cite book
|last=Hartle
|first=JBJ. B.
|year=2003
|title=Gravity: An Introduction to Einstein's General Relativity
|page=332
|publisher=[[Addison-Wesley]]
|isbn=978-981-02-2749-34
|url=https://archive.org/details/specialrelativit0000chan
}}</ref><ref name="Brügmann">The interpretation of observations on binary systems used to determine the speed of gravity is considered doubtful by some authors, leaving the experimental situation uncertain; see{{Cite book
|url-access=limited
|last1=Schäfer |first1=G
}}</ref> and observations of gravitational waves have been consistent with this prediction.<ref>See, for example:
|first2=MH |last2=Brügmann
* {{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}}
|editor1-first=H |editor1-last=Dittus
* {{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}}
|editor2-first=C |editor2-last=Lämmerzahl
* {{Cite journal |last1=Liu |first1=Xiaoshu |last2=He |first2=Vincent F. |last3=Mikulski |first3=Timothy M. |last4=Palenova |first4=Daria |last5=Williams |first5=Claire E. |last6=Creighton |first6=Jolien |last7=Tasson |first7=Jay D. |date=7 July 2020 |title=Measuring the speed of gravitational waves from the first and second observing run of Advanced LIGO and Advanced Virgo |url=https://link.aps.org/doi/10.1103/PhysRevD.102.024028 |journal=[[Physical Review D]] |volume=102 |issue=2 |pages=024028 |doi=10.1103/PhysRevD.102.024028 |arxiv=2005.03121 |bibcode=2020PhRvD.102b4028L |s2cid=220514677}}</ref> In [[non-inertial frame]]s of reference (gravitationally curved spacetime or [[accelerated reference frame]]s), the ''local'' speed of light is constant and equal to ''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.<ref name="Gibbs1997" />
|editor3-first=SG |editor3-last=Turyshev
|chapter=Propagation of light in the gravitational filed of binary systems to quadratic order in Newton's gravitational constant: Part 3: ‘On the speed-of-gravity controversy’
|url=http://books.google.com/?id=QYnfdXOI8-QC&pg=PA111
|title=Lasers, clocks and drag-free control: Exploration of relativistic gravity in space
|isbn=3-540-34376-8
|year=2008
|publisher=Springer
}}</ref> In [[non-inertial frame]]s of reference (gravitationally curved space or [[accelerated reference frame]]s), the ''local'' speed of light is constant and equal to ''c'', but the [[Propagation of light in non-inertial reference frames|speed of light along a trajectory of finite length]] can differ from ''c'', depending on how distances and times are defined.<ref name="Gibbs1997" />
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 [[Variable speed of light|speed of light may have changed over time]].<ref name=Ellis_Uzan>
{{citeCite journal
|lastlast1=Ellis |firstfirst1=GFRG. F. R. |last2=Uzan |first2=J.-P.
|year=2005
|title=‘c’'c' is the speed of light, isn’tisn't it?
|journal=[[American Journal of Physics]]
|volume=73
|issue=3 |pages=240–7240–227
|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 }}</ref><ref name|s2cid=Mota>An overview can be found in the dissertation of {{cite arxiv119530637
}}</ref><ref name=Mota>
|last=Mota |first=DF
{{Cite thesis |type=PhD
|last=Mota |first=D. F.
|year=2006
|title=Variations of the fineFine structureStructure constantConstant in spaceSpace and timeTime
|classarxiv=astro-ph/0401631
|bibcode=2004astro.ph..1631M
|eprint=astro-ph/0401631
}}</ref> No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.<ref name=Uzan>
{{citeCite journal
|last=Uzan |first=J.-P.
|year=2003
|title=The fundamental constants and their variation: observational status and theoretical motivations
Line 335 ⟶ 362:
|doi=10.1103/RevModPhys.75.403
|arxiv=hep-ph/0205340
|bibcode=2003RvMP...75..403U
|s2cid=118684485
}}</ref><ref name=Camelia>
{{citeCite arxivjournal
|last=Amelino-Camelia |first=G.
|year=20082013
|title=Quantum Gravity Phenomenology
|arxiv=0806.0339
|class=gr-qc
|doi=10.12942/lrr-2013-5
|eprint=0806.0339
|pmid=28179844
|pmc=5255913
|volume=16
|issue=1
|pages=5
|journal=Living Reviews in Relativity
|doi-access=free
|bibcode=2013LRR....16....5A
}}</ref>
It also is generally assumed that the two-way speed of light is [[isotropy|isotropic]], meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear [[energy level]]s as a function of the orientation of the emitting [[atomic nucleus|nuclei]] in a magnetic field (see [[Hughes–Drever experiment]]), and of rotating [[optical resonator]]s (see [[Michelson–Morley experiment#Recent experiments|Resonator experiments]]) have put stringent limits on the possible two-way [[anisotropy]].<ref name=Herrmann>
{{citeCite journal
|last1=Herrmann |first1=S. |last2=Senger |first2=A. |last3=Möhle |first3=K. |last4=Nagel |first4=M. |last5=Kovalchuk |first5=EVE. V. |last6=Peters |first6=A |display-authors=1.
|title=Rotating optical cavity experiment testing Lorentz invariance at the 10<sup>−17</sup> level
|journal=Physical Review D |volume=80 |issue=100 |pages=105011 |year=2009
|doi=10.1103/PhysRevD.80.105011 |arxiv=1002.1284 |bibcode = 2009PhRvD..80j5011H |s2cid=118346408 }}</ref><ref name=Lang>{{Cite book
|title=Astrophysical formulae
|first=KRK. R. |last=Lang
|url=httphttps://books.google.com/books?id=OvTjLcQ4MCQC&pg=PA152
|page=152
|isbn=978-3-540-29692-18
|publisher=Birkhäuser
|edition=3rd3
|year=1999
}}</ref>
=== Upper limit on speeds ===
According to special relativity, the energy of an object with [[rest mass]] ''m'' and speed ''v'' is given by {{nowrap|''γmc''<{{i sup>|2</sup>}}}}, where ''γ'' is the Lorentz factor defined above. When ''v'' is zero, ''γ'' is equal to one, giving rise to the famous {{nowrap|''E'' {{=}} ''mc''<{{i sup>|2</sup>}}}} formula for [[mass-energymass–energy equivalence]]. 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. The speed of light is the upper limit for the speeds of objects with positive rest mass., Thisand isindividual experimentallyphotons establishedcannot intravel manyfaster [[teststhan ofthe relativisticspeed energyof and momentum]]light.<ref>See, for example:
* {{Cite web |url=http://latimesblogs.latimes.com/technology/2011/07/time-travel-impossible.html |title=It's official: Time machines won't work |website=Los Angeles Times |date=25 July 2011}}
{{cite web
* {{Cite web |date=19 July 2011 |website=The Hong Kong University of Science and Technology |url=http://www.ust.hk/eng/news/press_20110719-893.html |title=HKUST Professors Prove Single Photons Do Not Exceed the Speed of Light}}
|last=Fowler |first=M
* {{Cite journal |title=Optical Precursor of a Single Photon |author1=Shanchao Zhang |author2=J. F. Chen |author3=Chang Liu |author4=M. M. T. Loy |author5=G. K. L. Wong |author6=Shengwang Du |journal=[[Physical Review Letters]] |volume=106 |issue=243602 |pages=243602 |date=16 June 2011 |doi=10.1103/physrevlett.106.243602|pmid=21770570 |bibcode=2011PhRvL.106x3602Z |url=http://repository.ust.hk/ir/bitstream/1783.1-7246/1/PhysRevLett.106.243602.pdf }}
</ref> This is experimentally established in many [[tests of relativistic energy and momentum]].<ref>
{{Cite web
|last=Fowler |first=M.
|date=March 2008
|title=Notes on Special Relativity
|url=http://galileo.phys.virginia.edu/classes/252/SpecRelNotes.pdf
|page=56
|publisher=[[University of Virginia]]
|accessdateaccess-date=7 May 2010-05-07
}}</ref>
[[File:Relativity of Simultaneity.svg|thumb|right|Event A precedes B in the red frame, is simultaneous with B in the green frame, and follows B in the blue frame.|alt=Three pairs of coordinate axes are depicted with the same origin A; in the green frame, the x axis is horizontal and the ct axis is vertical; in the red frame, the x′ axis is slightly skewed upwards, and the ct′ axis slightly skewed rightwards, relative to the green axes; in the blue frame, the x′′ axis is somewhat skewed downwards, and the ct′′ axis somewhat skewed leftwards, relative to the green axes. A point B on the green x axis, to the left of A, has zero ct, positive ct′, and negative ct′′.]]
More generally, it is normally impossible for informationsignals or energy to travel faster than ''c''. One argument for this follows from the counter-intuitive implication of special relativity known as the [[relativity of simultaneity]]. 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 (physics)|causality]] would be violated.{{#tag:ref|It ishas been thoughtspeculated that the [[Scharnhorst effect]] does allow signals to travel slightly faster than ''c'', but the specialvalidity conditionsof inthose whichcalculations thishas effectbeen can occur prevent one from using this effect to violate causality.questioned,<ref>See, for example:
* {{Cite journal|last=Ben-Menahem|first=Shahar|date=November 1990|title=Causality between conducting plates|url=https://linkinghub.elsevier.com/retrieve/pii/037026939091167A|journal=Physics Letters B|language=en|volume=250|issue=1–2|pages=133–138|doi=10.1016/0370-2693(90)91167-A|bibcode=1990PhLB..250..133B|osti=1449261}}
{{cite journal
* {{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}}
|last=Liberati |first=S |last2=Sonego |first2=S |last3=Visser |first3=M
* {{Cite journal |last=Fearn |first=H. |date=May 2007 |title=Can light signals travel faster than c in nontrivial vacua in flat space-time? Relativistic causality II |url=http://link.springer.com/10.1134/S1054660X07050155 |journal=Laser Physics |language=en |volume=17 |issue=5 |pages=695–699 |doi=10.1134/S1054660X07050155 |arxiv=0706.0553 |bibcode=2007LaPhy..17..695F |s2cid=61962 |issn=1054-660X}}</ref> and it appears the special conditions in which this effect might occur would prevent one from using it to violate causality.<ref>
{{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=167–85167–185
|doi=10.1006/aphy.2002.6233
|arxiv=gr-qc/0107091
|bibcode = 2002AnPhy.298..167L |s2cid=48166
}}</ref>|group="Note"}}<ref name="Taylor_p74">
{{Cite book
|last1=Taylor
|last=Taylor |first=EF |last2=Wheeler |first2=JA
|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]
|pages=74–5
|publisher=[[W. H. Freeman]]
|isbn=978-0-7167-2327-1
|url=https://archive.org/details/spacetimephysics00edwi_0/page/74
}}</ref> In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded,<ref name=Zhang/> and would lead to [[paradox]]es such as the [[tachyonic antitelephone]].<ref>
{{Cite book
|last=Tolman |first=RCR. C.
|year=2009 |origyearorig-year=1917
|chapter=Velocities greater than that of light
|title=The Theory of the Relativity of Motion
Line 401 ⟶ 453:
}}</ref>
== Faster-than-light observations and experiments ==
{{MainSee also|Faster-than-light|Superluminal motion}}
{{see|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 [[#In a medium|propagation of light in a medium]] section below, many wave velocities can exceed ''c''. For example, theThe [[phase velocity]] of [[X-ray]]s through most glasses can routinely exceed ''c'',<ref>
{{Cite book
|last=Hecht |first=E.
|year=1987
|title=Optics
|page=62
|edition=2nd2
|publisher=[[Addison-Wesley]]
|isbn=978-0-201-11609-X0
}}</ref> but phase velocity does not determine the velocity at which waves convey information.<ref>
{{citeCite book
|last=Quimby |first=RSR. S.
|title=Photonics and lasers: an introduction
|publisher=John Wiley and Sons
Line 421 ⟶ 473:
|page=9
|isbn=978-0-471-71974-8
|url=httphttps://books.google.com/books?id=yWeDVfaVGxsC&lpg=PA9&pg=PA9#v=onepage
}}</ref>
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.<ref>
{{citeCite news
|last=Wertheim |first=M.
|title=The Shadow Goes
|url=httphttps://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]]
|accessdateaccess-date=21 August 2009-08-21
|date=2007-06-20 June 2007
}}</ref> In neither case does any matter, energy, or information travel faster than light.<ref name=Gibbs>
{{citeCite web
|last=Gibbs
|first=P.
|year=1997
|title=Is Faster-Than-Light Travel or Communication Possible?
