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The speed of light in a vacuum is an important physical constant usually denoted by the symbol c. In free space, it is defined as exactly 299,792,458 metres per second (due to the definition of the metre). According to special relativity, it also is the speed of anything having zero rest mass. To within experimental accuracy it is the speed of all electromagnetic radiation in vacuum, and is anticipated to be the speed of gravitational waves as well.

Einstein's theory of relativity together with the principle of causality requires that no matter or information can travel with a speed larger than c. Speeds faster than that of light in a vacuum are encountered in physics but, in all such cases, no matter or information is transmitted faster than c.

For many practical purposes, the speed of light is so great that it can be regarded to travel instantaneously. An exception is where long distances or precise time measurements are involved. For example, in the Global Positioning System (GPS), a GPS receiver measures its distance to satellites based on how long it takes for a radio signal to arrive from the satellite. In astronomy, distances are often measured in light-years, the distance light travels in a year.

The speed of light when it passes through a transparent or translucent material medium, like glass or air, is less than its speed in free space. The speed is inversely proportional to the refractive index of the medium. In specially-prepared media, the speed can be tiny, or even zero.

For many years the speed of light was the subject of speculation, some believing it to be infinite. The first effective measurements of the speed of light were made in the seventeenth century, and these were progressively refined. Today measurement of light in vacuum continues, focusing upon setting limits upon any field or frequency dependence. In 1983, the standard for the speed of light in free space (an ideal vacuum) was fixed by definition.

Physics

Use of the symbol ‛c’ for the speed of light

The symbol c for 'constant' or the Latin celeritas (swiftness) is used for the speed of light in vacuum, and in this article c is used exclusively this way. Some authors, however, use c for the speed of light in any material media. The term "vacuum" in this context usually refers to a medium resembling free space. To avoid confusion, and for consistency with other constants of free space such as μ₀, ε₀ and Z₀, international bodies such as the International Bureau of Weights and Measures(BIPM) recommend using c₀ for the speed of light in free space.

In branches of physics in which the speed of light plays an important part, such as in relativity, it is common to use a system of units known as natural units in which c is 1; thus no symbol for the speed of light is required.

Constant velocity from all inertial reference frames

Most individuals are accustomed to the addition rule of velocities: if two cars approach each other from opposite directions, each traveling at a speed of 50 km/h, relative to the road surface, one expects that each car will measure the other as approaching at a combined speed of 50 + 50 = 100 km/h to a very high degree of accuracy. However, as speeds increase this rule becomes less accurate. Two spaceships approaching each other, each traveling at 90% the speed of light 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 Einstein velocity-addition formula:

${\displaystyle {\mathbf {u} }={\frac {{\mathbf {v} }_{1}+{\mathbf {v} }_{2}}{1+({\mathbf {v} }_{1}\cdot {\mathbf {v} }_{2})/{c^{2}}}}\,\!}$

where v₁ and v₂ are the velocities of the spaceships as measured by the third observer, and u is the measured velocity of either space ship 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 (v₁·v₂)/c approaches zero, reducing the denominator to 1.

If one of the velocities for the above formula (or both) are c, the final result is c, as is expected if the speed of light is the same in all reference frames. Another important result is that this formula always returns a value which is less than c whenever v₁ and v₂ are less than c: this shows that no acceleration in any frame of reference can cause one to exceed the speed of light with respect to another observer. Thus c acts as a speed limit for all objects with respect to all other objects in special relativity.

Doppler shift

Although the speed of light is measured to be the same by all inertial observers, the measured frequency of light depends on the relative velocity between the source and the observer. This is known as Doppler shift. An observer moving with respect to a collection of light sources would find that light from the sources ahead would be blueshifted while light from those behind was redshifted.

Spacetime

Experimental evidence has shown that the speed of light is independent of the motion of the source. It has also been confirmed by the Michelson-Morley experiment and others that the two-way speed of light (for example from a source, to a mirror, and back again) is constant. It is not, however, possible 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 receiver should be synchronized. Einstein (who was aware of this fact) postulated that the speed of light should be taken as constant in all cases, one-way and two-way. This postulate, together with the principle of relativity that all inertial frames are equivalent, forms the basis of Einstein's theory of special relativity.

According to relativity, space and time are viewed as a four dimensional unification of space and time, known as spacetime, in which the c plays the fundamental role of a conversion factor between space and time within spacetime, and between mass and energy.

