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Speed of light
The distance from the Sun to Earth is shown as 150 million kilometres, an approximate average. Sizes to scale.
Due to its finite speed, sunlight takes 8 minutes and 10 to 27 seconds to reach Earth, depending on the time of year.[1]
Exact value
metres per second299792458
Approximate values (to three significant digits)
kilometres per hour1080000000
miles per second186000
miles per hour[2]671000000
astronomical units per day173[Note 1]
parsecs per year0.307[Note 2]
Approximate light signal travel times
DistanceTime
one foot1.0 ns
one metre3.3 ns
from geostationary orbit to Earth119 ms
the length of Earth's equator134 ms
from Moon to Earth1.3 s
from Sun to Earth (1 AU)8.3 min
one light-year1.0 year
one parsec3.26 years
from the nearest star to Sun (1.3 pc)4.2 years
from the nearest galaxy to Earth70000 years
across the Milky Way87400 years
from the Andromeda Galaxy to Earth2.5 million years
Special relativity
The world line: a diagrammatic representation of spacetime
Foundations
Consequences
Spacetime
Dynamics
People

The speed of light in vacuum, commonly denoted c, is a universal physical constant exactly equal to 299,792,458 metres per second (approximately 1 billion kilometres per hour; 700 million miles per hour). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1⁄299792458 second. The speed of light is the same for all observers, no matter their relative velocity. It is the upper limit for the speed at which information, matter, or energy can travel through space.[3][4][5]

All forms of electromagnetic radiation, including visible light, travel at the speed of light. For many practical purposes, light and other electromagnetic waves will appear to propagate instantaneously, but for long distances and sensitive measurements, their finite speed has noticeable effects. Much starlight viewed on Earth is from the distant past, allowing humans to study the history of the universe by viewing distant objects. When communicating with distant space probes, it can take hours for signals to travel. In computing, the speed of light fixes the ultimate minimum communication delay. The speed of light can be used in time of flight measurements to measure large distances to extremely high precision.

Ole Rømer first demonstrated that light does not travel instantaneously by studying the apparent motion of Jupiter's moon Io. In an 1865 paper, James Clerk Maxwell proposed that light was an electromagnetic wave and, therefore, travelled at speed c.[6] Albert Einstein postulated that the speed of light c with respect to any inertial frame of reference is a constant and is independent of the motion of the light source.[7] He explored the consequences of that postulate by deriving the theory of relativity, and so showed that the parameter c had relevance outside of the context of light and electromagnetism.

Massless particles and field perturbations, such as gravitational waves, also travel at speed c in vacuum. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. Particles with nonzero rest mass can be accelerated to approach c but can never reach it, regardless of the frame of reference in which their speed is measured. In the theory of relativity, c interrelates space and time and appears in the famous mass–energy equivalence, E = mc2.[8]

In some cases, objects or waves may appear to travel faster than light. The expansion of the universe is understood to exceed the speed of light beyond a certain boundary. The speed at which light propagates through transparent materials, such as glass or air, is less than c; similarly, the speed of electromagnetic waves in wire cables is slower than c. The ratio between c and the speed v at which light travels in a material is called the refractive index n of the material (n = ⁠c/v⁠). For example, for visible light, the refractive index of glass is typically around 1.5, meaning that light in glass travels at ⁠c/1.5⁠ ≈ 200000 km/s (124000 mi/s); the refractive index of air for visible light is about 1.0003, so the speed of light in air is about 90 km/s (56 mi/s) slower than c.

Numerical value, notation, and units

[edit]

The speed of light in vacuum is usually denoted by a lowercase c. The origin of the letter choice is unclear, with guesses including "c" for "constant" or the Latin celeritas (meaning 'swiftness, celerity').[9] The "c" was used for "celerity" meaning a velocity in books by Leonhard Euler and others, but this velocity was not specifically for light; Isaac Asimov wrote a popular science article, "C for Celeritas", but did not explain the origin.[10] In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant that was later shown to equal √2 times the speed of light in vacuum. Historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1903, Max Abraham used c with its modern meaning in a widely read textbook on electromagnetism. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, which by then had become the standard symbol for the speed of light.[11][9]

Sometimes c is used for the speed of waves in any material medium, and c0 for the speed of light in vacuum.[12] This subscripted notation, which is endorsed in official SI literature,[13] has the same form as related electromagnetic constants: namely, μ0 for the vacuum permeability or magnetic constant, ε0 for the vacuum permittivity or electric constant, and Z0 for the impedance of free space. This article uses c exclusively for the speed of light in vacuum.

Use in unit systems

[edit]
Further information: Metre § Speed of light definition

Since 1983, the constant c has been defined in the International System of Units (SI) as exactly 299792458 m/s; this relationship is used to define the metre as exactly the distance that light travels in vacuum in 1⁄299792458 of a second. The second is, in turn, defined to be the length of time occupied by 9192631770 cycles of the radiation emitted by a caesium-133 atom in a transition between two specified energy states.[14] By using the value of c, as well as an accurate measurement of the second, one can establish a standard for the metre.[15]

The particular value chosen for the speed of light provided a more accurate definition of the metre that still agreed as much as possible with the definition used before 1983.[14][16]

As a dimensional physical constant, the numerical value of c is different for different unit systems. For example, in imperial units, the speed of light is approximately 186282 miles per second,[Note 3] or roughly 1 foot per nanosecond.[Note 4][17][18]

In branches of physics in which c appears often, such as in relativity, it is common to use systems of natural units of measurement or the geometrized unit system where c = 1.[19][20] Using these units, c does not appear explicitly because multiplication or division by 1 does not affect the result. Its unit of light-second per second is still relevant, even if omitted.

Fundamental role in physics

[edit]
See also: Special relativity

The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of the observer. This invariance of the speed of light was postulated by Einstein in 1905,[7] after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for motion against the luminiferous aether.[21] Experiments such as the Kennedy–Thorndike experiment and the Ives–Stilwell experiment have shown this postulate to match experimental observations.[22]

The special theory of relativity explores the consequences of this invariance of c with the assumption that the laws of physics are the same in all inertial frames of reference.[23][24] One consequence is that c is the speed at which all massless particles and waves, including light, must travel in vacuum.[25]

γ starts at 1 when v equals zero and stays nearly constant for small v, then it sharply curves upwards and has a vertical asymptote, diverging to positive infinity as v approaches c.
The Lorentz factor γ as a function of velocity. It starts at 1 and approaches infinity as v approaches c.

Special relativity has many counterintuitive and experimentally verified implications.[26] These include the equivalence of mass and energy (E = mc2), length contraction (moving objects shorten), Terrell rotation (apparent rotation),[27][28] and time dilation (moving clocks run more slowly). The factor γ by which lengths contract and times dilate is known as the Lorentz factor and is given by γ = (1 − v2/c2)−1/2, where v is the speed of the object. The difference of γ from 1 is negligible for speeds much slower than c, such as most everyday speeds – in which case special relativity is closely approximated by Galilean relativity – but it increases at relativistic speeds and diverges to infinity as v approaches c. For example, a time dilation factor of γ = 2 occurs at a relative velocity of 86.6% of the speed of light (v = 0.866c). Similarly, a time dilation factor of γ = 10 occurs at 99.5% the speed of light (v = 0.995c).

The results of special relativity can be summarized by treating space and time as a unified structure known as spacetime (with c relating the units of space and time), and requiring that physical theories satisfy a special symmetry called Lorentz invariance, whose mathematical formulation contains the parameter c.[29] Lorentz invariance is an almost universal assumption for modern physical theories, such as quantum electrodynamics, quantum chromodynamics, the Standard Model of particle physics, and general relativity. As such, the parameter c is ubiquitous in modern physics, appearing in many contexts that are unrelated to light. For example, general relativity predicts that c is also the speed of gravity and of gravitational waves,[30] and observations of gravitational waves have been consistent with this prediction.[31] In non-inertial frames of reference (gravitationally curved spacetime or accelerated reference frames), the local speed of light is constant and equal to c, but the speed of light can differ from c when measured from a remote frame of reference, depending on how measurements are extrapolated to the region.[32]

It is generally assumed that fundamental constants such as c have the same value throughout spacetime, meaning that they do not depend on location and do not vary with time. However, it has been suggested in various theories that the speed of light may have changed over time.[33][34] No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.[35][36]

It is generally assumed that the two-way speed of light is isotropic, meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear energy levels as a function of the orientation of the emitting nuclei in a magnetic field (see Hughes–Drever experiment), and of rotating optical resonators (see Resonator experiments) have put stringent limits on the possible two-way anisotropy.[37][38]

Upper limit on speeds

[edit]

An object with rest mass m and speed v relative to a laboratory has kinetic energy (γ − 1)mc2 with respect to that lab, where γ is the Lorentz factor defined above. The γ factor approaches infinity as v approaches c, and it would take an infinite amount of energy to accelerate an object with mass to the speed of light.[39]: 13.3  The speed of light is the upper limit for the speeds of objects with positive rest mass. Analysis of individual photons confirm that information cannot travel faster than the speed of light.[40][41] This is experimentally established in many tests of relativistic energy and momentum.[42]

Three pairs of coordinate axes are depicted with the same origin A; in the green frame, the x-axis is horizontal and the ct-axis is vertical; in the red frame, the x′-axis is slightly skewed upwards, and the ct′-axis slightly skewed rightwards, relative to the green axes; in the blue frame, the x″-axis is somewhat skewed downwards, and the ct″-axis somewhat skewed leftwards, relative to the green axes. A point B on the green x-axis, to the left of A, has zero ct, positive ct′, and negative ct″.
Event A precedes B in the red frame, is simultaneous with B in the green frame, and follows B in the blue frame.

