Speed of light

Sunlight takes approximately 8 minutes to reach Earth.

Speed of light in different units
metres per second	299,792,458 (exact)
kilometres per second	≈ 300 thousand
kilometres per hour	≈ 1,079 million
miles per second	≈ 186 thousand
miles per hour	≈ 671 million
astronomical units per day	≈ 173
natural units	1
Approximate length of time for light to travel:
One foot	1.0 nanoseconds
One metre	3.3 nanoseconds
One kilometre	3.3 microseconds
One statute mile	5.4 microseconds
To Earth from geostationary orbit	0.12 seconds
The length of Earth's equator	0.13 seconds
To Earth from the Moon	1.3 seconds
To Earth from the Sun	8.3 minutes
One parsec	3.26 years
To Earth from Alpha Centauri	4.4 years
Across the Milky Way	100,000 years
To Earth from the Andromeda Galaxy	2,500,000 years

The speed of light (usually denoted c) is a physical constant. Its exact value is 299,792,458 metres per second,^[1][2] often approximated as 300,000 kilometres per second or 186,000 miles per second. The name derives from the fact that electromagnetic radiation, such as light, travels at this speed in vacuum.

For much of human history, it was not known whether light was transmitted instantaneously or simply very quickly. In the 17th century, Ole Rømer demonstrated that it travelled at a finite speed by studying the apparent motion of Jupiter's moon Io. By 1975, the speed of light was known to be 299,792,458 m/s with a relative measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units (SI) as the distance travelled by light in vacuum in ¹⁄_299,792,458 of a second. As a result, the numerical value of c in metres per second is now fixed exactly by the definition of the metre.^[2][1]

According to the theory of special relativity, c connects space and time in the unified structure of spacetime, and its square is the constant of proportionality between mass and energy (E = mc²).^[3] In any inertial frame of reference, independently of the relative velocity of the emitter and the observer, c is the speed of all massless particles and associated fields, including all electromagnetic radiation in free space,^[4] and it is believed to be the speed of gravity and of gravitational waves.^[5][6] It is an upper bound on the speed at which energy, matter, and information can travel,^[7][8] as surpassing it would be equivalent to travelling backwards in time;^[9] its finite value is a limiting factor in the speed of operation of electronic devices.^[10]

The actual speed at which light propagates through transparent materials, such as glass or air, is less than c; the ratio between c and the speed v at which light travels in a material is called the refractive index n 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 at c / 1.5 ≈ 200,000 km/s; the refractive index of air is about 1.0003, so the speed of light in air is very close to c.

Numerical value, notation and units

The speed of light is a dimensional physical constant, so its numerical value depends upon the system of units used. In the International System of Units (SI), the metre is defined as the distance light travels in vacuum in ¹⁄_299,792,458 of a second (see "Redefinition of the metre", below). The effect of this definition is to fix the speed of light in vacuum at exactly 299,792,458 m/s.^{[11][12][13][Note 1]}

The speed of light in vacuum is usually denoted by c, for "constant" or the Latin celeritas (meaning "swiftness"). Originally, the symbol V was used, introduced by Maxwell in 1865; c was used in 1856 by Weber and Kohlrausch for a constant later shown to equal √2 times the speed of light in vacuum, and in 1894 Drude redefined it with the modern meaning. Einstein used V in his original 1905 German-language papers on special relativity, but in 1907 he switched to c, which by then had become the standard symbol.^[14]

Some authors use c for the speed of waves in any material medium, and c₀ for the speed of light in vacuum.^[15] This subscripted notation, which is endorsed in official SI literature,^[16] has the same form as other related constants: namely, μ₀ for the vacuum permeability or magnetic constant, ε₀ for the vacuum permittivity or electric constant, and Z₀ for the impedance of free space. However, in this article c will be exclusively used for the speed of light in vacuum.

In branches of physics in which the speed of light plays an important part, such as in relativity, it is common to use natural units, in which c = 1.^[17][18] Thus, no symbol for the speed of light is required.