In Einstein's general theory of relativity, spacetime is curved by the presence of matter and energy causing gravitation. Disturbances in this curvature, including gravitational waves, propagate at the speed of light.

Causality and information transfer

According to the theory of special relativity, causality would be violated if information could travel faster than c in one 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.

Information propagates to and from a point forming regions defined by a light cone. The 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").

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 simultaneity). However, there are also frames in which A precedes C (as shown) or in which C precedes A. Barring some way of traveling faster than light, it is not possible for any matter (or information) to travel from A to C or from C to A. Thus there is no causal connection between A and C.

Light as electromagnetic radiation

Light is a form of electromagnetic radiation. Its speed is determined by the electromagnetic wave equation, which is derived from Maxwell's equations.

In SI units the speed of all electromagnetic radiation in free space is related to the electric constant ε₀ (also called the permittivity of free space) and magnetic constant μ₀ (also called the permeability of free space) by the equation c₀=1/(ε₀ μ₀) . As speed of light in free space is now fixed by definition and the value of the magnetic constant is defined to be 4π×10 H/m the value of the electric constant is now also fixed.

In Gaussian units, the speed of light fixes the ratio between electrostatic and electromagnetic units.

According to classical electromagnetism, the speed of electromagnetic radiation in free space is the same for all frequencies. Any observation of dispersion in vacuum has bearing upon the mass of the photon, normally assumed to be zero, and established by measurement to be less than m_γ ≤ 1×10 g ≡ 6×10 eV. Measurements based on the arrival of electromagnetic radiation from distant astrophysical events put severe limits on the possible variation in the speed of light with frequency. Measurement of dispersion at very low frequencies is difficult, but experiments using radio waves generated by lightning have set a limit Δc/c ≤ 0.8% upon how much the speed of electromagnetic radiation varies with frequency at frequencies between 5 and 45 Hz.

Interaction with transparent materials

In passing through materials, the observed speed of light differs from c. 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 the microscopic level electromagnetic waves always travel at c. 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 ratio of c to the phase velocity of light in the material is called the refractive index. It is usually given the symbol n. In the case of light, it equals

${\displaystyle n={\sqrt {\epsilon _{r}\mu _{r}}}}$,

where ε_r is the material's relative permittivity, and μ_r is its relative permeability.

The speed of light in air is only slightly less than c (the refractive index of air is only slightly larger than one ). Denser media, such as water and glass, can slow light much more, to fractions such as 3⁄4 and 2⁄3 of c. Through diamond, light is much slower—only about 124,000 km/s, less than 1⁄2of c. This reduction in speed is also responsible for bending of light at an interface between two materials with different indices, a phenomenon known as refraction.

Since the speed of light in a material depends on the refractive index, and the refractive index may depend on the frequency of the light, light at different frequencies can travel at different speeds through the same material. This effect is called dispersion.

Classical and modern models of refractive index of "vacuum"

A simple model of a transparent electromagnetic medium often used to represent "vacuum" is one where electric permittivity and magnetic permeability are constants having the values ε₀ and μ₀. In this classical model, the speed of light c₀ is the same for all wavelengths, and there exists perfect isotropy, zero dispersion, perfect linearity and zero dichroism. The refractive index of this classical model is unity.

This classical model can be compared with modern models of a region of space containing only an electromagnetic field (a rather specific description of a particular "vacuum"). Some modern models predict variation in the electromagnetic behavior, a refractive index that departs from one: – for example, the independence of the speed of light from wavelength and from field strength is considered to be a "pre-quantum approximation". Experiments testing these new models, such as analysis of arrival times of light of different wavelengths from distant astronomical events, show that the divergences from the above classical model must be very small. Nonetheless, establishing departures from the classical model could be profound. (That is, departures not attributable to known imperfections, like cosmic dust or residual atoms.) Depending upon what was discovered, results might prove or disprove some theories of quantum gravity, provide insight into the short-distance structure of spacetime, impact our notion of how the universe is evolving, or test further the predictions of quantum electrodynamics.

Speed of light in astronomy

The 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 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 (near the beginning of the universe). 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, galaxies, and the universe itself.

Astronomical distances are sometimes measured in light-years, the distance light travels in one year. A light‑year is around 9 trillion km, 6 trillion miles, or 0.3 parsecs. Next to the Sun, the closest star to Earth, Proxima Centauri, is around 4.2 light‑years away.

Faster-than-light observations and experiments

It is generally considered that it is impossible for any information or matter to travel faster than c, because it would travel backwards in time relative to some observers. However, there are many physical situations in which speeds greater than c are encountered.