Line 440 ⟶ 493:
|publisher=[[University of California, Riverside]]
|work=Usenet Physics FAQ
|accessdateaccess-date=20 August 2008-08-20
|archive-date=10 March 2010
|archivedate=2009-11-17
|archive-url=https://web.archive.org/web/20100310205556/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html
|archiveurl=http://www.webcitation.org/5lLRguF0I
|url-status=dead
}}</ref>
The rate of change in the distance between two objects in a frame of reference with respect to which both are moving (their [[Faster-than-light#Closing speeds|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.<ref name="Gibbs" />
Certain quantum effects appear to be transmitted instantaneously and therefore faster than ''c'', as in the [[EPR paradox]]. An example involves the [[quantum state]]s of two particles that can be [[quantum entanglement|entangled]]. Until either of the particles is observed, they exist in a [[quantum superposition|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 (i.e., faster than light could travel from one particle to the other). 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.<ref name=Gibbs /><ref>See, for example:
* {{Cite book
|last=Sakurai |first=JJJ. J. |author-link=J. J. Sakurai
|year=1994
|editor-last=TTuan |editor-first=S. F.
|title=[[Modern Quantum Mechanics]] |edition=Revised |pages=[https://archive.org/details/modernquantummec00saku_488/page/n243 231]–232
|publisher=Addison-Wesley
|edition=Revised |pages=231–232
|isbn=978-0-201-53929-5
|publisher=[[Addison-Wesley]]
}}
|isbn=0-201-53929-2
* {{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 (society)|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.
}}</ref>
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 [[quantum tunnelling|tunnel]] through a barrier is constant, regardless of the thickness of the barrier.<ref name=Muga>
{{Cite book
|editor-last=Muga |editor-first=JGJ. G. |editor-last2=Mayato |editor-first2=RSR. S. |editor-last3=Egusquiza |editor-first3=IL,I. edsL.
|year=2007
|title=Time in Quantum Mechanics
|url=httphttps://books.google.com/books?id=InKru6zHQWgC&pg=PA48
|page=48
|publisher=[[Springer (publisher)|Springer]]
|isbn=978-3-540-73472-47
}}</ref><ref name=Recami>
{{Cite book
|lastlast1=Hernández-Figueroa |firstfirst1=HEH. E. |last2=Zamboni-Rached |first2=M. |last3=Recami |first3=E.
|year=2007
|title=Localized Waves
|url=httphttps://books.google.com/books?id=xxbXgL967PwC&pg=PA26
|page=26
|publisher=[[Wiley Interscience]]
|isbn=978-0-470-10885-17
}}</ref> This could result in a virtual particle crossing a large gap faster- than- light. However, no information can be sent using this effect.<ref name=Wynne>
{{citeCite journal
|last=Wynne
|first=K.
|year=2002
|title=Causality and the nature of information
|url=http://144.206.159.178/ft/809/64567/1101504.pdf
|journal=[[Optics Communications]]
|volume=209
|issue=1–3
|pages=84–100
|doi=10.1016/S0030-4018(02)01638-3
|bibcode=2002OptCo.209...85W
}} [http://web.archive.org/web/20090325093856/ |url=http://bcp.phys.strath.ac.uk/the_group/r/uf/2002-OC-causality.pdf archive]</ref>
|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
}}</ref>
So-called [[superluminal motion]] is seen in certain astronomical objects,<ref>
{{citeCite journal
|last=Rees |first=M. |author-link=Martin Rees
|year=1966
|title=The Appearance of Relativistically Expanding Radio Sources
Line 498 ⟶ 569:
|issue=5048 |page=468
|doi=10.1038/211468a0
|bibcode = 1966Natur.211..468R |s2cid=41065207
|bibcode = 1966Natur.211..468R }}</ref> such as the [[relativistic jet]]s of [[radio galaxy|radio galaxies]] and [[quasar]]s. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a [[graphical projection|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.<ref>
}}</ref> such as the [[relativistic jet]]s of [[radio galaxy|radio galaxies]] and [[quasar]]s. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a [[graphical projection|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.<ref>
{{cite web
{{Cite web
|last=Chase |first=IP
|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
|accessdate=2009-11-26
}}</ref>
A 2011 experiment where [[Faster-than-light neutrino anomaly|neutrinos were observed to travel faster than light]] turned out to be due to experimental error.<ref name=nature1204>{{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}}</ref><ref>{{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 }}</ref>
In models of the expanding universe, the farther galaxies are from each other, the faster they drift apart. This receding is not due to motion ''through'' space, but rather to the [[Metric expansion of space|expansion of space]] itself.<ref name="Gibbs" /> For example, galaxies far away from Earth appear to be moving away from the Earth with a speed 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.<ref name=Harrison>
In models of the [[expansion of the universe|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.<ref name=Harrison>
{{Cite book
|last= Harrison |first=ERE. R.
|year=2003
|title=Masks of the Universe
|url=httphttps://books.google.com/books?id=tSowGCP0kMIC&pg=PA206
|page=206
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-77351-25
}}</ref>
These recession rates, defined as the increase in [[comoving and proper distances|proper distance]] per [[cosmological time]], are not velocities in a relativistic sense. Faster-than-light cosmological recession speeds are only a [[coordinate conditions|coordinate]] artifact.
== Propagation of light ==
In September 2011, physicists working on the [[OPERA experiment]] published results that suggested beams of [[neutrino]]s had travelled from [[CERN]] (in Geneva, Switzerland) to [[Laboratori Nazionali del Gran Sasso|LNGS]] (at the Gran Sasso, Italy) faster than the speed of light.<ref name="OPERA">
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]] ''ε''<sub>0</sub> and the [[magnetic constant]] ''μ''<sub>0</sub>, by the equation<ref>{{Cite book
{{cite arxiv
|last1=Panofsky |first1=W. K. H.
|title=Measurement of the neutrino velocity with the OPERA detector in the CNGS beam
|last2=Phillips |first2=M.
|author=OPERA Collaboration
|author-link=OPERA experiment
|eprint=1109.4897 |class=hep-ex
|year=2011}}</ref> These findings, sometimes referred to as the [[faster-than-light neutrino anomaly]], were subsequently determined—subject to further confirmation—to be the result of a measurement error.<ref>{{cite news
| title = BREAKING NEWS: Error Undoes Faster-Than-Light Neutrino Results
| first = Edwin
| last = Cartlidge
| url = http://news.sciencemag.org/scienceinsider/2012/02/breaking-news-error-undoes-faster.html
| newspaper = Science
| accessdate = 2012-02-22
| date=2012-02-22}}</ref>
==Propagation of light==
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 through the vacuum is related to the [[electric constant]] ''ε''<sub>0</sub> and the [[magnetic constant]] ''μ''<sub>0</sub> by the equation {{nowrap|''c'' {{=}} 1/{{radic|''ε''<sub>0</sub>''μ''<sub>0</sub>}}}}.<ref>{{Cite book
|last=Panofsky |first=WKH
|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]
|page=182
|isbn=978-0-201-05702-7
}}</ref>
}}</ref> 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 [[photon]]s. In QED, photons are [[massless particle]]s and thus, according to special relativity, they travel at the speed of light in vacuum.
: <math> c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}. </math>
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 [[photon]]s. In QED, photons are [[massless particle]]s and thus, according to special relativity, they travel at the speed of light in vacuum.<ref name=":0" />
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.<ref name=Gibbs1997>
{{citeCite web
|last=Gibbs
|first=P.
|year=1997 |origyear=1996
|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
|accessdate=2009-11-26
|archive-url=https://web.archive.org/web/20100402090332/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html
|archiveurl=http://www.webcitation.org/5lLQD61qh
|archive-date=2 April 2010
|archivedate=2009-11-17
|url-status=dead
}}</ref> No variation of the speed of light with frequency has been observed in rigorous testing,<ref name=Schaefer>
}}</ref> No variation of the speed of light with frequency has been observed in rigorous testing, putting stringent limits on the mass of the photon.<ref>See, for example:
{{cite journal
* {{Cite journal
|last=Schaefer |first=BE
|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=4964–64964–4966
|doi=10.1103/PhysRevLett.82.4964
|arxiv=astro-ph/9810479
|bibcode=1999PhRvL..82.4964S
|s2cid=119339066
}}</ref><ref name=Sakharov>
}}
{{cite journal
* {{Cite journal
|last=Ellis |first=J
|last2last1=MavromatosEllis |first2first1=NEJ.
|last2=Mavromatos |first2=N. E. |author-link2=N. E. Mavromatos
|last3=Nanopoulos |first3=DV
|last3=Nanopoulos |first3=D. V.
|last4=Sakharov |first4=AS
|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=409–24409–424
|doi=10.1051/0004-6361:20030263
|arxiv=astro-ph/0210124 |bibcode=2003A&A...402..409E
|s2cid=15388873
}}</ref><ref name="Füllekrug">
}}
{{cite journal
* {{Cite journal
|last=Füllekrug |first=M
|last=Füllekrug |first=M.
|year=2004
|title=Probing the Speed of Light with Radio Waves at Extremely Low Frequencies
Line 590 ⟶ 659:
|issue=4 |page=043901
|doi=10.1103/PhysRevLett.93.043901
|bibcode=2004PhRvL..93d3901F
|pmid=15323762
}}</ref> putting stringent limits on the mass of the photon. The limit obtained depends on the model used: if the massive photon is described by [[Proca action|Proca theory]],<ref name="adelberger">{{cite journal
}}
|last=Adelberger |first=E
* {{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}}</ref> The limit obtained depends on the model used: if the massive photon is described by [[Proca action|Proca theory]],<ref name="adelberger">
|last2=Dvali |first2=G
{{Cite journal
|last3=Gruzinov |first3=A
|last1=Adelberger |first1=E.
|last2=Dvali |first2=G.
|last3=Gruzinov |first3=A.
|year=2007
|title=Photon Mass Bound Destroyed by Vortices
Line 602 ⟶ 674:
|doi=10.1103/PhysRevLett.98.010402
|arxiv=hep-ph/0306245 |pmid=17358459 |bibcode=2007PhRvL..98a0402A
|s2cid=31249827
}}</ref> the experimental upper bound for its mass is about 10<sup>−57</sup> [[gram]]s;<ref name=Sidharth>
{{Cite book
|last=Sidharth |first=BGB. G.
|year=2008
|title=The Thermodynamic Universe
|url=httphttps://books.google.com/books?id=OUfHR36wSfAC&pg=PA134
|page=134
|publisher=[[World Scientific]]
|isbn=978-981-281-234-26
}}</ref> if photon mass is generated by a [[Higgs mechanism]], the experimental upper limit is less sharp, {{nowrap|''m'' ≤ 10<sup>−14</sup> [[Electron volt#As a unit of mass{{val|e=-14|ul=eV/c<sup>2</sup>]]c2}}}} <ref name="adelberger"/> (roughly 2 × 10<sup>−47</sup> g).<ref name="adelberger" />
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 the spectrum of [[gamma-ray burst]] [[GRB 090510]] didfound notno findevidence anyfor differencea in the speedsdependence of photonsphoton ofspeed differenton energiesenergy, confirmingsupporting thattight Lorentzconstraints invariancein isspecific verifiedmodels atof leastspacetime downquantization toon thehow scalethis ofspeed theis [[Planckaffected length]]by (''l''<sub>P</sub> = {{radic|[[reducedphoton Planckenergy for energies approaching the constant|''ħ'']][[gravitationalPlanck constant|''G''scale]]/''c''<sup>3</sup>}} ≈ {{val|1.6163|e=-35|u=m}}) divided by 1.2.<ref>{{cite journal
{{Cite journal
|last=Amelino-Camelia |first=G
|last=Amelino-Camelia |first=G.
|year=2009
|title=Astrophysics: Burst of support for relativity
Line 620 ⟶ 694:
|volume=462 |pages=291–292
|doi=10.1038/462291a
|laysummary=http://www.nature.com/nature/journal/v462/n7271/edsumm/e091119-06.html
|laysource=Nature |laydate=19 November 2009
|pmid=19924200
|issue=7271
|bibcode = 2009Natur.462..291A }}</ref>|s2cid=205051022
|doi-access=free
}}</ref>
=== In a medium ===
{{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''<sub>p</sub>. An actualA physical signal with a finite extent (a pulse of light) travels at a different speed. The largestoverall part[[Envelope (waves)|envelope]] of the pulse travels at the [[group velocity]] ''v''<sub>g</sub>, and its earliest part travels at the [[front velocity]] ''v''<sub>f</sub>.<ref name="Milonni">{{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}}</ref>
[[File:frontgroupphase.gif|thumb|left|The blue dot moves at the speed of the ripples, the phase velocity; the green dot moves with the speed of the envelope, the group velocity; and the red dot moves with the speed of the foremost part of the pulse, the front velocity.|alt=A modulated wave moves from left to right. There are three points marked with a dot: A blue dot at a node of the carrier wave, a green dot at the maximum of the envelope, and a red dot at the front of the envelope.]]
