Thespeed of light invacuum, commonly denotedc, is a universalphysical constant that is exactly equal to 299,792,458 metres per second (approximately 300,000 kilometres per second; 186,000 miles per second; 671 million miles per hour). It is exact because, by a 1983 international agreement, ametre is defined as the length of the path travelled bylight in vacuum during a time interval of1⁄299792458second. According to thespecial theory of relativity,c is the upper limit for the speed at which conventionalmatter orenergy (and thus anysignal carryinginformation) can travel throughspace.[2][3][4]
All forms ofelectromagnetic radiation, includingvisible 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. Muchstarlight viewed onEarth is from the distant past, allowing humans to study the history of the universe by viewing distant objects. Whencommunicating with distantspace probes, it can take minutes to hours for signals to travel. Incomputing, the speed of light fixes the ultimate minimumcommunication delay. The speed of light can be used intime of flight measurements to measure large distances to extremely high precision.
Ole Rømer firstdemonstrated in 1676 that light does not travel instantaneously by studying the apparent motion ofJupiter's moonIo. Progressively more accurate measurements of its speed came over the following centuries. In apaper published in 1865,James Clerk Maxwell proposed that light was anelectromagnetic wave and, therefore, travelled at speedc.[5] In 1905,Albert Einstein postulated that the speed of lightc with respect to anyinertial frame of reference is a constant and is independent of the motion of the light source.[6] He explored the consequences of that postulate by deriving thetheory of relativity and, in doing so, showed that the parameterc had relevance outside of the context of light and electromagnetism.
Massless particles andfield perturbations, such asgravitational waves, also travel at speedc in vacuum. Such particles and waves travel atc regardless of the motion of the source or the inertial reference frame of theobserver. Particles with nonzerorest mass can be accelerated to approachc but can never reach it, regardless of the frame of reference in which their speed is measured. In thetheory of relativity,c interrelatesspace and time and appears in the famousmass–energy equivalence,E =mc2.[7]
The speed at which light propagates throughtransparent materials, such as glass or air, is less thanc; similarly, the speed ofelectromagnetic waves in wire cables is slower thanc. The ratio betweenc and the speedv at which light travels in a material is called therefractive indexn of the material (n =c/v). For example, for visible light, the refractive index of glass is typically around 1.5, meaning that light in glass travels atc/1.5 ≈200000 km/s (124000 mi/s); therefractive index of air for visible light is about 1.0003, so the speed of light in air is about 90 km/s (56 mi/s) slower thanc.
Numerical value, notation, and units
The speed of light in vacuum is usually denoted by a lowercasec. The origin of the letter choice is unclear, with guesses including "c" for "constant" or the Latinceleritas (meaning 'swiftness, celerity').[8] The "c" was used for "celerity" meaning a velocity in books byLeonhard Euler and others, but this velocity was not specifically for light;Isaac Asimov wrote a popular science article,. "C for Celeritas", but did not explain the origin.[9] In 1856,Wilhelm Eduard Weber andRudolf Kohlrausch had usedc for a different constant that was later shown to equal√2 times the speed of light in vacuum. Historically, the symbolV was used as an alternative symbol for the speed of light, introduced byJames Clerk Maxwell in 1865. In 1903,Max Abraham usedc with its modern meaning in a widely read textbook on electromagnetism.Einstein usedV in hisoriginal German-language papers on special relativity in 1905, but in 1907 he switched toc, which by then had become the standard symbol for the speed of light.[10][8]
Sometimesc is used for the speed of waves in any material medium, andc0 for the speed of light in vacuum.[11] This subscripted notation, which is endorsed in official SI literature,[12] has the same form as related electromagnetic constants: namely,μ0 for thevacuum permeability or magnetic constant,ε0 for thevacuum permittivity or electric constant, andZ0 for theimpedance of free space. This article usesc exclusively for the speed of light in vacuum.