Some of these situations involve entities that actually travel faster than c in a particular reference frame but none involves either matter, energy, or information traveling faster than light.

Wave velocities and synchronized events

It is possible for the "group velocity" of light to exceed c and in an experiment in 2000 laser beams traveled for extremely short distances through caesium atoms with a group velocity of 300 times c. It is not, however, possible to use this technique to transfer information faster than c since the velocity of information transfer depends on the front velocity, which is always less than c.

Exceeding the group velocity of light in this manner is comparable to exceeding the speed of sound by arranging people distantly spaced in a line, and asking them all to shout "I'm here!", one after another with short intervals, each one timing it by looking at their own wristwatch so they don't have to wait until they hear the previous person shouting. Another example can be seen when watching ocean waves washing up on shore. With a narrow enough angle between the wave and the shoreline, the breakers travel along the waves' length much faster than the waves' movement inland.

If a laser is swept across a distant object, the spot of light can easily be made to move at a speed greater than c. Similarly, a shadow projected onto a distant object can be made to move faster than c. In neither case does any matter or information travel faster than light.

Quantum mechanics

In quantum mechanics, certain quantum effects may be transmitted at speeds greater than c. For example, the quantum states of two particles can be entangled. Until the particles are observed, they exist in a superposition of two quantum states. If the particles are separated and one of them is observed to determine its quantum state then the quantum state of the second particle is determined automatically and faster than a light signal could travel between the two particles. However, it is impossible to control which quantum state the first particle will take on when it is observed, so no information can be transmitted in this manner.

Speeds not representing that of an object measured in a single inertial frame

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.

Superluminal motion of astronomical objects

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.

Čerenkov radiation

It is possible for shock waves to be formed with electromagnetic radiation. If a charged particle travels through an insulating medium faster than the speed of light in that medium then radiation is emitted which is analogous to a sonic boom and is known as Čerenkov radiation.

Galaxies moving faster than light

In models of the expanding universe, the further 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. Hubble's law states the recessional velocity in terms of comoving distance to the object as:

${\displaystyle v=H(t_{0})d\ ,}$

where v = recessional velocity of object due to expansion of the universe, H = value of the Hubble constant at the time of observation t₀, and d is the distance to the object. At a boundary called the Hubble sphere, the recessional velocity is the speed of light.

At distances beyond the Hubble sphere, objects move away faster than the speed of light. This speed does not contradict special relativity because the motion occurs outside any observer's inertial frame. Light from such objects moves away from the receding source, toward us, while the Hubble sphere expands toward the light, as described shortly. The Hubble sphere can overtake the photons, the light enters the Hubble sphere and eventually becomes observable on Earth, even though the originating sources are receding at a rate faster than light.

The Hubble "constant" H(t) decreases with time, at a rate that depends upon the cosmological model assumed (for example, the ΛCDM model), causing the radius r_HS of the Hubble sphere to expand with time.

${\displaystyle r_{HS}={\frac {c}{H(t)}}={\frac {a(t)c}{{\overset {\cdot }{a}}(t)}}\ ,}$

In a Friedmann universe the scale factor a(t) increases with time, but its rate of change å(t) increases more slowly, causing r_HS to increase.

So it happens that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light. The most distant objects that we can see now were outside the Hubble sphere when they emitted the photons we see now. The current recession velocity of the points from which the cosmic microwave background was emitted is v = 3.2c. We routinely see radiation from objects that lie outside the Hubble sphere.

Ancient, medieval and early modern history

Until relatively recent times, the speed of light was largely a matter of conjecture. Empedocles maintained that light was something in motion, and therefore there had to be some time elapsed in traveling. Aristotle said that, on the contrary, "light is due to the presence of something, but it is not a movement".

Euclid proposed the emission theory of vision, (also advanced by Ptolemy) where light was emitted from the eye, instead of entering the eye from another source. Using this 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 Muslim philosophers initially agreed with the Aristotelian view of the speed of light being infinite. In 1021, however, the Iraqi physicist, Ibn al-Haytham (Alhazen), published the Book of Optics, in which he used experiments to support the intromission theory of vision, where light moves from an object into the eye, making use of instruments such as the camera obscura. This led to Alhazen proposing 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". This debate continued in Europe and the Middle East throughout the Middle Ages.

