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
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

In physics, the speed of light (usually denoted c) is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space (i.e., perfect vacuum). Its value is 299,792,458 metres per second (see the table on the right for conversions). In the theory of special relativity, c is an important constant connecting space and time in the unified structure of spacetime and an upper bound on the speed at which energy, matter, and information can travel, and its square is the constant of proportionality between mass and energy (E = mc²). It is significant in the understanding and study of astronomy, space travel, and other fields.

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, and it is believed to be the speed of gravity and of gravitational waves.

For much of human history, it was not known whether light was transmitted instantaneously or simply very quickly. In the 17th century, Ole Rømer first demonstrated that it traveled at a finite speed by studying the apparent motion of Jupiter's moon Io; using these observations, Christiaan Huygens estimated the speed of light to be about 220,000 kilometres per second. Since then, scientists have devised increasingly sophisticated techniques to improve the precision of measurement. 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 (4×10⁻⁹), mostly due to the uncertainty in the length of the metre. In 1983, the metre was redefined in the International System of Units (SI) as the distance travelled by light in vacuum in 1⁄299792458 of a second. As a result, the numerical value of c in metres per second is now a fixed, exact value by definition.

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, the refractive index of glass is typically around 1.5, meaning that light in glass travel 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 that in vacuum.

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 1⁄299792458 of a second (see "Redefinition of the metre § Notes", below). The effect of this definition is to fix the speed of light in vacuum at exactly 299792458 m/s.

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.

Some authors use c for the speed of waves in any material medium, and c₀ for the speed of light in vacuum. This subscripted notation, which is endorsed in official SI literature, 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. Thus, no symbol for the speed of light is required.

The nature of light

Light as electromagnetic radiation

See also: Electromagnetic spectrum and Wave–particle duality

Light is a form of electromagnetic radiation and as such can be described in classical electromagnetism by the electromagnetic wave equation, which is derived from Maxwell's equations. According to this theory, the speed of light in vacuum is related to the electric constant ε₀ and the magnetic constant μ₀ by the equation c² = 1/(ε₀μ₀). In Gaussian units, the speed of light fixes the ratio between electrostatic and electromagnetic units.

An important consequence of Maxwell's equations is that the speed of light in vacuum is independent of the frequency and wavelength of the waves, unlike many other types of waves in physics, including light travelling through a transparent material such as water or glass. However, for light in vacuum, it has been experimentally verified to a very high degree of precision (see "Variations with time and frequency § Notes", below).

Light as photons

It is also possible to describe light as a stream of particles, called photons. In this case, the energy E of each photon is related to the frequency f of the light by the Planck constant h: E = hf. The speed of light is simply the speed at which these particles travel but, if it is independent of the frequency, it must also be independent of the energy of the particles. This is paradoxical in Newtonian mechanics, where the kinetic energy of a particle depends on its speed: the paradox is resolved in relativistic mechanics if the photon has no rest mass. The interaction of photons with matter is described by quantum electrodynamics (QED). Here the speed of light enters the theory via the dimensionless fine structure constant α.

Fundamental importance in physics

Constant speed in inertial frames

See also: Introduction to special relativity and Special relativity

The speed of light has been shown experimentally to be independent of the motion of the source. 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.

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. 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. Although the speed of propagation is independent of motion of the source, the 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

See also: Spacetime, Minkowski space, and Lorentz group

Fusion of deuterium with tritium creating helium-4, freeing a neutron, and releasing 17.59 MeV of energy by conversion of mass in accord with E = mc².

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

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.

In Einstein's theory of general relativity, c is still an important constant of spacetime. This spacetime is curved by the presence of matter and energy causing gravitation. Disturbances of this curvature, including gravitational waves, are predicted by the theory to propagate at the speed of light, although they have not been directly observed yet.

Causality and information transfer

See also: Causal contact, Tachyonic antitelephone, and Horizon problem

A double cone, with the "time" axis passing in it through its vertex and the "space" axis perpendicular to it. Point A is the vertex of the cone, point B is inside the upper cone and point C is outside the cone.

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, and would lead to paradoxes.

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.