Since 1983, the constantc has been defined in theInternational System of Units (SI) asexactly299792458 m/s; this relationship is used to define the metre as exactly the distance that light travels in vacuum in1⁄299792458 of a second. The second is, in turn, defined to be the length of time occupied by9192631770 cycles of the radiation emitted by acaesium-133atom in a transition between two specifiedenergy states.[13] By using the value ofc, as well as an accurate measurement of the second, one can establish a standard for the metre.[14]
The particular value chosen for the speed of light provided a more accurate definition of the metre that still agreed as much as possible with the definition used before 1983.[13][15]
In branches of physics in whichc appears often, such as in relativity, it is common to use systems ofnatural units of measurement or thegeometrized unit system wherec = 1.[18][19] Using these units,c does not appear explicitly because multiplication or division by1 does not affect the result. Its unit oflight-second per second is still relevant, even if omitted.
The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of theinertial frame of reference of the observer.[Note 5] This invariance of the speed of light was postulated by Einstein in 1905,[6] after being motivated byMaxwell's theory of electromagnetism and the lack of evidence for motion against theluminiferous aether.[20] It has since been consistently confirmed by many experiments.[Note 6] 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 theone-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.[21][22]
By adoptingEinstein synchronization for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition.[21][22] Thespecial theory of relativity explores the consequences of this invariance ofc with the assumption that the laws of physics are the same in all inertial frames of reference.[23][24] One consequence is thatc is the speed at which all massless particles and waves, including light, must travel in vacuum.[25][Note 7]
TheLorentz factorγ as a function of velocity. It starts at1 and approaches infinity asv approaches c.
Special relativity has many counterintuitive and experimentally verified implications.[27] These include theequivalence of mass and energy(E =mc2),length contraction (moving objects shorten),[Note 8] andtime dilation (moving clocks run more slowly). The factor γ by which lengths contract and times dilate is known as theLorentz factor and is given byγ = (1 −v2/c2)−1/2, wherev is the speed of the object. The difference ofγ from1 is negligible for speeds much slower than c, such as most everyday speeds – in which case special relativity is closely approximated byGalilean relativity – but it increases at relativistic speeds and diverges to infinity asv approachesc. For example, a time dilation factor ofγ = 2 occurs at a relative velocity of 86.6% of the speed of light (v = 0.866 c). Similarly, a time dilation factor ofγ = 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 asspacetime (with c relating the units of space and time), and requiring that physical theories satisfy a specialsymmetry calledLorentz invariance, whose mathematical formulation contains the parameter c.[30] Lorentz invariance is an almost universal assumption for modern physical theories, such asquantum electrodynamics,quantum chromodynamics, theStandard Model ofparticle physics, andgeneral 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 thespeed of gravity and ofgravitational waves,[31] and observations of gravitational waves have been consistent with this prediction.[32] Innon-inertial frames of reference (gravitationally curved spacetime oraccelerated reference frames), thelocal 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.[33]
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 thespeed of light may have changed over time.[34][35] No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.[36][37]
It is generally assumed that the two-way speed of light isisotropic, meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclearenergy levels as a function of the orientation of the emittingnuclei in a magnetic field (seeHughes–Drever experiment), and of rotatingoptical resonators (seeResonator experiments) have put stringent limits on the possible two-wayanisotropy.[38][39]
Upper limit on speeds
According to special relativity, the energy of an object withrest massm and speedv is given byγmc2, whereγ is the Lorentz factor defined above. Whenv is zero,γ is equal to one, giving rise to the famousE =mc2 formula for mass–energy equivalence. Theγ factor approaches infinity asv 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.[40] This is experimentally established in manytests of relativistic energy and momentum.[41]
Event A precedes B in the red frame, is simultaneous with B in the green frame, and follows B in the blue frame.
More generally, it is impossible for signals or energy to travel faster than c. One argument for this follows from the counter-intuitive implication of special relativity known as therelativity 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, andcausality would be violated.[Note 9][44] In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded,[22] and would lead toparadoxes such as thetachyonic antitelephone.[45]
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 thepropagation of light in a medium section below, many wave velocities can exceed c. Thephase velocity ofX-rays through most glasses can routinely exceedc,[46] but phase velocity does not determine the velocity at which waves convey information.[47]
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.[48] In neither case does any matter, energy, or information travel faster than light.[49]
The rate of change in the distance between two objects in a frame of reference with respect to which both are moving (theirclosing 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.[49]
Certain quantum effects appear to be transmitted instantaneously and therefore faster thanc, as in theEPR paradox. An example involves thequantum states of two particles that can beentangled. Until either of the particles is observed, they exist in asuperposition of two quantum states. If the particles are separated and one particle's quantum state is observed, the other particle's quantum state is determined instantaneously. However, it is impossible to control which quantum state the first particle will take on when it is observed, so information cannot be transmitted in this manner.[49][50]
Another quantum effect that predicts the occurrence of faster-than-light speeds is called theHartman effect: under certain conditions the time needed for avirtual particle totunnel through a barrier is constant, regardless of the thickness of the barrier.[51][52] This could result in a virtual particle crossing a large gap faster than light. However, no information can be sent using this effect.[53]
So-calledsuperluminal motion is seen in certain astronomical objects,[54] such as therelativistic jets ofradio galaxies andquasars. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is aprojection 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.[55]
In models of theexpanding 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 theHubble sphere, the rate at which their distance from Earth increases becomes greater than the speed of light.[58]These recession rates, defined as the increase inproper distance percosmological time, are not velocities in a relativistic sense. Faster-than-light cosmological recession speeds are only acoordinate artifact.