More generally, it is impossible for signals or energy to travel faster than c. One argument for this is known as causality. If the spatial distance between two events A and B is greater than the time interval between them multiplied by c then there are frames of reference in which A precedes B, others in which B precedes A, and others in which they are simultaneous. As a result, if something were travelling faster than c relative to an inertial frame of reference, it would be travelling backwards in time relative to another frame, and causality would be violated.[43]: 497 [44][45] In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded,[46] and would lead to paradoxes such as the tachyonic antitelephone.[47]

In some theoretical treatments, the Scharnhorst effect allows signals to travel faster than c, by one part in 1036.[48] However other approaches to the same physical set up show no such effect.[49] and it appears the special conditions in which this effect might occur would prevent one from using it to violate causality.

One-way speed of light

[edit]
Main article: One-way speed of light

It is only possible to verify experimentally that the two-way speed of light (for example, from a source to a mirror and back again) is frame-independent, because it is impossible 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 at the detector should be synchronized. By adopting Einstein synchronization for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition.[50][46]

Faster-than-light observations and experiments

[edit]
See also: Faster-than-light and Superluminal motion

There are situations in which it may seem that matter, energy, or information-carrying signal travels at speeds greater than c, but they do not. For example, as is discussed in the propagation of light in a medium section below, many wave velocities can exceed c. The phase velocity of X-rays through most glasses can routinely exceed c,[51] but phase velocity does not determine the velocity at which waves convey information.[52]

If a laser beam is swept quickly across a distant object, the spot of light can move faster than c, although the initial movement of the spot is delayed because of the time it takes light to get to the distant object at the speed c. However, the only physical entities that are moving are the laser and its emitted light, which travels at the speed c from the laser to the various positions of the spot. Similarly, a shadow projected onto a distant object can be made to move faster than c, after a delay in time.[53] In neither case does any matter, energy, or information travel faster than light.[54]

The rate of change in the distance between two objects in a frame of reference with respect to which both are moving (their closing speed) may have a value in excess of c. However, this does not represent the speed of any single object as measured in a single inertial frame.[54]

Certain quantum effects appear to be transmitted instantaneously and therefore faster than c, as in the EPR paradox. An example involves the quantum states of two particles that can be entangled. Until either of the particles is observed, they exist in a superposition of two quantum states. If the particles are separated and one particle's quantum state is observed, the other particle's quantum state is determined instantaneously. However, it is impossible to control which quantum state the first particle will take on when it is observed, so information cannot be transmitted in this manner.[54][55]

Another quantum effect that predicts the occurrence of faster-than-light speeds is called the Hartman effect: under certain conditions the time needed for a virtual particle to tunnel through a barrier is constant, regardless of the thickness of the barrier.[56][57] This could result in a virtual particle crossing a large gap faster than light. However, no information can be sent using this effect.[58]

So-called superluminal motion is seen in certain astronomical objects,[59] such as the relativistic 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 approaching Earth at a small angle to the line of sight: since the light which was emitted when the jet was farther away took longer to reach the Earth, the time between two successive observations corresponds to a longer time between the instants at which the light rays were emitted.[60]

A 2011 experiment where neutrinos were observed to travel faster than light turned out to be due to experimental error.[61][62]

In models of the expanding universe, the farther galaxies are from each other, the faster they drift apart. For example, galaxies far away from Earth are inferred to be moving away from the Earth with speeds proportional to their distances. Beyond a boundary called the Hubble sphere, the rate at which their distance from Earth increases becomes greater than the speed of light.[63] These recession rates, defined as the increase in proper distance per cosmological time, are not velocities in a relativistic sense. Faster-than-light cosmological recession speeds are only a coordinate artifact.

Propagation of light

[edit]

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 in vacuum is related to the distributed capacitance and inductance of vacuum, otherwise respectively known as the electric constant ε0 and the magnetic constant μ0, by the equation[64]

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 travel at the speed of light in vacuum.[25]

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.[32] No variation of the speed of light with frequency has been observed in rigorous testing, putting stringent limits on the mass of the photon.[65] The limit obtained depends on the model used: if the massive photon is described by Proca theory,[66] the experimental upper bound for its mass is about ×10−57 grams;[67] if photon mass is generated by a Higgs mechanism, the experimental upper limit is less sharp, m ≤ 10−14 eV/c2 (roughly 2×10−47 g).[66]

Another reason for the speed of light to vary with its frequency would be the failure of special relativity to apply to arbitrarily small scales, as predicted by some proposed theories of quantum gravity. In 2009, the observation of gamma-ray burst GRB 090510 found no evidence for a dependence of photon speed on energy, supporting tight constraints in specific models of spacetime quantization on how this speed is affected by photon energy for energies approaching the Planck scale.[68]

In a medium

[edit]
See also: Refractive index

In a medium, light usually does not propagate at a speed equal to c; further, different types of light wave will travel at different speeds. The speed at which the individual crests and troughs of a plane wave (a wave filling the whole space, with only one frequency) propagate is called the phase velocity vp. A physical signal with a finite extent (a pulse of light) travels at a different speed. The overall envelope of the pulse travels at the group velocity vg, and its earliest part travels at the front velocity vf.[69]

A modulated wave moves from left to right. There are three points marked with a dot: A blue dot at a node of the carrier wave, a green dot at the maximum of the envelope, and a red dot at the front of the envelope.
The blue dot moves at the speed of the ripples, the phase velocity; the green dot moves with the speed of the envelope, the group velocity; and the red dot moves with the speed of the foremost part of the pulse, the front velocity.

The phase velocity is important in determining how a light wave travels through a material or from one material to another. It is often represented in terms of a refractive index. The refractive index of a material is defined as the ratio of c to the phase velocity vp in the material: larger indices of refraction indicate lower 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 of air is approximately 1.0003.[70] Denser media, such as water,[71] glass,[72] and diamond,[73] have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light.

In exotic materials like Bose–Einstein condensates near absolute zero, the effective speed of light may be only a few metres per second. However, this represents absorption and re-radiation delay between atoms, as do all slower-than-c speeds in material substances. As an extreme example of light "slowing" in matter, two independent teams of physicists claimed to bring light to a "complete standstill" by passing it through a Bose–Einstein condensate of the element rubidium. The popular description of light being "stopped" in these experiments refers only to light being stored in the excited states of atoms, then re-emitted at an arbitrarily later time, as stimulated by a second laser pulse. During the time it had "stopped", it had ceased to be light. This type of behaviour is generally microscopically true of all transparent media which "slow" the speed of light.[74]

In transparent materials, the refractive index generally is greater than 1, meaning that the phase velocity is less than c. In other materials, it is possible for the refractive index to become smaller than 1 for some frequencies; in some exotic materials it is even possible for the index of refraction to become negative.[75] The requirement that causality is not violated implies that the real and imaginary parts of the dielectric constant of any material, corresponding respectively to the index of refraction and to the attenuation coefficient, are linked by the Kramers–Kronig relations.[76][77] In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly.[78]

A pulse with different group and phase velocities (which occurs if the phase velocity is not the same for all the frequencies of the pulse) smears out over time, a process known as dispersion. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called slow light.[79] The opposite, group velocities exceeding c, was proposed theoretically in 1993 and achieved experimentally in 2000.[80] It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.[69]

None of these options allow information to be transmitted faster than c. It is impossible to transmit information with a light pulse any faster than the speed of the earliest part of the pulse (the front velocity). It can be shown that this is (under certain assumptions) always equal to c.[69]

It is possible for a particle to travel through a medium faster than the phase velocity of light in that medium (but still slower than c). When a charged particle does that in a dielectric material, the electromagnetic equivalent of a shock wave, known as Cherenkov radiation, is emitted.[81]

Practical effects of finiteness

[edit]

The speed of light is of relevance to telecommunications: the one-way and round-trip delay time are greater than zero. This applies from small to astronomical scales. On the other hand, some techniques depend on the finite speed of light, for example in distance measurements.

Small scales

[edit]

In computers, the speed of light imposes a limit on how quickly data can be sent between processors. If a processor operates at 1 gigahertz, a signal can travel only a maximum of about 30 centimetres (1 ft) in a single clock cycle – in practice, this distance is even shorter since the printed circuit board refracts and slows down signals. Processors must therefore be placed close to each other, as well as memory chips, to minimize communication latencies, and care must be exercised when routing wires between them to ensure signal integrity. If clock frequencies continue to increase, the speed of light may eventually become a limiting factor for the internal design of single chips.[82][83]

Large distances on Earth

[edit]
Acoustic representation of the speed of light: in the period between beeps, light travels the circumference of Earth at the equator.