Composition of velocities

See also: Velocity addition formula and Fizeau–Foucault apparatus

If two cars approach each other from opposite directions, each traveling at a speed of 50 km/h relative to the road surface, according to intuitive Galilean relativity, one expects that an observer in one car will measure the speed of approach of the other car as 50 + 50 = 100 km/h. However, as speeds increase, the simple addition of speeds becomes less accurate. Two spaceships approaching each other, each traveling at 90% of the speed of light in opposite directions relative to some third observer, would not measure each other as approaching at 90% + 90% = 180% the speed of light; instead they each measure the other as approaching at slightly less than 99.5% the speed of light. This last result is given by the velocity-addition formula, which in the case of two velocities in opposite directions is:

$u={\frac {v_{1}+v_{2}}{1+v_{1}v_{2}/{c^{2}}}},$

where v₁ and v₂ are the speeds of the spaceships as measured by the third observer, and u is the measured speed of either spaceship as observed by the other. This reduces to u = v₁ + v₂ for sufficiently small values of v₁ and v₂, such as those typically encountered in common daily experiences, as the term v₁v₂/c² becomes negligible, reducing the denominator to 1.

If one of the two speeds v_i in the above formula is c, the final result is c, as is expected if the speed of light is the same in all inertial reference frames. Another important result is that this formula never returns a value greater than c. This is consistent with c being the maximum possible speed.

Variations with time and frequency

Main article: Variable speed of light

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. These questions remain an interest of ongoing research.

Others have suggested that the speed of light may exhibit a small dispersion—that is, different frequencies may travel at slightly different velocities. (The invariant speed c of special relativity would the be the upper limit of the speed of light in vacuum.)

However, to date, such variations, if any, are smaller than experimental errors of observation—the measured speed of electromagnetic radiation is the same for different frequencies to within very stringent experimental limits.

A reason for light to travel at a frequency-dependent speed would be a non-zero rest mass of the photon, the carrier of the electromagnetic force or field. At present, the rest mass of the photon is taken to be zero, implying that it always travels at speed c. If the hypothetically massive photon is described by Proca theory, the experimental upper bound for its mass is about 10⁻⁵⁷ grams. If photon mass is generated by a Higgs mechanism, the experimental upper limit is less sharp, m ≤ 10⁻¹⁴ eV/c² (equivalent to of the order of 10⁻⁴⁷ g).

Light in transparent media

Refractive index and refraction

See also: Refractive index and Snell's law

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. 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. 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.

Classically, when an electromagnetic wave meets the surface of a dielectric material at an angle, the leading edge is slowed while the trailing edge continues normally. This causes the wave to change direction. This is called refraction and the change in direction may be calculated from the refractive index using Snell's law.

Dispersion

See also: Dispersion (optics), Electric susceptibility, and Permittivity

A white beam of light enters a prism and a spectrum comes out of it.

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

The dependence of the speed of light in a material on its frequency is known as dispersion. When a light pulse consisting of multiple frequencies passes through transparent materials, the speed of light is characterized by two speeds: the phase velocity and the group velocity. The phase velocity of light may be found from knowledge of the frequency-dependent refractive index: $n={\sqrt {\epsilon _{r}\mu _{r}}}=c/v_{p}$ where ε_r is the material's relative permittivity, and μ_r is its relative permeability.

The group velocity of the wave is the speed at which the envelope of the pulse travels through the medium, and is dependent on the frequency content of the pulse as well as the properties of the medium. A wave with different group and phase velocities is said to undergo dispersion.

General media

In real materials the current and charge densities appearing in Maxwell's equations are determined by the way charges respond to the electromagnetic fields. One approach to describing this behaviour is the calculation of constitutive equations, which describe how charges behave within the medium, usually using a linear approximation. (Ohm's law is an example.) In simplest form, the constitutive equations replace the electric constant and magnetic constant by the electric permittivity and magnetic permeability of the medium. However, non-linear behaviour also occurs, as in ferroelectric materials, for example. A consequence of the charge response to fields is that the speed of light may become dependent upon field amplitudes, polarization, direction of propagation, and frequency. One field concerned with the calculation of these relationships is condensed matter physics.

Slow light

Main article: Slow light

Certain materials have an exceptionally high group index and a correspondingly low group velocity for light waves. 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); in 2001, they were able to momentarily stop a beam.

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.

Faster-than-light observations and experiments

Main article: Faster-than-light

See also: Scharnhorst effect, Hartman effect, and Chronology protection conjecture

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 traveling faster than c relative to an inertial frame of reference, it would be traveling backwards in time relative to another frame. (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.) 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.

