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{{Use dmy dates|date=August 2019}}{{short description|Speed at which all massless particles and associated fields travel in a vacuum}}{{Redirect|Lightspeed|other uses|Speed of light (disambiguation)|and|Lightspeed (disambiguation)}}{{pp-protected|small=yes}}{{featured article}}

factoids

## Numerical value, notation, and units

The speed of light in vacuum is usually denoted by a lowercase {{math|c}}, for "constant" or the Latin (meaning "swiftness, celerity"). In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used {{math|c}} for a different constant 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. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to {{math|c}}, which by then had become the standard symbol for the speed of light.WEB
, Gibbs
, P
, 2004
, 1997
, Why is c the symbol for the speed of light?
, Usenet Physics FAQ
, University of California, Riverside
, 2009-11-16
, 2010-03-25
, yes
,
, "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'."JOURNAL
, Mendelson, KS
, 2006
, The story of c
, American Journal of Physics
, 74, 11, 995â€“97
, 10.1119/1.2238887, 2006AmJPh..74..995M,
Sometimes {{math|c}} is used for the speed of waves in any material medium, and {{math|c}}0 for the speed of light in vacuum.See for example:
• BOOK

, Lide, DR
, 2004
, CRC Handbook of Chemistry and Physics
, 2â€“9
, CRC Press
, 978-0-8493-0485-9
,
• BOOK

, Harris, JW, 2002
, Handbook of Physics
, 499
, Springer
, 978-0-387-95269-7, etal,
• BOOK

, Whitaker, JC
, 2005
, The Electronics Handbook
, 235
, CRC Press
, 978-0-8493-1889-4
,
• BOOK

, Cohen, ER, 2007
, Quantities, Units and Symbols in Physical Chemistry
, 184
, 3rd
, Royal Society of Chemistry
, 978-0-85404-433-7
page=112}} has the same form as other related constants: namely, Î¼0 for the vacuum permeability or magnetic constant, Îµ0 for the vacuum permittivity or electric constant, and Z0 for the impedance of free space. This article uses {{math|c}} exclusively for the speed of light in vacuum.Since 1983, the metre has been defined in the International System of Units (SI) as the distance light travels in vacuum in {{frac|1|{{val|299792458}}}} of a second. This definition fixes the speed of light in vacuum at exactly {{val|299792458|u=m/s}}.BOOK
, Sydenham, PH
, 2003
, Measurement of length
, Boyes, W
, Instrumentation Reference Book
, 3rd
, 56
, Butterworthâ€“Heinemann
, 978-0-7506-7123-1
, ... if the speed of light is defined as a fixed number then, in principle, the time standard will serve as the length standard ...
, WEB
, CODATA value: Speed of Light in Vacuum
, The NIST reference on Constants, Units, and Uncertainty
National Institute of Standards and Technology>NIST
, 2009-08-21
, BOOK
, Jespersen, J, Fitz-Randolph, J, Robb, J
, 1999
, From Sundials to Atomic Clocks: Understanding Time and Frequency
, 280
, Reprint of National Bureau of Standards 1977, 2nd
, Courier Dover
, 978-0-486-40913-9
, As a dimensional physical constant, the numerical value of {{math|c}} is different for different unit systems.{{#tag:ref|The speed of light in imperial units and US units is based on an inch of exactly {{val|2.54|u=cm}} and is exactly {{val|186,282}} miles, 698 yards, 2 feet, and {{sfrac|5|21|127}} inches per second.WEB
, Savard
, J
, From Gold Coins to Cadmium Light
, John Savard
, 2009-11-14
, 2009-11-20
, yes
,
, |group="Note"|name="imperial"}}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}}.BOOK
, Lawrie, ID
, 2002
, Appendix C: Natural units
, A Unified Grand Tour of Theoretical Physics
, 540
, 2nd
, CRC Press
, 978-0-7503-0604-1
, BOOK
, Hsu, L
, 2006
, Appendix A: Systems of units and the development of relativity theories
, A Broader View of Relativity: General Implications of Lorentz and PoincarÃ© Invariance
, 427â€“28
, 2nd
, World Scientific
, 978-981-256-651-5
, Using these units, {{math|c}} does not appear explicitly because multiplication or division by 1 does not affect the result.