Given that the equatorial circumference of the Earth is about 40075 km and that c is about 300000 km/s, the theoretical shortest time for a piece of information to travel half the globe along the surface is about 67 milliseconds. When light is traveling in optical fibre (a transparent material) the actual transit time is longer, in part because the speed of light is slower by about 35% in optical fibre with an refractive index n around 1.52.[84] Straight lines are rare in global communications and the travel time increases when signals pass through electronic switches or signal regenerators.[85]

Although this distance is largely irrelevant for most applications, latency becomes important in fields such as high-frequency trading, where traders seek to gain minute advantages by delivering their trades to exchanges fractions of a second ahead of other traders. For example, traders have been switching to microwave communications between trading hubs, because of the advantage which radio waves travelling at near to the speed of light through air have over comparatively slower fibre optic signals.[86][87]

Spaceflight and astronomy

[edit]
The diameter of the moon is about one quarter of that of Earth, and their distance is about thirty times the diameter of Earth. A beam of light starts from the Earth and reaches the Moon in about a second and a quarter.
A beam of light is depicted travelling between the Earth and the Moon in the time it takes a light pulse to move between them: 1.255 seconds at their mean orbital (surface-to-surface) distance. The relative sizes and separation of the Earth–Moon system are shown to scale.

Similarly, communications between the Earth and spacecraft are not instantaneous. There is a brief delay from the source to the receiver, which becomes more noticeable as distances increase. This delay was significant for communications between ground control and Apollo 8 when it became the first crewed spacecraft to orbit the Moon: for every question, the ground control station had to wait at least three seconds for the answer to arrive.[88]

The communications delay between Earth and Mars can vary between five and twenty minutes depending upon the relative positions of the two planets. As a consequence of this, if a robot on the surface of Mars were to encounter a problem, its human controllers would not be aware of it until approximately 4–24 minutes later. It would then take a further 4–24 minutes for commands to travel from Earth to Mars.[89][90]

Receiving light and other signals from distant astronomical sources takes much longer. 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.[91][92] 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.[91] The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars, of galaxies, and of the universe itself.[93]

Astronomical distances are sometimes expressed in light-years, especially in popular science publications and media.[94] A light-year is the distance light travels in one Julian year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs. In round figures, a light year is nearly 10 trillion kilometres or nearly 6 trillion miles. Proxima Centauri, the closest star to Earth after the Sun, is around 4.2 light-years away.[95]

Distance measurement

[edit]
Main article: Distance measurement

Radar systems measure the distance to a target by the time it takes a radio-wave pulse to return to the radar antenna after being reflected by the target: the distance to the target is half the round-trip transit time multiplied by the speed of light. A Global Positioning System (GPS) receiver measures its distance to GPS satellites based on how long it takes for a radio signal to arrive from each satellite, and from these distances calculates the receiver's position. Because light travels about 300000 kilometres (186000 miles) in one second, these measurements of small fractions of a second must be very precise. The Lunar Laser Ranging experiment, radar astronomy and the Deep Space Network determine distances to the Moon,[96] planets[97] and spacecraft,[98] respectively, by measuring round-trip transit times.

Measurement

[edit]

There are different ways to determine the value of c. One way is to measure the actual speed at which light waves propagate, which can be done in various astronomical and Earth-based setups. It is also possible to determine c from other physical laws where it appears, for example, by determining the values of the electromagnetic constants ε0 and μ0 and using their relation to c. Historically, the most accurate results have been obtained by separately determining the frequency and wavelength of a light beam, with their product equalling c. This is described in more detail in the "Interferometry" section below.

In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of 1⁄299792458 of a second",[99] fixing the value of the speed of light at 299792458 m/s by definition, as described below. Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value of c.

Astronomical measurements

[edit]
Measurement of the speed of light from the time it takes Io to orbit Jupiter, using eclipses of Io by Jupiter's shadow to precisely measure its orbit.

Outer space is a convenient setting for measuring the speed of light because of its large scale and nearly perfect vacuum. Typically, one measures the time needed for light to traverse some reference distance in the Solar System, such as the radius of the Earth's orbit. Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units.

Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676.[100][101] When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it. The difference is small, but the cumulative time becomes significant when measured over months. The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the diameter of the Earth's orbit around the Sun. The observed change in the moon's orbital period is caused by the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect for Jupiter's innermost major moon Io and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.[100]

A star emits a light ray that hits the objective of a telescope. While the light travels down the telescope to its eyepiece, the telescope moves to the right. For the light to stay inside the telescope, the telescope must be tilted to the right, causing the distant source to appear at a different location to the right.
Aberration of light: light from a distant source appears to be from a different location for a moving telescope due to the finite speed of light.

Another method is to use the aberration of light, discovered and explained by James Bradley in the 18th century.[102] This effect results from the vector addition of the velocity of light arriving from a distant source (such as a star) and the velocity of its observer (see diagram on the right). A moving observer thus sees the light coming from a slightly different direction and consequently sees the source at a position shifted from its original position. Since the direction of the Earth's velocity changes continuously as the Earth orbits the Sun, this effect causes the apparent position of stars to move around. From the angular difference in the position of stars (maximally 20.5 arcseconds)[103] it is possible to express the speed of light in terms of the Earth's velocity around the Sun, which with the known length of a year can be converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled 10210 times faster than the Earth in its orbit (the modern figure is 10066 times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.[102]

Astronomical unit

[edit]
Main article: Astronomical unit

Historically the speed of light was used together with timing measurements to determine a value for the astronomical unit (AU).[104] It was redefined in 2012 as exactly 149597870700 m.[105][106] This redefinition is analogous to that of the metre and likewise has the effect of fixing the speed of light to an exact value in astronomical units per second (via the exact speed of light in metres per second).[107]

Time of flight techniques

[edit]
One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930–1935 experiment used a rotating mirror and a one-mile (1.6 km) long vacuum chamber which the light beam traversed 10 times. It achieved accuracy of ±11 km/s.
A light ray passes horizontally through a half-mirror and a rotating cog wheel, is reflected back by a mirror, passes through the cog wheel, and is reflected by the half-mirror into a monocular.
Diagram of the Fizeau apparatus:
  1. Light source
  2. Beam-splitting semi-transparent mirror
  3. Toothed wheel-breaker of the light beam
  4. Remote mirror
  5. Telescopic tube

A method of measuring the speed of light is to measure the time needed for light to travel to a mirror at a known distance and back. This is the working principle behind experiments by Hippolyte Fizeau and Léon Foucault.

The setup as used by Fizeau consists of a beam of light directed at a mirror 8 kilometres (5 mi) away. On the way from the source to the mirror, the beam passes through a rotating cogwheel. At a certain rate of rotation, the beam passes through one gap on the way out and another on the way back, but at slightly higher or lower rates, the beam strikes a tooth and does not pass through the wheel. Knowing the distance between the wheel and the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light can be calculated.[108]

The method of Foucault replaces the cogwheel with a rotating mirror. Because the mirror keeps rotating while the light travels to the distant mirror and back, the light is reflected from the rotating mirror at a different angle on its way out than it is on its way back. From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated.[109] Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion by François Arago.[110]

Today, using oscilloscopes with time resolutions of less than one nanosecond, the speed of light can be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror. This method is less precise (with errors of the order of 1%) than other modern techniques, but it is sometimes used as a laboratory experiment in college physics classes.[111]

Electromagnetic constants

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An option for deriving c that does not directly depend on a measurement of the propagation of electromagnetic waves is to use the relation between c and the vacuum permittivity ε0 and vacuum permeability μ0 established by Maxwell's theory: c2 = 1/(ε0μ0). The vacuum permittivity may be determined by measuring the capacitance and dimensions of a capacitor, whereas the value of the vacuum permeability was historically fixed at exactly 4π×10−7 H⋅m−1 through the definition of the ampere. Rosa and Dorsey used this method in 1907 to find a value of 299710±22 km/s. Their method depended upon having a standard unit of electrical resistance, the "international ohm", and so its accuracy was limited by how this standard was defined.[112][113]

Cavity resonance

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A box with three waves in it; there are one and a half wavelength of the top wave, one of the middle one, and a half of the bottom one.
Electromagnetic standing waves in a cavity

Another way to measure the speed of light is to independently measure the frequency f and wavelength λ of an electromagnetic wave in vacuum. The value of c can then be found by using the relation c = λf. One option is to measure the resonance frequency of a cavity resonator. If the dimensions of the resonance cavity are also known, these can be used to determine the wavelength of the wave. In 1946, Louis Essen and A. C. Gordon-Smith established the frequency for a variety of normal modes of microwaves of a microwave cavity of precisely known dimensions. The dimensions were established to an accuracy of about ±0.8 μm using gauges calibrated by interferometry.[112] 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.[112][114]

The Essen–Gordon-Smith result, 299792±9 km/s, was substantially more precise than those found by optical techniques.[112] By 1950, repeated measurements by Essen established a result of 299792.5±3.0 km/s.[115]

A household demonstration of this technique is possible, using a microwave oven and food such as marshmallows or margarine: if the turntable is removed so that the food does not move, it will cook the fastest at the antinodes (the points at which the wave amplitude is the greatest), where it will begin to melt. The distance between two such spots is half the wavelength of the microwaves; by measuring this distance and multiplying the wavelength by the microwave frequency (usually displayed on the back of the oven, typically 2450 MHz), the value of c can be calculated, "often with less than 5% error".[116][117]

Interferometry

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Schematic of the working of a Michelson interferometer.
An interferometric determination of length. Left: constructive interference; Right: destructive interference.