In modernquantum physics, the electromagnetic field is described by the theory ofquantum electrodynamics (QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, calledphotons. In QED, photons aremassless particles and thus, according to special relativity, they travel at the speed of light in vacuum.[25]
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.[33] No variation of the speed of light with frequency has been observed in rigorous testing, putting stringent limits on the mass of the photon.[60] The limit obtained depends on the model used: if the massive photon is described byProca theory,[61] the experimental upper bound for its mass is about 10−57grams;[62] if photon mass is generated by aHiggs mechanism, the experimental upper limit is less sharp,m ≤10−14eV/c2 (roughly 2 × 10−47 g).[61]
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 ofquantum gravity. In 2009, the observation ofgamma-ray burstGRB 090510 found no evidence for a dependence of photon speed on energy, supporting tight constraints in specific models of spacetime quantization on how this speed is affected by photon energy for energies approaching thePlanck scale.[63]
In a medium, light usually does not propagate at a speed equal toc; further, different types of light wave will travel at different speeds. The speed at which the individual crests and troughs of aplane wave (a wave filling the whole space, with only onefrequency) propagate is called thephase velocityvp. A physical signal with a finite extent (a pulse of light) travels at a different speed. The overallenvelope of the pulse travels at thegroup velocityvg, and its earliest part travels at thefront velocityvf.[64]
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.
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 arefractive index. The refractive index of a material is defined as the ratio ofc 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. Therefractive index of air is approximately 1.0003.[65] Denser media, such aswater,[66]glass,[67] anddiamond,[68] have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light.
In exotic materials likeBose–Einstein condensates nearabsolute 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 elementrubidium. 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.[69]
In transparent materials, the refractive index generally is greater than 1, meaning that the phase velocity is less thanc. In other materials, it is possible for the refractive index to become smaller than1 for some frequencies; in some exotic materials it is even possible for the index of refraction to become negative.[70] The requirement that causality is not violated implies that thereal and imaginary parts of thedielectric constant of any material, corresponding respectively to the index of refraction and to theattenuation coefficient, are linked by theKramers–Kronig relations.[71][72] In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly.[73]
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 asdispersion. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon calledslow light.[74]The opposite, group velocities exceedingc, was proposed theoretically in 1993 and achieved experimentally in 2000.[75] It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.[64]
None of these options allow information to be transmitted faster thanc. 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 toc.[64]
It is possible for a particle to travel through a medium faster than the phase velocity of light in that medium (but still slower thanc). When acharged particle does that in adielectric material, the electromagnetic equivalent of ashock wave, known asCherenkov radiation, is emitted.[76]
Practical effects of finiteness
The speed of light is of relevance totelecommunications: the one-way andround-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
Incomputers, the speed of light imposes a limit on how quickly data can be sent betweenprocessors. If a processor operates at 1gigahertz, a signal can travel only a maximum of about 30 centimetres (1 ft) in a single clock cycle – in practice, this distance is even shorter since theprinted circuit board refracts and slows down signals. Processors must therefore be placed close to each other, as well asmemory chips, to minimize communication latencies, and care must be exercised when routing wires between them to ensuresignal integrity. If clock frequencies continue to increase, the speed of light may eventually become a limiting factor for the internal design of singlechips.[77][78]
Large distances on Earth
Acoustic representation of the speed of light: in the period between beeps, light travels the circumference of Earth at the equator.