## Fundamental role in physics

{{See also|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.However, the frequency of light can depend on the motion of the source relative to the observer, due to the Doppler effect. This invariance of the speed of light was postulated by Einstein in 1905, after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous aether;JOURNAL
, Einstein, A
, 1905
, Zur Elektrodynamik bewegter KÃ¶rper
, Annalen der Physik
, 17, 10
, 890â€“921
, 10.1002/andp.19053221004
, German
, Submitted manuscript
, English translation: WEB
, Perrett, W
, Jeffery, GB
, Walker, J
, On the Electrodynamics of Moving Bodies
, Fourmilab
, 2009-11-27
, it has since been consistently confirmed by many experiments. 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. However, by adopting Einstein synchronization for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition.BOOK
, Hsu, J-P, Zhang, YZ
, 2001
, World Scientific
, Advanced Series on Theoretical Physical Science
, 8, 543ff
, 978-981-02-4721-8
, BOOK
, Zhang
, YZ
, 1997
, Special Relativity and Its Experimental Foundations
, World Scientific
, Advanced Series on Theoretical Physical Science
, 4
, 172â€“73
, 978-981-02-2749-4
, 23 July 2009
, 2012-05-19
, yes
,
, 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.BOOK
, d'Inverno, R
, 1992
, Introducing Einstein's Relativity
, 19â€“20
, Oxford University Press
, 978-0-19-859686-8
, BOOK
, Sriranjan, B
, 2004
, Postulates of the special theory of relativity and their consequences
, The Special Theory to Relativity
, PHI Learning Pvt. Ltd.
, 978-81-203-1963-9
, 20ff
, One consequence is that c is the speed at which all massless particles and waves, including light, must travel in vacuum.File:Lorentz factor.svg|thumb|left|upright|alt=Î³ 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 factorLorentz factorSpecial relativity has many counterintuitive and experimentally verified implications.WEB
, Roberts
, T
, Schleif
, S
, Dlugosz
, JM
, 2007
, What is the experimental basis of Special Relativity?
, Usenet Physics FAQ
, University of California, Riverside
, 27 November 2009
, 2009-10-15
, yes
,
, These include the equivalence of mass and energy {{nowrap|(E {{=}} mc2)}}, 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.JOURNAL
, Terrell, J
, 1959
, Invisibility of the Lorentz Contraction
, Physical Review
, 116
, 4, 1041â€“5
, 10.1103/PhysRev.116.1041, 1959PhRv..116.1041T,
JOURNAL
, Penrose, R
, 1959
, The Apparent Shape of a Relativistically Moving Sphere
, Proceedings of the Cambridge Philosophical Society
, 55
, 1, 137â€“39
, 10.1017/S0305004100033776
group="Note"}} and time dilation (moving clocks run more slowly). The factor Î³ by which lengths contract and times dilate is known as the Lorentz factor and is given by {{nowrap|Î³ {{=}} (1 âˆ’ v2/c2)âˆ’1/2}}, where v is the speed of the object. The difference of Î³ from 1 is negligible for speeds much slower than c, such as most everyday speedsâ€”in which case special relativity is closely approximated by Galilean relativityâ€”but it increases at relativistic speeds and diverges to infinity as v approaches c. For example, a time dilation factor of Î³ = 2 occurs at a relative velocity of 86.6% of the speed of light (v = .866c). Similarly, a time dilation factor of Î³ = 10 occurs at v = 99.5% c.The results of special relativity can be summarized by treating space and time as a unified structure known as spacetime (with c relating the units of space and time), and requiring that physical theories satisfy a special symmetry called Lorentz invariance, whose mathematical formulation contains the parameter c.BOOK
, Hartle, JB
, 2003
, Gravity: An Introduction to Einstein's General Relativity
, 52â€“59
, 978-981-02-2749-4
, 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.BOOK
, Hartle, JB
, 2003
, Gravity: An Introduction to Einstein's General Relativity
, 332
, 978-981-02-2749-4
, {{#tag:ref|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.BOOK
, SchÃ¤fer, G
, MH, BrÃ¼gmann
, H, Dittus
, C, LÃ¤mmerzahl
, SG, Turyshev
, Propagation of light in the gravitational field of binary systems to quadratic order in Newton's gravitational constant: Part 3: â€˜On the speed-of-gravity controversyâ€™
, Lasers, clocks and drag-free control: Exploration of relativistic gravity in space
, 978-3-540-34376-9
, 2008
, Springer
, |group="Note"}} In non-inertial frames of reference (gravitationally curved spacetime or accelerated reference frames), the local speed of light is constant and equal to c, but the speed of light along a trajectory of finite length can differ from c, depending on how distances and times are defined.It is generally assumed that fundamental constants such as c have the same value throughout spacetime, meaning that they do not depend on location and do not vary with time. However, it has been suggested in various theories that the speed of light may have changed over time.JOURNAL
, Ellis, GFR, Uzan, J-P
, 2005
, 'c' is the speed of light, isn't it?
, American Journal of Physics
, 73
, 3, 240â€“27
, 10.1119/1.1819929
, gr-qc/0305099
, 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.
, Mota, DF
, 2006
, Variations of the Fine Structure Constant in Space and Time
, astro-ph/0401631, 2004astro.ph..1631M, No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.
JOURNAL
, Uzan, J-P
, 2003
, The fundamental constants and their variation: observational status and theoretical motivations
, Reviews of Modern Physics
, 75
, 2, 403
, 10.1103/RevModPhys.75.403
, hep-ph/0205340, 2003RvMP...75..403U
, JOURNAL
, Amelino-Camelia, G
, 2013
, Quantum Gravity Phenomenology
, 0806.0339
, 10.12942/lrr-2013-5
, 28179844
, 5255913
, 16
, 1
, 5
, Living Reviews in Relativity, 2013LRR....16....5A
,
It also is generally assumed that the speed of light is isotropic, meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear energy levels as a function of the orientation of the emitting nuclei in a magnetic field (see Hughesâ€“Drever experiment), and of rotating optical resonators (see Resonator experiments) have put stringent limits on the possible two-way anisotropy.JOURNAL
, Herrmann, S, Senger, A, MÃ¶hle, K, Nagel, M, Kovalchuk, EV, Peters, A, 1
, Rotating optical cavity experiment testing Lorentz invariance at the 10âˆ’17 level
, Physical Review D, 80, 100, 105011, 2009
, 10.1103/PhysRevD.80.105011, 1002.1284, 2009PhRvD..80j5011H, BOOK
, Astrophysical formulae
, KR, Lang
, 152
, 978-3-540-29692-8
, BirkhÃ¤user
, 3rd
, 1999,