Interferometry is another method to find the wavelength of electromagnetic radiation for determining the speed of light.[118] A coherent beam of light (e.g. from a laser), with a known frequency f, is split to follow two paths and then recombined. By adjusting the path length while observing the interference pattern and carefully measuring the change in path length, the wavelength of the light λ can be determined. The speed of light is then calculated using the equation c = λf.

Before the advent of laser technology, coherent radio sources were used for interferometry measurements of the speed of light.[119] Interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~4 mm [0.16 in]) of the radiowaves. The precision can be improved by using light with a shorter wavelength, but then it becomes difficult to directly measure the frequency of the light.[120]

One way around this problem is to start with a low frequency signal of which the frequency can be precisely measured, and from this signal progressively synthesize higher frequency signals whose frequency can then be linked to the original signal. A laser can then be locked to the frequency, and its wavelength can be determined using interferometry.[120] This technique was due to a group at the National Bureau of Standards (which later became the National Institute of Standards and Technology). They used it in 1972 to measure the speed of light in vacuum with a fractional uncertainty of 3.5×10−9.[120][121]

History

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Until the early modern period, it was not known whether light travelled instantaneously or at a very fast finite speed. The first extant recorded examination of this subject was in ancient Greece. The ancient Greeks, Arabic scholars, and classical European scientists long debated this until Rømer provided the first calculation of the speed of light. Einstein's theory of special relativity postulates that the speed of light is constant regardless of one's frame of reference. Since then, scientists have provided increasingly accurate measurements.

History of measurements of c (in m/s)
Year Experiment Value Deviation from 1983 value
<1638 Galileo, covered lanterns inconclusive[122][123][124]: 1252 
<1667 Accademia del Cimento, covered lanterns inconclusive[124]: 1253 [125]
1675 Rømer and Huygens, moons of Jupiter 220000000[101][126] −27%
1729 James Bradley, aberration of light 301000000[108] +0.40%
1849 Hippolyte Fizeau, toothed wheel 315000000[108] +5.1%
1862 Léon Foucault, rotating mirror 298000000±500000[108] −0.60%
1875 Werner Siemens 260 000 000[127] −13.3%
1893 Heinrich Hertz 200 000 000[128] −33.3%
1907 Rosa and Dorsey, EM constants 299710000±30000[112][113] −280 ppm
1926 Albert A. Michelson, rotating mirror 299796000±4000[129] +12 ppm
1950 Essen and Gordon-Smith, cavity resonator 299792500±3000[115] +0.14 ppm
1958 K. D. Froome, radio interferometry 299792500±100[119] +0.14 ppm
1972 Evenson et al., laser interferometry 299792456.2±1.1[121] −0.006 ppm
1983 17th CGPM, definition of the metre 299792458 (exact)[99] —

Early history

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Empedocles (c. 490–430 BCE) was the first to propose a theory of light[130] and claimed that light has a finite speed.[131] He 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".[132] Euclid and Ptolemy advanced Empedocles' emission theory of vision, where light is emitted from the eye, thus enabling sight. Based on that theory, Heron of Alexandria argued that the speed of light must be infinite because distant objects such as stars appear immediately upon opening the eyes.[133]

Early Islamic philosophers initially agreed with the Aristotelian view that light had no speed of travel. In 1021, Alhazen (Ibn al-Haytham) published the Book of Optics, in which he presented a series of arguments dismissing the emission theory of vision in favour of the now accepted intromission theory, in which light moves from an object into the eye.[134] This led Alhazen to propose that light must have a finite speed,[132][135][136] and that the speed of light is variable, decreasing in denser bodies.[136][137] He argued that light is substantial matter, the propagation of which requires time, even if this is hidden from the senses.[138] 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.[139]

In the 13th century, Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle.[140][141] In the 1270s, Witelo considered the possibility of light travelling at infinite speed in vacuum, but slowing down in denser bodies.[142]

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 to be finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. Although this argument fails when aberration of light is taken into account, the latter was not recognized until the following century.[143] 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.[132] Despite this, in his derivation of Snell's law, Descartes assumed that some kind of motion associated with light was faster in denser media.[144][145] Pierre de Fermat derived Snell's law using the opposing assumption, the denser the medium the slower light travelled. Fermat also argued in support of a finite speed of light.[146]

First measurement attempts

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In 1629, Isaac Beeckman proposed an experiment in which a person observes 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 instantaneous or not, but concluded that if it were not, it must nevertheless be extraordinarily rapid.[122][123] According to Galileo, the lanterns he used were "at a short distance, less than a mile". Assuming the distance was not too much shorter than a mile, and that "about a thirtieth of a second is the minimum time interval distinguishable by the unaided eye", Boyer notes that Galileo's experiment could at best be said to have established a lower limit of about 60 miles per second for the velocity of light.[123] In 1667, the Accademia del Cimento of Florence reported that it had performed Galileo's experiment, with the lanterns separated by about one mile, but no delay was observed.[147] The actual delay in this experiment would have been about 11 microseconds.

A diagram of a planet's orbit around the Sun and of a moon's orbit around another planet. The shadow of the latter planet is shaded.
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 Rømer.[100][101] 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 it, he concluded that light travels at a finite speed, and estimated that it takes 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 220000 km/s, which is 27% lower than the actual value.[126]

In his 1704 book Opticks, Isaac Newton reported Rømer's calculations of the finite speed of light and gave 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).[148] Newton queried whether Rømer's eclipse shadows were coloured. Hearing that they were not, he concluded the different colours travelled at the same speed. In 1729, James Bradley discovered stellar aberration.[102] From this effect he determined that light must travel 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 12 seconds to travel from the Sun to the Earth.[102]

Connections with electromagnetism

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See also: History of electromagnetic theory and History of special relativity

In the 19th century Hippolyte Fizeau developed a method to determine the speed of light based on time-of-flight measurements on Earth and reported a value of 315000 km/s.[149] His method was improved upon by Léon Foucault who obtained a value of 298000 km/s in 1862.[108] In the year 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch measured the ratio of the electromagnetic and electrostatic units of charge, 1/√ε0μ0, by discharging a Leyden jar, and found that its numerical value was very close to the speed of light as measured directly by Fizeau. The following year Gustav Kirchhoff calculated that an electric signal in a resistanceless wire travels along the wire at this speed.[150]

In the early 1860s, Maxwell showed that, according to the theory of electromagnetism he was working on, electromagnetic waves propagate in empty space[151] at a speed equal to the above Weber/Kohlrausch ratio, and drawing attention to the numerical proximity of this value to the speed of light as measured by Fizeau, he proposed that light is in fact an electromagnetic wave.[152] Maxwell backed up his claim with his own experiment published in the 1868 Philosophical Transactions which determined the ratio of the electrostatic and electromagnetic units of electricity.[153]

"Luminiferous aether"

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Main article: Luminiferous aether

The wave properties of light were well known since Thomas Young. In the 19th century, physicists believed light was propagating in a medium called aether (or ether). After Maxwell's theory unified light and electric and magnetic waves, it was favored that both light and electric magnetic waves propagate in the same aether medium (or called the luminiferous aether).[154]

Hendrik Lorentz (right) with Albert Einstein (1921)

Some physicists thought that this aether acted as a preferred frame of reference for the propagation of light and therefore it should be possible to measure the motion of the Earth with respect to this medium, by measuring the isotropy of the speed of light. Beginning in the 1880s several experiments were performed to try to detect this motion, the most famous of which is the experiment performed by Albert A. Michelson and Edward W. Morley in 1887.[155][156] The detected motion was found to always be nil (within observational error). Modern experiments indicate that the two-way speed of light is isotropic (the same in every direction) to within 6 nanometres per second.[157]

Because of Michelson-Morley experiment Hendrik Lorentz proposed that the motion of the apparatus through the aether may cause the apparatus to contract along its length in the direction of motion, and he further assumed that the time variable for moving systems must also be changed accordingly ("local time"), which led to the formulation of the Lorentz transformation. Based on Lorentz's aether theory, Henri Poincaré (1900) showed that this local time (to first order in v/c) is indicated by clocks moving in the aether, which are synchronized under the assumption of constant light speed. In 1904, he speculated that the speed of light could be a limiting velocity in dynamics, provided that the assumptions of Lorentz's theory are all confirmed. In 1905, Poincaré brought Lorentz's aether theory into full observational agreement with the principle of relativity.[158][159]

Special relativity

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In 1905 Einstein postulated from the outset that the speed of light in vacuum, measured by a non-accelerating observer, is independent of the motion of the source or observer. Using this and the principle of relativity as a basis he derived the special theory of relativity, in which the speed of light in vacuum c featured as a fundamental constant, also appearing in contexts unrelated to light. This made the concept of the stationary aether (to which Lorentz and Poincaré still adhered) useless and revolutionized the concepts of space and time.[160][161]