It is possible for the group velocity of light waves to exceed c. In an experiment in 2000, laser beams travelled for extremely short distances through caesium atoms with a group velocity of 300 times c. 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.

If a laser beam is swept quickly across a distant object, the spot of light can move faster than c. Similarly, a shadow projected onto a distant object can be made to move faster than c. In neither case does 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.

Another prediction of faster-than-light speeds occurs for quantum tunneling and is called the Hartman effect. However, no information can be sent using these effects.

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, 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

Main article: Cherenkov radiation

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

It is possible for shock waves to be formed with electromagnetic radiation. 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

See also: Metric expansion of space, Hubble's Law, Observable universe, and Cosmological horizon

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.

Transit time

See also: Transit time and Light year

A beam of light is depicted traveling 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 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. 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.

Doppler effect

Main article: Relativistic Doppler effect

See also: Redshift and Blue shift

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

Main article: 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.

Light from location 1 appears to come from location 2 in a moving telescope.

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 photon 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

Main article: 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. 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". 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. This led Alhazen to propose that light must therefore have a finite speed, and that the speed of light is variable, with its speed decreasing in denser bodies. He argued that light is a "substantial matter", the propagation of which requires time "even if this is hidden to our senses".

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. Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle. In the 1270s, Witelo considered the possibility of light traveling at infinite speed in a vacuum but slowing down in denser bodies. A comment on a verse in the Rigveda by the 14th century Indian scholar Sayana may be interpreted as suggesting an estimate for the speed of light that is in good agreement with its actual speed. 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.

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.

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. 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. One of the group's discoveries, announced by Cassini in 1675, was that the periods of the moons appeared to be shorter when the Earth was approaching Jupiter than when it was receding from it. Rømer suggested that this occurred because the speed of light was finite. In September 1676, on the basis of this assumption, he predicted that on the following ninth of November, Io, Jupiter's innermost moon, would emerge from Jupiter's shadow 10 minutes later than had been expected from calculations based on its mean motion and the times of previous emersions observed during the preceding August.

After Rømer's prediction was fulfilled (to the second, according to Rømer himself), he read a report to a meeting of the French Academy of Sciences in which he explained how he had derived it. The procedure was illustrated with the aid of a diagram reproduced at the right. The points L and K represent positions of the Earth (moving away from Jupiter at B) at which observations are made of Io emerging from Jupiter's shadow at D. Because light takes a finite time to travel from D to L or K, these observations will occur later than the actual times at which the emersions themselves did. If the actual time interval between the two emersions is p, then the time interval between their observations at L and K will be p + ?, where ? is the time taken by light to travel the distance from L to K. But, for the same reason, the time interval between two observations, at F and G, of Io entering into Jupiter's shadow at C, as the Earth moves towards it, will be p - ? (asssuming the distance from G to F is the same as that from L to K).

If only one revolution of Io occurs between the two observed emersions or immersions, then ? will only be about 20 seconds—too small to obtain a reliable estimate of the speed of light. But Rømer realized that when many revolutions of Io occurred between two observations, the time lag ? would be proportionately larger, and would therefore make a useful estimate possible. Using the available observations, he estimated that it would take 22 minutes for light to cross the diameter of the earth's orbit.

Rømer did not express his estimate in terms of conventional units of length, except to say that light took less than a second to travel a distance of 3,000 leagues—a distance on the order of the diameter of the earth. Christiaan Huygens, however, on reading the English translation of Rømer's account, wrote to him for confirmation of the estimate, and using it and the then estimated size of the astronomical unit (roughly, the Earth-to-Sun distance), estimated the speed of light to be about 110,000,000 toises per second. This is about 220,000 kilometres per second (136,000 miles per second), 26% lower than the currently accepted value, but still very much faster than any physical phenomenon then known.

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). 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 ? 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, 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 ? Draconis, and almost zero for stars on the ecliptic. Bradley was able to predict the aberration for several other stars, and confirm his predictions by observation. His observations on ? 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.

Earth-bound techniques

See also: Fizeau–Foucault apparatus

Diagram of the Fizeau apparatus

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 313000 km/s. Léon Foucault improved on Fizeau's method by replacing the cogwheel with a rotating mirror. Foucault's estimate, published in 1862, was 298000 km/s.

Cavity resonance

Electromagnetic standing waves in a cavity. Sinusoidal waves at the top have larger frequencies than below and are shifted upward for clarity. The conductive walls require the nodes of the standing wave to be at the wall surfaces, so the allowed wavelengths are: λ/2 = W, λ = W, 3λ/2 = W where W = width of cavity.