Given that the equatorial circumference of the Earth is about40075 km and thatc is about300000 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 inoptical fibre (atransparent 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 indexn.[Note 10] Straight lines are rare in global communications and the travel time increases when signals pass through electronic switches or signal regenerators.[80]
Although this distance is largely irrelevant for most applications, latency becomes important in fields such ashigh-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 tomicrowave communications between trading hubs, because of the advantage which radio waves travelling at near to the speed of light through air have over comparatively slowerfibre optic signals.[81][82]
Spaceflight and astronomy
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.
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 betweenground control andApollo 8 when it became the first crewed spacecraft to orbit theMoon: for every question, the ground control station had to wait at least three seconds for the answer to arrive.[83]
The communications delay between Earth andMars 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 approximately4–24 minutes later. It would then take a further4–24 minutes for commands to travel from Earth to Mars.[84][85]
Receiving light and other signals from distant astronomical sources takes much longer. For example, it takes 13 billion (13×109) years for light to travel to Earth from the faraway galaxies viewed in theHubble Ultra-Deep Field images.[86][87] 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.[86] The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer theevolution of stars,of galaxies, andof the universe itself.[88]
Astronomical distances are sometimes expressed inlight-years, especially inpopular science publications and media.[89] A light-year is the distance light travels in oneJulian year, around 9461 billion kilometres, 5879 billion miles, or 0.3066parsecs. 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.[90]
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-triptransit time multiplied by the speed of light. AGlobal Positioning System (GPS) receiver measures its distance toGPS 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 about300000 kilometres (186000 miles) in one second, these measurements of small fractions of a second must be very precise. TheLunar Laser Ranging experiment,radar astronomy and theDeep Space Network determine distances to the Moon,[91] planets[92] and spacecraft,[93] respectively, by measuring round-trip transit times.
Measurement
There are different ways to determine the value ofc. One way is to measure the actual speed at which light waves propagate, which can be done in various astronomical and Earth-based setups. It is also possible to determinec from other physical laws where it appears, for example, by determining the values of the electromagnetic constantsε0 andμ0 and using their relation toc. Historically, the most accurate results have been obtained by separately determining the frequency and wavelength of a light beam, with their product equallingc. This is described in more detail in the"Interferometry" section below.
In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of1⁄299792458 of a second",[94] fixing the value of the speed of light at299792458 m/s by definition, asdescribed below. Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value ofc.
Astronomical measurements
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 convenient setting for measuring the speed of light because of its large scale and nearly perfectvacuum. Typically, one measures the time needed for light to traverse some reference distance in theSolar System, such as theradius of the Earth's orbit. Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units.
Ole Rømer used an astronomical measurement to makethe first quantitative estimate of the speed of light in the year 1676.[95][96] 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 thediameter of the Earth's orbit around the Sun. The observed change in the moon's orbital period is caused by the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect forJupiter's innermost major moon Io and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.[95]
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.
Another method is to use theaberration of light, discovered and explained byJames Bradley in the 18th century.[97] This effect results from thevector 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 20.5arcseconds)[98] 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 travelled10210 times faster than the Earth in its orbit (the modern figure is10066 times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.[97]
Astronomical unit
An astronomical unit (AU) is approximately the average distance between the Earth and Sun. It was redefined in 2012 as exactly149597870700 m.[99][100] Previously the AU was not based on theInternational System of Units but in terms of the gravitational force exerted by the Sun in the framework of classical mechanics.[Note 11] The current definition uses the recommended value in metres for the previous definition of the astronomical unit, which was determined by measurement.[99] 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).[102]
Previously, the inverse of 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, abest fit value for the light time per unit distance could be obtained. For example, in 2009, the best estimate, as approved by theInternational Astronomical Union (IAU), was:[103][104]
light time for unit distance:tau = 499.004783836(10) s,
c = 0.00200398880410(4) AU/s = 173.144632674(3) AU/d.
The relative uncertainty in these measurements is 0.02 parts per billion (2×10−11), equivalent to the uncertainty in Earth-based measurements of length by interferometry.[105] 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.[Note 12]
Time of flight techniques
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.Diagram of theFizeau apparatus:
Light source
Beam-splitting semi-transparent mirror
Toothed wheel-breaker of the light beam
Remote mirror
Telescopic tube
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 byHippolyte Fizeau andLéon Foucault.