Increased accuracy of c and redefinition of the metre and second

[edit]
See also: History of the metre

In the second half of the 20th century, much progress was made in increasing the accuracy of measurements of the speed of light, first by cavity resonance techniques and later by laser interferometer techniques. These were aided by new, more precise, definitions of the metre and second. In 1950, Louis Essen determined the speed as 299792.5±3.0 km/s, using cavity resonance.[115] This value was adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. In 1960, the metre was redefined in terms of the wavelength of a particular spectral line of krypton-86, and, in 1967, the second was redefined in terms of the hyperfine transition frequency of the ground state of caesium-133.[162]

In 1972, using the laser interferometer method and the new definitions, a group at the US National Bureau of Standards in Boulder, Colorado determined the speed of light in vacuum to be c = 299792456.2±1.1 m/s. This was 100 times less uncertain than the previously accepted value. The remaining uncertainty was mainly related to the definition of the metre.[163][121] As similar experiments found comparable results for c, the 15th General Conference on Weights and Measures in 1975 recommended using the value 299792458 m/s for the speed of light.[164]

Defined as an explicit constant

[edit]

In 1983 the 17th meeting of the General Conference on Weights and Measures (CGPM) found that wavelengths from frequency measurements and a given value for the speed of light are more reproducible than the previous standard. They kept the 1967 definition of second, so the caesium hyperfine frequency would now determine both the second and the metre. To do this, they redefined the metre as "the length of the path traveled by light in vacuum during a time interval of 1/299792458 of a second".[99]

As a result of this definition, the value of the speed of light in vacuum is exactly 299792458 m/s[165][166] and has become a defined constant in the SI system of units.[15] Improved experimental techniques that, prior to 1983, would have measured the speed of light no longer affect the known value of the speed of light in SI units, but instead allow a more precise realization of the metre by more accurately measuring the wavelength of krypton-86 and other light sources.[167][168]

In 2011, the CGPM stated its intention to redefine all seven SI base units using what it calls "the explicit-constant formulation", where each "unit is defined indirectly by specifying explicitly an exact value for a well-recognized fundamental constant", as was done for the speed of light. It proposed a new, but completely equivalent, wording of the metre's definition: "The metre, symbol m, is the unit of length; its magnitude is set by fixing the numerical value of the speed of light in vacuum to be equal to exactly 299792458 when it is expressed in the SI unit m s−1."[169] This was one of the changes that was incorporated in the 2019 revision of the SI, also termed the New SI.[170]

See also

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Notes

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References

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Further reading

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Speed of light

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The speed of light in vacuum, commonly denoted as c, is a fundamental physical constant that represents the propagation speed of electromagnetic radiation, including visible light, in empty space, fixed exactly at 299,792,458 meters per second (approximately 186,282 miles per second or 983,571,056 feet per second, more precisely 186,282.397 miles per second or 983,571,056.43 feet per second, exactly 1,079,252,848.8 kilometers per hour (often approximated as 1.08 billion km/h or 1.079 × 10^9 km/h)) since its definition in 1983 to establish the meter as the distance light travels in vacuum during 1/299,792,458 of a second.[1][2] This value is invariant, independent of the motion of the source or observer, and serves as the universal speed limit for matter, energy, and information transfer in the cosmos, as no particle or signal can exceed it according to established physics.[3][4] First estimated in 1676 by Danish astronomer Ole Rømer through observations of Jupiter's moon Io, which revealed delays in eclipses attributable to light's finite travel time across Earth's orbit, the speed of light was later refined via terrestrial experiments, such as Hippolyte Fizeau's 1849 toothed-wheel method yielding approximately 313,000 km/s and Léon Foucault's 1862 rotating-mirror apparatus measuring about 298,000 km/s in air.[5][6][7] By the late 19th century, electromagnetic theory by James Clerk Maxwell predicted c as the ratio of electromagnetic constants, aligning with measurements and confirming light as an electromagnetic wave.[8] In the 20th century, laser interferometry and cavity resonator techniques achieved precision to within parts per billion, culminating in the 1983 redefinition that eliminated measurement uncertainty for c itself.[9] Central to Albert Einstein's 1905 theory of special relativity, c underpins the equivalence of mass and energy via E = mc² and the relativity of simultaneity, time dilation, and length contraction for objects approaching this speed, while general relativity extends its role in spacetime curvature and gravitational effects on light paths.[3][10] In modern physics, c appears in quantum field theory, cosmology (e.g., horizon distances), and engineering applications like GPS, where relativistic corrections for satellite clocks are essential, and it remains a cornerstone constant in the International System of Units (SI).[11][12]

Definition and Numerical Value

Exact Value and Notation

The speed of light in vacuum, commonly denoted by the symbol $ c $, is a fundamental physical constant with an exact value of 299792458 meters per second (approximately 186,282 miles per second or 983,571,056 feet per second, more precisely 186,282.397 miles per second or 983,571,056.43 feet per second).[13][14] This precise numerical value was established by the 17th Conférence Générale des Poids et Mesures (CGPM) in 1983, which redefined the metre as the distance traveled by light in vacuum during a time interval of 1/299792458 of a second, thereby fixing $ c $ exactly as part of the International System of Units (SI).[15] Prior to this redefinition, $ c $ was determined through experimental measurements that carried uncertainties, but the 1983 convention assigned zero uncertainty to its value, making it a defining constant of the SI.[16] The notation $ c $ specifically refers to the speed of light in vacuum, an invariant universal constant, and is distinguished from other wave propagation speeds such as phase velocity ($ v_p )orgroupvelocity() or group velocity ( v_g $), which can differ in contexts involving dispersive media or materials.[17] In classical electromagnetism, $ c $ arises directly from Maxwell's equations as the propagation speed of electromagnetic waves in vacuum. The relevant vacuum equations (in SI units, with no charges or currents) are Faraday's law,
∇×E=−∂B∂t, \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},
and Ampère's law with Maxwell's displacement current correction,
∇×B=μ0ϵ0∂E∂t, \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t},
where $ \mathbf{E} $ is the electric field, $ \mathbf{B} $ is the magnetic field, $ \epsilon_0 $ is the vacuum permittivity, and $ \mu_0 $ is the vacuum permeability.[18] To derive the wave nature, take the curl of Faraday's law:
∇×(∇×E)=−∂∂t(∇×B)=−μ0ϵ0∂2E∂t2. \nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) = -\mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}.
Using the vector identity $ \nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} $ and noting that $ \nabla \cdot \mathbf{E} = 0 $ in vacuum (from Gauss's law), this simplifies to the wave equation
∇2E=μ0ϵ0∂2E∂t2. \nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}.
A similar equation holds for $ \mathbf{B} $. The general solution describes waves propagating at speed $ v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} $, which matches the measured speed of light, identifying electromagnetic waves as light and yielding $ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} $.[18][19]

Units and Constants

In the International System of Units (SI), the speed of light in vacuum, denoted cc, plays a defining role in establishing the metre as the base unit of length. The metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299 792 4581/299\,792\,458 of a second. This definition, adopted by the 17th General Conference on Weights and Measures in 1983 and retained in the 2019 revision of the SI, fixes the numerical value of cc at exactly 299,792,458 metres per second (approximately 186,282 miles per second or 983,571,056 feet per second, more precisely 186,282.397 miles per second or 983,571,056.43 feet per second, exactly 1,079,252,848.8 kilometres per hour (often approximated as 1.08 billion km/h or 1.079 × 10^9 km/h)). As a result, the metre derives its realization from the second—the SI base unit of time, defined by the frequency of the caesium-133 hyperfine transition—and cc, ensuring the metric system's internal consistency and universality without reliance on physical artefacts.[20][21] In natural unit systems prevalent in theoretical and particle physics, cc is conventionally set to 1, simplifying equations by equating units of length and time. This choice, often combined with the reduced Planck's constant ℏ=1\hbar = 1, expresses physical quantities like energy and momentum in units of inverse length (e.g., electronvolts or GeV), facilitating calculations in quantum field theory and relativity. Planck units extend this approach by incorporating the gravitational constant GG, setting c=1c = 1, ℏ=1\hbar = 1, and G=1G = 1 to define scales where quantum mechanics, gravity, and relativity intersect, such as the Planck length (≈1.616×10−35\approx 1.616 \times 10^{-35} m). These systems highlight cc's role as a conversion factor between spatial and temporal dimensions, underscoring the spacetime unity in modern physics.[22][23] The speed of light is integral to the fine-structure constant α\alpha, a dimensionless fundamental constant that measures the electromagnetic force's strength. Expressed as
α=e24πϵ0ℏc, \alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c},
where ee is the elementary charge, ϵ0\epsilon_0 is the vacuum permittivity, and ℏ\hbar is the reduced Planck's constant, α\alpha emerges as a pure number because the dimensions of its components cancel: e2/(4πϵ0)e^2 / (4\pi \epsilon_0) has units of action times velocity, which ℏc\hbar c matches. With a value of approximately 1/137.0359991771/137.035999177 (relative uncertainty 1.6×10−101.6 \times 10^{-10}), α\alpha governs atomic spectra fine structure and quantum electrodynamics processes, with cc's inclusion reflecting the theory's relativistic foundations.[24][25] For practical applications outside SI, cc is often expressed in context-specific units. In engineering, particularly for electromagnetic signals, c≈0.983571c \approx 0.983571 feet per nanosecond (corresponding to approximately 983,571,056 feet per second), commonly approximated as 1 foot per nanosecond to estimate propagation delays in circuits and cables. Astronomically, light traverses approximately 173 AU per day, derived from the 499-second travel time across one astronomical unit (the mean Earth-Sun distance), emphasizing cc's scale in solar system dynamics. These representations aid intuitive understanding without altering cc's fundamental value.[26][27]

