Speed of light

"Lightspeed" redirects here. For other uses, see Lightspeed (disambiguation).

For other uses, see Speed of light (disambiguation).

Here, laser light in air is traveling at 99.97% the speed of light in a vacuum.

The term speed of light is used to denote two related but distinct quantities:

The speed of light in free space is a physical constant defined as 299,792,458 metres per second. It is often denoted by the symbol c - originating from the Latin word celeritās (speed). The speed of light was officially included in the International System of Units on October 21, 1983, leaving the metre defined in terms of both c and the second.

The speed of light in a material is the speed that light travels in that material and may be very different than c.

Description

Speed of light in different units
metres per second	299,792,458 (exact)
km per hour	1,079,252,848.8 (exact)
miles per hour	≈ 670,616,629.4
miles per second	≈ 186,282.397
Approximate length of time for light to travel...
One foot	1.0 nanosecond
One metre	3.3 nanoseconds
One km	3.3 microseconds
One mile	5.4 microseconds
Around Earth's equator	0.13 seconds
From Earth to geostationary orbit and back	0.24 seconds
From Earth to the moon	1.3 seconds
From Earth to the sun	8.3 minutes
To Earth from Alpha Centauri	4.4 years
From edge to edge of the Milky Way	100,000 years

As light propagates down the telescope, the telescope moves requiring a tilt to the telescope that depends on the speed of light. The apparent angle of the star φ differs from its true angle θ, a phenomenon called stellar aberration

Light is a type of electromagnetic radiation. According to electromagnetic theory, all electromagnetic radiation travels in free space at the speed of light in free space: 299,792,458 m/s. This applies regardless of the light's color, brightness, or direction. Experimentally, no variations have been found in the speed of light in any realizable approximation to free space. Laboratory measurements show that light of different colors travels at the same speed to within one part in 10¹⁴. Other experiments show the variation as the universe ages is less than two parts in 10¹⁶/year, for both microwaves and visible light.

For many practical purposes, the speed of light is so great that it can be regarded to travel instantaneously. However, the finite speed of light becomes noticeable when very long distances, very short time intervals, or precise time measurements are involved. For instance, the speed of light is a critical factor in astronomy, modern electronics, and navigation systems such as global positioning systems.

Notation and units

The symbol "c" for "constant" or the Latin celeritas (meaning “swiftness”) is used for the speed of light in free space, and in this article c is used exclusively this way. Some authors, however, use c for the speed of light in any material media. 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.

Practical impacts of the finite speed of light

The speed of light plays an important part in many modern sciences and technologies. 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 distances to the moon, planets, and spacecraft are determined by measuring the round-trip travel time.

Another effect of the finite speed of light is stellar aberration. Suppose we look at a star with a telescope idealized as a narrow tube. The light enters the tube from a star at angle θ and travels at speed c taking a time a time h/c to reach the bottom of the tube, where our eye detects the light. Suppose observations are made from Earth, which is moving with a speed v. During the transit of the light, the tube moves a distance vh/c. Consequently, for the photon to reach the bottom of the tube, the tube must be inclined at an angle φ different from θ , resulting in an apparent position of the star at angle φ.

In astronomy beyond the solar system, distances are often measured in light-years, the distance light travels in a year.

In electronic systems, despite their small size, the speed of light can become a limiting factor in their maximum speed of operation.

Speed of light in various systems

Light in free space

Main article: Free space

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 c₀²=1/(ε₀ μ₀) .

According to classical electromagnetism, the speed of electromagnetic radiation in free space is the same for all frequencies. That is, light in free space exhibits zero dispersion. The speed of light in free space is polarization independent and isotropic.

Free space is a reference state. Like absolute zero, it is an idealized state that only can be approximated in the physical world. Measurements in any real-world medium, such as air or a medium perturbed by gravity), must account for the medium's refractive index to relate them to the reference standard of free space.

Light in realizable vacuum

Although free space is an idealized reference state unattainable in practice, some media accurately approximate free space, such as outer space and terrestrial ultra high vacuum.

A simple model often used to represent free-space-like vacuum is one where electric permittivity and magnetic permeability are constants have the values ε₀ and μ₀. In this classical model, the speed of light 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.

Quantum electrodynamic theory predicts deviations from a unitary refractive index in the vacuum state for extremely strong electromagnetic fields. To date, there has been no experimental confirmation of that effect.

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 in outer space.

Light in transparent media

See also: Dispersion (optics), Refractive index, Snell's law, and Permittivity

File:PrismAndLight.jpg

The refractive index of a transparent material indicates how much slower light propagates in that medium than in a vacuum. This slowing causes refraction. 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, as demonstrated by this prism (in the case of a prism splitting white light into a spectrum of colours, the refraction 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:

$n={\sqrt {\epsilon _{r}\mu _{r}}}=c_{0}/v_{p}$

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. If the light passing through the medium is monochromatic, the phase velocity is often referred to as the "speed of light".