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, 299792±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 299792.5±1 km/s, which became the value adopted by the 12th General Assembly of the Radio-Scientific Union in 1957.

Heterodyne laser measurements

An idealized interferometric determination of wavelength obtained by looking at interference fringes between two coherent beams recombined after traveling 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. 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. 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). 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.

Redefinition of the metre

See also: Metre

The 1972 measurement, 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 realization of the metre.

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, but it turned out that the chosen spectral line was not perfectly symmetrical. 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 299792458 m/s for "the speed of propagation of electromagnetic waves in vacuum". 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 1⁄299792458 of a second".

The effect of this definition gives the speed of light the exact value 299792458 m/s, the value obtained in the 1972 experiment. This number was chosen in order to ensure that the change in the value of the metre was minimised, being limited to the elimination of the measurement uncertainty (then four parts per billion, 4×10⁻⁹). As a result, within the SI system of units, the speed of light is now a defined constant and no longer something to be measured. Improved experimental techniques do not affect the value of the speed of light in SI units, but do result in a better realization of the SI metre.

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. Practical realizations 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.

Modern astronomical measurements

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

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 (2×10⁻¹¹), equivalent to the uncertainty in Earth-based measurements of length by interferometry.

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", 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.

See also

• Mathematical descriptions of the electromagnetic field
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Light second

References

Historical references

• Rømer, Ole (1676). "Démonstration touchant le mouvement de la lumière". Journal des sçavans: 223–236.. Translated as "A Demonstration concerning the Motion of Light". Philosophical Transactions of the Royal Society (136): 893–894. 1677.
• Halley, Edmund (1694). "Monsieur Cassini, his New and Exact Tables for the Eclipses of the First Satellite of Jupiter, reduced to the Julian Stile and Meridian of London". Philosophical Transactions of the Royal Society. 18 (214): 237–256.
• Fizeau, H. L. (1849). Sur une expérience relative à la vitesse de propagation de la lumière. Comptes rendus de l'Académie des sciences (Paris). Vol. 29. pp. 90–92, 132. Template:Fr icon
• Foucault, J. L. (1862). Détermination expérimentale de la vitesse de la lumière: parallaxe du Soleil. Comptes rendus de l'Académie des sciences (Paris). Vol. 55. pp. 501–503, 792–796.
• Michelson, A. A. (1878). Experimental Determination of the Velocity of Light. Proceedings of the American Association of Advanced Science. Vol. 27. pp. 71–77.
• Newcomb, Simon (1886). "The Velocity of Light". Nature: 29–32.
• Perrotin, Joseph (1900). "Sur la vitesse de la lumière". Comptes rendus de l'Académie des sciences. 131: 731–734. Template:Fr icon
• Michelson, A. A.; Pease, F. G.; Pearson, F. (1935). "Measurement of the Velocity of Light in a Partial Vacuum". Astrophysical Journal. 82: 26–61. doi:10.1086/143655.

Modern references

• Brillouin, Léon (1960). Wave propagation and group velocity. Academic Press.
• Jackson, John David (1975). Classical electrodynamics (2nd ed.). John Wiley & Sons. ISBN 0-471-30932-X.
• MacKay, R. J.; Oldford, R. W. (2000). "Scientific Method, Statistical Method and the Speed of Light". Statistical Science. 15 (3): 254–278. doi:10.1214/ss/1009212817.
• Keiser, Gerd (2000). Optical Fiber Communications (3rd ed.). McGraw-Hill. p. 32. ISBN 0072321016.
• Y Jack Ng (2004). "Quantum Foam and Quantum Gravity Phenomenology". In Giovanni Amelino-Camelia & Jerzy Kowalski-Glikman (editors) (ed.). Planck Scale Effects in Astrophysics and Cosmology. Springer. pp. 321ff. ISBN 3540252630. {{cite book}}: |editor= has generic name (help)
• J Helmcke & F Riehle (2001). "Physics behind the definition of the meter". In T. J. Quinn, S. Leschiutta, P. Tavella (ed.). Recent advances in metrology and fundamental constants. IOS Press. p. 453. ISBN 1586031678.{{cite book}}: CS1 maint: multiple names: editors list (link)
• M. J. Duff (2004). "Comment on time-variation of fundamental constants". ArXiv preprint.

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