Fundamental role in physics

Constant speed in inertial frames

The speed of light has been shown experimentally to be independent of the motion of the source.^[18][19] It has also been confirmed by the Michelson–Morley experiment and other experiments that the two-way speed of light (for example from a source to a mirror and back again) in an inertial frame is constant.^{[19][20][21][22]}

It is not possible, however, 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 detector should be synchronized.^[19] Einstein (who was aware of this fact) adopted a synchronization method, the Einstein synchronization, by which the one-way speed of light becomes equal to the two-way speed of light and assumed that the synchronization method could be applied consistently all over the frame. Later it was proven that the assumption of consistency could be removed provided the two-way speed of light on the given inertial frame was a constant (see Einstein synchronisation) as experiments have indeed confirmed to be. Given this possibility, the clocks once synchronized measure a one-way speed which is a constant over the entire frame (homogeneity and isotropy). This is exactly the starting point which appeared as an assumption in Einstein's first works.

The constancy of the two-way speed, not only in the given inertial frame, but also with respect to changes of frame was then used by him to deduce the Lorentz transformations and the theory of special relativity.^[23][24] Although the speed of propagation of light is independent of the motion of the source, its observed frequency can change due to the Doppler effect.

In non-inertial frames (gravitationally curved space or accelerated frames), the local speed of light is constant and equal to c, however the speed of light along a trajectory of finite length can differ from c depending on how distances and times are defined.

Spacetime constant

In Einstein's theory of special relativity, space and time are viewed as a four-dimensional unification of space and time, known as spacetime,^[25] with c playing the fundamental role of a conversion factor between the units of space and time,^[26] and also between mass and energy according to the famous mass–energy equation E = mc².^[27]

The finite speed of light in relativity leads to some counter-intuitive consequences, which include length contraction, time dilation and the relativity of simultaneity, this last item contradicting the classical notion that the duration of the time interval between two events is equal for all observers. These effects are very small when the speeds involved are much slower than c, in which case special relativity is closely approximated by Galilean relativity.

According to the theory of general relativity, gravitational waves and changes in gravity also propagate at the speed of light, although this has not yet been directly observed.^[28][29]

Causality and information transfer

A light cone defines locations that are in causal contact and those that are not.

According to the theory of special relativity, causality would be violated if information could travel faster than c in some reference frame: in some other reference frames, the information would be received before it had been sent, so the "effect" could be observed before the "cause". Such a violation of causality has never been recorded,^[19] and would lead to paradoxes (see Tachyonic antitelephone for an example).^[30]

Information propagates to and from a point forming regions defined by a light cone. The spacetime interval AB in the diagram to the right is "time-like" (that is, there is a frame of reference in which event A and event B occur at the same location in space, separated only by their occurring at different times, and if A precedes B in that frame then A precedes B in all frames: there is no frame of reference in which event A and event B occur simultaneously). Thus, it is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the "cause" and B the "effect"). In other words, c represents the maximum speed at which matter, energy, or information can be transmitted.

On the other hand, the interval AC in the diagram to the right is "space-like" (that is, there is a frame of reference in which event A and event C occur simultaneously, separated only in space; see Relativity of simultaneity). In some frames A precedes C (as shown), but in others C precedes A. Therefore there can be no causal connection between A and C, and in particular it is not possible for any matter (or information) to travel from A to C or from C to A.^[31]

Variations with time

Some scientists have questioned why the fundamental constants of nature, including c, have the values they do, and whether they are changing as the universe evolves.^[32][33][34] These questions remain an interest of ongoing research.^{[35][36][37][38][39][40]}

Propagation of light

In classical physics, light is described as a type of electromagnetic wave. The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the electric constant ε₀ and the magnetic constant μ₀ by the equation c = 1/√ε₀μ₀).^[41]

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 must travel at the speed of light.

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.^[42] To date no such effects have been observed^[43][44][45] putting stringent limits on the photon mass. The limit obtained depends on the used model: if the massive photon is described by Proca theory,^[46] the experimental upper bound for its mass is about 10⁻⁵⁷ grams.^[47] If photon mass is generated by a Higgs mechanism, the experimental upper limit is less sharp, m ≤ 10⁻¹⁴ eV/c^2 [46] (roughly 2 × 10⁻⁴⁷ g).