### Upper limit on speeds

According to special relativity, the energy of an object with rest mass m and speed v is given by {{nowrap|Î³mc2}}, where Î³ is the Lorentz factor defined above. When v is zero, Î³ is equal to one, giving rise to the famous {{nowrap|E {{=}} mc2}} formula for massâ€“energy equivalence. The Î³ 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, and individual photons cannot travel faster than the speed of light.WEB,weblink It's official: Time machines won't work, Los Angeles Times, 25 July 2011, WEB, 2011-07-19, The Hong Kong University of Science and Technology,weblink HKUST Professors Prove Single Photons Do Not Exceed the Speed of Light, JOURNAL, Optical Precursor of a Single Photon, Shanchao Zhang, J.F. Chen, Chang Liu, M.M.T. Loy, G.K.L. Wong, Shengwang Du, Phys. Rev. Lett., 106, 243602, 243602, 16 June 2011, 10.1103/physrevlett.106.243602, 21770570, 2011PhRvL.106x3602Z, This is experimentally established in many tests of relativistic energy and momentum.WEB
, Fowler, M
, March 2008
, Notes on Special Relativity
, 56
, University of Virginia
, 2010-05-07
, (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 xâ€² axis is slightly skewed upwards, and the ctâ€² axis slightly skewed rightwards, relative to the green axes; in the blue frame, the xâ€²â€² axis is somewhat skewed downwards, and the ctâ€²â€² 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 ctâ€², and negative ctâ€²â€².)More generally, it is normally impossible for information 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 would be violated.{{#tag:ref|It is thought that the Scharnhorst effect does allow signals to travel slightly faster than c, but the special conditions in which this effect can occur prevent one from using this effect to violate causality.JOURNAL
, Liberati, S, Sonego, S, Visser, M
, 2002
, Faster-than-c signals, special relativity, and causality
, Annals of Physics
, 298
, 1, 167â€“85
, 10.1006/aphy.2002.6233
, gr-qc/0107091
group="Note"}}BOOK
, Taylor
, EF
, Wheeler
, JA
, 1992
, Spacetime Physics
, 74â€“75
, W.H. Freeman
, 978-0-7167-2327-1
, In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded, and would lead to paradoxes such as the tachyonic antitelephone.
BOOK
, Tolman, RC
, 2009, 1917
, Velocities greater than that of light
, The Theory of the Relativity of Motion
, Reprint, 54
, BiblioLife
, 978-1-103-17233-7
,

## Faster-than-light observations and experiments

{{Further|Superluminal motion}}There are situations in which it may seem that matter, energy, or information travels at speeds greater than c, but they do not. For example, as is discussed in the propagation of light in a medium section below, many wave velocities can exceed c. For example, the phase velocity of X-rays through most glasses can routinely exceed c,BOOK
, Hecht, E
, 1987
, Optics
, 62
, 2nd
, 978-0-201-11609-0
, but phase velocity does not determine the velocity at which waves convey information.BOOK
, Quimby, RS
, Photonics and lasers: an introduction
, John Wiley and Sons
, 2006
, 9
, 978-0-471-71974-8
, 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.NEWS
, Wertheim, M
, The New York Times
, 2009-08-21
, 2007-06-20
, In neither case does any matter, energy, or information travel faster than light.WEB
, Gibbs
, P
, 1997
, Is Faster-Than-Light Travel or Communication Possible?
, University of California, Riverside
, Usenet Physics FAQ
, 2008-08-20
, 2010-03-10
, yes
,
, The rate of change in the distance between two objects in a frame of reference with respect to which both are moving (their closing speed) may have a value in excess of c. However, this does not represent the speed of any single object as measured in a single inertial frame.Certain quantum effects appear to be transmitted instantaneously and therefore faster than c, as in the EPR paradox. An example involves the quantum states of two particles that can be entangled. Until either of the particles is observed, they exist in a superposition of two quantum states. If the particles are separated and one particle's quantum state is observed, the other particle's quantum state is determined instantaneously (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.BOOK
, Sakurai, JJ
, 1994
, Tuan, SF
, Modern Quantum Mechanics
, Revised, 231â€“32
, 978-0-201-53929-5
, Another quantum effect that predicts the occurrence of faster-than-light speeds is called the Hartman effect: under certain conditions the time needed for a virtual particle to tunnel through a barrier is constant, regardless of the thickness of the barrier.BOOK
, Muga, JG, Mayato, RS, Egusquiza, IL
, 2007
, Time in Quantum Mechanics
, 48
, Springer
, 978-3-540-73472-7
, BOOK
, HernÃ¡ndez-Figueroa, HE, Zamboni-Rached, M, Recami, E
, 2007
, Localized Waves
, 26
, Wiley Interscience
, 978-0-470-10885-7
, This could result in a virtual particle crossing a large gap faster-than-light. However, no information can be sent using this effect.JOURNAL
, Wynne
, K
, 2002
, Causality and the nature of information
, Optics Communications
, 209
, 1â€“3
, 84â€“100
, 10.1016/S0030-4018(02)01638-3
, 2002OptCo.209...85W
, Rees, M
, 1966
, The Appearance of Relativistically Expanding Radio Sources
, Nature (journal), Nature
, 211
, 5048, 468
, 10.1038/211468a0
relativistic jets of radio galaxy>radio galaxies and quasars. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection effect caused by objects moving near the speed of light and approaching Earth at a small angle to the line of sight: since the light which was emitted when the jet was farther away took longer to reach the Earth, the time between two successive observations corresponds to a longer time between the instants at which the light rays were emitted.WEB
, Chase, IP
, Apparent Superluminal Velocity of Galaxies
, University of California, Riverside
, Usenet Physics FAQ
, 2009-11-26
, 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 expansion of space itself. 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.BOOK
, Harrison, ER
, 2003
, 206
, Cambridge University Press
, 978-0-521-77351-5
,

