In classical physics, light is described as a type of electromagnetic wave. The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the electric constant ε0 and the magnetic constant μ0 by the equation c = 1/√ε0μ0.[45]
In modern quantum physics, the electromagnetic field is described by the theory of quantum electrodynamics (QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, called photons. In QED, photons are massless particles and thus, according to special relativity, they must travel at the speed of light.
Extensions of QED in which the photon has a mass have been considered. In such a theory, its speed would depend on its frequency, and the invariant speed c of special relativity would then be the upper limit of the speed of light in vacuum.[29] To date no such effects have been observed[46][47][48] putting stringent limits on the photon mass. The limit obtained depends on the used model: if the massive photon is described by Proca theory,[49] the experimental upper bound for its mass is about 10−57 grams.[50] If photon mass is generated by a Higgs mechanism, the experimental upper limit is less sharp, m ≤ 10−14 eV/c2 [49] (roughly 2 × 10−47 g).
In a medium
When light enters materials, its energy is absorbed. In the case of transparent materials, this energy is quickly re-radiated. However, this absorption and re-radiation introduces a delay. As light propagates through dielectric material it undergoes continuous absorption and re-radiation. Therefore when the speed of light in a medium is said to be less than c, this should be read as the speed of energy propagation at the macroscopic level. At an atomic level, electromagnetic waves always travel at c in the empty space between atoms. Two factors influence this slowing; stronger absorption leading to shorter path length between each re-radiation cycle and longer delays. The slowing is therefore the product of these two factors.[51] The refractive index of a transparent material is defined as the ratio of c to the speed of light v in the material. Larger indexes of refraction indicate smaller speeds. The refractive index of a material may depend on the light's frequency, intensity, polarization, or direction of propagation. In many cases, though, it can be treated as a material-dependent constant. The refractive index in air is approximately 1.0003.[52] Denser media, such as water and glass, have refractive indexes of around 1.3 and 1.5 respectively for visible light. Diamond has a refractive index of about 2.4.
If the refractive index of a material dependence on the frequency of the light passing through the medium, there exist two notions of speed of light in the medium. The first is speed of a wave of a single frequency f. This is called the phase velocity vp(f), and is related to (frequency dependent) refractive index n(f) by
The second is average velocity of a pulse of light consisting of different frequencies of light. This is the called the group velocity and not only depends on the properties of the medium but also distribution of frequencies in the pulse. A pulse with different group and phase velocities is said to undergo dispersion.
Certain materials have an exceptionally high group index and a correspondingly low group velocity for light waves, a phenomenon called slow light. In 1999, a team of scientists led by Lene Hau were able to slow the speed of a light pulse to about 17 metres per second (61 km/h; 38 mph);[53] in 2001, they were able to momentarily stop a beam.[54] In 2003, scientists at Harvard University and the Lebedev Physical Institute in Moscow, succeeded in completely halting light by directing it into a Bose–Einstein condensate of the element rubidium, the atoms of which, in Lukin's words, behaved "like tiny mirrors" due to an interference pattern in two "control" beams.[55][56]
It is also possible for the group velocity of light pulses to exceed c.[57][58] In an experiment in 2000, laser beams travelled for extremely short distances through caesium atoms with a group velocity of 300 times c.[59] It is not possible to transmit information faster than c by this means because the speed of information transfer cannot exceed the front velocity of the wave pulse, which is always less than c.[60]
Cherenkov radiation
It is possible for shock waves to be formed with electromagnetic radiation.[61][62] If a charged particle travels through an electrical insulator faster than the speed of light in that medium (but always slower than the speed of light in vacuum), then Cherenkov radiation, the electromagnetic analogue of a sonic boom, is emitted.
Practical effect of the finite speed of light
The speed of light plays an important part in many modern sciences and technologies. In electronic systems, despite their small size, the speed of light can become a limiting factor in their maximum speed of operation.[10]
Transit time
Radar systems measure the distance to a target by measuring the time taken for an echo of the light pulse to return. Similarly, a Global Positioning System (GPS) receiver measures its distance to satellites based on how long it takes for a radio signal to arrive from the satellite. The Lunar Laser Ranging Experiment, radar astronomy and the Deep Space Network determine the distances to the Moon, planets and spacecraft respectively by measuring the round-trip travel time.
The finite speed of light is particularly important in astronomy. Due to the vast distances involved it can take a very long time for light to travel from its source to Earth. For example, it takes 13 billion (13 × 109) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field images. Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when the universe was less than a billion years old. The fact that farther-away objects appear younger (due to the finite speed of light) is crucial in astronomy, allowing astronomers to infer the evolution of stars, of galaxies, and of the universe itself.