In a medium

The light passing through this prism demonstrates refraction and, by the splitting of white light into a spectrum of colours, dispersion.

When light enters materials, its energy is absorbed. In the case of transparent materials (dielectrics), this energy is quickly re-radiated. However, this absorption and re-radiation introduces a delay. As light propagates through dielectric material it undergoes continuous absorption and re-radiation. Therefore when the speed of light in a medium is said to be less than c, this should be read as the speed of energy propagation at the macroscopic level. At an atomic level, electromagnetic waves always travel at c in the empty space between atoms. Two factors influence this slowing; stronger absorption leading to shorter path length between each re-radiation cycle and longer delays. The slowing is therefore the product of these two factors.^[48] The refractive index of a transparent material is defined as the ratio of c to the speed of light v in the material. Larger indexes of refraction indicate smaller 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 in air is approximately 1.0003.^[49] Denser media, such as water and glass, have refractive indexes of around 1.3 and 1.5 respectively for visible light. Diamond has a refractive index of about 2.4.

If the refractive index of a material dependence on the frequency of the light passing through the medium, there exist two notions of speed of light in the medium. The first is speed of a wave light of a single frequency f. This is called the phase velocity v_p(f), and is related to (frequency dependent) refractive index n(f) by

The second is average velocity of a pulse of light consisting of different frequencies of light. This is the called the group velocity and not only depends on the properties of the medium but also distribution of frequencies in the pulse. A pulse with different group and phase velocities is said to undergo dispersion.

Certain materials have an exceptionally high group index and a correspondingly low group velocity for light waves, a phenomenon called slow light. In 1999, a team of scientists led by Lene Hau were able to slow the speed of a light pulse to about 17 metres per second (61 km/h; 38 mph);^[50] in 2001, they were able to momentarily stop a beam.^[51] In 2003, scientists at Harvard University and the Lebedev Physical Institute in Moscow, succeeded in completely halting light by directing it into a Bose–Einstein condensate of the element rubidium, the atoms of which, in Lukin's words, behaved "like tiny mirrors" due to an interference pattern in two "control" beams.^[52][53]

It is also possible for the group velocity of light pulses to exceed c.^[54][55] In an experiment in 2000, laser beams travelled for extremely short distances through caesium atoms with a group velocity of 300 times c.^[56] It is not possible to transmit information faster than c by this means because the speed of information transfer cannot exceed the front velocity of the wave pulse, which is always less than c.^[57]

Faster-than-light observations and experiments

It is widely accepted that it is normally impossible for information or matter to travel faster than c for several reasons. One reason is that if an object were travelling faster than c relative to an inertial frame of reference, it would be travelling backwards in time relative to another frame.^[9] (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.^[58]) However, there are many physical situations in which speeds greater than c are encountered. In some of these, entities travel faster than c in a particular reference frame. Even in these situations, however, neither matter, energy, nor information travels faster than the speed of light in a vacuum.

If a laser beam is swept quickly across a distant object, the spot of light can move faster than c.^[59] Similarly, a shadow projected onto a distant object can be made to move faster than c.^[60] In neither case does any matter or information travel faster than light.

In some interpretations of quantum mechanics, certain quantum effects may be transmitted faster than c. For example, the quantum states of two particles 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 immediately (i.e., faster than light could travel between the particles). 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.^[61][59]

Another prediction of faster-than-light speeds occurs for quantum tunnelling and is called the Hartman effect.^[62][63] However, no information can be sent using these effects.^[64]

Closing speeds and proper speeds are examples of calculated speeds that may have value in excess of c but that do not represent the speed of an object as measured in a single inertial frame.

So-called superluminal motion is seen in certain astronomical objects,^[65] such as the jets of radio galaxies and quasars. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection effect caused by objects moving near the speed of light and at a small angle to the line of sight.

Cherenkov radiation

The blue glow in this "swimming pool" nuclear reactor is Cherenkov radiation, emitted as a result of electrons travelling faster than the speed of light in water.