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. 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 agreed with Alhazen 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 millions of kilometres away 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. Francis Bacon argued that the speed of light was not necessarily infinite, since something can travel too fast to be perceived. 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 was found to be finite, his whole system of philosophy might be demolished.

Measurement of the speed of light

Early attempts

Isaac Beeckman proposed an experiment (1629) in which a person would observe the flash of a cannon reflecting off a mirror about one mile (1.6 km) away. Galileo Galilei proposed an experiment (1638), 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 concluded that the speed of light is ten times faster than the speed of sound (in reality, light is around a million times faster than sound). This experiment was carried out by the Accademia del Cimento of Florence in 1667, with the lanterns separated by about one mile (1.6 km). No delay was observed. 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.

Astronomical techniques

The first quantitative estimate of the speed of light was made in 1676 by Ole Christensen Rømer, who was studying the motions of Jupiter's moon, Io, with a telescope. It is possible to time the orbital revolution of Io because it enters and exits Jupiter's shadow at regular intervals (at C or D). Rømer observed that Io revolved around Jupiter once every 42.5 hours when Earth was closest to Jupiter (at H). He also observed that, as Earth and Jupiter moved apart, (as from L to K), Io's exit from the shadow would begin progressively later than predicted. It was clear that these exit "signals" took longer to reach Earth, as Earth and Jupiter moved further apart. This was as a result of the extra time it took for light to cross the extra distance between the planets, time which had accumulated in the interval between one signal and the next. The opposite is the case when they are approaching (as from F to G). Rømer observed 40 orbits of Io when Earth was approaching Jupiter to be 22 minutes shorter than 40 orbits of Io when Earth was moving away from Jupiter. On the basis of those observations, Rømer concluded that it took light 22 minutes to cross the distance the Earth traversed in 80 orbits of Io. This means that in travelling from L to K and F to G, whereas the earth took 80 periods of Io's orbits (42.5 hours), the light only took 22 minutes. This corresponds to a ratio between the speed of light and the speed at which the Earth travels in its orbit around the sun of:

${\displaystyle 80\times {\frac {42.5\,{\rm {hours}}}{22\,{\rm {minutes}}}}\approx 9,300.}$

In comparison the modern value is about 10,100.

Around the same time, the astronomical unit was estimated to be about 140 million kilometres. The astronomical unit and Rømer's time estimate were combined by Christiaan Huygens, who estimated the speed of light to be 1,000 Earth diameters per minute, based on having misinterpreted Rømer's value of 22 minutes to mean the time it would take light to cross the diameter of the orbit of the Earth. 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 reports the value of 16.6 Earth diameters per second (210,000 kilometres per second, 30% less than the actual value), which it seems he inferred for himself (whether from Rømer's data, or otherwise, is not known). The same effect was subsequently observed by Rømer for a "spot" rotating with the surface of Jupiter. And 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.

Even if, by these observations, the finite speed of light may not have been established to everyone's satisfaction (notably Jean-Dominique Cassini's), after the observations of James Bradley (1728), the hypothesis of infinite speed was considered discredited. Bradley deduced that starlight falling on the Earth should appear to come from a slight angle, which could be calculated by comparing the speed of the Earth in its orbit to the speed of light. This "aberration of light", as it is called, was observed to be about 1/200 of a degree. Bradley calculated the speed of light as about 298,000 kilometres per second (186,000 miles per second). This is only slightly less than the currently accepted value (less than one percent). The aberration effect has been studied extensively over the succeeding centuries, notably by Friedrich Georg Wilhelm Struve and de:Magnus Nyrén.

Earth-bound techniques

The first successful measurement of the speed of light using an earthbound apparatus was carried out by Hippolyte Fizeau in 1849. (This measures the speed of light in air, which is slower than the speed of light in vacuum by a factor of the refractive index of air, about 1.0003.) Fizeau's experiment was conceptually similar to those proposed by Beeckman and Galileo. A beam of light was directed at a mirror several thousand metres 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. If α is the angle between two consecutive openings and d the distance between the toothed wheel and the mirror, then the tooth wheel must rotate with the angular speed (ω):

${\displaystyle \omega ={\frac {\alpha c}{2d}}}$

in order for the light to pass through. Fizeau chose d = 8 km.

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 kilometres per second. Fizeau's method was later refined by Marie Alfred Cornu (1872) and Joseph Perrotin (1900).