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 refractive index in air is only slightly larger than one . Denser media, such as water and glass, have refractive indices of between 1.3 and 1.5 for visible light. Diamond has a refractive index of about 2.4. This reduction in speed is also responsible for bending of light at an interface between two materials with different refractive indices, a phenomenon known as refraction.

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.

In 1861 Maxwell proposed a theory which linked the speed of light to the electromagnetic field. In the early 20th century c assumed an even greater importance as a pivotal constant in Einstein's theory of special relativity, which holds that the speed of light has a special role connecting space and time in the structure of spacetime. As one result, the speed of light sets an absolute speed limit to how fast matter or information can move. As another result, energy and mass are connected by the speed of light in the famous mass-energy equation E = mc² underlying nuclear energy. General relativity goes further to explain gravity as an effect of curved space time and, of course, reduces locally to the special theory with speed of light c. Today, the speed of light continues to be a subject of research, for example, in cosmology and quantum gravity.

Light travels very rapidly by everyday standards - light travels roughly a million times faster than sound, and can circle the Earth more than 7 times in one second. Such a rapid speed is very hard to measure without specialized techniques, and in ancient times 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 were progressively refined. With modern technology the speed of light can be measured so accurately that it is now used to define other standards of measurements. For example, in 1983 the metre was re-defined to be the distance light travels in a perfect vacuum in 1 ⁄ 299,792,458 of a second. (Previously, it was defined as the distance between two scratches on a specific metal bar kept in a vault in Paris). As a result, the speed of light in vacuum is now exactly 299,792,458 metres per second.

Measurements of the speed of light

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

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, 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:

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

In comparison the modern value is about 10,100.

Around the same time, the astronomical unit (roughly, the Earth-to-Sun distance) 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.

In 1864 Maxwell refers to three measurements of c: Fizeau (314,858,000 m/s) and Foucault (298,000,000 m/s) based on light in air; and a value based on reception of light on Earth at different positions in its orbit (stellar aberration) (308,000,000 m/s). He compared these values with the ratio of esu to emu units by Weber and Kohlrausch (310,740,000 m/s) to support the connection between light and electromagnetic phenomena.

Earth-bound techniques

Diagram of the Fizeau apparatus.

The first successful entirely earthbound measurement of the speed of light 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):

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

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):

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

in order for the light to pass through.

Michelson–Morley experiment and the 'luminiferous aether'

Main articles: Aether theories, Luminiferous aether, and Michelson–Morley experiment

Interference pattern produced with a Michelson interferometer using a mercury lamp

A variety of possible interference patterns

A schematic representation of a Michelson interferometer, as used for the Michelson-Morley experiment. The point of reflection on the tilted beam splitter is shown as two separated points for clarity. Both beams travel equal length paths. If light speed is anisotropic, the interference pattern seen at the detector will vary with orientation of the apparatus.

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 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. Some interesting details of these experiments are found in Hollberg et al. 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.

The null result also led Lorentz to propose that motion through the aether contracts the Michelson interferometer due to Fitzgerald-Lorentz contraction, and to incorporate this effect into what now are called the Lorentz transformations, which play an important role in the mathematics 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 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.

Other laboratory-based methods

Electromagnetic standing waves in a cavity. Sinusoidal waves at the top have larger frequencies than below and are shifted upward for clarity. The walls require nodes in the waves, so only multiple half- wavelengths λ/2 fit: 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 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. In the special case of electromagnetic waves moving through vacuum:

$c={\lambda }\,f\ ,$

where λ = wavelength and f = frequency. In the example shown in the figure, the theory insists that the waves have nodes at the cavity boundaries. Consequently, the width of the cavity is some multiple of a half-wavelength of a supported mode. To make a measurement, one approach is to make the cavity dimension precisely controllable to within a fractional wavelength so the wave can be seen to appear and disappear as the dimension is adjusted.

The Essen-Gordon-Smith 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.

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); Bottom: Destructive interference (out of phase). (The source is symbolized as a light bulb, which would not work. However, any coherent source would work, for example, a laser, or monochromatic light from an emission line of a mercury-vapor lamp passed through a pinhole.)

An alternative to the cavity resonator method to find the wavelength λ for determining c is to use a form of interferometer, indicated schematically in the figure. A coherent light beam (symbolized as a light bulb, but really a laser) is split to follow two paths and then recombined. Top: If the difference in path length is a multiple of a wavelength, the recombined beams support one another and reconstitute the original beam. Bottom: 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. Thus, by carefully changing the path length and observing the interference pattern, the wavelength of the light can be determined. For a striking animation of some patterns, see vibrations.