It is possible for shock waves to be formed with electromagnetic radiation.^[66][67] If a charged particle travels through an electrical insulator faster than the speed of light in that medium (but always slower than the speed of light in vacuum) then electromagnetic radiation is emitted which is analogous to a sonic boom and is known as Cherenkov radiation.

Galaxies moving faster than light

In models of the expanding universe, the farther things are from Earth, the faster they move away from us. This movement is not considered to be a straightforward travel, like a rocket for example, but a movement due to the expansion of space itself. This expansion moves distant objects away from us faster and faster the further away they are. At a boundary called the Hubble sphere, the recessional velocity is the speed of light.

Practical effect of the finite speed of light

The speed of light plays an important part in many modern sciences and technologies. In electronic systems, despite their small size, the speed of light can become a limiting factor in their maximum speed of operation.^[10]

Transit time

A beam of light is depicted travelling between the Earth and the Moon in the same time it takes light to scale the distance between them: 1.255 seconds at its mean orbital (surface to surface) distance. The relative sizes and separation of the Earth–Moon system are shown to scale.

Radar systems measure the distance to a target by measuring the time taken for an echo of the light pulse to return. Similarly, a Global Positioning System (GPS) receiver measures its distance to satellites based on how long it takes for a radio signal to arrive from the satellite. The Lunar Laser Ranging Experiment, radar astronomy and the Deep Space Network determine the distances to the Moon, planets and spacecraft respectively by measuring the round-trip travel time.

The finite speed of light is particularly important in astronomy. Due to the vast distances involved it can take a very long time for light to travel from its source to Earth. For example, it takes 13 billion (13 × 10⁹) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field images. 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 farther-away objects appear younger (due to the finite speed of light) is crucial in astronomy, allowing 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.^[68] A light‑year is the distance light travels in one year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs. Next to the Sun, the closest star to Earth, Proxima Centauri, is around 4.2 light‑years away.^[69]

Doppler effect

Two light spectra with some black lines on them. The pattern of the line is the same for both spectra, but in the lower spectrum the lines are shifted toward the red end.

The Doppler effect for electromagnetic waves such as light is of great use in astronomy and results in either a so-called redshift or blue shift. It has been used to measure the speed at which stars and galaxies are approaching or receding from us, that is, the radial velocity. This is used to detect if an apparently single star is, in reality, a close binary star and even to measure the rotational speed of stars and galaxies.

Stellar aberration

Stellar aberration is the apparent motion of celestial objects about their real locations due to the finite speed of light and the motion of Earth. It was discovered and later explained by the third Astronomer Royal, James Bradley, in 1725.^[70]

Light from location 1 will appear to be coming from location 2 for a moving telescope due to the finite speed of light, a phenomenon known as the aberration of light.

The figure to the left examines how light from a star (at location 1) travels down a telescope idealized as a narrow tube and moving at a speed v to the right. (This motion is largely due to the Earth as it orbits the Sun.) The light enters the tube from a star at angle θ and travels at speed c taking a time h/c to reach the bottom of the tube, where the light is detected by an observer. During the transit of the light, the tube moves a distance vh/c. Consequently, for the light ray to reach the bottom of the tube, where the tube must be inclined at an angle φ different from θ, resulting in an apparent position of the star at angle φ.

The maximum amount of the aberrational displacement of a star is approximately 20 arcseconds. Although this is a relatively small value, it was well within the observational capability of the instruments available in the early eighteenth century.

Terrell rotation

Whereas objects passing rapidly by an observer will be measured to have shrunk along the line of relative motion due to Lorentz contraction, they will be actually seen as being rotated. This is due to the differences in time that it takes light to reach the eye of the observer from different parts of the object.^[71][72] This effect is known as Terrell rotation.

History

Ancient, medieval and early modern speculation

Until relatively recent times, it was not known whether light travelled instantaneously or at a finite speed. The first extant recorded examination of this subject was in ancient Greece. Empedocles 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".^[73] Euclid and Ptolemy advanced the emission theory of vision, where light is emitted from the eye, thus enabling sight. Using that theory, Heron of Alexandria advanced the argument that the speed of light must be infinite, since distant objects such as stars appear immediately upon opening the eyes.