Fundamental Role in Physics

Invariance in Relativity

One of the foundational postulates of special relativity, as formulated by Albert Einstein, states that the speed of light in vacuum, denoted cc, is constant and the same for all inertial observers, irrespective of the motion of the light source or the observer.[28] This invariance implies that classical notions of absolute time and space must be abandoned, leading to the relativity of simultaneity and the unification of space and time into spacetime.[28] To reconcile this postulate with the principle of relativity—which asserts that the laws of physics are identical in all inertial frames—Einstein derived the Lorentz transformations, which replace the Galilean transformations of classical mechanics.[28] Consider two inertial frames SS and S′S', where S′S' moves at velocity vv along the xx-axis relative to SS. The transformations for coordinates and time are:
x′=γ(x−vt),y′=y,z′=z,t′=γ(t−vxc2), x' = \gamma (x - vt), \quad y' = y, \quad z' = z, \quad t' = \gamma \left(t - \frac{vx}{c^2}\right),
where γ=11−v2c2\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} is the Lorentz factor.[28] These equations ensure that if a light pulse emitted at the origin of SS at t=0t=0 satisfies x=ctx = ct (and similarly for other directions), then in S′S' it satisfies x′=ct′x' = ct', preserving the invariance of cc.[28] The invariance of cc directly implies time dilation and length contraction. For time dilation, imagine a light clock in S′S' where two mirrors separated by proper distance L0L_0 reflect a light pulse back and forth; the proper time Δτ\Delta \tau for one tick is Δτ=2L0c\Delta \tau = \frac{2L_0}{c}.[28] In SS, the light path elongates due to the frame's motion, yielding dilated time Δt=γΔτ\Delta t = \gamma \Delta \tau.[28] Similarly, length contraction follows from the requirement that speeds measured in SS match those in S′S'; a rod of proper length L0L_0 parallel to the motion appears contracted to L=L0γL = \frac{L_0}{\gamma} in SS.[28] These effects arise solely from enforcing cc's constancy across frames. Hermann Minkowski reformulated special relativity in 1908 using four-dimensional spacetime, where the invariant interval is
ds2=c2dt2−dx2−dy2−dz2. ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.
This metric remains unchanged under Lorentz transformations, highlighting cc's role as a conversion factor between time and space units.[29] Events connected by null intervals (ds=0ds = 0) lie on light cones, which define the causal structure: future-directed light cones bound the region where signals can propagate at or below cc, ensuring no faster-than-light causation.[29] Worldlines of massive particles remain within these cones, while light rays trace the cone surfaces. Experimental support for cc's invariance includes the 1887 Michelson-Morley experiment, which sought to detect Earth's motion through a hypothetical luminiferous ether by measuring light speed differences in perpendicular directions but yielded a null result, with the observed shift less than one-fortieth of the expected ether drift.[30] In the context of special relativity, this isotropy confirms that light speed is independent of the observer's velocity relative to any medium, providing indirect evidence for the postulate.[28] Subsequent tests, such as Kennedy-Thorndike experiments, further validated the Lorentz transformations by ruling out alternative explanations involving variable cc.[28]

Causal Limit on Speeds

In special relativity, the speed of light in vacuum, denoted as cc, serves as the ultimate speed limit for any object with nonzero rest mass. Accelerating a massive particle to approach cc requires progressively greater energy input, as described by the relativistic total energy formula E=γmc2E = \gamma m c^2, where mm is the rest mass and γ=11−v2c2\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} is the Lorentz factor. As the particle's speed vv nears cc, γ\gamma diverges to infinity, implying that an infinite amount of energy would be needed to reach or exceed cc. This prohibition arises directly from the Lorentz transformations and the invariance of cc, ensuring that no massive object can attain superluminal speeds.[31] Hypothetical particles known as tachyons, which would always travel faster than cc with imaginary rest mass (m2<0m^2 < 0), have been proposed but face significant theoretical challenges. Introduced in the context of quantum field theory, tachyons would require a spacelike four-momentum, leading to issues with Lorentz invariance and particle creation processes. Moreover, their presence in a quantum field theory typically signals an unstable vacuum state, where the field's potential has a maximum rather than a minimum, causing spontaneous decay and rendering the theory unphysical for stable systems. No experimental evidence for tachyons exists, and their incorporation into consistent frameworks remains problematic.[32][33] The causal limit imposed by cc is essential for preserving the principle of causality, which dictates that cause precedes effect in all inertial frames. Superluminal signaling would allow information to propagate along spacelike paths, potentially enabling paradoxes such as sending messages backward in time relative to some observers, including the formation of closed timelike curves in curved spacetime extensions of relativity. By restricting information transfer to speeds at or below cc, special relativity ensures a consistent ordering of events without such violations, aligning with the observed unidirectional flow of cause and effect.[31][34] This limit manifests in the relativistic velocity addition formula, which prevents speeds from combining to exceed cc. For two objects moving at speeds uu and v′v' relative to an observer, the combined speed vv in the observer's frame is given by
v=u+v′1+uv′c2. v = \frac{u + v'}{1 + \frac{u v'}{c^2}}.
Even if both uu and v′v' approach cc, vv asymptotes to cc but never surpasses it, reinforcing the universal prohibition on superluminal motion for causal propagation.[31]

Propagation and Variations

In Vacuum

In vacuum, light propagates as an electromagnetic wave, consisting of mutually perpendicular oscillating electric and magnetic fields that are transverse to the direction of travel. The speed of light cc in this medium relates the wave's frequency ff and vacuum wavelength λ\lambda through the fundamental relation c=fλc = f \lambda, which holds for all electromagnetic radiation regardless of frequency.[35] This propagation speed is isotropic, meaning cc remains constant and identical in all directions for any inertial observer, independent of the source's or observer's motion—a postulate central to special relativity that resolves apparent asymmetries in classical electrodynamics.[28] James Clerk Maxwell's equations in vacuum predict electromagnetic wave propagation via the derived wave equation for the electric field E\mathbf{E}:
∇2E−1c2∂2E∂t2=0, \nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0,
where c=1ϵ0μ0c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}, with ϵ0\epsilon_0 as the electric permittivity and μ0\mu_0 as the magnetic permeability of vacuum; this derivation unifies electricity, magnetism, and light as manifestations of the same phenomenon. This expression for cc is analogous to the speed of mechanical waves v=stiffnessdensityv = \sqrt{\frac{\text{stiffness}}{\text{density}}}, where "stiffness" corresponds to the resistance of free space to changes in the electric field (1/ϵ01/\epsilon_0), and "heaviness" or inertia corresponds to the resistance to changes in the magnetic field (μ0\mu_0). The high value of cc in vacuum thus results from its high electric stiffness (low ϵ0\epsilon_0) and low magnetic heaviness (low μ0\mu_0).[36] Physically, vacuum denotes the absence of matter, enabling unimpeded wave travel at cc, though quantum field theory describes it as permeated by fleeting virtual particle-antiparticle pairs from zero-point fluctuations; these do not alter light's speed but manifest in effects like the Casimir force, an attraction between nearby uncharged conductors arising from restricted fluctuation modes between them.[37]