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''<sub>p</sub> 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 (waves)|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.<ref name=Podesta>
{{Cite book
|last=de Podesta |first=M.
|year=2002
|title=Understanding the Properties of Matter
|url=httphttps://books.google.com/books?id=h8BNvnR050cC&pg=PA131&lpg=PA131
|page=131
|publisher=CRC Press
|isbn=978-0-415-25788-36
}}</ref> Denser media, such as [[Optical properties of water and ice|water]],<ref>
{{citeCite web
|title=RefractiveOptical indexconstants of H<sub>2</sub>O, D<sub>2</sub>O (Water, H20heavy water, [Liquids]ice)
|url=httphttps://refractiveindex.info/?groupshelf=LIQUIDSmain&materialbook=WaterH2O&page=Hale
|publisher=Mikhail Polyanskiy
|work=refractiveindex.info
|access-date=7 November 2017
|accessdate =2010-03-14
}}</ref> [[glass]],<ref>
{{citeCite web
|title=RefractiveOptical indexconstants of FusedSoda Silicalime [Glasses]glass
|url=httphttps://refractiveindex.info/?groupshelf=GLASSESglass&materialbook=F_SILICAsoda-lime&page=Rubin-clear
|publisher=Mikhail Polyanskiy
|work=refractiveindex.info
|access-date=7 November 2017
|accessdate =2010-03-14
}}</ref> and [[Material properties of diamond#Optical properties|diamond]],<!--there must be a way to make it clearer where these links go--><ref>
{{citeCite web
|title=RefractiveOptical indexconstants of C [Crystals(Carbon, etc.]diamond, graphite)
|url=httphttps://refractiveindex.info/?groupshelf=CRYSTALSmain&materialbook=C&page=Phillip
|publisher=Mikhail Polyanskiy
|work=refractiveindex.info
|access-date =7 November 2017
|accessdate =2010-03-14
}}</ref> have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light.
}}</ref> have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light. In exotic materials like [[Bose–Einstein condensate]]s near absolute zero, the effective speed of light may be only a few meters 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 this, 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]], one team at [[Harvard University]] and the [[Rowland Institute for Science]] in Cambridge, Mass., and the other at the [[Harvard–Smithsonian Center for Astrophysics]], also in Cambridge. However, 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.<ref>{{cite web|author=Harvard News Office |url=http://www.news.harvard.edu/gazette/2001/01.24/01-stoplight.html |title=Harvard Gazette: Researchers now able to stop, restart light |publisher=News.harvard.edu |date=2001-01-24 |accessdate=2011-11-08}}</ref>
In exotic materials like [[Bose–Einstein condensate]]s 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.<ref>{{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 }}</ref>
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 1 for some frequencies; in some exotic materials it is even possible for the index of refraction to become negative.<ref>{{Cite book
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.<ref>
{{Cite book
|title=Fast light, slow light and left-handed light
|last=Milonni |first=PWP. W.
|author-link1=Peter W. Milonni
|url=http://books.google.com/?id=kE8OUCvt7ecC&pg=PA25
|url=https://books.google.com/books?id=kE8OUCvt7ecC&pg=PA25
|page=25
|isbn=978-0-7503-0926-14
|year=2004
|publisher=CRC Press
}}</ref> 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 relation]]s.<ref>
{{citeCite journal
|last=Toll |first=JSJ. S.
|year=1956
|title=Causality and the Dispersion Relation: Logical Foundations
Line 676 ⟶ 758:
|issue=6 |pages=1760–1770
|doi=10.1103/PhysRev.104.1760
|bibcode = 1956PhRv..104.1760T
|bibcode = 1956PhRv..104.1760T }}</ref> In practical terms, this means that in a material with refractive index less than 1, the absorption of the wave is so quick that no signal can be sent faster than ''c''.
}}</ref><ref>{{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}}</ref> In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly.<ref>{{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}}</ref>
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 (optics)|dispersion]]. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called [[slow light]], which has been confirmed in various experiments.<ref>{{citeSee, for journalexample:
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|url=http://www.nature.com/nature/journal/v409/n6819/pdf/409490a0.pdf
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|bibcode = 2001Natur.409..490L }}<|s2cid=1894748 |url=http:/ref><ref>/www.seas.harvard.edu/haulab/publications/pdf/Stopped_Light_2001.pdf
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The opposite, group velocities exceeding ''c'', haswas alsoproposed beentheoretically shownin 1993 and achieved experimentally in experiment2000.<ref>See, for example:
* {{Cite journal
{{cite news
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|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
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|issue=1 |pages=R34–R37 |doi=10.1103/PhysRevA.48.R34
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* {{Cite journal
|first1=L. J. |last1=Wang
|first2=A. |last2=Kuzmich
|first3=A. |last3=Dogariu
|title=Gain-assisted superluminal light propagation
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|volume=406
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* {{Cite news
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|date=19 July 2000
|title=Beam Smashes Light Barrier
|url=http://news.bbc.co.uk/2/hi/science/nature/841690.stm
|publisherwork=BBC News
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}}</ref> It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.<ref name="MilonniCh2">{{Cite book
* {{Cite web
|title=Fast light, slow light and left-handed light
|first=Greg |last=Gbur |author-link=Greg Gbur
|author=Milonni, PW
|title=Light breaking its own speed limit: how 'superluminal' shenanigans work
|url=http://books.google.com/?id=kE8OUCvt7ecC&pg=PA25
|url=https://skullsinthestars.com/2008/02/26/light-breaking-its-own-speed-limit-how-superluminal-shenanigans-work/
|chapter=2
|date=26 February 2008
|isbn=0-7503-0926-1
|access-date=9 February 2022
|year=2004
}}</ref> It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.<ref name="Milonni" />
|publisher=CRC Press
}}</ref>
None of these options, however, 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''.<ref name="MilonniCh2Milonni" /> {{clrClear}}
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.<ref>{{citeCite journal | last=Cherenkov | first=Pavel A. | authorlinkauthor-link=Pavel Alekseyevich Cherenkov | year=1934 |title=Видимое свечение чистых жидкостей под действием γ-радиации| trans_title|trans-title=Visible emission of cleanpure liquids by action of γ radiation | journal=[[Doklady Akademii Nauk SSSR]] | volume=2 | page=451}} Reprinted: in{{Cite [http://ufnjournal |last=Cherenkov |first=P.ru/ru/articles/ A. |date=1967/10/n/ ''|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 (1967)|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 (eds.), |location=Moscow, |publisher=Nauka (|date=1999) pp. |pages=149–153.}}</ref>
== Practical effects of finiteness ==
The speed of light is of relevance to [[telecommunication|communicationstelecommunications]]: 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.
=== Small scales ===
In [[supercomputercomputer]]s, the speed of light imposes a limit on how quickly data can be sent between [[central processing unit|processorprocessors]]s. If a processor operates at 1 {{nbsp}}[[gigahertz]], a signal can only 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 [[Computer memory|memory]] chips, to minimize communication latencies;, thisand cancare causemust difficultybe withexercised coolingwhen routing wires between them to ensure [[signal integrity]]. If clock frequencies continue to increase, the speed of light willmay eventually become a limiting factor for the internal design of single [[integrated circuit|chips]].<ref name="processorlimit">
{{Cite book
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}}</ref><ref name="processorlimit2">
}} and {{cite conference
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|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 31 – 4 September 4, 2009
|editor=Malyshkin, V.
|publisher=Springer
|booktitle=Parallel Computing Technologies
|isbn=978-3-642-03274-5
|publisher=Springer
|page=26
|isbn=978-3-642-03274-5
|page=26
}}</ref>
=== Large distances on Earth ===
[[File:Light world trip.ogg|thumb|Acoustic representation of the speed of light, at every beep the light makes a full circle around the equator]]
For example, given the equatorial circumference of the Earth is about {{nowrap|40,075 km}} and ''c'' about {{nowrap|300,000 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 travelling around the globe in an [[optical fibre]], the actual transit time is longer, in part because the speed of light is slower by about 35% in an optical fibre, depending on its refractive index ''n''.<ref name=Midwinter>A typical value for the refractive index of optical fibre is between 1.518 and 1.538: {{Cite book
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 [[Transparency and translucency|transparent material]]) the actual transit time is longer, in part because the speed of light is slower by about 35% in optical fibre, depending on its refractive index ''n''.{{#tag:ref|A typical value for the refractive index of optical fibre is between 1.518 and 1.538.<ref name=Midwinter>
| last = Midwinter |first=JE
{{Cite book
| last = Midwinter |first=J. E.
| year = 1991
| title = Optical Fibers for Transmission
| edition = 2nd2
| publisher = [[Krieger Publishing Company]]
| isbn = 978-0-89464-595-12
}}</ref>|group="Note"}} Furthermore, straightStraight lines rarelyare occurrare in global communications situations, and delaysthe aretravel time createdincreases when thesignals signal passespass through an electronic switchswitches or signal regeneratorregenerators.<ref>{{cite web
{{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/
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|url-status=dead
}}</ref>
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.<ref>{{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}}</ref><ref>{{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 }}</ref>
===Spaceflights and astronomy===
[[File:Speed of light from Earth to Moon.gif|thumb|right|upright=1.6|alt=The diameter of the moon is about one quarter of that of Earth, and their distance is about thirty times the diameter of Earth. A beam of light starts from the Earth and reaches the Moon in about a second and a quarter.|A beam of light is depicted travelling between the Earth and the Moon in the time it takes a light pulse to move between them: 1.255 seconds at their mean orbital (surface-to-surface) distance. The relative sizes and separation of the Earth–Moon system are shown to scale.]]
=== Spaceflight and astronomy ===
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 [[Mission Control Center|ground control]] and [[Apollo 8]] when it became the first manned spacecraft to orbit the Moon: for every question, the ground control station had to wait at least three seconds for the answer to arrive.<ref>{{cite web
[[File:Earth and Moon speed of light by James O'Donoghue.gif|thumb|upright=3|alt=The diameter of the moon is about one quarter of that of Earth, and their distance is about thirty times the diameter of Earth. A beam of light starts from the Earth and reaches the Moon in about a second and a quarter.|A beam of light is depicted travelling between the Earth and the Moon in the time it takes a light pulse to move between them: 1.255 seconds at their mean orbital (surface-to-surface) distance. The relative sizes and separation of the Earth–Moon system are shown to scale.]]
|url=http://history.nasa.gov/ap08fj/15day4_orbits789.htm
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 [[Mission Control Center|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.<ref>
|title=Day 4: Lunar Orbits 7, 8 and 9
{{Cite web
|work=The Apollo 8 Flight Journal
|url = https://history.nasa.gov/ap08fj/15day4_orbits789.htm
|publisher=NASA
|title = Day 4: Lunar Orbits 7, 8 and 9
|accessdate=2010-12-16
|work = The Apollo 8 Flight Journal
}}</ref> 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 at least five minutes later, and possibly up to twenty minutes later; it would then take a further five to twenty minutes for instructions to travel from Earth to Mars.
|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
}}</ref>
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.<ref>{{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}}</ref><ref>{{Cite journal |last=Parisi |first=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}}</ref>
NASA must wait several hours for information from a probe orbiting Jupiter, and if it needs to correct a navigation error, the fix will not arrive at the spacecraft for an equal amount of time, creating a risk of the correction not arriving in time.