Early Islamic philosophers initially agreed with the Aristotelian view that light had no speed of travel. In 1021, Iraqi physicist Alhazen (Ibn al-Haytham) published the Book of Optics, in which he used experiments related to the camera obscura to support the now accepted intromission theory of vision, where light moves from an object into the eye.^[74] This led Alhazen to propose that light must therefore have a finite speed,^[73][75][76] and that the speed of light is variable, decreasing in denser bodies.^[76][77] He argued that light is a "substantial matter", the propagation of which requires time "even if this is hidden to our senses".^[78]

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.^[79] Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle.^[80][81] In the 1270s, Witelo considered the possibility of light travelling at infinite speed in a vacuum but slowing down in denser bodies.^[82] A comment on a verse in the Rigveda by the 14th century Indian scholar Sayana mentioned a speed of light, about 186,400 miles per second, that was chosen so that light would encircle the Puranic universe in one day, making it "the most astonishing 'blind hit' in the history of science!"^[83][84] In 1574, the Ottoman astronomer and physicist Taqi al-Din concluded that the speed of light is constant, but variable in denser bodies, and suggested that it would take a long time for light from the stars, which are very distant, to reach the Earth.^[85]

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 finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. 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.^[73]

First measurement attempts

In 1629, Isaac Beeckman proposed an experiment in which a person would observe 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 instaneous or not, but concluded that if it weren't, it must nevertheless be extraordinarily rapid.^[86][87] Galileo's experiment was carried out by the Accademia del Cimento of Florence in 1667, with the lanterns separated by about one mile, but no delay was observed. Based on the modern value of the speed of light, the actual delay in this experiment would be about 11 microseconds. Robert Hooke explained the negative results as Galileo had by pointing out that such observations did not establish the infinite speed of light, but only that the speed must be very great.

Early astronomical techniques

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 Christensen Rømer, one of a group of astronomers of the French Royal Academy of Sciences who were studying the motion of Jupiter's moons.^{[88][Note 2][89]} From the observation that the periods of Jupiter's innermost moon Io appeared to be shorter when the earth was approaching Jupiter than when receding from Jupiter he concluded that light travels at a finite speed, and was able to estimate that would take light 22 minutes to cross the diameter of Earth's orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth's orbit to obtain an estimate of speed of light of 220,000 km/s, 26% lower than the actual value.^[90]

Isaac Newton also accepted the finite speed. In his 1704 book Opticks he gives 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).^[91] The same effect was subsequently observed by Rømer for a "spot" rotating with the surface of Jupiter. Later observations also showed the effect with the three other Galilean moons, where it was more difficult to observe, thus laying to rest some further objections that had been raised.

Between 1725 and 1728, James Bradley, while searching for stellar parallax, observed the apparent motion of the star γ Draconis (Eltanin) depending on the season of the year. He realized that the motion (about 39 arcseconds) could not be a parallax (it was in the wrong direction at any given time) and, after ruling out several other possible causes, produced the theory of the aberration of light,^[92] a vector addition of the velocity of light arriving from the star and the velocity of the Earth around its orbit. The effect is that an observer on the Earth will see the light coming from a slightly different angle than the "true" value which, for a star in the sky, means a slightly different position. The effect is greatest near the orbital pole which, for the Earth, is close to γ Draconis. Bradley was able to predict the aberration for several other stars, and confirm his predictions by observation.^[92] His observations on γ Draconis gave a ratio of the speed of light to the mean linear speed of the Earth's orbital motion: Bradley's figure was that light travelled 10,210 times faster than the Earth in its orbit (the modern figure is 10,066 times faster) or, equivalently, that it would take light 8 minutes and 12 seconds to travel from the Sun to the Earth.^[92]

Diagram of the Fizeau apparatus

Earth-bound techniques

The first successful entirely earthbound measurement of the speed of light was carried out by Hippolyte Fizeau in 1849. Fizeau's experiment was conceptually similar to those proposed by Beeckman and Galileo. A beam of light was directed at a mirror 8 km away. On the way from the source to the mirror, the beam passed through a rotating cog wheel. At a certain rate of rotation, the beam could pass through one gap on the way out and another on the way back. But at slightly higher or lower rates, the beam would strike a tooth and not pass through the wheel. Knowing the distance to the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light could be calculated. Fizeau reported the speed of light as 313,000 km/s. Léon Foucault improved on Fizeau's method by replacing the cogwheel with a rotating mirror. Foucault's estimate, published in 1862, was 298,000 km/s.