## 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 distributed capacitance and inductance of the vacuum, otherwise respectively known as the electric constant Îµ0 and the magnetic constant Î¼0, by the equationBOOK
, Panofsky, WKH
, Phillips, M
, 1962
, Classical Electricity and Magnetism
, 182
, 978-0-201-05702-7
,
c =frac {1}{sqrt{varepsilon_0 mu_0}} .
In modern quantum physics, the electromagnetic field is described by the theory of quantum electrodynamics (QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, called photons. In QED, photons are massless particles and thus, according to special relativity, they travel at the speed of light in vacuum.Extensions of QED in which the photon has a mass have been considered. In such a theory, its speed would depend on its frequency, and the invariant speed c of special relativity would then be the upper limit of the speed of light in vacuum.WEB
, Gibbs
, P
, 1997
, 1996
, Is The Speed of Light Constant?
, Carlip
, S
, Usenet Physics FAQ
, University of California, Riverside
, 2009-11-26
, 2010-04-02
, yes
,
, No variation of the speed of light with frequency has been observed in rigorous testing,JOURNAL
, Schaefer, BE
, 1999
, Severe limits on variations of the speed of light with frequency
, Physical Review Letters
, 82
, 25, 4964â€“66
, 10.1103/PhysRevLett.82.4964
, astro-ph/9810479, 1999PhRvL..82.4964S
, JOURNAL
, Ellis, J
, Mavromatos, NE
, Nanopoulos, DV
, Sakharov, AS
, 2003
, Quantum-Gravity Analysis of Gamma-Ray Bursts using Wavelets
, Astronomy & Astrophysics
, 402
, 2, 409â€“24
, 10.1051/0004-6361:20030263
, astro-ph/0210124, 2003A&A...402..409E
, JOURNAL
, FÃ¼llekrug, M
, 2004
, Probing the Speed of Light with Radio Waves at Extremely Low Frequencies
, Physical Review Letters
, 93
, 4, 043901
, 10.1103/PhysRevLett.93.043901, 2004PhRvL..93d3901F
, 15323762
, 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 theory,JOURNAL
, Dvali, G
, Gruzinov, A
, 2007
, Photon Mass Bound Destroyed by Vortices
, Physical Review Letters
, 98
, 1, 010402
, 10.1103/PhysRevLett.98.010402
, hep-ph/0306245, 17358459, 2007PhRvL..98a0402A
, the experimental upper bound for its mass is about 10âˆ’57 grams;BOOK
, Sidharth, BG
, 2008
, The Thermodynamic Universe
, 134
, World Scientific
, 978-981-281-234-6
, if photon mass is generated by a Higgs mechanism, the experimental upper limit is less sharp, {{nowrap|m â‰¤ 10âˆ’14 eV/c2}}  (roughly 2 Ã— 10âˆ’47 g).Another reason for the speed of light to vary with its frequency would be the failure of special relativity to apply to arbitrarily small scales, as predicted by some proposed theories of quantum gravity. In 2009, the observation of the spectrum of gamma-ray burst GRB 090510 did not find any difference in the speeds of photons of different energies, confirming that Lorentz invariance is verified at least down to the scale of the Planck length (lP = {{radic|Ä§ G/c3}} â‰ˆ {{val|1.6163|e=-35|u=m}}) divided by 1.2.JOURNAL
, Amelino-Camelia, G
, 2009
, Astrophysics: Burst of support for relativity
, Nature (journal), Nature
, 462, 291â€“92
, 10.1038/462291a
, Nature, 19 November 2009
, 19924200
, 7271, 2009Natur.462..291A,