Receiving light and other signals from distant astronomical sources can even taketakes much longer. For example, it has takentakes 13 billion (13{{e|9}}) years for light to travel to Earth from the faraway galaxies viewed in the [[Hubble Ultra -Deep Field]] images.<ref name=Hubble>
{{citeCite 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
|url=http://hubblesite.org/newscenter/archive/releases/2010/02/full/
|publisher=[[Space Telescope Science Institute]]
}}</ref><ref>
{{citeCite web
|title=The Hubble Ultra Deep Field Lithograph
|url=http://www.nasa.gov/pdf/283957main_Hubble_Deep_Field_Lithograph.pdf
|publisher=NASA
|format=PDF
|access-date=4 February 2010
|publisher=[[NASA]]
}}</ref> 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.<ref name=Hubble/> 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]], [[Galaxy formation and evolution|of galaxies]], and [[history of the universe|of the universe]] itself.<ref>{{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)}}</ref>
|accessdate=2010-02-04
}}</ref> 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.<ref name=Hubble/> 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]], [[Galaxy formation and evolution|of galaxies]], and [[history of the universe|of the universe]] itself.
Astronomical distances are sometimes expressed in [[light-year]]s, especially in [[popular science]] publications and media.<ref>
{{citeCite web
|title=The IAU and astronomical units
|url=http://www.iau.org/public/measuring/
|publisher=[[International Astronomical Union]]
|access-date=11 October 2010
|accessdate=2010-10-11
}}</ref> A light-year is the distance light travels in one [[Julian year (astronomy)|Julian year]], around 9461 billion kilometres, 5879 billion miles, or 0.3066 [[parsec]]s. 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.<ref name=starchild>Further discussion can be found at
{{citeCite 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]]
|accessdateaccess-date=22 August 2009-08-22
}}</ref>
=== Distance measurement ===
{{mainMain|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 [[Radar#Transit time|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 300,000 {{val|300000|u=kilometres}} (186,000 {{val|186000|u=miles}}) in one second, these measurements of small fractions of a second must be very precise. The [[Lunar Laser Ranging Experimentexperiment]], [[radar astronomy]] and the [[Deep Space Network]] determine distances to the Moon,<ref name=science265_5171_482>
{{citeCite journal
|last=Dickey |first=JOJ. O.
|coauthors=''et al.''
|title=Lunar Laser Ranging: A Continuing Legacy of the Apollo Program
|journal=Science | volume=265 | issue=5171
|pages=482–490 | monthdate=July | year=1994
|doi=10.1126/science.265.5171.482
|bibcode=1994Sci...265..482D | pmid=17781305}}</ref> planets<ref name|s2cid=cm26_181>{{cite journal10157934
|display-authors=etal|url=https://trs.jpl.nasa.gov/bitstream/2014/32452/1/94-0193.pdf}}</ref> planets<ref name=cm26_181>{{Cite journal
|last=Standish |first=EM
|last=Standish |first=E. M.
|title=The JPL planetary ephemerides
|journal=Celestial Mechanics |volume=26 |monthdate=February 1982
|issue=2 |pages=181–186 |doi=10.1007/BF01230883
|issue=2
|bibcode=1982CeMec..26..181S |s2cid=121966516
|year=1982 |pages=181–186 |doi=10.1007/BF01230883
|bibcode=1982CeMec..26..181S }}</ref> and spacecraft,<ref name=pieee95_11_2202>
{{citeCite journal
|last1=Berner |first1=JBJ. B.
|last2=Bryant |first2=SHS. H.
|last3=Kinman |first3=PWP. W.
|title=Range Measurement as Practiced in the Deep Space Network
|journal=Proceedings of the IEEE |monthdate=November 2007 |volume=95 |issue=11 |pages=2202–2214
|doi=10.1109/JPROC.2007.905128 |s2cid=12149700
|year=2007 |volume=95 |issue=11 |pages=2202–2214
|url=https://trs.jpl.nasa.gov/bitstream/2014/40972/1/07-0166.pdf}}
|doi=10.1109/JPROC.2007.905128 }}</ref> respectively, by measuring round-trip transit times.
</ref> respectively, by measuring round-trip transit times.
== Measurement ==
<!--- The article Galileo Galileo links to this section. Please do not change the title of the section without amending the articles which link to it. --->
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 earthEarth-based setups. However, itIt is also possible to determine ''c'' from other physical laws where it appears, for example, by determining the values of the electromagnetic constants [[relative permittivity|''ε''<sub>0</sub>]] and [[permeability (electromagnetism)|''μ''<sub>0</sub>]] 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|"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 1⁄299,792,458{{frac|1|{{val|299792458}}}} of a second",<ref name=Resolution_1/> fixing the value of the speed of light at {{val|299792458|u=m/s}} by definition, as [[#Increased accuracy of c and redefinition of the metre and second|described below]]. Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value of ''c''.
=== Astronomical measurements ===
[[File:Io eclipse speed of light measurement.svg|thumb|upright=2|Measurement of the speed of light from the time it takes Io to orbit Jupiter, using eclipses of Io by Jupiter's shadow to precisely measure its orbit.]]
[[Outer space]] is a natural 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. It is customary to express the results in [[astronomical unit]]s (AU) per day. An astronomical unit is approximately the average distance between the Earth and Sun; it is not based on the [[International System of Units]].{{#tag:ref|The astronomical unit is defined as the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an [[angular frequency]] of {{gaps|0.017|202|098|95}} [[radian]]s (approximately {{frac|{{val|365.256898}}}} of a revolution) per day.<ref>{{SIbrochure8th|page=126}}.
[[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.
</ref> It may be noted that the astronomical unit increases at a rate of about (15 ± 4) cm/yr, probably due to the changing mass of the Sun.<ref name=Nieto>{{cite journal
|arxiv=0907.2469
|title=Astrometric solar-system anomalies
|author=John D. Anderson and Michael Martin Nieto
|journal=Proceedings of the International Astronomical Union
|year=2009 |volume=5
|issue=S261 |pages=189–197
|publisher=Cambridge University Press
|doi=10.1017/S1743921309990378 }}</ref> This unit has the advantage that the [[gravitational constant]] multiplied by the Sun's mass has a fixed, exact value in cubic astronomical units per day squared.|group=Note}} Because the AU determines an actual length, and is not based upon time-of-flight like the SI units, modern measurements of the speed of light in astronomical units per day can be compared with the defined value of ''c'' in the International System of Units.
[[Ole Christensen Rømer]] used an astronomical measurement to make [[Rømer's determination of the speed of light|the first quantitative estimate of the speed of light]] in the year 1676.<ref name=cohen>
{{citeCite journal
|last=Cohen |first=IBI. B. |author-link=I. Bernard Cohen
|year=1940
|title=Roemer and the first determination of the velocity of light (1676)
|journal=[[Isis (journal)|Isis]]
|volume=31 |issue=2 |pages=327–79327–379
|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
}}</ref><ref name=roemer>
{{citeCite journal
|year=1676
|title=TouchantDemonstration 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=French
|language=fr
|url=http://www-obs.univ-lyon1.fr/labo/fc/ama09/pages_jdsc/pages/jdsc_1676_lumiere.pdf
|journal=[[Journal des sçavans]]
|pages=233–36233–236
|ref=roemer-1676
}}<br />Translated in
{{citeCite 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
|title=On the Motion of Light by M. Romer
|url=http://www.archive.org/stream/philosophicaltra02royarich#page/397/mode/1up
|journal=[[Philosophical Transactions of the Royal Society]]
|volume=12 |issue=136 |pages=893–95893–895
|ref=roemer-1676-EnglishTrans
|bibcode=1677RSPT...12..893.|doi-access=free
}} (As reproduced in {{Cite book
}}<br />Reproduced in
|last1=Hutton |first1=C
{{Cite book
|last2=Shaw |first2=G
|editor1-last=Hutton |editor1-first=C.
|last3=Pearson |first3=R eds.
|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
|chapterurlchapter-url=httphttps://www.archive.org/stream/philosophicaltra02royarich#page/397/mode/1up
|location=London |publisher=C. & R. Baldwin
|volume=II. 2|From 1673 to 1682 |pages=397–98397–398
}})<br />
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|(Cohen, 1940, p. 346)]].</ref> 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 actuallycaused 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 (moon)|Io]] and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.<ref name="cohen" />
[[File:SoL AbberationAberration.svg|thumb|rightupright|Aberration of light: light from a distant source appears to be from a different location for a moving telescope due to the finite speed of light.|alt=A star emits a light ray whichthat hits the objective of a telescope. While the light travels down the telescope to its eyepiece, the telescope moves to the right. For the light to stay inside the telescope, the telescope must be tilted to the right, causing the distant source to appear at a different location to the right.]]
Another method is to use the [[aberration of light]], discovered and explained by [[James Bradley]] in the 18th century.<ref name="Bradley1729">
{{Cite journal
|last=Bradley |first=J.
|year=1729
|title=Account of a new discoveddiscovered 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
|doi=
}}</ref> 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 [[arcsecond]]s)<ref>
{{Cite book
|last=Duffett-Smith
|first=P.
|year=1988
|title=Practical Astronomy with your Calculator
|url=https://archive.org/details/practicalastrono0000duff
|url=http://books.google.com/?id=DwJfCtzaVvYC
|url-access=registration
|page=62
|page=[https://archive.org/details/practicalastrono0000duff/page/62 62]
|publisher=[[Cambridge University Press]]
|publisher=Cambridge University Press
|isbn=0-521-35699-7}}, [http://books.google.com/books?id=DwJfCtzaVvYC&pg=PA62 Extract of page 62]</ref> 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 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.<ref name="Bradley1729"/>
|isbn=978-0-521-35699-2
}} [https://archive.org/details/practicalastrono0000duff/page/62 Extract of page 62].</ref> 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.<ref name="Bradley1729"/>
==== Astronomical unit ====
An [[astronomical unit]] (AU) is approximately the average distance between the Earth and Sun. It was redefined in 2012 as exactly {{val|149597870700|u=m}}.<ref name=AU_redef /><ref>{{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}}</ref> Previously the AU was not based on the [[International System of Units]] but in terms of the gravitational force exerted by the Sun in the framework of classical mechanics.{{#tag:ref|The astronomical unit was defined as the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an [[angular frequency]] of {{gaps|0.017|202|098|95}} [[radian]]s (approximately {{frac|{{val|365.256898}}}} of a revolution) per day.<ref>{{SIbrochure8th|page=126}}</ref>|group="Note"}} The current definition uses the recommended value in metres for the previous definition of the astronomical unit, which was determined by measurement.<ref name=AU_redef>{{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}}</ref> 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).<ref>{{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 (journal)|Nature]]|language=en|doi=10.1038/nature.2012.11416|s2cid=123424704|issn=1476-4687}}</ref>
NowadaysPreviously, the "light time for unit distance"—the inverse of {{math|''c'',}} expressed in seconds per astronomical unit—isunit was measured by comparing the time for radio signals to reach different spacecraft in the Solar System, with their position calculated from the gravitational effects of the Sun and various planets. By combining many such measurements, a [[best fit]] value for the light time per unit distance iscould be obtained. {{AsFor example, in of|2009}}, the best estimate, as approved by the [[International Astronomical Union]] (IAU), iswas:<ref name="Pitjeva09">See the following:
* {{citeCite journal
|last1=Pitjeva |first1=EVE. V.
|last2=Standish |first2=EME. M.
|year=2009
|title=Proposals for the masses of the three largest asteroids, the Moon-EarthMoon–Earth mass ratio and the Astronomical Unit
|journal=[[Celestial Mechanics and Dynamical Astronomy]]
|volume=103 |issue=4 |pages=365–372
|doi=10.1007/s10569-009-9203-8
|bibcode = 2009CeMDA.103..365P }}</ref><ref name|s2cid="IAU">121374703
|url=https://zenodo.org/record/1000691
{{cite web
}}
* {{Cite web |url=https://ssd.jpl.nasa.gov/?constants |title=Astrodynamic Constants |publisher=Jet Propulsion Laboratory |work=Solar System Dynamics |access-date=10 February 2022}}</ref><ref name="IAU">
{{Cite web
|author=IAU Working Group on Numerical Standards for Fundamental Astronomy
|title=IAU WG on NSFA Current Best Estimates
|url=http://maia.usno.navy.mil/NSFA/CBE.html
|publisher=[[US Naval Observatory]]
|access-date=25 September 2009
|accessdate=2009-09-25
|url-status=dead
|archive-url=https://web.archive.org/web/20091208011235/http://maia.usno.navy.mil/NSFA/CBE.html
|archive-date=8 December 2009
}}</ref>
: light time for unit distance: ''t''<sub>au</sub> = {{val|499.004783836|(10)|u=s}},
: ''c'' = {{val|0.00200398880410|(4)|u=AU/s}} = {{val|173.144632674|(3)|u=AU/day.}}.