Leon Foucault improved on Fizeau's method by replacing the cogwheel with a rotating mirror. Foucault's estimate, published in 1862, was 298,000 kilometres per second. Foucault's method was also used by Simon Newcomb and Albert A. Michelson. Michelson began his lengthy career by replicating and improving on Foucault's method. If α is the angle between the normals to two consecutive facets and d the distance between the light source and the mirror, then the mirror must rotate with the angular speed (ω):

${\displaystyle \omega ={\frac {\alpha c}{2d}}}$

in order for the light to pass through.

Luminiferous aether and the Michelson–Morley experiment

After the work of many physicists in the 19th century, it was believed that light travelled through the "luminiferous aether", the medium that was then thought to be necessary for its transmission, its speed being determined by the aether's permittivity and permeability. Because light travels with immense speed and is a transverse wave, the aether was assumed to be extremely rigid and solid rather than fluid. On the other hand, it apparently offered no resistance to the motions of the moon and planets. Maxwell’s equations allow the speed of light to be calculated, in much the same way as the speed of sound can be calculated in normal matter. The speed of sound in a medium is relative to the the medium itself, and the speed of sound with respect to an observer may be changed if the observer is moving with respect to the medium. The speed of light was believed to be relative to the medium of transmission for light (the aether), which acted in the same way that a solid does for the transmission of sound.

In 1887, the physicists Albert Michelson and Edward Morley performed the influential Michelson–Morley experiment to measure the velocity of the Earth through the aether. As the Earth is in orbit round the sun, and the aether was assumed to be fixed, the Earth would be expected to be in motion with respect to the aether for at least some of the time. As shown in the diagram of a Michelson interferometer, a half-silvered mirror was used to split a beam of monochromatic light into two beams traveling at right angles to one another. After leaving the splitter, each beam was reflected back and forth between mirrors several times (the same number for each beam to give a long but equal path length; the actual Michelson-Morley experiment used more mirrors than shown) then recombined to produce a pattern of constructive and destructive interference. Any slight change in speed of light along one arm of the interferometer compared with its speed along the other arm (because the apparatus was moving with the Earth through the proposed "aether") would then be observed as a change in the pattern of interference. In the event, the experiment gave a null result. Later experiments confirmed this result to a much higher accuracy.

The Michelson–Morley null result disproved the original rigid fixed aether theory and no scientist has since succeeded in elaborating a mechanical model for the aether which would furnish a satisfactory mechanical interpretation of Maxwell's laws of the electromagnetic field. Ernst Mach was among the first physicists to suggest that the experiment amounted to a disproof of the aether theory.

The null result also led Lorentz to propose that motion through the aether contracts the Michelson interferometer due to Fitzgerald-Lorentz contraction, and later to propose his Lorentz transformations which formed the mathematical basis of Einstein's special theory of relativity. Although it uses the Lorentz transformations, Einstein's theory explains the null result of the Michelson–Morley experiment by postulating that the speed of light is always the same for all inertial observers. This means that the speed of light speed will always be the same in both arms of the interferometer, regardless of their orientation or state of inertial motion, thus no changes in the observed fringes would be expected when it was rotated. The postulates that the speed of light is the same for all inertial observers and the equivalence of inertial frames, are the the fundamental postulates of special relativity.

It is uncertain whether Albert Einstein knew the results of the Michelson-Morley experiment when he developed his theory, but the null result of the experiment greatly assisted the acceptance of his theory of relativity. After Einstein published his general theory of relativity, which extended his special theory to include gravitation, the concept of aether rapidly fell into disuse and it forms no part of physics today.

Laboratory-based methods

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 in collaboration with A.C. Gordon-Smith used a microwave cavity of precisely known dimensions to establish the frequency for a variety of normal modes of microwaves—which, in common with all electromagnetic radiation, travels at the speed of light in vacuum. 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. Their result, 299,792 ± 3km/s, was substantially greater 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 km/s; this became the value adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. Most subsequent measurements have been consistent with this value.

With modern electronics (and most particularly the availability of oscilloscopes with time resolutions in the sub-nanosecond regime) the speed of light can now be directly measured by timing the delay of a light pulse from a laser or a LED in reflecting from a mirror, and this kind of experiment is now routine in undergraduate physics laboratories.

Speed of light set by definition

In 1983, the 17th Conférence Générale des Poids et Mesures defined the metre in terms of the distance traveled by light in a given amount of time in so-called "vacuum", which amounts to adopting a standard value for the speed of light in vacuum:

The metre is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second.