Astronomical distances are sometimes expressed in light-years, especially in popular science publications.[63] A light‑year is the distance light travels in one year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs. Next to the Sun, the closest star to Earth, Proxima Centauri, is around 4.2 light‑years away.[64]
Stellar aberration
Stellar aberration is the apparent motion of celestial objects about their real locations due to the finite speed of light and the motion of Earth. It was discovered and later explained by the third Astronomer Royal, James Bradley, in 1725.[65]
The figure to the left examines how light from a star (at location 1) travels down a telescope idealized as a narrow tube and moving at a speed v to the right. (This motion is largely due to the Earth as it orbits the Sun.) The light enters the tube from a star at angle θ and travels at speed c taking a time h/c to reach the bottom of the tube, where the light is detected by an observer. During the transit of the light, the tube moves a distance vh/c. Consequently, for the light ray to reach the bottom of the tube, where the tube must be inclined at an angle φ different from θ, resulting in an apparent position of the star at angle φ.
The maximum amount of the aberrational displacement of a star is approximately 20 arcseconds. Although this is a relatively small value, it was well within the observational capability of the instruments available in the early eighteenth century.
History
Ancient, medieval and early modern speculation
Until relatively recent times, it was not known whether light travelled instantaneously or at a finite speed. The first extant recorded examination of this subject was in ancient Greece. Empedocles maintained that light was something in motion, and therefore must take some time to travel. Aristotle argued, to the contrary, that "light is due to the presence of something, but it is not a movement".[66] Euclid and Ptolemy advanced the emission theory of vision, where light is emitted from the eye, thus enabling sight. Using that theory, Heron of Alexandria advanced the argument that the speed of light must be infinite, since distant objects such as stars appear immediately upon opening the eyes.
Early Islamic philosophers initially agreed with the Aristotelian view that light had no speed of travel. In 1021, Islamic physicist Alhazen (Ibn al-Haytham) published the Book of Optics, in which he used experiments related to the camera obscura to support the now accepted intromission theory of vision, where light moves from an object into the eye.[67] This led Alhazen to propose that light must therefore have a finite speed,[66][68][69] and that the speed of light is variable, decreasing in denser bodies.[69][70] He argued that light is a "substantial matter", the propagation of which requires time "even if this is hidden to our senses".[71]
Also in the 11th century, Abū Rayhān al-Bīrūnī agreed that light has a finite speed, and observed that the speed of light is much faster than the speed of sound.[72] Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle.[73][74] In the 1270s, Witelo considered the possibility of light travelling at infinite speed in a vacuum but slowing down in denser bodies.[75] A comment on a verse in the Rigveda by the 14th century Indian scholar Sayana mentioned a speed of light, about 186,400 miles per second, that was chosen so that light would encircle the Puranic universe in one day, making it "the most astonishing 'blind hit' in the history of science!"[76][77] In 1574, the Ottoman astronomer and physicist Taqi al-Din concluded that the speed of light is constant, but variable in denser bodies, and suggested that it would take a long time for light from the stars, which are very distant, to reach the Earth.[78]
In the early 17th century, Johannes Kepler believed that the speed of light was infinite since empty space presents no obstacle to it. René Descartes argued that if the speed of light were finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. Since such misalignment had not been observed, Descartes concluded the speed of light was infinite. Descartes speculated that if the speed of light were found to be finite, his whole system of philosophy might be demolished.[66]
First measurement attempts
In 1629, Isaac Beeckman proposed an experiment in which a person would observe the flash of a cannon reflecting off a mirror about one mile (1.6 km) away. In 1638, Galileo Galilei proposed an experiment, with an apparent claim to having performed it some years earlier, to measure the speed of light by observing the delay between uncovering a lantern and its perception some distance away. He was unable to distinguish whether light travel was instaneous or not, but concluded that if it weren't, it must nevertheless be extraordinarily rapid.[79][80] Galileo's experiment was carried out by the Accademia del Cimento of Florence in 1667, with the lanterns separated by about one mile, but no delay was observed. Based on the modern value of the speed of light, the actual delay in this experiment would be about 11 microseconds. Robert Hooke explained the negative results as Galileo had by pointing out that such observations did not establish the infinite speed of light, but only that the speed must be very great.