The relative uncertainty in these measurements is 0.02 parts per billion (2{{eval|2|e=-11}}), equivalent to the uncertainty in Earth-based measurements of length by interferometry.<ref>
{{citeCite web
|title=NPL's Beginner's Guide to Length
|archive-url=https://web.archive.org/web/20100831214704/http://www.npl.co.uk/educate-explore/posters/length/length-%28poster%29
|archive-date=31 August 2010
|url=http://www.npl.co.uk/educate-explore/posters/length/length-%28poster%29
|publisher=[[National Physical Laboratory (United Kingdom)|UK National Physical Laboratory]]
|access-date=28 October 2009
|accessdate=2009-10-28
}}</ref>{{#tag:ref|The value of the speed of light in [[Astronomical system of units|astronomical units]] has a measurement uncertainty, unlike the value in SI units, because of the different definitions of the unit of length.|group=Note}} Since the metre is defined to be the length travelled by light in a certain time interval, the measurement of the light time forin terms of the previous unitdefinition distanceof the astronomical unit can also be interpreted as measuring the length of an AU (old definition) in metres.{{#tag:ref|Nevertheless, at this degree of precision, the effects of [[general relativity]] must be taken into consideration when interpreting the length. The metre is considered to be a unit of [[proper length]], whereas the AU is usually used as a unit of observed length in a given frame of reference. The values cited here follow the latter convention, and are [[Barycentric Dynamical Time|TDB]]-compatible.<ref name="IAU"/>|group=Note}}
=== Time of flight techniques ===
[[File:Michelson speed of light measurement 1930.jpg|thumb|center|One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930–1935 experiment used a rotating mirror and a one-mile (1.6 km) long vacuum chamber which the light beam traversed 10 times. It achieved accuracy of ±11 km/s.|600x600px]]
[[File:Fizeau-int.svg|thumb|Diagram of the [[Fizeau's measurement of the speed of light in air|Fizeau apparatus]]:{{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]].
===Time of flight techniques===
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 the [[Fizeau–Foucault apparatus]] developed by [[Hippolyte Fizeau]] and [[Léon Foucault]].
The [[Fizeau's measurement of the speed of light in air|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.<ref name=How>
[[File:Fizeau.JPG|thumb|right|Diagram of the [[Fizeau–Foucault apparatus|Fizeau apparatus]]|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.]]
{{Cite web
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.<ref name=How>{{cite web
|last=Gibbs
|first=P.
|year=1997
|title=How is the speed of light measured?
Line 962 ⟶ 1,114:
|work=Usenet Physics FAQ
|publisher=University of California, Riverside
|access-date=13 January 2010
|accessdate=2010-01-13
|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
}}</ref>
The [[Foucault's measurements of the speed of light|method of Foucault]] replaces the cogwheel bywith 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.<ref>
{{citeCite web
|last=Fowler |first=M.
|date=
|title=The Speed of Light
|url=http://galileoandeinstein.physics.virginia.edu/lectures/spedlite.html
|publisher=[[University of Virginia]]
|accessdateaccess-date=21 April 2010-04-21
}}</ref> Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion by [[François Arago]].<ref name="Hughes2012">{{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}}</ref>
}}</ref>
NowadaysToday, 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.<ref>See, for example:
* {{citeCite journal
|lastlast1=Cooke |firstfirst1=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
Line 985 ⟶ 1,140:
|volume=36 |issue=9 |page=847
|doi=10.1119/1.1975166
|bibcode = 1968AmJPh..36..847C
}}</ref><ref>
* {{citeCite journal
|lastlast1=Aoki |firstfirst1=K. |last2=Mitsui |first2=T.
|year=2008
|title=A small tabletop experiment for a direct measurement of the speed of light
Line 994 ⟶ 1,150:
|doi=10.1119/1.2919743
|arxiv=0705.3996
|bibcode = 2008AmJPh..76..812A }}</ref><ref>|s2cid=117454437
}}
{{cite journal
* {{Cite journal
|last=James |first=MB |last2=Ormond |first2=RB |last3=Stasch |first3=AJ
|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
Line 1,002 ⟶ 1,159:
|volume=67 |issue=8 |pages=681–714
|doi=10.1119/1.19352
|bibcode = 1999AmJPh..67..681J
}}</ref>{{clr}}
=== Electromagnetic constants ===
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]] ''ε''<sub>0</sub> and [[vacuum permeability]] ''μ''<sub>0</sub> established by Maxwell's theory: ''c''<sup>2</sup> = 1/(''ε''<sub>0</sub>''μ''<sub>0</sub>). The vacuum permittivity may be determined by measuring the [[capacitance]] and dimensions of a [[capacitor]], whereas the value of the [[vacuum permeability]] iswas historically fixed at exactly {{val|4|end=π|e=-7|u=H*.m-1}} through the definition of the [[ampere (unit)|ampere]]. [[Edward Bennett Rosa|Rosa]] and [[Noah Ernest Dorsey|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.<ref name="Essen1948"/><ref name="RosaDorsey">{{citeCite 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}}</ref>
|last=Rosa |first=EB |last2=Dorsey |first2=NE
|year=1907
|title=The Ratio of the Electromagnetic and Electrostatic Units|journal=[[Bulletin of the Bureau of Standards]]
|volume=3
|issue=6 |page=433
|doi=10.1103/PhysRevSeriesI.22.367
|bibcode = 1906PhRvI..22..367R }}</ref>
=== Cavity resonance ===
[[File:Waves in Box.svg|thumb|right|Electromagnetic [[standing waves]] 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 mode]]s 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.<ref name="Essen1948"/> 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.<ref name="Essen1948">
{{citeCite journal
|lastlast1=Essen |firstfirst1=L.
|last2=Gordon-Smith |first2=ACA. C.
|year=1948
|title=The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator
Line 1,027 ⟶ 1,179:
|bibcode=1948RSPSA.194..348E
|jstor=98293
|doi-access=free
}}</ref><ref>
{{citeCite journal
|last=Essen |first=L.
|year=1947
|title=Velocity of Electromagnetic Waves
Line 1,036 ⟶ 1,189:
|doi=10.1038/159611a0
|bibcode=1947Natur.159..611E
|s2cid=4101717
}}</ref>
The Essen–Gordon-Smith result, {{val|299792|9|u=km/s}}, was substantially more precise than those found by optical techniques.<ref name="Essen1948" /> By 1950, repeated measurements by Essen established a result of {{val|299792.5|3.0|u=km/s}}.<ref name="Essen1950">
{{citeCite 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
Line 1,048 ⟶ 1,202:
|bibcode=1950RSPSA.204..260E
|jstor=98433
|s2cid=121261770
}}</ref>
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 [[antinode]]s (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".<ref>
{{citeCite journal
| last = Stauffer | first = RHR. H.
|date=April 1997
| year = 1997 | month = April
| title = Finding the Speed of Light with Marshmallows
| journal = [[The Physics Teacher]]
| volume = 35
| pageissue = 2314
|page = 231
| publisher = American Association of Physics Teachers
| url = httphttps://www.physics.umd.edu/icpe/newsletters/n34/marshmal.htm
|access-date = 15 February 2010
| accessdate = 2010-02-15
|bibcode = 1997PhTea..35..231S |doi = 10.1119/1.2344657
| issue = 4 }}</ref><ref>{{cite web
{{Cite web
| url =http://www.bbc.co.uk/norfolk/features/ba_festival/bafestival_speedoflight_experiment_feature.shtml
|url =http://www.bbc.co.uk/norfolk/features/ba_festival/bafestival_speedoflight_experiment_feature.shtml
| title = BBC Look East at the speed of light
|title = BBC Look East at the speed of light
| work = BBC Norfolk website
|work = BBC Norfolk website
|publisher=BBC
|access-date = 15 February 2010
| accessdate = 2010-02-15
}}</ref>
=== Interferometry ===
[[File:Interferometer sol.svg|thumb|upright=1.4|An interferometric determination of length. Left: [[constructive interference]]; 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.{{#tag:ref |A detailed discussion of the interferometer and its use for determining the speed of light can be found in Vaughan (1989).<ref name=Vaughan>
{{Cite book
A detailed discussion of the interferometer and its use for determining the speed of light can be found in {{Cite book
|last=Vaughan |first=JMJ. M.
|year=1989
|title=The Fabry-Perot interferometer
|url=httphttps://books.google.com/books?id=mMLuISueDKYC&printsec=frontcover#PPA47,M1
|pagepages=47, pp. 384–391
|publisher=CRC Press
|isbn=978-0-85274-138-32
}}</ref>|group="Note"}} A [[Coherence (physics)|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 (wave propagation)|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 [[radiowave|radio]] sources were used for interferometry measurements of the speed of light.<ref name=Froome1858>
{{citeCite journal
|doi=10.1098/rspa.1958.0172
|title=A New Determination of the Free-Space Velocity of Electromagnetic Waves
|first=KDK. 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
|publisher=The Royal Society
|jstor=100591 |s2cid=121444888
|bibcode = 1958RSPSA.247..109F
|jstor=100591 }}</ref> However interferometricInterferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~0.{{cvt|4 cm|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.<ref name="NIST_pub"/>
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.<ref name="NIST_pub">
{{Cite book
|title = A Century of Excellence in Measurements, Standards, and Technology
|editor-last =Lide Lide
|editor-first =DR D. R.
|contribution = Speed of Light from Direct Frequency and Wavelength Measurements
|last=Sullivan |first =DB Sullivan
|first = D. B.
|year=2001
|year = 2001
|pages=191–193
|pages = 191–193
|publisher=CRC Press
|publisher = CRC Press
|isbn=0-8493-1247-7
|isbn = 978-0-8493-1247-2
|url=http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf
|url = http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf
}}</ref> This technique was due to a group at the National Bureau of Standards (NBS) (which later became [[National Institute of Standards and Technology|NIST]]). They used it in 1972 to measure the speed of light in vacuum with a [[Measurement uncertainty|fractional uncertainty]] of {{val|3.5|e=-9}}.<ref name="NIST_pub"/><ref name="NIST heterodyne">
|url-status = dead
{{cite journal
|archive-url = https://web.archive.org/web/20090813061018/http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf
|last1=Evenson |first1=KM |coauthors=''et al.''
|archive-date = 13 August 2009
|year=1972
}}</ref> 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 [[Measurement uncertainty|fractional uncertainty]] of {{val|3.5|e=-9}}.<ref name="NIST_pub"/><ref name="NIST heterodyne">
{{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=1346–491346–1349
|doi=10.1103/PhysRevLett.29.1346
|bibcode=1972PhRvL..29.1346E
|s2cid=120300510 |display-authors=etal
}}</ref>
== History ==
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.