Here, the term vacuum is meant in the technical sense of free space. This definition of the metre relies on the definition of the second, which is:

The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

The consequence of this definition is that no experimental measurement can change the speed of light from its exact value of 299,792,458 metres per second in the hypothetical medium "vacuum" or free space. That leaves experiment to answer the question, however, of how closely any realizable physical vacuum approximates the ideal of free space. 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.

Speed of light and cosmology

Weyl, Eddington and Dirac suggested the questions of just why the fundamental constants of nature have the values they do, and whether they are changing as the universe evolves. See Dirac large numbers hypothesis, an interview with Dirac, and the review by Prestage et al. A change with time of the speed of light also affects the fine structure constant:

${\displaystyle \alpha \ =\ {\frac {e^{2}}{\hbar c\ 4\pi \epsilon _{0}}}\ =\ {\frac {e^{2}c\mu _{0}}{2h}}\ ,}$

so theories describing an evolution of α have much in common with theories involving the evolution of c. Kafatos et al. have explored the possibility that the the speed of light is identical to the rate of change of the scale of the universe, and summarize some recent work of this type. Experiment and theory continue to explore these ideas.

Quantum gravity models suggest that the speed of light exhibits dispersion. While being smooth at large distances, space-time might show a complex, foamy, structure due to quantum fluctuations at short distances on the order of the Planck length ℓ_P  :

${\displaystyle \ell _{P}={\sqrt {\frac {\hbar G}{c^{3}}}}\thickapprox 1.616252(81)\times 10^{-35}{\mbox{ metres}}}$, where:

• G is the gravitational constant;
• ${\displaystyle \hbar }$ (pronounced "h-bar") ${\displaystyle \ =h/2\pi }$ is the reduced Planck constant.

An energy dependence of the speed of light in vacuum may arise from photon propagation through such a gravitational medium. Lehnert and Roy also discussed as a possible effect of fluctuations of permittivity and permeability in vacuum that photons may be gaining mass, if indeed photons have non-zero masses. Recently, Rañada proposed that due to variation of physical constants, there will be change of permittivity and permeability of quantum vacuum causing a change of refractive index of the vacuum. There should be an effect upon the rest mass of a photon as well, because such a vacuum can shift the frequency of a photon propagating through it.

Observations of astrophysical events at high redshifts can be used to place severe limits on the variation of the speed of light itself Δc/c, as well as on the photon mass m_γ. Schaefer presented limits on Δc/c < 6.3 × 10 and a limit on m_γ < 4.2 × 10g based upon the difference in arrival times on Earth of distant, explosive events that simultaneously emit radiation at multiple frequencies. A different experimental approach is to compare the energy level separations of atomic transitions in distant objects from those near at hand. At higher redshifts, a possible time dependence of α will be registered in the form of small shifts in the absorption line spectra seen in distant quasars because the energy of the atomic transitions depend on α. Interesting experimental observations using absorption systems in the spectra of distant quasars may suggest time evolution of the fine structure constant. An overview of time variation of fundamental constants is provided by Landau et al. Recent analysis of experimental data suggests −0.050 ≤ Δα/α ≤ 0.042, an experimental uncertainty that includes the possibility that α is constant. This limit applies to a change occurring in the time between today and the so-called recombination epoch when the Universe became cool enough for protons to capture electrons and form neutral hydrogen (5–10 billion years, or a redshift of z_* = 1078 ± 11). Laboratory measurements based upon precision comparisons of different atomic frequency standards over a period of a few years set a rate of variation as dℓn α /dt = (−0.26 ± 0.39) × 10 /yr.

See also

References

Footnotes

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5. Brian Greene (2003). The Elegant Universe. WW Norton & Co. p. 56. ISBN 0393058581.
6. PCW Davies (1979). The Forces of Nature. Cambridge University Press. p. 128. ISBN 052122523X.
7. Astronomers also use the parsec as a unit of length, which is unrelated to the speed of light.
8. P. Gibbs (1997). "Why is c the symbol for the speed of light?". University of California, Riverside. Retrieved 2008-08-20.
9. See, for example, some handbooks:
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10. ^ The International System of Units (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022, p. 112, ISBN 978-92-822-2272-0
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Historical references

• Rømer, Ole (1676). "Démonstration touchant le mouvement de la lumière". Journal des sçavans: 223–236. Template:Fr icon. 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", C. R. Acad. Sci. Paris, 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", Astrophys. J., 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)

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

Template:Link FA Template:Link FA Template:Link FA

• Light
• Units of velocity
• Special relativity
• Fundamental physics concepts