Thesetup as used by Fizeau consists of a beam of light directed at a mirror 8 kilometres (5 mi) 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.[106]
Themethod of Foucault replaces the cogwheel with a rotating mirror. Because the mirror keeps rotating while the light travels to the distant mirror and back, the light is reflected from the rotating mirror at a different angle on its way out than it is on its way back. From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated.[107] Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion byFrançois Arago.[108]
Today, usingoscilloscopes 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.[109]
Electromagnetic constants
An option for derivingc that does not directly depend on a measurement of the propagation of electromagnetic waves is to use the relation betweenc and thevacuum permittivityε0 andvacuum permeabilityμ0 established by Maxwell's theory:c2 = 1/(ε0μ0). The vacuum permittivity may be determined by measuring thecapacitance and dimensions of acapacitor, whereas the value of the vacuum permeability was historically fixed at exactly4π×10−7 H⋅m−1 through the definition of theampere.Rosa andDorsey used this method in 1907 to find a value of299710±22 km/s. Their method depended upon having a standard unit of electrical resistance, the "internationalohm", and so its accuracy was limited by how this standard was defined.[110][111]
Another way to measure the speed of light is to independently measure the frequencyf and wavelengthλ of an electromagnetic wave in vacuum. The value ofc can then be found by using the relationc = fλ. One option is to measure the resonance frequency of acavity 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 ofnormal modes of microwaves of amicrowave cavity of precisely known dimensions. The dimensions were established to an accuracy of about ±0.8 μm using gauges calibrated by interferometry.[110] As the wavelength of the modes was known from the geometry of the cavity and fromelectromagnetic theory, knowledge of the associated frequencies enabled a calculation of the speed of light.[110][112]
The Essen–Gordon-Smith result,299792±9 km/s, was substantially more precise than those found by optical techniques.[110] By 1950, repeated measurements by Essen established a result of299792.5±3.0 km/s.[113]
A household demonstration of this technique is possible, using amicrowave 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 theantinodes (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 ofc can be calculated, "often with less than 5% error".[114][115]
Interferometry is another method to find the wavelength of electromagnetic radiation for determining the speed of light.[Note 13] Acoherent beam of light (e.g. from alaser), with a known frequency (f), is split to follow two paths and then recombined. By adjusting the path length while observing theinterference 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, coherentradio sources were used for interferometry measurements of the speed of light.[117] Interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~4 mm (0.16 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.[118]
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.[118] This technique was due to a group at the National Bureau of Standards (which later became theNational Institute of Standards and Technology). They used it in 1972 to measure the speed of light in vacuum with afractional uncertainty of3.5×10−9.[118][119]
History
Until theearly 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 inancient Greece. The ancient Greeks, Arabic scholars, and classical European scientists long debated this until Rømer provided the first calculation of the speed of light. Einstein's theory of special relativity postulates that the speed of light is constant regardless of one's frame of reference. Since then, scientists have provided increasingly accurate measurements.
Empedocles (c. 490–430 BCE) was the first to propose a theory of light[128] and claimed that light has a finite speed.[129] 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".[130]Euclid andPtolemy advanced Empedocles'emission theory of vision, where light is emitted from the eye, thus enabling sight. Based on that theory,Heron of Alexandria argued that the speed of light must beinfinite because distant objects such as stars appear immediately upon opening the eyes.[131]
Early Islamic philosophers initially agreed with theAristotelian view that light had no speed of travel. In 1021,Alhazen (Ibn al-Haytham) published theBook of Optics, in which he presented a series of arguments dismissing the emission theory ofvision in favour of the now accepted intromission theory, in which light moves from an object into the eye.[132] This led Alhazen to propose that light must have a finite speed,[130][133][134] and that the speed of light is variable, decreasing in denser bodies.[134][135] He argued that light is substantial matter, the propagation of which requires time, even if this is hidden from the senses.[136] 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.[137]
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.[138][139] In the 1270s,Witelo considered the possibility of light travelling at infinite speed in vacuum, but slowing down in denser bodies.[140]
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 alunar eclipse. Although this argument fails when aberration of light is taken into account, the latter was not recognized until the following century.[141] 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.[130] Despite this, in his derivation ofSnell's law, Descartes assumed that some kind of motion associated with light was faster in denser media.[142][143]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.[144]
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.[120][121] In 1667, theAccademia 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.[145] The actual delay in this experiment would have been about 11microseconds.