### 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 vp. An actual physical signal with a finite extent (a pulse of light) travels at a different speed. The largest part of the pulse travels at the group velocity vg, and its earliest part travels at the front velocity vf.(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 vp in the material: larger indices of refraction indicate lower speeds. The refractive index of a material may depend on the light's frequency, intensity, polarization, or direction of propagation; in many cases, though, it can be treated as a material-dependent constant. The refractive index of air is approximately 1.0003.BOOK
, de Podesta, M
, 2002
, Understanding the Properties of Matter
, 131
, CRC Press
, 978-0-415-25788-6
, Denser media, such as water,WEB
, Optical constants of H2O, D2O (Water, heavy water, ice)
, Mikhail Polyanskiy
, refractiveindex.info
, 2017-11-07
, glass,WEB
, Optical constants of Soda lime glass
, Mikhail Polyanskiy
, refractiveindex.info
, 2017-11-07
, and diamond,WEB
, Optical constants of C (Carbon, diamond, graphite)
, Mikhail Polyanskiy
, refractiveindex.info
, 2017-11-07
, have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light. In exotic materials like Boseâ€“Einstein condensates near absolute zero, the effective speed of light may be only a few metres per second. However, this represents absorption and re-radiation delay between atoms, as do all slower-than-c speeds in material substances. As an extreme example of light "slowing" in matter, two independent teams of physicists claimed to bring light to a "complete standstill" by passing it through a Boseâ€“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 Harvardâ€“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.WEB, Cromie, William J.,weblink Researchers now able to stop, restart light, Harvard University Gazette, 2001-01-24, 2011-11-08, yes,weblink" title="web.archive.org/web/20111028041346weblink">weblink 2011-10-28, 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.BOOK
, Fast light, slow light and left-handed light
, Milonni, PW
, Peter W. Milonni
, 25
, 978-0-7503-0926-4
, 2004
, CRC Press
, The requirement that causality is not violated implies that the real and imaginary parts of the dielectric constant of any material, corresponding respectively to the index of refraction and to the attenuation coefficient, are linked by the Kramersâ€“Kronig relations.JOURNAL
, Toll, JS
, 1956
, Causality and the Dispersion Relation: Logical Foundations
, Physical Review
, 104
, 6, 1760â€“70
, 10.1103/PhysRev.104.1760, 1956PhRv..104.1760T, 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.
A pulse with different group and phase velocities (which occurs if the phase velocity is not the same for all the frequencies of the pulse) smears out over time, a process known as dispersion. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called slow light, which has been confirmed in various experiments.JOURNAL
, Hau, LV
, Harris, SE
, Dutton, Z
, Behroozi, CH
, 1999
, Light speed reduction to 17 metres per second in an ultracold atomic gas
, Nature
, 397
, 6720, 594â€“98
, 10.1038/17561
url=http://www.seas.harvard.edu/haulab/publications/pdf/Slow_Light_1999.pdf
,
JOURNAL
, Liu, C, Dutton, Z, Behroozi, CH, Hau, LV
, 2001
, Observation of coherent optical information storage in an atomic medium using halted light pulses
, Nature
, 409, 6819, 490â€“93
, 10.1038/35054017
, 11206540
url=http://www.seas.harvard.edu/haulab/publications/pdf/Stopped_Light_2001.pdf, JOURNAL
, Bajcsy, M, Zibrov, AS, Lukin, MD
, 2003
, Stationary pulses of light in an atomic medium
, Nature
, 426, 6967, 638â€“41
, 10.1038/nature02176
, 14668857
bibcode = 2003Natur.426..638B, WEB
, 2003
, Switching light on and off
, Physics World
, Institute of Physics, 2008-12-08
, The opposite, group velocities exceeding c, has also been shown in experiment.NEWS
, Whitehouse, D
, 19 July 2000
, Beam Smashes Light Barrier
, BBC News
, 2008-12-08
, It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.BOOK
, Fast light, slow light and left-handed light
, Milonni, PW
, Peter W. Milonni
, 2
, 978-0-7503-0926-4
, 2004
, CRC Press
, 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. {{clear}}It is possible for a particle to travel through a medium faster than the phase velocity of light in that medium (but still slower than c). When a charged particle does that in a dielectric material, the electromagnetic equivalent of a shock wave, known as Cherenkov radiation, is emitted.JOURNAL, Cherenkov, Pavel A., Pavel Alekseyevich Cherenkov, 1934, Ð’Ð¸Ð´Ð¸Ð¼Ð¾Ðµ ÑÐ²ÐµÑ‡ÐµÐ½Ð¸Ðµ Ñ‡Ð¸ÑÑ‚Ñ‹Ñ… Ð¶Ð¸Ð´ÐºÐ¾ÑÑ‚ÐµÐ¹ Ð¿Ð¾Ð´ Ð´ÐµÐ¹ÑÑ‚Ð²Ð¸ÐµÐ¼ Î³-Ñ€Ð°Ð´Ð¸Ð°Ñ†Ð¸Ð¸, Visible emission of pure liquids by action of Î³ radiation, Doklady Akademii Nauk SSSR, 2, 451, Reprinted: JOURNAL, Cherenkov, P.A., 1967, Ð’Ð¸Ð´Ð¸Ð¼Ð¾Ðµ ÑÐ²ÐµÑ‡ÐµÐ½Ð¸Ðµ Ñ‡Ð¸ÑÑ‚Ñ‹Ñ… Ð¶Ð¸Ð´ÐºÐ¾ÑÑ‚ÐµÐ¹ Ð¿Ð¾Ð´ Ð´ÐµÐ¹ÑÑ‚Ð²Ð¸ÐµÐ¼ Î³-Ñ€Ð°Ð´Ð¸Ð°Ñ†Ð¸Ð¸, Visible emission of pure liquids by action of Î³ radiation, Usp. Fiz. Nauk, 93, 10, 385, 10.3367/ufnr.0093.196710n.0385, , and in BOOK, Pavel Alekseyevich ÄŒerenkov: Chelovek i Otkrytie, Pavel Alekseyevich ÄŒerenkov: Man and Discovery, A.N. Gorbunov, E.P. ÄŒerenkova, Moscow, Nauka, 1999, 149â€“53,

## Practical effects of finiteness

The speed of light is of relevance to communications: 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 supercomputers, the speed of light imposes a limit on how quickly data can be sent between processors. If a processor operates at 1 gigahertz, a signal can only travel a maximum of about {{convert|30|cm|ft|0}} in a single cycle. Processors must therefore be placed close to each other to minimize communication latencies; this can cause difficulty with cooling. If clock frequencies continue to increase, the speed of light will eventually become a limiting factor for the internal design of single chips.BOOK
, Parhami, B
, 1999
, Introduction to parallel processing: algorithms and architectures
, 5
, Plenum Press
, 978-0-306-45970-2
, CONFERENCE
, Software Transactional Memories: An Approach for Multicore Programming
last1=Imbslast2=Raynal, 2009, 10th International Conference, PaCT 2009, Novosibirsk, Russia, 31 August â€“ 4 September 2009, Malyshkin, V, Springer, 978-3-642-03274-5, 26,