Early astronomical techniques
The first quantitative estimate of the speed of light was made in 1676 by Ole Christensen Rømer, one of a group of astronomers of the French Royal Academy of Sciences who were studying the motion of Jupiter's moons.[Note 8][81][82] From the observation that the periods of Jupiter's innermost moon Io appeared to be shorter when the earth was approaching Jupiter than when receding from Jupiter he concluded that light travels at a finite speed, and was able to estimate that would take light 22 minutes to cross the diameter of Earth's orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth's orbit to obtain an estimate of speed of light of 220,000 km/s, 26% lower than the actual value.[83]
Isaac Newton also accepted the finite speed. In his 1704 book Opticks he gives a value of "seven or eight minutes" for the time taken for light to travel from the Sun to the Earth (the modern value is 8 minutes 19 seconds).[84] The same effect was subsequently observed by Rømer for a "spot" rotating with the surface of Jupiter. Later observations also showed the effect with the three other Galilean moons, where it was more difficult to observe, thus laying to rest some further objections that had been raised.
Between 1725 and 1728, James Bradley, while searching for stellar parallax, observed the apparent motion of the star γ Draconis (Eltanin) depending on the season of the year. He realized that the motion (about 39 arcseconds) could not be a parallax (it was in the wrong direction at any given time) and, after ruling out several other possible causes, produced the theory of the aberration of light,[85] a vector addition of the velocity of light arriving from the star and the velocity of the Earth around its orbit. The effect is that an observer on the Earth will see the light coming from a slightly different angle than the "true" value which, for a star in the sky, means a slightly different position. The effect is greatest near the orbital pole which, for the Earth, is close to γ Draconis. Bradley was able to predict the aberration for several other stars, and confirm his predictions by observation.[85] His observations on γ Draconis gave a ratio of the speed of light to the mean linear speed of the Earth's orbital motion: Bradley's figure was that light travelled 10,210 times faster than the Earth in its orbit (the modern figure is 10,066 times faster) or, equivalently, that it would take light 8 minutes and 12 seconds to travel from the Sun to the Earth.[85]
Earth-bound techniques
The first successful entirely earthbound measurement of the speed of light was carried out by Hippolyte Fizeau in 1849. Fizeau's experiment was conceptually similar to those proposed by Beeckman and Galileo. A beam of light was directed at a mirror 8 km away. On the way from the source to the mirror, the beam passed through a rotating cog wheel. At a certain rate of rotation, the beam could pass through one gap on the way out and another on the way back. But at slightly higher or lower rates, the beam would strike a tooth and not pass through the wheel. Knowing the distance to the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light could be calculated. Fizeau reported the speed of light as 313,000 km/s. Léon Foucault improved on Fizeau's method by replacing the cogwheel with a rotating mirror. Foucault's estimate, published in 1862, was 298,000 km/s.
In 1879, Albert Michelson performed a similar experiment at the U.S. Naval Academy. He measured the speed of light in air to be 299,864±51 kilometres per second, and estimated the speed of light in vacuum as 299,940 km/s, or 186,380 mps.[86]
In 1887, Michelson and Edward Morley performed an experiment to detect differences in the speed of light due to the Earth's motion through the luminiferous ether, at what is now Case Western Reserve University.[87] Its failure is generally considered to be the first strong evidence against the ether theory.
Cavity resonance
During World War II, the development of the cavity resonance wavemeter for use in radar, together with precision timing methods, opened the way to laboratory-based measurements of the speed of light. In 1946, Louis Essen and A.C. Gordon-Smith used a microwave cavity of precisely known dimensions to establish the frequency for a variety of normal modes of microwaves. As the wavelength of the modes was known from the geometry of the cavity and from electromagnetic theory, knowledge of the associated frequencies enabled a calculation of the speed of light.
The Essen–Gordon-Smith result, 299,792±3 km/s, was substantially more precise than those found by optical techniques, and prompted much controversy. However, by 1950 repeated measurements by Essen established a result of 299,792.5±1.0 km/s, which became the value adopted by the 12th General Assembly of the Radio-Scientific Union in 1957.
Heterodyne laser measurements
An alternative to the cavity resonator method to find the wavelength for determining the speed of light is to use a form of interferometer, indicated schematically in the figure.[88] A coherent light beam with a known frequency (f), as from a laser, is split to follow two paths and then recombined. By carefully changing the path length and observing the interference pattern, the wavelength of the light (λ) can be determined, which is related to the speed of light by the equation c = λf.