{| class="infobox wikitable" style="width:40%; margin:0 0 0.5em 1em; text-align:left;"
|+History of measurements of ''c'' (in km/s)
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+ History of measurements of ''c'' (in m/s)
|-
| <1638 || [[Galileo Galilei|Galileo]], covered lanterns ||colspan="2"| inconclusive<ref name=2newsciences /><ref name=boyer /><ref name=foschi&leone>
|1675||[[Ole Rømer|Rømer]] and [[Christiaan Huygens|Huygens]], moons of Jupiter||{{val|220000}}<ref name=roemer/><ref name="Huygens 1690 8–9"/>
{{Citation
|first1=Renato
|last1=Foschi
|first2=Matteo
|last2=Leone
|title=Galileo, measurement of the velocity of light, and the reaction times
|journal=Perception
|volume=38
|issue=8
|year=2009
|pages=1251–1259
|doi=10.1068/p6263
|pmid=19817156
|hdl=2318/132957
|s2cid=11747908
|hdl-access=free
}}</ref>{{rp|1252}}{{#tag:ref|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.<ref name="boyer"/>|group="Note"}}
|-
| <1667 || [[Accademia del Cimento]], covered lanterns ||colspan="2"| inconclusive<ref name=foschi&leone />{{rp|1253}}<ref name=magalotti>
|1729||[[James Bradley]], aberration of light||{{val|301000}}<ref name=How/>
{{Citation
|first1=Lorenzo
|last1=Magalotti
|author-link=Lorenzo Magalotti
|title=Saggi di Naturali Esperienze fatte nell' Accademia del Cimento
|url=http://brunelleschi.imss.fi.it/cimentosite/indice.asp?xmlFile=Indice00.xml
|edition=digital, online
|publisher=Istituto e Museo di Storia delle Scienze
|place=Florence
|year=2001
|orig-year=1667
|access-date=25 September 2015
|pages=[http://brunelleschi.imss.fi.it/cimentosite/ShowFullSize.asp?Image=FullSize/A0000283.JPG&Title=Pagina:%20265 265]–[http://brunelleschi.imss.fi.it/cimentosite/ShowFullSize.asp?Image=FullSize/A0000284.JPG&Title=Pagina:%20266 266]
}}</ref>
|-
|1849 1675 || [[HippolyteOle FizeauRømer|Rømer]], toothedand [[Christiaan wheelHuygens|Huygens]], moons of Jupiter || {{val|315000220000000}}<ref name=Howroemer/><ref name="Huygens 1690 8–9"/> || −27%
|-
|1862 1729 || [[LéonJames FoucaultBradley]], rotating mirroraberration of light || {{val|298000|500301000000}}<ref name=How/> || +0.40%
|-
|1907 1849 ||Rosa [[Hippolyte and DorseyFizeau]], <abbrtoothed wheel title="electromagnetic">EM</abbr> constants|| {{val|299710|30315000000}}<ref name="Essen1948"How/><ref name="RosaDorsey"/>|| +5.1%
|-
|1926 1862 || [[AlbertLéon MichelsonFoucault]], rotating mirror || {{val|299796298000000|4500000}}<ref name=How/>{{cite doi|10| −0.1086/143021}}</ref>60%
|-
| 1875 || Werner Siemens || 260 000 000<ref>{{Cite journal |last1=Buchwald |first1=Jed |last2=Yeang |first2=Chen-Pang |last3=Stemeroff |first3=Noah |last4=Barton |first4=Jenifer |last5=Harrington |first5=Quinn |date=2021-03-01 |title=What Heinrich Hertz discovered about electric waves in 1887–1888 |url=https://doi.org/10.1007/s00407-020-00260-1 |journal=Archive for History of Exact Sciences |language=en |volume=75 |issue=2 |pages=125–171 |doi=10.1007/s00407-020-00260-1 |s2cid=253895826 |issn=1432-0657}}</ref> ||
|1950||{{nowrap|Essen and Gordon-Smith}}, cavity resonator||{{val|299792.5|3.0}}<ref name="Essen1950"/>
|-
| 1893 || [[Heinrich Hertz]] || 200 000 000<ref>{{Cite book |last=Hertz |first=Heinrich |title=Electric Waves |publisher=Macmillan and Co. |year=1893 |location=London}}</ref> ||
|1958||K.D. Froome, radio interferometry||{{val|299792.50|0.10}}<ref name="Froome1858"/>
|-
| 1907 || Rosa and Dorsey, <abbr title="electromagnetic">EM</abbr> constants || {{val|299710000|30000}}<ref name="Essen1948"/><ref name="RosaDorsey"/> || −280 [[Parts-per notation|ppm]]
|1972||Evenson ''et al.'', laser interferometry||{{val|299792.4562|0.0011}}<ref name="NIST heterodyne"/>
|-
| 1926 || [[Albert A. Michelson]], rotating mirror || {{val|299796000|4000}}<ref>{{Cite journal |doi = 10.1086/143021 |title = Measurement of the Velocity of Light Between Mount Wilson and Mount San Antonio |year = 1927 |last1 = Michelson |first1 = A. A. |journal = The Astrophysical Journal| volume = 65 |pages = 1 |bibcode=1927ApJ....65....1M}}</ref> || +12 ppm
|1983||17th CGPM, definition of the metre||{{val|299792.458}} (exact)<ref name=Resolution_1/>
|-
| 1950 || {{nowrap|Essen and Gordon-Smith}}, cavity resonator || {{val|299792500|3000}}<ref name="Essen1950"/> || +0.14 ppm
|-
| 1958 || K. D. Froome, radio interferometry || {{val|299792500|100}}<ref name="Froome1858"/> || +0.14 ppm
|-
| 1972 || Evenson ''et al.'', laser interferometry || {{val|299792456.2|1.1}}<ref name="NIST heterodyne"/> || −0.006 ppm
|-
| 1983 || 17th CGPM, definition of the metre ||colspan="2"| {{val|299792458}} (exact)<ref name=Resolution_1/>
|}
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, Muslim 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 concluded that the speed of light is constant regardless of one's frame of reference. Since then, scientists have provided increasingly accurate measurements.
=== Early history ===
[[Empedocles]] (c. 490–430 BCE) was the first to claim that light haspropose a finitetheory speed.of light<ref>
{{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].</ref> and claimed that light has a finite speed.<ref>
{{Cite book
|last=Sarton |first=G. |author-link=George Sarton
|year=1993
|title=Ancient science through the golden age of Greece
|url=httphttps://books.google.com/books?id=VcoGIKlHuZcC&pg=PA248
|page=248
|publisher=[[Courier Dover]]
|isbn=978-0-486-27495-09
}}</ref> 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".<ref name=Statistics>
{{citeCite journal
|lastlast1=MacKay |firstfirst1=RHR. H. |last2=Oldford |first2=RWR. 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 (journal)|Statistical Science]]
|volume=15 |issue=3 |pages=254–78254–278
|doi=10.1214/ss/1009212817
|doi-access=free
}} (click on "Historical background" in the table of contents)</ref> [[Euclid]] and [[Ptolemy]] advanced the [[Emission theory (vision)|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 [[Infinity|infinite]] because distant objects such as stars appear immediately upon opening the eyes.
}} (click on "Historical background" in the table of contents)</ref> [[Euclid]] and [[Ptolemy]] advanced Empedocles' [[Emission theory (vision)|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 [[Infinity|infinite]] because distant objects such as stars appear immediately upon opening the eyes.<ref>{{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]</ref>
[[Early Islamic philosophy|Early Islamic philosophers]] initially agreed with the [[Aristotelian physics|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 [[Visual perception|vision]] in favour of the now accepted intromission theory of [[Visual perception|vision]], in which light moves from an object into the eye.<ref>
{{Cite journal
| last1 = Gross | first1 = CGC. G.
| title = The Fire That Comes from the Eye
| journal = Neuroscientist
| volume = 5
| pages = 58–49 58–64
| year = 1999
| doi = 10.1177/107385849900500108
| s2cid = 84148912
}}</ref> This led Alhazen to propose that light must have a finite speed,<ref name=Statistics/><ref name=Hamarneh>
{{citeCite journal
|last=Hamarneh |first=S.
|year=1972
|title=Review: Hakim Mohammed Said, ''Ibn al-Haitham''
Line 1,184 ⟶ 1,401:
}}</ref><ref name=Lester>
{{Cite book
|last=Lester |first=PMP. M.
|year=2005
|title=Visual Communication: Images With Messages
|pages=10–11
|publisher=[[Thomson Wadsworth]]
|isbn=978-0-534-63720-50
}}</ref> and that the speed of light is variable, decreasing in denser bodies.<ref name=Lester/><ref>
{{citeCite web
|first1=JJJ. J.
|last1=O'Connor
|authorlink1author-link1=John J. O'Connor (mathematician)
|first2=EFE. F.
|last2=Robertson
|authorlink2author-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
|accessdate=2010-01-12
}}</ref> He argued that light is substantial matter, the propagation of which requires time, even if this is hidden from ourthe senses.<ref>
{{citeCite conference
|last=Lauginie |first =P Lauginie
|first = P.
|year=2005
|year = 2004
|title=Measuring: Why? How? What?
|title = Measuring Speed of Light: Why? Speed of what?
|url=http://www.ihpst2005.leeds.ac.uk/papers/Lauginie.pdf
|url = http://sci-ed.org/documents/Lauginie-M.pdf
|booktitle=Proceedings of the 8th International History, Philosophy, Sociology & Science Teaching Conference
|conference = Fifth International Conference for History of Science in Science Education
|accessdate=2008-07-18
|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
}}</ref> 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.<ref>
{{citeCite web
|first1=JJJ. J.
|last1=O'Connor
|first2=EFE. F.
|last2=Robertson
|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Biruni.html
Line 1,221 ⟶ 1,444:
|work=MacTutor History of Mathematics archive
|publisher=University of St Andrews
|access-date=12 January 2010
|accessdate=2010-01-12
}}</ref>
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.<ref name=Lindberg>
{{Cite book
|last=Lindberg |first=DCD. 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=httphttps://books.google.com/books?id=jSPHMKbjYkQC&pg=PA143
|page=143
|isbn=978-0-19-823992-05
|publisher=[[Oxford University Press]]
}}</ref><ref>
{{Cite book
|last=Lindberg |first=DCD. C.
|year=1974
|chapter=Late Thirteenth-Century Synthesis in Optics
|editor=Edward Grant
|title=A source book in medieval science
|chapter-url=httphttps://books.google.com/books?id=fAPN_3w4hAUC&pg=RA1-PA395&dqq=roger-bacon+speed%20speed-of-light&qpg=rogerRA1-bacon%20speed-of-lightPA395
|page=396
|publisher=[[Harvard University Press]]
|isbn=978-0-674-82360-0
}}</ref> In the 1270s, [[Witelo]] considered the possibility of light travelling at infinite speed in vacuum, but slowing down in denser bodies.<ref name=Marshall>
{{citeCite journal
|last=Marshall |first=P.
|year=1981
|title=Nicole Oresme on the Nature, Reflection, and Speed of Light
|journal=[[Isis (journal)|Isis]]
|volume=72 |issue=3 |pages=357–74357–374 [367–74367–374]
|doi=10.1086/352787
|s2cid=144035661
}}</ref>
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.<ref>{{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}}</ref> 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.<ref name=Statistics /> Despite this, in his derivation of [[Snell's law]], Descartes assumed that some kind of motion associated with light was faster in denser media.<ref>{{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}}</ref><ref>{{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}}</ref> [[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.<ref>{{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}}</ref>
=== First measurement attempts ===
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.<ref name=boyer2newsciences>
{{citeCite journalbook
|last=Boyer |first=CBGalilei
|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
}}</ref><ref name=boyer>
{{Cite journal
|last=Boyer |first=C. B.
|year=1941
|title=Early Estimates of the Velocity of Light
Line 1,266 ⟶ 1,507:
|volume=33 |issue=1 |page=24
|doi=10.1086/358523
|s2cid=145400212
|ref=boyer-1941
}}</ref> 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.<ref>{{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}}</ref> The actual delay in this experiment would have been about 11 [[microsecond]]s.