Rømer's observations of the occultations of Io from Earth
The first quantitative estimate of the speed of light was made in 1676 by Ole Rømer.[95][96] From the observation that the periods of Jupiter's innermost moonIo 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 of220000 km/s, which is 27% lower than the actual value.[124]
In his 1704 bookOpticks,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).[146] Newton queried whether Rømer's eclipse shadows were coloured. Hearing that they were not, he concluded the different colours travelled at the same speed. In 1729,James Bradley discoveredstellar aberration.[97] 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.[97]
In the 19th centuryHippolyte Fizeau developed a method to determine the speed of light based on time-of-flight measurements on Earth and reported a value of315000 km/s.[147] His method was improved upon byLéon Foucault who obtained a value of298000 km/s in 1862.[106] In the year 1856,Wilhelm Eduard Weber andRudolf Kohlrausch measured the ratio of the electromagnetic and electrostatic units of charge, 1/√ε0μ0, by discharging aLeyden jar, and found that its numerical value was very close to the speed of light as measured directly by Fizeau. The following yearGustav Kirchhoff calculated that an electric signal in aresistanceless wire travels along the wire at this speed.[148]
In the early 1860s, Maxwell showed that, according to the theory of electromagnetism he was working on, electromagnetic waves propagate in empty space[149] 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.[150] 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.[151]
The wave properties of light were well known sinceThomas 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 theluminiferous aether).[152]
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 apreferred 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 theisotropy of the speed of light. Beginning in the 1880s several experiments were performed to try to detect this motion, the most famous of which isthe experiment performed byAlbert A. Michelson andEdward W. Morley in 1887.[153][154] The detected motion was found to always be nil (within observational error). Modern experiments indicate that the two-way speed of light isisotropic (the same in every direction) to within 6 nanometres per second.[155]
Because of this experimentHendrik Lorentz proposed that the motion of the apparatus through the aether may cause the apparatus tocontract 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 theLorentz transformation. Based onLorentz's aether theory,Henri Poincaré (1900) showed that this local time (to first order inv/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 theprinciple of relativity.[156][157]
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 thespecial theory of relativity, in which the speed of light in vacuumc 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.[158][159]
Increased accuracy ofc and redefinition of the metre and second
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 as299792.5±3.0 km/s, using cavity resonance.[113] This value was adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. In 1960, themetre was redefined in terms of the wavelength of a particular spectral line ofkrypton-86, and, in 1967, the second was redefined in terms of the hyperfine transition frequency of the ground state ofcaesium-133.[160]
In 1972, using the laser interferometer method and the new definitions, a group at the USNational Bureau of Standards inBoulder, Colorado determined the speed of light in vacuum to bec = 299792456.2±1.1 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.[Note 15][119] As similar experiments found comparable results forc, the 15thGeneral Conference on Weights and Measures in 1975 recommended using the value299792458 m/s for the speed of light.[163]
Defined as an explicit constant
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 morereproducible than the previous standard. They kept the 1967 definition of second, so thecaesiumhyperfine 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 1/299792458 of a second".[94]
As a result of this definition, the value of the speed of light in vacuum is exactly299792458 m/s[164][165] and has become a defined constant in the SI system of units.[14] Improved experimental techniques that, prior to 1983, would have measured the speed of light no longer affect the known value of the speed of light in SI units, but instead allow a more precise realization of the metre by more accurately measuring the wavelength of krypton-86 and other light sources.[166][167]
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 exactly299792458 when it is expressed in the SI unitm s−1."[168] This was one of the changes that was incorporated in the2019 revision of the SI, also termed theNew SI.[169]
^Becauseneutrinos have a small but non-zero mass, they travel through empty spacevery 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, likesupernovae.[26]
^Whereas moving objects aremeasured to be shorter along the line of relative motion, they are alsoseen as being rotated. This effect, known asTerrell rotation, is due to the different times that light from different parts of the object takes to reach the observer.[28][29]
^It has been speculated that theScharnhorst effect does allow signals to travel slightly faster than c, but the validity of those calculations has been questioned,[42] and it appears the special conditions in which this effect might occur would prevent one from using it to violate causality.[43]
^A typical value for the refractive index of optical fibre is between 1.518 and 1.538.[79]
^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 anangular frequency of0.01720209895radians (approximately1⁄365.256898 of a revolution) per day.[101]
^Nevertheless, at this degree of precision, the effects ofgeneral relativity must be taken into consideration when interpreting the length. The metre is considered to be a unit ofproper 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 areTDB-compatible.[104]
^A detailed discussion of the interferometer and its use for determining the speed of light can be found in Vaughan (1989).[116]
^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.[121]
^Between 1960 and 1983 the metre was defined as "the length equal to1650763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom".[161] It was 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.[162]
^Gibbs, P. (2004) [1997]."Why isc the symbol for the speed of light?".Usenet Physics FAQ.University of California, Riverside. Archived fromthe original on 25 March 2010. Retrieved16 November 2009."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'."