The main problem with interferometry is to measure the frequency of light in or near the optical region. This was first overcome by a group at the NIST laboratories in Boulder, Colorado, in 1972.[89] By a series of photodiodes and specially constructed metal–insulator–metal diodes, they succeeded in linking the frequency of the caesium transition used in atomic clocks to the frequency of a methane-stabilized laser (nearly 10,000 times higher).[90] Their results were
- f = 88.376181627(50) THz
- λ = 3.392231376(12) μm
- c = 299792456.2(1.1) m/s
nearly a hundred times more precise than previous measurements of the speed of light.
Redefinition of the metre
The 1972 measurement of the speed of light, with a relative uncertainty of 4×10−9, was not only a feat of experimental precision, it also demonstrated a fundamental limit to how precisely the speed of light could be measured at that time using any technique. The remaining uncertainty in the value was almost completely attributable to uncertainty in the length of the metre.[89][90][91]
Since 1960, the metre had been defined as a given number of wavelengths of the light of one of the spectral lines of a krypton lamp,[Note 9] but it turned out that the chosen spectral line was not perfectly symmetrical.[90] This gave an uncertainty in its wavelength, and hence in the length of the metre. By analogy with a metal measuring stick, it was as if the stick were slightly fuzzy at each end, although if it were a real measuring stick, the fuzziness at the ends of a one-metre stick would only be apparent at the atomic scale.
To get round this problem, the 15th Conférence Générale des Poids et Mesures (CGPM) in 1975 recommended the use of the value 299,792,458 m/s for "the speed of propagation of electromagnetic waves in vacuum".[91] The 17th CGPM in 1983 decided to redefine the metre to be "the length of the path travelled by light in vacuum during a time interval of 1⁄299,792,458 of a second".[93]
The effect of this definition gives the speed of light the exact value 299,792,458 m/s, which is nearly the same as the value 299,792,456.2(1.1) m/s obtained in the 1972 experiment. This number was chosen so that the change in the actual length of the metre was minimised, being similar to the measurement uncertainty.[94][95] As a result, within the SI system of units, the speed of light is now a defined constant[14] and no longer something to be measured.[90] Improved experimental techniques do not affect the value of the speed of light in SI units, but do result in a more precise realisation of the SI metre.[96][97]
Rather than measure a time-of-flight, one implementation of this definition is to use a recommended source with established frequency f, and delineate the metre in terms of the wavelength λ of this light as determined using the defined numerical value of c and the relationship λ = c / f.[98] Practical realisations of the metre use recommended wavelengths of visible light in a laboratory vacuum with corrections being applied to take account of actual conditions such as diffraction, gravitation or imperfection in the vacuum.[99][100]
Modern astronomical measurements
The overriding problem with any modern measurement of the speed of light (c) is the definition of a precise standard of length. For practical length measurements on Earth, c is the length standard, through the 1983 definition of the metre, but it is still possible to define other standards and hence to measure c against those standards.
In astronomy and satellite communication, it is useful to use standards based on the mass of either the Sun or the Earth. This is transformed into a length standard by saying that the standard length is the distance from the centre of the body at which a planet or satellite would have a given orbital velocity. The method was first used by Carl Friedrich Gauss in 1801 to calculate the orbit of Ceres, and was refined by Simon Newcomb in his Tables of the Sun (1895).
The astronomical unit is one example of such a length standard, based on the solar mass and approximately equal to the average distance between the Earth and the Sun. The "light time per unit distance" is an essential parameter in calculating planetary ephemerides, and is simply the inverse of c in astronomical units per second. It is measured by comparing the time taken for radio signals to reach different spacecraft in the Solar System with their position as caculated from the gravitational effects of the Sun and the various planets. By combining many such measurements, a "best fit" value for the light time per unit distance can be obtained. The 2009 best estimate, as approved by the International Astronomical Union (IAU), is:[101][102][103]
- light time per unit distance: 499.004783836(10) s
- c = 0.00200398880410(4) AU/s = 173.144632674(3) AU/day
The relative uncertainty in these measurements is 0.02 parts per billion (0.02 × 10−9), equivalent to the uncertainty in Earth-based measurements of length by interferometry.
The light time per unit distance is effectively the same quantity that was measured by Rømer and Cassini in the late 17th century, where they gave a value of "ten to eleven minutes",[106] slightly longer than the currently accepted value of 8 minutes 19 seconds.
Laboratory demonstration
With modern electronics, particularly oscilloscopes with time resolutions of less than one nanosecond, the speed of light can now be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror, although this method is less precise than either the cavity resonator or the interferometric methods
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