### Large distances on Earth

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 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.{{#tag:ref|A typical value for the refractive index of optical fibre is between 1.518 and 1.538.BOOK
, Midwinter, JE
, 1991
, Optical Fibers for Transmission
, 2nd
, Krieger Publishing Company
, 978-0-89464-595-2
, |group="Note"}} Furthermore, straight lines rarely occur in global communications situations, and delays are created when the signal passes through an electronic switch or signal regenerator.WEB
, June 2007
, Theoretical vs real-world speed limit of Ping
, Pingdom, 2010-05-05
,

### Spaceflights and astronomy

(File:Speed of light from Earth to Moon.gif|thumb|right|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 Earthâ€“Moon system are shown to scale.)Similarly, communications between the Earth and spacecraft are not instantaneous. There is a brief delay from the source to the receiver, which becomes more noticeable as distances increase. This delay was significant for communications between ground control and Apollo 8 when it became the first 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.WEB
, Day 4: Lunar Orbits 7, 8 and 9
, The Apollo 8 Flight Journal
, NASA
, 16 December 2010
, yes
, 2011-01-04
,
, 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.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 take much longer. For example, it has taken 13 billion (13{{e|9}}) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field images.PRESS
, 5 January 2010
, Hubble Reaches the "Undiscovered Country" of Primeval Galaxies
, Space Telescope Science Institute
, WEB
, The Hubble Ultra Deep Field Lithograph
, NASA
, 2010-02-04
, Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when the universe was less than a billion years old. The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars, of galaxies, and of the universe itself.Astronomical distances are sometimes expressed in light-years, especially in popular science publications and media.WEB
, The IAU and astronomical units
, International Astronomical Union
, 2010-10-11
, A light-year is the distance light travels in one year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs. In round figures, a light year is nearly 10 trillion kilometres or nearly 6 trillion miles. Proxima Centauri, the closest star to Earth after the Sun, is around 4.2 light-years away.Further discussion can be found at WEB
, 2000
, StarChild Question of the Month for March 2000
, StarChild
, NASA
, 2009-08-22
,

### Distance measurement

Radar systems measure the distance to a target by the time it takes a radio-wave pulse to return to the radar antenna after being reflected by the target: the distance to the target is half the round-trip transit time multiplied by the speed of light. A Global Positioning System (GPS) receiver measures its distance to GPS satellites based on how long it takes for a radio signal to arrive from each satellite, and from these distances calculates the receiver's position. Because light travels about {{val|300000|u=kilometres}} ({{val|186000|u=miles}}) in one second, these measurements of small fractions of a second must be very precise. The Lunar Laser Ranging Experiment, radar astronomy and the Deep Space Network determine distances to the Moon,JOURNAL
, Dickey, JO
, Lunar Laser Ranging: A Continuing Legacy of the Apollo Program
, Science, 265, 5171
, 482â€“90, July 1994
, 10.1126/science.265.5171.482
, Standish, EM
, The JPL planetary ephemerides
, Celestial Mechanics, 26, February 1982
, 2, 181â€“86, 10.1007/BF01230883
, 1982CeMec..26..181S, and spacecraft,JOURNAL
, Berner, JB
, Bryant, SH
, Kinman, PW
, Range Measurement as Practiced in the Deep Space Network
, Proceedings of the IEEE, November 2007, 95, 11, 2202â€“2214
, 10.1109/JPROC.2007.905128,weblink respectively, by measuring round-trip transit times.

## Measurement

There are different ways to determine the value of c. One way is to measure the actual speed at which light waves propagate, which can be done in various astronomical and earth-based setups. However, it is also possible to determine c from other physical laws where it appears, for example, by determining the values of the electromagnetic constants Îµ0 and Î¼0 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.In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of {{frac|{{val|299792458}}}} of a second", fixing the value of the speed of light at {{val|299792458|u=m/s}} by definition, as described below. Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value of c.