Electromagnetic standing waves in a cavity.

Cavity resonance

During World War II, the development of the cavity resonance wavemeter for use in radar, together with precision timing methods, opened the way to laboratory-based measurements of the speed of light. In 1946, Louis Essen and A.C. Gordon-Smith used a microwave cavity of precisely known dimensions to establish the frequency for a variety of normal modes of microwaves. 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.

The Essen–Gordon-Smith result, 299,792±3 km/s, was substantially more precise than those found by optical techniques, and prompted much controversy. However, by 1950 repeated measurements by Essen established a result of 299,792.5±1.0 km/s, which became the value adopted by the 12th General Assembly of the Radio-Scientific Union in 1957.^{[citation needed]}

Heterodyne laser measurements

An idealized interferometric determination of wavelength obtained by looking at interference fringes between two coherent beams recombined after travelling different distances. Top: Constructive interference (in phase); If the difference in path length is a multiple of a wavelength, the recombined beams support one another and reconstitute the original beam. Bottom: Destructive interference (out of phase); If the two paths differ by half a wavelength, the recombined beams are out of phase and cancel each other. The bottom panel in the figure suggests the path length has been increased by half a wavelength by moving the right-hand point of reflection further out.

An alternative to the cavity resonator method to find the wavelength for determining the speed of light is to use a form of interferometer, indicated schematically in the figure.^[93] A coherent light beam with a known frequency (f), as from a laser, is split to follow two paths and then recombined. By carefully changing the path length and observing the interference pattern, the wavelength of the light (λ) can be determined, which is related to the speed of light by the equation c = λf.

The main problem with interferometry is to measure the frequency of light in or near the optical region. This was first overcome by a group at the NIST laboratories in Boulder, Colorado, in 1972.^[94] By a series of photodiodes and specially constructed metal–insulator–metal diodes, they succeeded in linking the frequency of the caesium transition used in atomic clocks to the frequency of a methane-stabilized laser (nearly 10,000 times higher).^[95] Their results were

f = 88.376181627(50) THz

λ = 3.392231376(12) μm

c = 299792456.2(1.1) m/s

nearly a hundred times more precise than previous measurements of the speed of light.^[94][95]

Redefinition of the metre

The 1972 measurement of the speed of light, with a relative uncertainty of 4×10⁻⁹, was not only a feat of experimental precision, it also demonstrated a fundamental limit to how precisely the speed of light could be measured at that time using any technique. The remaining uncertainty in the value was almost completely attributable to uncertainty in the length of the metre.^[94][95][96]

Since 1960, the metre had been defined as a given number of wavelengths of the light of one of the spectral lines of a krypton lamp,^{[Note 3][97]} but it turned out that the chosen spectral line was not perfectly symmetrical.^[95] This gave an uncertainty in its wavelength, and hence in the length of the metre. By analogy with a metal measuring stick, it was as if the stick were slightly fuzzy at each end, although if it were a real measuring stick, the fuzziness at the ends of a one-metre stick would only be apparent at the atomic scale.