}}</ref><ref name=2newsciences>
{{Cite book
|last=Galilei |first=G
|year=1954 |origyear=1638
|title=Dialogues Concerning Two New Sciences
|url=http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&layout=html#a_2288356
|page=43
|others=Crew, H; de Salvio A (trans.)
|publisher=[[Dover Publications]]
|isbn=0-486-60099-8
|ref=Reference-Galileo-1954
}}</ref> Galileo's experiment was carried out by the [[Accademia del Cimento]] of Florence, Italy, in 1667, with the lanterns separated by about one mile, but no delay was observed. The actual delay in this experiment would have been about 11 [[microsecond]]s.
[[File:Illustration from 1676 article on Ole Rømer's measurement of the speed of light.jpgpng|thumb|left|upright=0.8|Rømer's observations of the occultations of Io from Earth|alt=A diagram of a planet's orbit around the Sun and of a moon's orbit around another planet. The shadow of the latter planet is shaded.]]
[[Rømer's determination of the speed of light|The first quantitative estimate of the speed of light]] was made in 1676 by RømerOle (see [[Rømer's determination of the speed of light]]).<ref name="cohen"/><ref name="roemer"/> From the observation that the periods of Jupiter's innermost moon [[Io (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}}, 26which is 27% lower than the actual value.<ref name="Huygens 1690 8–9">
{{Cite book
|last=Huygens |first=C.
|year=1690
|title=Traitée de la Lumière |language=Frenchfr
|url=https://archive.org/details/bub_gb_kVxsaYdZaaoC
|url=http://books.google.com/?id=No8PAAAAQAAJ&pg=PA9
|pages=[https://archive.org/details/bub_gb_kVxsaYdZaaoC/page/n19 8]–9
|pages=8–9
|publisher=[[Pierre van der Aa]]
}}</ref>
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).<ref>
{{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.</ref> Newton queried whether Rømer's eclipse shadows were coloured;. hearingHearing that they were not, he concluded the different colours travelled at the same speed. In 1729, [[James Bradley]] discovered the [[aberration of light|stellar aberration]].<ref name="Bradley1729"/> 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.<ref name="Bradley1729"/>
=== Connections with electromagnetism ===
{{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}}.<ref name="guarnieri 7-1">{{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}}</ref> His method was improved upon by [[Léon Foucault]] who obtained a value of {{val|298000|u=km/s}} in 1862.<ref name="How"/> In the year 1856, [[Wilhelm Eduard Weber]] and [[Rudolf Kohlrausch]] measured the ratio of the electromagnetic and electrostatic units of charge, 1/√{{radic|''ε''<sub>0</sub>''μ''<sub>0</sub>}}, 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 [[electrical resistance|resistanceless]] wire travels along the wire at this speed.<ref>
{{citeCite journal
|last1=GraneauKirchhoff |first1=PG.
|title=Über die Bewegung der Elektricität
|last2=Assis |first2=AKT
|url=http://gallica.bnf.fr/ark:/12148/bpt6k15187j/f549.item.r=
|title=Kirchhoff on the motion of electricity in conductors
|journal=[[ApeironAnnalen (physicsder journal)Physik|ApeironAnn. Phys.]]
|volume=19178
|yearissue=199412
|year=1857
|pages=19–25
|pages=529–244
|url=http://www.physics.princeton.edu/~mcdonald/examples/EM/kirchhoff_apc_102_529_57_english.pdf
|doi=10.1002/andp.18571781203
|accessdate=2010-10-21
|bibcode=1857AnP...178..529K
}}</ref> In the early 1860s, Maxwell showed that according to the theory of electromagnetism which he was working on, that electromagnetic waves propagate in empty space<ref>{{cite book
}}</ref>
|title=College physics: reasoning and relationships
|first1=Nicholas J.
In the early 1860s, Maxwell showed that, according to the theory of electromagnetism he was working on, electromagnetic waves propagate in empty space<ref>See, for example:
|last1=Giordano
* {{Cite book
|publisher=Cengage Learning
|title=College physics: reasoning and relationships
|year=2009
|first1=Nicholas J.
|isbn=0-534-42471-6
|last1=Giordano
|page=787
|publisher=Cengage Learning
|url=http://books.google.com/books?id=BwistUlpZ7cC}}, [http://books.google.com/books?id=BwistUlpZ7cC&pg=PA787 Extract of page 787]
|year=2009
</ref><ref>{{cite book
|isbn=978-0-534-42471-8
|title=The riddle of gravitation
|page=787
|first1=Peter Gabriel
|url=https://books.google.com/books?id=BwistUlpZ7cC
|last1=Bergmann
}} [https://books.google.com/books?id=BwistUlpZ7cC&pg=PA787 Extract of page 787]
|publisher=Courier Dover Publications
* {{Cite book
|year=1992
|title=The riddle of gravitation
|isbn=0-486-27378-4
|first1=Peter Gabriel
|page=17
|last1=Bergmann
|url=http://books.google.com/books?id=WYxkrwMidp0C}}, [http://books.google.com/books?id=WYxkrwMidp0C&pg=PA17 Extract of page 17]
|publisher=Courier Dover Publications
</ref><ref>{{cite book
|year=1992
|title=The equations: icons of knowledge
|isbn=978-0-486-27378-5
|first1=Sander
|page=17
|last1=Bais
|url=https://books.google.com/books?id=WYxkrwMidp0C
|publisher=Harvard University Press
}} [https://books.google.com/books?id=WYxkrwMidp0C&pg=PA17 Extract of page 17]
|year=2005
* {{Cite book
|isbn=0-674-01967-9
|title=The equations: icons of knowledge
|page=40
|first1=Sander
|url=http://books.google.com/books?id=jKbVuMSlJPoC}}, [http://books.google.com/books?id=jKbVuMSlJPoC&pg=PA40 Extract of page 40]
|last1=Bais
</ref> at a speed equal to the above Weber/Kohrausch 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.<ref name=maxwellbio>
|publisher=Harvard University Press
{{cite web
|year=2005
|last1=O'Connor |first1=JJ |last2=Robertson |first2=EF
|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]
</ref> 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.<ref name=maxwellbio>
{{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
|accessdate=2010-10-13
|url-status=dead
}}</ref>
|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
}}</ref> 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.<ref>Campbell, Lewis; Garnett, William; Rautio, James C. "The Life of James Clerk Maxwell", p. 544, {{ISBN|978-1-77375-139-9}}.</ref>
=== "Luminiferous aether" ===
{{main|Luminiferous aether}}
[[File:Einstein en Lorentz.jpg|thumb|upright|Hendrik Lorentz with Albert Einstein.]]
The wave properties of light were well known since [[Thomas Young (scientist)|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]]).<ref>{{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}}</ref>
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 [[Michelson–Morley experiment|the experiment]] performed by [[Albert Michelson]] and [[Edward Morley]] in 1887.<ref>
[[File:Einstein en Lorentz.jpg|thumb|upright|Hendrik Lorentz (right) with Albert Einstein (1921)]]
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 [[Michelson–Morley experiment|the experiment]] performed by [[Albert A. Michelson]] and [[Edward W. Morley]] in 1887.<ref>{{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}}</ref><ref>
{{Cite journal
|last1=Michelson |first1=AAA. A. |last2=Morley |first2=EWE. W.
|year=1887
|title=[[s:On the Relative Motion of the Earth and the Luminiferous Ether|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
|doi=
|s2cid=98374065
}}</ref> The detected motion was always less than the 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.<ref>
|url=https://zenodo.org/record/1450078
}}</ref> 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.<ref>
{{Cite book
| last = French | first = APA. P.
| year = 1983
| title = Special relativity
| pages = 51–57
| publisher = Van Nostrand Reinhold
| isbn = 978-0-442-30782-90
}}</ref>
Because of this experiment [[Hendrik Lorentz]] proposed that the motion of the apparatus through the aether may cause the apparatus to [[Lorentz contraction|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 ether theory|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]].<ref>
Because of this experiment [[Hendrik Lorentz]] proposed that the motion of the apparatus through the aether may cause the apparatus to [[Lorentz contraction|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 ether theory|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]].<ref>
{{Cite book
|last=Darrigol
|first=O.
|year=2000
|title= Electrodynamics from Ampére to Einstein
|publisher=Clarendon Press
|isbn=978-0-19-850594-9}}</ref><ref>{{Cite book5
|url-access=registration
|last=Galison |first=P
|url=https://archive.org/details/electrodynamicsf0000darr
|authorlink=Peter Galison
}}</ref><ref>
|year=2003
{{Cite book
|title= Einstein's Clocks, Poincaré's Maps: Empires of Time
|last=Galison |first=P.
|publisher=W.W. Norton
|author-link=Peter Galison
|isbn=0-393-32604-7}}</ref>
|year=2003
|title= Einstein's Clocks, Poincaré's Maps: Empires of Time
|publisher=W. W. Norton
|isbn=978-0-393-32604-8
}}</ref>
=== Special relativity ===
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.<ref>{{Cite book
|last=Miller |first=AI
|year=1981
|title= Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911)
|publisher=Addison–Wesley
|isbn=0-201-04679-2}}</ref><ref>
{{Cite book
|last=Pais |first=AMiller
|first=A. I.
|authorlink=Abraham Pais
|year=19821981
|title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911)
|title= Subtle is the Lord: The Science and the Life of Albert Einstein
|publisher=Oxford University PressAddison–Wesley
|isbn=978-0-19201-52043804679-7}}</ref>3
|url-access=registration
|url=https://archive.org/details/alberteinsteinss0000mill
}}</ref><ref>
{{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
}}</ref>
=== Increased accuracy of ''c'' and redefinition of the metre and second ===
{{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. In 1972, using the latter method and the [[history of the metre#Krypton standard|1960 definition of the metre]] in terms of a particular spectral line of krypton-86, a group at [[National Institute of Standards and Technology|NBS]] 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 [[Measurement uncertainty|uncertain]] than the previously accepted value. The remaining uncertainty was mainly related to the definition of the metre.{{#tag:ref|Since 1960 the metre was defined as: "The metre is the length equal to {{val|1650763.73}} wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p<sub>10</sub> and 5d<sub><sub>5</sub></sub> of the krypton 86 atom."<ref name="11thCGPM">
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.<ref name="Essen1950"/> This value was adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. In 1960, the [[history of the metre#Krypton standard|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]].<ref name="13thCGPMr1">
{{cite web
{{Cite web
|year=1967
|title=Resolution 1 of the 15th CGPM
|url=https://www.bipm.org/en/CGPM/db/13/1/
|publisher=[[International Bureau of Weights and Measures|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
}}</ref>
In 1972, using the laser interferometer method and the new definitions, a group at the US [[National Institute of Standards and Technology|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.{{#tag:ref|Between 1960 and 1983 the metre was defined as "the length equal to {{val|1650763.73}} wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p<sup>10</sup> and 5d<sup>5</sup> of the krypton-86 atom".<ref name="11thCGPM">
{{Cite web
|year=1967
|title=Resolution 6 of the 15th CGPM
|url=http://www.bipm.org/en/CGPM/db/11/6/
|publisher=[[International Bureau of Weights and Measures|BIPM]]
|access-date=13 October 2010
|accessdate=2010-10-13
}}</ref> It was later discovered in the 1970s that this spectral line was not symmetric, which put a limit on the precision with which the definition could be realized in interferometry experiments.<ref>{{citeCite journal doi| volume = 22 | pages = 196 | year = 1973 | doi = 10.1063/1.1654608 | last1 = Barger | journal = Applied Physics Letters | first1 = R. | title = Wavelength of the 3.39-μm laser-saturated absorption line of methane | last2 = Hall | first2=J. | issue = 4 |bibcode = 1973ApPhL..22..196B | s2cid = 1841238 }}
</ref>|group="Note"}}<ref name="NIST heterodyne"/> As similar experiments found comparable results for ''c'', the 15th [[ConférenceGeneral GénéraleConference deson PoidsWeights etand MesuresMeasures]] (CGPM) in 1975 recommended using the value {{val|299792458|u=m/s}} for the speed of light.<ref name="15thCGPM">
{{citeCite 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
|accessdate=2009-09-09
}}</ref>
=== Defined as an explicit constant ===
In 1983 the 17th CGPM redefined the metre thus, "The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second."<ref name=Resolution_1>
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 [[reproducibility|reproducible]] than the previous standard. They kept the 1967 definition of second, so the [[caesium]] [[Hyperfine structure#Use in defining the SI second and meter|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".<ref name=Resolution_1>
{{cite web
{{Cite web
|year=1983
|title=Resolution 1 of the 17th CGPM
|url=httphttps://www.bipm.org/en/CGPMcommittees/dbcg/cgpm/17-1983/resolution-1/
|publisher=BIPM
|accessdateaccess-date=23 August 2009-08-23
}}</ref>
}}</ref> As a result of this definition, the value of the speed of light in vacuum is exactly {{val|299792458|u=m/s}}<ref name=Wheeler>
As a result of this definition, the value of the speed of light in vacuum is exactly {{val|299792458|u=m/s}}<ref name=Wheeler>
{{Cite book
|lastlast1=Taylor |firstfirst1=EFE. F. |author-link1=Edwin F. Taylor |last2=Wheeler |first2=JAJ. A. |author-link2=John Archibald Wheeler
|year=1992
|title=Spacetime Physics: Introduction to Special Relativity
|url=httphttps://books.google.com/books?id=PDA8YcvMc_QC&pg=PA59#v=onepage&q=
|edition=2nd2
|publisher=[[Macmillan Publishers|Macmillan]]
|isbn=978-0-7167-2327-1
|page=59
}}</ref><ref name=timeline>
{{citeCite web
|last=Penzes |first=WBW. B.