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 (although not of state-of-the-art accuracy) 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 such a 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. The realization of "vacuum" is obtained by measurement in a real medium (which may simply be a controlled volume of air) and employing various corrections to reduce the measurements to what they are expected to be in ideal vacuum. The assessment and nature of these corrections is administered by international standards organizations such as NIST and the BIPM. 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.

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.

Although this definition sets an exact value of 299,792,458 metres per second in the hypothetical medium "vacuum" or free space, that leaves experiment to answer the question of how closely any realizable physical vacuum approximates the ideal of free space.

An exact value for the speed of light in ideal vacuum as 299 792 458 m/s combined with the definition of the second as 9 192 631 770 periods of a particular cesium 133 atomic transition implies a definition of the meter in terms of the relation:

$\lambda ={\frac {c}{f}}\ =c\,T\ ,$

(with the period T = 1 / f), which means the interferometer can be used to establish a standard meter, rather than following the time-of-flight approach seemingly implied by the official definition. Of course, one must be able to establish how one's measurements in the laboratory vacuum relate to ideal vacuum and, more significantly, establish the accuracy of the laboratory realization of the frequency. In the optical range of frequencies, the frequency is assessed using an optical frequency comb.

Speed of light in scientific fields

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

Doppler shift

Main article: 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

Main article: Spacetime

Although at speeds small compared to c a runner in a train appears to a ground observer to travel at a speed that is the sum of the train and the runner speeds, a moving light source propagates light that travels the same speed whether observed from the train or from the ground. The light frequency changes, however.

According to relativity, space and time are viewed as a four dimensional unification of space and time, known as spacetime, in which c plays the fundamental role of a conversion factor between space and time within spacetime, and between mass and energy. The ideas and observations leading to this theory are tied to an understanding of the role of the speed of light.

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, which immediately impacts the synchronization of clocks and the notion of simultaneity. This postulate, together with the principle of relativity that all inertial frames are equivalent, forms the basis of Einstein's theory of special relativity.

In Einstein's general theory of relativity, spacetime is curved by the presence of matter and energy causing gravitation. The propagation of disturbances in this curvature, including gravitational waves, theoretically propagate at or near the speed of light The failure of experimental observation of pulsars to detect gravitational waves suggests any difference in speed is less than 0.4%.

Speed of light in astronomy

The relative sizes and separation of the Earth–Moon system are shown to scale above. The beam of light is depicted traveling between the Earth and the Moon in the same time it actually takes light to scale the real distance between them: 1.255 seconds at its mean orbital distance (surface to surface). The light beam helps provide the sense of scale of the Earth-Moon system relative to the Sun, which is 8.28 light-minutes away (photosphere to Earth surface).

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.

Speed of light and cosmology

See also: Primordial fluctuations, Cosmological constant, and Lambda-CDM model

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. According to Ellis and Uzan:

The possibility that the fundamental constants may vary during the evolution of the universe offers an exceptional window onto higher dimensional theories and is probably linked with the nature of the dark energy that makes the universe accelerate today.

A change with time of the speed of light also affects the fine structure constant:

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

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 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  :

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

• G is the gravitational constant;
• $\hbar$ (pronounced "h-bar") $\ =h/2\pi$ 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 × 10⁻²¹ and a limit on m_γ < 4.2 × 10⁻⁴⁴g 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) × 10⁻¹⁵ /yr.

Speed of light as the maximum speed of information transfer

Causality and information transfer

See also: Causal contact and Horizon problem

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, which frame would then become a frame privileged above all others. 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.

Faster-than-light observations and experiments

Main article: Faster-than-light

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

Only zero-rest mass particles can travel at the speed of light. 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 some interpretations of 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.

Another prediction of faster-than-light speeds occurs for tunneling and is called the Hartman effect. However, no information can be sent using these effects. One ‘‘bit’’ of information is received when a detector has received a sufficient number of photons to be sufficiently sure that an on-bit rather than an off-bit was received.

Curved space time

Quantum field theory predicts an apparent superluminal propagation of photons due to vacuum polarization. This prediction raises the question of whether causality may be violated by quantum effects in curved spacetime. This matter is a subject of ongoing research.

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

See also: Metric expansion of space, Cosmological principle, Observable universe, and Cosmological horizon

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:

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

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. One view is that this speed does not contradict special relativity because each observer is the center of their own Hubble sphere, so the motion occurs outside any particular observer's inertial frame. A different explanation is that the "velocity" calculated this way does not correspond to a velocity seen in any single inertial frame, but is concatenated from distances observed in an infinite sequence of local inertial frames between the observer and the object (there are no global inertial frames), and special relativity refers to observations made in a single inertial frame, not an assembly of such frames.

We can see such objects because 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.

In more detail, 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.

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

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.

See also

• Electromagnetic wave equation
• Mathematical descriptions of the electromagnetic field
• Maxwell's equations

• Quantum electrodynamics
• Sinusoidal plane-wave solutions of the electromagnetic wave equation

References

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