To get round this problem, the 15th Conférence Générale des Poids et Mesures (CGPM) in 1975 recommended the use of the value 299,792,458 m/s for "the speed of propagation of electromagnetic waves in vacuum".^[96] The 17th CGPM in 1983 decided to redefine the metre to be "the length of the path travelled by light in vacuum during a time interval of ¹⁄_299,792,458 of a second".^[98]

The effect of this definition gives the speed of light the exact value 299,792,458 m/s, which is nearly the same as the value 299,792,456.2(1.1) m/s obtained in the 1972 experiment. This number was chosen so that the change in the actual length of the metre was minimised, being similar to the measurement uncertainty.^[99][100] As a result, within the SI system of units, the speed of light is now a defined constant^[13] and no longer something to be measured.^[95] Improved experimental techniques do not affect the value of the speed of light in SI units, but do result in a more precise realisation of the SI metre.^[101][102]

Rather than measure a time-of-flight, one implementation of this definition is to use a recommended source with established frequency f, and delineate the metre in terms of the wavelength λ of this light as determined using the defined numerical value of c and the relationship λ = c / f.^[103] Practical realisations of the metre use recommended wavelengths of visible light in a laboratory vacuum with corrections being applied to take account of actual conditions such as diffraction, gravitation or imperfection in the vacuum.^[104][105]

Modern astronomical measurements

The overriding problem with any modern measurement of the speed of light (c) is the definition of a precise standard of length. For practical length measurements on Earth, c is the length standard, through the 1983 definition of the metre, but it is still possible to define other standards and hence to measure c against those standards.

In astronomy and satellite communication, it is useful to use standards based on the mass of either the Sun or the Earth. This is transformed into a length standard by saying that the standard length is the distance from the centre of the body at which a planet or satellite would have a given orbital velocity. The method was first used by Carl Friedrich Gauss in 1801 to calculate the orbit of Ceres, and was refined by Simon Newcomb in his Tables of the Sun (1895).

The astronomical unit is one example of such a length standard, based on the solar mass and approximately equal to the average distance between the Earth and the Sun. The "light time per unit distance" is an essential parameter in calculating planetary ephemerides, and is simply the inverse of c in astronomical units per second. It is measured by comparing the time taken for radio signals to reach different spacecraft in the Solar System with their position as caculated from the gravitational effects of the Sun and the various planets. By combining many such measurements, a "best fit" value for the light time per unit distance can be obtained. The 2009 best estimate, as approved by the International Astronomical Union (IAU), is:^{[106][107][108]}

light time per unit distance: 499.004783836(10) s

c = 0.00200398880410(4) AU/s = 173.144632674(3) AU/day

The relative uncertainty in these measurements is 0.02 parts per billion (0.02 × 10⁻⁹), equivalent to the uncertainty in Earth-based measurements of length by interferometry.^[109][110]

The light time per unit distance is effectively the same quantity that was measured by Rømer and Cassini in the late 17th century, where they gave a value of "ten to eleven minutes"^[111], slightly longer than the currently accepted value of 8 minutes 19 seconds.

Laboratory demonstration

With modern electronics, particularly oscilloscopes with time resolutions of less than one nanosecond, the speed of light can now be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror, although this method is less precise than either the cavity resonator or the interferometric methods.^{[112][113][114]}

See also

• Mathematical descriptions of the electromagnetic field
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Light second
• Photon (See section Photons in matter)

Notes

1. ^ The speed of light can also be expressed exactly in imperial units and US units, based on an inch of exactly 2.54 cm, as 186,282 miles, 698 yards, 2 feet, and 5+²¹⁄₁₂₇ inches per second. Savard, John. "From Gold Coins to Cadmium Light". John Savard's Home Page. http://www.quadibloc.com/other/cnv03.htm. Retrieved 2009-11-14.  Archived version 2009-11-14
2. ^ Besides Rømer, the group included Giovanni Domenico Cassini and Jean Picard.
3. ^ The metre was defined (1960–1983) as "the length equal to 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p₁₀ and 5d₅ of the krypton-86 atom."

References

Citations

Historical references

Modern references

External links

• Speed of light in vacuum (at NIST)
• Definition of the metre (BIPM)
• Data Gallery: Michelson Speed of Light (Univariate Location Estimation) (download data gathered by A.A. Michelson)
• Subluminal (Java applet demonstrating group velocity information limits)
• De Mora Luminis at MathPages
• Light discussion on adding velocities
• Speed of Light (University of Colorado Department of Physics)
• How is the speed of light measured?
• The Fizeau "Rapidly Rotating Toothed Wheel" Method