|year=2009
|title=Time Line for the Definition of the Meter
|url=httphttps://www.nist.gov/pml/div683/upload/museum-timeline.pdf
|publisher=[[National Institute of Standards and Technology|NIST]]
|access-date=11 January 2010
|accessdate=2010-01-11
}}</ref> and has become a defined constant in the SI system of units.<ref name="Jespersenfixes"/> Improved experimental techniques dothat, prior to 1983, would have measured the speed of light no notlonger 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.<ref name=Adams>
{{Cite book
|last=Adams |first=S.
|year=1997
|title=Relativity: An Introduction to Space-TimeSpace–Time Physics
|url=httphttps://books.google.com/books?id=1RV0AysEN4oC&pg=PA140
|page=140
|publisher=CRC Press
|isbn=978-0-7484-0621-20
|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!
}}</ref><ref name=W_Rindler>
{{Cite book
|last=Rindler |first=W.
|year=2006
|title=Relativity: Special, General, and Cosmological
|url=httphttps://books.google.com/books?id=MuuaG5HXOGEC&pg=PT41
|page=41
|edition=2nd2
|publisher=[[Oxford University Press]]
|isbn=978-0-19-856731-68
|quote=Note that [...] improvements in experimental accuracy will modify the meter relative to atomic wavelengths, but not the value of the speed of light!
}}</ref>
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<sup>−1</sup>}}."<ref>{{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}}</ref> This was one of the changes that was incorporated in the [[2019 revision of the SI]], also termed the ''New SI''.<ref>See, for example:
==See also==
* {{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}}
*[[Light-second]]
* {{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}}
</ref>
==Notes See also ==
{{Portal|Physics|Astronomy|Outer space}}
{{cols |colwidth=20em}}
* [[Light-second]]
* [[Speed of electricity]]
* [[Speed of gravity]]
* [[Speed of sound]]
* [[Velocity factor]]
* [[Warp drive|Warp factor]] (fictional)
{{colend}}
== Notes ==
{{reflist|group="Note"|30em}}
== References ==
{{Reflist|30emreflist}}
== Further reading ==
=== Historical references ===
{{refbegin}}
===Historical references===
* {{Cite journal
{{Refbegin}}
|first=O.
*{{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://dbhswww.wvusdffn.k12ub.ca.us/webdocses/Chem-Historyluisnavarro/Roemer-1677nuevo_maletin/Roemer-1677Roemer_1676.htmlpdf
|journal=[[Journal des sçavans]]
|pages=223–36223–236
|language=Frenchfr
|access-date=10 March 2020
|archiveurl=http://web.archive.org/web/20070729214326/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html
|archive-date=8 September 2022
|archivedate=2007-07-29
|archive-url=https://web.archive.org/web/20220908221513/http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Roemer_1676.pdf
|url-status=dead
}}
** Translated as {{citeCite 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 =893–4 136
|pages = 893–894
|archiveurl = http://web.archive.org/web/20070729214326/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html
|archive-url = https://web.archive.org/web/20070729214326/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html
|archivedate = 2007-07-29
|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=237–56237–256
|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
* {{Cite journal
|first=HL |last=Fizeau |author-link=Hippolyte Fizeau
|yearfirst=1849J. L.
|last=Foucault
|title=Sur une expérience relative à la vitesse de propagation de la lumière
|author-link=Léon Foucault
|url=http://web.archive.org/web/20110613224002/http://www.academie-sciences.fr/membres/in_memoriam/Fizeau/Fizeau_pdf/CR1849_p90.pdf
|journal=[[Comptes rendus de l'Académie des sciences]]
|volume=29 |pages=90–92, 132
|language=French
}}
*{{Cite journal
|first=JL |last=Foucault |author-link=Léon Foucault
|year=1862
|title=Détermination expérimentale de la vitesse de la lumière: parallaxe du Soleil
|url=httphttps://books.google.cacom/books?id=yYIIAAAAMAAJ&pg=PA216&lpg=PA216&dq
|journal=[[Comptes rendus de l'Académie des sciences]]
|volume=55
|pages=501–503, 792–796
|language=Frenchfr
}}
* {{Cite journal
|first=AAA. A.
|last=Michelson
|author-link=Albert AbrahamA. Michelson
|year=1878
|title=Experimental Determination of the Velocity of Light
|url=http://www.gutenberg.org/ebooks/11753
|journal=[[Proceedings of the American Association offor Advancedthe Advancement of Science]]
|volume=27
|pages=71–77
}}
* {{Cite journal
|first1=AAA. A. |last1=Michelson
|first2=FGF. G. |last2=Pease |author2-link=Francis GladheimG. Pease
|first3=F. |last3=Pearson |author3-link=F. 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 |bibcode=1935ApJ....82...26M
|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
Line 1,544 ⟶ 1,849:
|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=731–4731–734
|language=Frenchfr
}}
{{Refendrefend}}
=== Modern references ===
{{refbegin}}
* {{Cite book
{{Refbegin}}
|first=L. |last=Brillouin |author-link=Léon Brillouin
*{{Cite book
|first=L |last=Brillouin |author-link=Léon Brillouin
|year=1960
|title=Wave propagation and group velocity
|publisher=[[Academic Press]]
|isbn=
}}
* {{Cite book
|first=JDJ. D. |last=Jackson |author-link=J.John D.David Jackson (physicist)
|year=1975
|title=Classical Electrodynamics
|edition=2nd2
|publisher=[[John Wiley & Sons]]
|isbn=978-0-471-30932-X1
}}
* {{Cite book
|first=G .
|last=Keiser
|year=2000
|title=Optical Fiber Communications
|page=[https://archive.org/details/opticalfibercomm00gerd/page/32 32]
|page=32 |edition=3rd
|edition=3
|publisher=[[McGraw-Hill]]
|publisher=McGraw-Hill
|isbn=0-07-232101-6
|isbn=978-0-07-232101-2
|url=https://archive.org/details/opticalfibercomm00gerd/page/32
}}
* {{Cite book
|last=Ng |first=YJY. J.
|year=2004
|chapter=Quantum Foam and Quantum Gravity Phenomenology
|chapter-url=httphttps://books.google.com/books?id=RntpN7OesBsC
|editor=Amelino-Camelia, G; |editor2=Kowalski-Glikman, J
|title=Planck Scale Effects in Astrophysics and Cosmology
|pages=321''ff''321ff
|publisher=[[Springer (publisher)|Springer]]
|isbn=978-3-540-25263-04
}}
* {{Cite book
|lastlast1=Helmcke |firstfirst1=J. |last2=Riehle |first2=F.
|year=2001
|chapter=Physics behind the definition of the meter
|chapter-url=httphttps://books.google.com/books?id=WE22Fez60EcC&pg=PA453
|editor=Quinn, TJ;T. J. |editor2=Leschiutta, S;. |editor3=Tavella, P.
|title=Recent advances in metrology and fundamental constants
|page=453
|publisher=[[IOS Press]]
|isbn=978-1-58603-167-81
}}
* {{citeCite arxivarXiv
|last=Duff |first=MJM. J. |author-link=Michael James Duff
|year=2004
|title=Comment on time-variation of fundamental constants
|class=hep-th
|eprint=hep-th/0208093
}}
{{Refendrefend}}
== External links ==
{{wikiquote}}
* [http://books.google.com/books?id=UigDAAAAMBAJ&pg=PA17&dq=1930+plane+%22Popular&hl=en&ei=bfiPTs-NGInE0AHC_4k_&sa=X&oi=book_result&ct=result&resnum=8&ved=0CEwQ6AEwBzgK#v=onepage&q=1930%20plane%20%22Popular&f=true "Test Light Speed in Mile Long Vacuum Tube."] ''Popular Science Monthly'', September 1930, p. 17–18.
* [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://physics.nist.gov/cgi-bin/cuu/Value?c Speed of light in vacuum] (National Institute of Standards and Technology, NIST)
* [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)
*[http://www.itl.nist.gov/div898/bayesian/datagall/michelso.htm Data Gallery: Michelson Speed of Light (Univariate Location Estimation)] (download data gathered by [[Albert Abraham Michelson|A.A. Michelson]])
* [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]])
*[http://gregegan.customer.netspace.net.au/APPLETS/20/20.html Subluminal] (Java applet demonstrating group velocity information limits)
* [https://www.gregegan.net/APPLETS/20/20.html Subluminal] (Java applet by [[Greg Egan]] demonstrating group velocity information limits)
*[http://www.mathpages.com/rr/s3-03/3-03.htm De Mora Luminis] at MathPages
* [http://www.ertin.com/sloan_on_speed_of_light.html Light discussion on adding velocities]
* [http://wwwsixtysymbols.colorado.educom/physicsvideos/2000/waves_particles/lightspeed-1light.htmlhtm Speed of Light] (Sixty Symbols, University of ColoradoNottingham Department of Physics [video])
* [httphttps://sixtysymbolswww.combbc.co.uk/videosprogrammes/light.htm c:p0038x9h Speed of Light], (SixtyBBC Symbols,Radio{{nbsp}}4 Universitydiscussion of(''In NottinghamOur DepartmentTime'', of30 PhysicsNovember [video]2006)
* [http://www.live-counter.com/speed-of-light/ Speed of Light] (Live-Counter – Illustrations)
*[http://math.ucr.edu/home/baez/physics/ Usenet Physics FAQ]
<!-- en-GB-oed, -ize -->
* [https://www.sciencealert.com/simple-animations-by-a-nasa-scientist-prove-the-speed-of-light-is-torturously-slow Speed of Light – animated demonstrations]
{{extreme motion}}
* "[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}}
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{{Relativity}}
{{featuredSpecial articlerelativity}}
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{{DEFAULTSORT:Speed Of Light}}
[[Category:Fundamental constants]]
[[Category:ConceptsPhysical in physicsquantities]]
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[[Category:Special relativity]]
[[Category:Units of velocityVelocity|Light]]
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