### Astronomical measurements

(File:Io eclipse speed of light measurement.svg|thumb|upright=1.5|Measurement of the speed of light using the eclipse of Io by Jupiter)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. It is customary to express the results in astronomical units (AU) per day.Ole Christensen RÃ¸mer used an astronomical measurement to make the first quantitative estimate of the speed of light.JOURNAL
, Cohen, IB
, 1940
, Roemer and the first determination of the velocity of light (1676)
, Isis (journal), Isis
, 31, 2, 327â€“79
, 10.1086/347594
, cohen-1940, 2027/uc1.b4375710
,
JOURNAL
, 1676
, Demonstration tovchant le mouvement de la lumiere trouvÃ© par M. RÅmer de l'AcadÃ©mie Royale des Sciences
, Demonstration to the movement of light found by Mr. RÃ¶mer of the Royal Academy of Sciences
, French
, Journal des sÃ§avans
, 233â€“36
, roemer-1676
, Translated in JOURNAL
, 10.1098/rstl.1677.0024
, 1677
, A demonstration concerning the motion of light, communicated from Paris, in the Journal des SÃ§avans, and here made English
, Philosophical Transactions of the Royal Society
, 12, 136, 893â€“95
, roemer-1676-EnglishTrans
, 1677RSPT...12..893., Reproduced in BOOK
, Hutton, C
, Shaw, G
, Pearson, R
, 1809
, The Philosophical Transactions of the Royal Society of London, from Their Commencement in 1665, in the Year 1800: Abridged
, On the Motion of Light by M. Romer
, London, C. & R. Baldwin
, Vol. II. From 1673 to 1682, 397â€“98
, The account published in Journal des sÃ§avans was based on a report that RÃ¸mer read to the French Academy of Sciences in November 1676 (Cohen, 1940, p. 346). When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it. The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the diameter of the Earth's orbit around the Sun. The observed change in the moon's orbital period is caused by the difference in the time it takes light to traverse the shorter or longer distance. RÃ¸mer observed this effect for Jupiter's innermost moon Io and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.(File:SoL Aberration.svg|thumb|right|upright|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 which 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.JOURNAL
, 1729
, Account of a new discoved Motion of the Fix'd Stars
, Philosophical Transactions
, 35, 637â€“60
,
, This effect results from the vector addition of the velocity of light arriving from a distant source (such as a star) and the velocity of its observer (see diagram on the right). A moving observer thus sees the light coming from a slightly different direction and consequently sees the source at a position shifted from its original position. Since the direction of the Earth's velocity changes continuously as the Earth orbits the Sun, this effect causes the apparent position of stars to move around. From the angular difference in the position of stars (maximally 20.5 arcseconds)BOOK
, Duffett-Smith
, P
, 1988
, Practical Astronomy with your Calculator
, 62
, Cambridge University Press
, 978-0-521-35699-2, Extract of page 62 it is possible to express the speed of light in terms of the Earth's velocity around the Sun, which with the known length of a year can be converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled {{val|10,210}} times faster than the Earth in its orbit (the modern figure is {{val|10,066}} times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.

#### 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}}.JOURNAL, The International System of Units, Supplement 2014: Updates to the 8th edition (2006) of the SI Brochure,weblink 2014, International Bureau of Weights and Measures, 14, 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}} radians (approximately {{frac|{{val|365.256898}}}} of a revolution) per day.{{SIbrochure8th|page=126}}|group="Note"}} The current definition uses the recommended value in metres for the previous definition of the astronomical unit, which was determined by measurement.JOURNAL, Resolution B2 on the re-definition of the astronomical unit of length,weblink 2012, International Astronomical Union, This redefinition is analogous to that of the metre, and likewise has the effect of fixing the speed of light to an exact value in astronomical units per second (via the exact speed of light in metres per second).Previously, the inverse of {{math|c}} expressed in seconds per astronomical unit 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 could be obtained. For example, in 2009, the best estimate, as approved by the International Astronomical Union (IAU), was:JOURNAL
, Pitjeva, EV
, Standish, EM
, 2009
, Proposals for the masses of the three largest asteroids, the Moonâ€“Earth mass ratio and the Astronomical Unit
, Celestial Mechanics and Dynamical Astronomy
, 103, 4, 365â€“72
, 10.1007/s10569-009-9203-8
url=https://zenodo.org/record/1000691, WEB
, IAU Working Group on Numerical Standards for Fundamental Astronomy
, IAU WG on NSFA Current Best Estimates
, US Naval Observatory
, 2009-09-25
, yes
, 2009-12-08
,
, WEB,weblink Astrodynamic Constants, Jet Propulsion Laboratory, Solar System Dynamics, 2017-11-08,
light time for unit distance: tau = {{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 ({{val|2|e=-11}}), equivalent to the uncertainty in Earth-based measurements of length by interferometry.WEB
, NPL's Beginner's Guide to Length
, 2010-08-31
, National Physical Laboratory (United Kingdom), UK National Physical Laboratory
, 2009-10-28
, Since the metre is defined to be the length travelled by light in a certain time interval, the measurement of the light time in terms of the previous definition of 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 TDB-compatible.|group=Note}}

### Time of flight techniques

(File:Michelson speed of light measurement 1930.jpg|thumb|center|upright=4.0|One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930â€“35 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 Â±11 km/s.)File:Fizeau.JPG|thumb|right|Diagram of the Fizeau apparatusFizeau apparatusA 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 Fizeauâ€“Foucault apparatus developed by Hippolyte Fizeau and LÃ©on Foucault.The setup as used by Fizeau consists of a beam of light directed at a mirror {{convert|8|km|mi|0}} away. On the way from the source to the mirror, the beam passes through a rotating cogwheel. At a certain rate of rotation, the beam passes through one gap on the way out and another on the way back, but at slightly higher or lower rates, the beam strikes a tooth and does not pass through the wheel. Knowing the distance between the wheel and the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light can be calculated.WEB
, Gibbs
, P
, 1997
, How is the speed of light measured?
, Usenet Physics FAQ
, University of California, Riverside
, 2010-01-13
, yes
, 2015-08-21
,
, The method of Foucault replaces the cogwheel by 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.WEB
, Fowler, M
, The Speed of Light
, University of Virginia
, 2010-04-21
, Nowadays, 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.JOURNAL
, Cooke, J
, Martin, M
, McCartney, H
, Wilf, B
, 1968
, Direct determination of the speed of light as a general physics laboratory experiment
, American Journal of Physics
, 36, 9, 847
, 10.1119/1.1975166, 1968AmJPh..36..847C,
JOURNAL
, Aoki, K, Mitsui, T
, 2008
, A small tabletop experiment for a direct measurement of the speed of light
, American Journal of Physics
, 76, 9, 812â€“15
, 10.1119/1.2919743
, 0705.3996, 2008AmJPh..76..812A,
JOURNAL
, James, MB, Ormond, RB, Stasch, AJ
, 1999
, Speed of light measurement for the myriad
, American Journal of Physics
, 67, 8, 681â€“714
, 10.1119/1.19352, 1999AmJPh..67..681J,