In Media and Effective Speeds

When light propagates through a material medium, its speed $ v $ is reduced relative to the vacuum speed $ c $, quantified by the refractive index $ n = \frac{c}{v} $.[38] This reduction arises from the interaction of the electromagnetic wave with the charged particles in the medium, which become polarized and radiate secondary waves that interfere constructively with the incident wave but with a phase delay, resulting in a net propagation speed lower than $ c $.[39] پانی میں روشنی کی رفتار خلا سے کم ہوتی ہے کیونکہ پانی کا ریفریکٹو انڈیکس (n ≈ 1.33) 1 سے زیادہ ہے، جس سے رفتار v = c / n کم ہو جاتی ہے (خلا میں c ≈ 3 × 10^8 m/s، پانی میں ≈ 2.26 × 10^8 m/s)۔ سائنسی وضاحت: روشنی ایک الیکٹرومیگنیٹک لہر ہے۔ پانی جیسے ڈائی الیکٹرک میں اس کی برقی میدان الیکٹران کو ہلاتی ہے، جو ثانوی لہریں خارج کرتے ہیں۔ یہ ثانوی لہریں اصل لہر کے ساتھ سپرپوزیشن میں آ کر فیز میں تاخیر (تقریباً 90°) پیدا کرتی ہیں، جس سے مجموعی لہر کی فیز ویلوسٹی کم ہو جاتی ہے۔ یہ شفاف مواد میں ہوتا ہے جہاں روشنی جذب نہیں ہوتی بلکہ coherent طور پر ری-ایمیٹ ہوتی ہے۔ The refractive index typically exceeds 1 for common media like water ($ n \approx 1.33 )orglass() or glass ( n \approx 1.5 $), meaning light travels at about 75% or 67% of $ c $, respectively. A common question concerns the energy source for light to resume its speed $ c $ when exiting a medium like glass back into vacuum. However, light does not gain energy in this process; the apparent slowing within the medium results from phase delays due to atomic interactions, such as electric polarization or absorption and re-emission cycles, and upon exiting, the electromagnetic wave propagates at $ c $ without net energy loss or gain, as the process conserves energy overall. Classically, this can be understood via Huygens' principle, where wavefronts propagate freely in vacuum upon exit, while quantum mechanically, it involves superpositions of photon and matter states that resolve into free photons at $ c $.[40] The refractive index governs refraction at interfaces between media, as described by Snell's law: $ n_1 \sin \theta_1 = n_2 \sin \theta_2 $, where $ \theta_1 $ and $ \theta_2 $ are the angles of incidence and refraction.[41] This law explains phenomena such as the bending of light rays when entering a denser medium, leading to effects like mirages or the apparent depth of objects in water.[42] In non-vacuum environments, the phase velocity $ v_p = \frac{\omega}{k} = \frac{c}{n} $ represents the speed of constant-phase surfaces, but it does not necessarily carry information.[43] For signal propagation, the relevant quantity is the group velocity $ v_g = \frac{d\omega}{dk} $, which is the velocity of the wave packet's envelope and determines how energy or information travels.[44] In dispersive media, where $ n $ varies with frequency, $ v_g $ differs from $ v_p $ and remains subluminal ($ v_g \leq c $), ensuring no violation of causality.[45] Dispersion, the wavelength dependence of $ n $, causes shorter wavelengths (e.g., blue light) to experience higher $ n $ and slower speeds than longer ones (e.g., red light) in materials like glass.[46] This dispersive effect separates white light into a spectrum when passing through a prism, producing rainbows by differential refraction and total internal reflection in water droplets.[47] In optical fibers, dispersion leads to pulse broadening, where different wavelengths in a signal spread out over distance, limiting high-speed data transmission unless compensated by dispersion-shifted fibers.[48] Cherenkov radiation occurs when a charged particle traverses a medium at a speed exceeding the local phase velocity ($ v > \frac{c}{n} $) but below $ c $, analogous to a sonic boom for light.[49] The emitted coherent shock wave of blue light forms a cone with angle $ \cos \theta = \frac{c}{n v} $, observable in nuclear reactors or particle detectors using water or aerogel.[50] This phenomenon was first observed experimentally in 1934 by Pavel Cherenkov and theoretically explained in 1937 by Igor Tamm and Ilya Frank, providing a method to measure particle velocities without exceeding the universal speed limit.[51]

Apparent Faster-Than-Light Phenomena

Observational Examples

One prominent astronomical observation of apparent faster-than-light motion occurs in the relativistic jets emanating from quasars, where radio-emitting components appear to traverse angular distances on the sky at speeds exceeding cc. For instance, in the quasar 3C 279, very long baseline interferometry (VLBI) observations have revealed knot-like features moving with apparent transverse velocities up to approximately 4.3 times the speed of light, as measured over multiple epochs between 1979 and 1981.[52] This illusion arises from relativistic beaming, where the jet is oriented close to our line of sight; the apparent speed is given by $ v_{\rm app} = \frac{v \sin \theta}{1 - \frac{v \cos \theta}{c}} $, with vv the actual jet speed, θ\theta the angle to the line of sight, and cc the speed of light, amplifying the projected motion without any material exceeding cc.[53] Similar superluminal ejections, with vappv_{\rm app} reaching 10cc or more, have been documented in other quasars like 3C 345 and BL Lac objects, confirming the effect in over 100 sources through coordinated global VLBI campaigns. In binary pulsar systems, timing delays due to light travel time across the orbit provide another example where the effect can lead to misinterpretations of superluminal motion if not properly modeled. For the relativistic binary pulsar PSR B1913+16, pulsar timing observations spanning decades show periodic delays in pulse arrival times due to the varying light path length as the pulsar orbits its companion, with delays up to tens of seconds corresponding to the 7.75-hour orbital period. Without accounting for this geometric light travel time effect—known as the Roemer delay—the observed timing variations could erroneously suggest orbital velocities exceeding cc, as the signal's propagation delay mimics an impossibly rapid positional shift.[54] This effect has been precisely quantified in PSR B1913+16, where the orbital inclination and eccentricity confirm the delays align with special relativistic predictions, avoiding any true superluminal inference. Terrestrial optical illusions further illustrate apparent superluminal phenomena without violating causal limits. A classic demonstration involves rapidly closing a pair of long scissors, where the intersection point of the blades sweeps along a distant wall or screen at speeds far exceeding cc, as the local motion of each blade remains subluminal.[55] Similarly, projecting a shadow from a moving object, such as sweeping a laser pointer across the Moon, can produce a spot that traverses the lunar surface at apparent speeds orders of magnitude greater than cc, since no information or energy travels faster than light—the shadow is merely an absence of illumination propagating geometrically.[55] These effects highlight how phase velocities or image projections can exceed cc locally while preserving relativity's prohibition on signal transmission. The expansion of the universe provides a cosmological example of apparent recession speeds surpassing cc, observed in the redshift of distant galaxies. According to Hubble's law, galaxies beyond about 14 billion light-years recede at velocities greater than cc due to the metric expansion of spacetime itself, rather than local motion through space; for instance, galaxies at redshift z>1.5z > 1.5 exhibit recession speeds up to several times cc. This superluminal recession does not allow causal influence beyond the cosmic horizon, as the expansion stretches the light paths without enabling faster-than-light communication. Observations from the Hubble Space Telescope confirm this in surveys of thousands of supernovae and galaxies, showing the effect increases with distance, consistent with the Friedmann-Lemaître-Robertson-Walker metric.

Experimental Results

Laboratory experiments exploring evanescent waves in quantum tunneling have reported apparent superluminal group velocities, where the peak of a microwave or optical pulse traverses a barrier in less time than expected for light in vacuum. For instance, experiments using frustrated total internal reflection or undersized waveguides demonstrated pulse peaks emerging before the time light would take to travel the same distance, suggesting group velocities exceeding cc. However, detailed analysis reveals that these effects arise from the non-local nature of evanescent fields and do not permit superluminal information transfer, as the signal's leading edge propagates at or below cc, and reshaping of the pulse ensures causality is preserved.[56] In waveguide structures, phase velocities greater than cc are routinely observed for electromagnetic waves propagating below the cutoff frequency, where the wave's phase advances faster than light to compensate for the evanescent field in the transverse direction. This superluminal phase velocity vp=ωβv_p = \frac{\omega}{\beta}, with β<ωc\beta < \frac{\omega}{c}, has been confirmed in microwave and optical experiments, but the associated group velocity vg=dωdβv_g = \frac{d\omega}{d\beta}, which determines the speed of energy and information transport, remains subluminal (vg<cv_g < c). Such configurations highlight that while phase propagation can exceed cc, no violation of relativity occurs for observable signals.[57][58] During the early 2000s, several experiments investigated light pulse propagation in dispersive media with anomalous absorption or gain, reporting apparent faster-than-light travel. A prominent example is the 2000 study by Wang et al., where a laser pulse in a cesium vapor cell with coherent population oscillation emerged with its peak advanced relative to cc, implying a group velocity over 300 times cc. Subsequent analyses, including direct pulse shape measurements, resolved these observations by demonstrating pulse distortion and reformation: the superluminal advance is an artifact of the medium's response, with no part of the pulse or information carrier exceeding cc, thus upholding the causal limit.[59] Quantum entanglement experiments, such as those involving Bell state measurements on photon pairs, exhibit correlations that appear instantaneous across large separations, raising questions about faster-than-light influences. The no-communication theorem rigorously demonstrates that such entanglement cannot enable superluminal signaling, as the local density operator for one party's subsystem remains unchanged by the distant measurement, preventing any controllable information transfer without classical communication. This theorem, foundational to quantum information theory, has been verified in numerous entanglement distribution and measurement protocols.

Historical Development

The measurement of the speed of light represented a major advancement in the science of metrology, as it involved quantifying a phenomenon vastly faster than any previously measured speed. In preceding centuries, humans estimated wind speeds qualitatively or with early mechanical devices like anemometers (conceptualized by Leonardo da Vinci in the 15th century and refined in the 19th), while the speed of sound in air—approximately 343 m/s—was first quantitatively determined in the early 17th century. Pioneering measurements of sound speed employed methods such as timing the delay between a visual signal (e.g., a cannon flash) and its auditory arrival over measured distances, feasible with human senses and rudimentary timing tools. In comparison, light's speed (~3 × 10^8 m/s) is roughly 874,000 times greater than that of sound, making direct timing over accessible terrestrial distances impossible with the era's technology and requiring sophisticated astronomical observations or high-speed optical apparatuses to capture the minuscule time-of-flight differences.