Sydenham, P. H. (2003)."Measurement of length". In Boyes, W (ed.).Instrumentation Reference Book (3 ed.).Butterworth–Heinemann. p. 56.ISBN978-0-7506-7123-1....if the speed of light is defined as a fixed number then, in principle, the time standard will serve as the length standard...
^Ellis, G. F. R.; Uzan, J.-P. (2005). "'c' is the speed of light, isn't it?".American Journal of Physics.73 (3):240–227.arXiv:gr-qc/0305099.Bibcode:2005AmJPh..73..240E.doi:10.1119/1.1819929.S2CID119530637.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.
^Tolman, R. C. (2009) [1917]. "Velocities greater than that of light".The Theory of the Relativity of Motion (Reprint ed.).BiblioLife. p. 54.ISBN978-1-103-17233-7.
Caves, Carlton M. (2015). "Quantum Information Science: Emerging No More".OSA Century of Optics.Optica. pp. 320–326.arXiv:1302.1864.ISBN978-1-943-58004-0.[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.
^Cherenkov, Pavel A. (1934). "Видимое свечение чистых жидкостей под действием γ-радиации" [Visible emission of pure liquids by action of γ radiation].Doklady Akademii Nauk SSSR.2: 451. Reprinted:Cherenkov, P. A. (1967). "Видимое свечение чистых жидкостей под действием γ-радиации" [Visible emission of pure liquids by action of γ radiation].Usp. Fiz. Nauk.93 (10): 385.doi:10.3367/ufnr.0093.196710n.0385., and inA. N. Gorbunov; E. P. Čerenkova, eds. (1999).Pavel Alekseyevich Čerenkov: Chelovek i Otkrytie [Pavel Alekseyevich Čerenkov: Man and Discovery]. Moscow: Nauka. pp. 149–153.
^Hertz, Heinrich (1893).Electric Waves. London: Macmillan and Co.
^Michelson, A. A. (1927). "Measurement of the Velocity of Light Between Mount Wilson and Mount San Antonio".The Astrophysical Journal.65: 1.Bibcode:1927ApJ....65....1M.doi:10.1086/143021.
^Lauginie, P. (2004).Measuring Speed of Light: Why? Speed of what?(PDF). Fifth International Conference for History of Science in Science Education. Keszthely, Hungary. pp. 75–84. Archived fromthe original(PDF) on 4 July 2015. Retrieved12 August 2017.
^O'Connor, J. J.; Robertson, E. F."Abu han Muhammad ibn Ahmad al-Biruni".MacTutor History of Mathematics archive. University of St Andrews. Retrieved12 January 2010.
^Newton, I. (1704). "Prop. XI".Optiks. The text of Prop. XI is identical between the first (1704) and second (1719) editions.
^Guarnieri, M. (2015). "Two Millennia of Light: The Long Path to Maxwell's Waves".IEEE Industrial Electronics Magazine.9 (2):54–56, 60.doi:10.1109/MIE.2015.2421754.S2CID20759821.
^Adams, S. (1997).Relativity: An Introduction to Space–Time Physics. CRC Press. p. 140.ISBN978-0-7484-0621-0.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!
^Rindler, W. (2006).Relativity: Special, General, and Cosmological (2 ed.). Oxford University Press. p. 41.ISBN978-0-19-856731-8.Note that [...] improvements in experimental accuracy will modify the meter relative to atomic wavelengths, but not the value of the speed of light!