### 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 Îµ0 and vacuum permeability Î¼0 established by Maxwell's theory: c2 = 1/(Îµ0Î¼0). The vacuum permittivity may be determined by measuring the capacitance and dimensions of a capacitor, whereas the value of the vacuum permeability is fixed at exactly {{val|4|end=Ï€|e=-7|u=H.m-1}} through the definition of the ampere. Rosa and Dorsey used this method in 1907 to find a value of {{val|299710|22|u=km/s}}.JOURNAL
, Rosa, EB, Dorsey, NE
, 1907
, A new determination of the ratio of the electromagnetic to the electrostatic unit of electricity., Bulletin of the Bureau of Standards
, 3

### Cavity resonance

File:Waves in Box.svg|thumb|right|Electromagnetic 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 Î» of an electromagnetic wave in vacuum. The value of c can then be found by using the relation c = fÎ». One option is to measure the resonance frequency of a cavity resonator. If the dimensions of the resonance cavity are also known, these can be used to determine the wavelength of the wave. In 1946, Louis Essen and A.C. Gordon-Smith established the frequency for a variety of normal modes of microwaves of a microwave cavity of precisely known dimensions. The dimensions were established to an accuracy of about Â±0.8 Î¼m using gauges calibrated by interferometry. As the wavelength of the modes was known from the geometry of the cavity and from electromagnetic theory, knowledge of the associated frequencies enabled a calculation of the speed of light.JOURNAL
, Essen, L
, Gordon-Smith, AC
, 1948
, The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator
, Proceedings of the Royal Society of London A
, 194, 1038, 348â€“61
, 10.1098/rspa.1948.0085
, 1948RSPSA.194..348E
, 98293
, JOURNAL
, Essen, L
, 1947
, Velocity of Electromagnetic Waves
, Nature
, 159, 4044, 611â€“12
, 10.1038/159611a0
, 1947Natur.159..611E
, The Essenâ€“Gordon-Smith result, {{val|299792|9|u=km/s}}, was substantially more precise than those found by optical techniques. By 1950, repeated measurements by Essen established a result of {{val|299792.5|3.0|u=km/s}}.JOURNAL
, Essen, L
, 1950
, The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator
, Proceedings of the Royal Society of London A
, 204, 1077, 260â€“77
, 10.1098/rspa.1950.0172
, 1950RSPSA.204..260E
, 98433
, A household demonstration of this technique is possible, using a microwave oven and food such as marshmallows or margarine: if the turntable is removed so that the food does not move, it will cook the fastest at the antinodes (the points at which the wave amplitude is the greatest), where it will begin to melt. The distance between two such spots is half the wavelength of the microwaves; by measuring this distance and multiplying the wavelength by the microwave frequency (usually displayed on the back of the oven, typically 2450 MHz), the value of c can be calculated, "often with less than 5% error".STAUFFER > FIRST = RH, April 1997, Finding the Speed of Light with Marshmallows, The Physics Teacher, 35, 231,weblink 15 February 2010doi = 10.1119/1.2344657, 4, WEB,weblink BBC Look East at the speed of light, BBC Norfolk website
, 15 February 2010
,

### Interferometry

File:Interferometer sol.svg|thumb|An interferometric determination of length. Left: constructive interference; Right: 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).BOOK
, Vaughan, JM
, 1989
, The Fabry-Perot interferometer
, 47, 384â€“91
, CRC Press
, 978-0-85274-138-2
, |group="Note"}} A coherent beam of light (e.g. from a laser), with a known frequency (f), is split to follow two paths and then recombined. By adjusting the path length while observing the interference pattern and carefully measuring the change in path length, the wavelength of the light (Î») can be determined. The speed of light is then calculated using the equation c = Î»f.Before the advent of laser technology, coherent radio sources were used for interferometry measurements of the speed of light.JOURNAL
, 10.1098/rspa.1958.0172
, A New Determination of the Free-Space Velocity of Electromagnetic Waves
, KD
, Froome
, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
, 247
, 1958
, 109â€“22
, 1248
, 1958RSPSA.247..109F
, 100591, However interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~{{cvt|4|mm|in}}) of the radiowaves. The precision can be improved by using light with a shorter wavelength, but then it becomes difficult to directly measure the frequency of the light. One way around this problem is to start with a low frequency signal of which the frequency can be precisely measured, and from this signal progressively synthesize higher frequency signals whose frequency can then be linked to the original signal. A laser can then be locked to the frequency, and its wavelength can be determined using interferometry.
BOOK
, A Century of Excellence in Measurements, Standards, and Technology
, Lide
, DR
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, Sullivan
, DB
, 2001
, 191â€“93
, CRC Press
, 978-0-8493-1247-2
, yes
, 13 August 2009
,
, This technique was due to a group at the National Bureau of Standards (NBS) (which later became NIST). They used it in 1972 to measure the speed of light in vacuum with a fractional uncertainty of {{val|3.5|e=-9}}.JOURNAL
, Evenson, KM, 1972
, Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser
, Physical Review Letters
, 29
, 19, 1346â€“49
, 10.1103/PhysRevLett.29.1346, 1972PhRvL..29.1346E, etal,
c (in km/s)">

## History{| class"wikitable floatright"|+History of measurements of c (in km/s)

|

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