Ancient and Early Concepts

In ancient Greece during the 5th century BCE, Empedocles was the first philosopher to propose that light travels at a finite speed, suggesting that the interval between a stimulus and its perception by the eye indicated a non-instantaneous propagation, though he did not quantify it.[60] This view contrasted sharply with that of Aristotle in the 4th century BCE, who argued that light moves instantaneously, likening it to a presence rather than a motion that requires time, a position that dominated philosophical thought for centuries.[60] During the medieval Islamic Golden Age, the polymath Ibn al-Haytham (known as Alhazen in the Latin West), writing in the early 11th century, advanced the understanding by proposing light propagates in straight lines at a large but finite velocity, which varies depending on the medium's density, laying foundational ideas for later optics without providing a numerical value.[60][61] In his seminal Book of Optics, he described these concepts based on optical observations. In the 17th century, Galileo Galilei attempted the first experimental measurement of light's speed in 1638, using lanterns positioned on distant hilltops where one observer would uncover a light and time the moment another observer, miles away, responded by uncovering theirs; however, he concluded that the speed was too great to detect with human reaction times and terrestrial distances.[62] This effort marked a shift toward empirical approaches, though unsuccessful in yielding a value. Shortly after, in 1676, Danish astronomer Ole Rømer provided the first quantitative estimate by observing discrepancies in the predicted eclipses of Jupiter's moon Io, attributing them to the finite time light takes to cross varying Earth-Jupiter distances; his calculation yielded approximately 220,000 km/s, remarkably close to the modern value despite assumptions about orbital sizes.[5][60] In 1728, English astronomer James Bradley refined astronomical measurements using stellar aberration, estimating the speed at approximately 301,000 km/s.[63]

19th-Century Advances

In the early 19th century, Thomas Young conducted the double-slit experiment in 1801, which provided key evidence for the wave nature of light through observed interference patterns.[64] By passing sunlight through two closely spaced slits onto a screen, Young demonstrated that light produces alternating bright and dark fringes, consistent with wave superposition rather than particle behavior.[64] Building on Christiaan Huygens' 17th-century wave theory, Augustin-Jean Fresnel refined the model in the 1810s and 1820s, incorporating the Huygens-Fresnel principle to explain diffraction and polarization.[65] These advancements mathematically predicted a finite speed for light waves propagating through space, aligning with emerging experimental evidence against instantaneous transmission.[65] The first terrestrial measurement of light's speed came in 1849 from Hippolyte Fizeau, who used a toothed-wheel apparatus over an 8.6 km path in air.[6] Light from a source passed through gaps in a rapidly rotating wheel, reflected off a distant mirror, and returned; by adjusting the wheel's 720-tooth rotation to 12.6 turns per second for the first minimum visibility, Fizeau calculated a speed of approximately 313,000 km/s.[6] James Clerk Maxwell's 1865 electromagnetic theory unified electricity, magnetism, and optics, deriving light as an electromagnetic wave with speed given by
c=1ϵ0μ0 c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}
where ϵ0\epsilon_0 and μ0\mu_0 are the permittivity and permeability of free space.[66] Using contemporary values, Maxwell obtained approximately 310,000 km/s, closely matching astronomical and Fizeau's results, thus confirming light's electromagnetic character.[67] In the 1880s, Albert A. Michelson improved the rotating-mirror method, achieving higher precision over longer baselines.[68] His 1879–1880 experiments at the U.S. Naval Academy yielded a value of about 299,850 km/s in air, refined further in subsequent work to within 0.01% accuracy, setting a benchmark for vacuum measurements.[68]

Modern Measurements

Terrestrial Techniques

Terrestrial techniques for measuring the speed of light have evolved from mechanical and optical methods to highly precise laser-based approaches, enabling laboratory determinations with exceptional accuracy. One foundational modern variant of the time-of-flight method involves sending a light pulse over a known distance and timing its return, with significant improvements in the mid-20th century. In 1941, W. C. Anderson employed a Kerr cell electro-optic modulator as a high-speed shutter to precisely capture the transit time of light pulses, achieving a value of $ c = 299776 \pm 14 $ km/s, corresponding to an accuracy better than 0.01%. Contemporary laser time-of-flight experiments build on this principle using picosecond or femtosecond pulsed lasers over short baselines, such as 20 cm paths with beam splitters and fast photodetectors, yielding results consistent with the defined value of $ c $ to within parts per million.[69] Cavity resonance techniques represent a cornerstone of modern terrestrial measurements, leveraging the relationship $ c = f \lambda $, where frequency $ f $ is determined from microwave or optical cavity resonances and wavelength $ \lambda $ from interferometric calibration. In the 1970s, researchers at the National Institute of Standards and Technology (NIST) pioneered stabilized laser systems in Fabry-Pérot cavities to measure optical frequencies directly against cesium time standards. A seminal 1973 experiment by Evenson et al. used a methane-stabilized helium-neon laser, measuring its frequency and wavelength to obtain $ c = 299792456.2 \pm 1.1 $ m/s, reducing uncertainty by two orders of magnitude compared to prior values and establishing a benchmark for subsequent refinements. These methods, refined with frequency combs in later decades, have achieved relative uncertainties below $ 10^{-11} $, tying the speed of light to atomic clocks without relying on physical length standards.[70] Interferometric approaches, particularly variants of the Michelson interferometer, have been instrumental in terrestrial measurements by precisely determining light wavelengths, which, when multiplied by measured frequencies, yield $ c $. Albert A. Michelson adapted his interferometer in the early 20th century for baseline determinations in time-of-flight setups, but later applications focused on wavelength metrology in vacuum paths. For instance, interferometers with evacuated arms and stabilized light sources measure fringe shifts to calibrate $ \lambda $, enabling $ c $ computations with accuracies approaching $ 10^{-8} $ in pre-laser eras.[71] In the context of length standards, these techniques historically linked the meter to light wavelengths (e.g., the krypton-86 line), indirectly affirming $ c $ through the relation $ c = f \lambda $, though post-1983 definitions fix $ c $ exactly and derive length from it.[72] Electromagnetic methods derive $ c $ from independent measurements of the vacuum permittivity $ \epsilon_0 $ and permeability $ \mu_0 $ via the relation $ c = 1 / \sqrt{\epsilon_0 \mu_0} $, using capacitance and inductance standards in controlled laboratory conditions. A key early 20th-century implementation by E. B. Rosa and Louis B. Dorsey in 1907 at the Bureau of Standards involved electrostatic and electromagnetic unit comparisons with precision capacitors and coils, yielding $ c = 299710 \pm 22 $ km/s and validating Maxwell's theoretical prediction to within 0.03%. Modern codifications, as in CODATA recommendations, incorporate such measurements into adjusted values; prior to the 1983 SI redefinition, these contributed to $ c = 299792458 $ m/s with uncertainties around $ 10^{-8} $, now exact by convention while $ \epsilon_0 $ is derived accordingly.[73]

Astronomical Methods

The explosion of Supernova 1987A in the Large Magellanic Cloud provided a modern astronomical confirmation that light and neutrinos travel at effectively the same speed. Neutrino detectors, including Kamiokande-II and the Irvine-Michigan-Brookhaven (IMB) experiment, recorded a burst of about 20 electron antineutrinos on February 23, 1987, UTC, while the first visible light photons arrived roughly three hours later on February 24. Over the supernova's distance of approximately 168,000 light-years, any significant speed difference would have produced a arrival time gap of years or more; the observed short delay instead reflects the physics of the explosion, where neutrinos escape the collapsing core almost immediately, whereas photons are trapped and diffuse slowly through the dense stellar envelope before emerging. This near-simultaneous arrival, with neutrinos traveling at more than 99.999% of light speed (consistent with their tiny mass), validates that both massless photons and near-massless particles propagate at the universal speed limit c in vacuum.[74] Gravitational lensing time delays offer a contemporary method to probe the speed of light in the curved spacetime predicted by general relativity. In strong lensing systems, such as quasars multiply imaged by foreground galaxies, the differing geometric paths cause light rays to arrive at Earth with measurable delays, typically days to weeks, between images. These delays arise from the Shapiro time delay effect, where light slows in the gravitational potential well near the lens, combined with path length differences; the formula incorporates c explicitly, assuming it remains constant locally even in curved regions. Observations of systems like the Einstein Cross or COSMOGRAIL-monitored lenses have yielded time delays precise to hours, enabling tests of general relativity by comparing predicted delays (using known distances and lens models) to measurements, thereby confirming that light travels at c without variation in strong gravitational fields. For instance, analyses of lensed supernovae or quasars constrain potential deviations in c to less than 1 part in 10^5, supporting the theory's invariance of light speed across cosmic scales.[75] The detection of gravitational waves (GWs) from binary neutron star mergers, such as GW170817 observed by LIGO/Virgo in 2017, provides a precise astronomical test of c's invariance through multi-messenger astronomy. The GW signal arrived at Earth 1.7 seconds before the gamma-ray burst counterpart detected by Fermi and INTEGRAL, over a distance of about 40 megaparsecs. This near-coincidence, combined with subsequent kilonova observations, constrains the speed of GWs to equal c within 10^{-15} (at 1σ confidence), confirming that massless gravitons (if they exist) propagate at the same speed as photons in vacuum and ruling out significant deviations predicted by some quantum gravity models.[76]

Practical Implications

Terrestrial Effects

The finite speed of light manifests in various terrestrial applications, where propagation delays and related effects influence technology and natural observations on human and planetary scales. In the Global Positioning System (GPS), electromagnetic signals from satellites orbiting at approximately 20,200 km altitude take about 67 milliseconds to reach ground-based receivers, as the one-way distance corresponds to this travel time at the speed of light.[77] This propagation delay is fundamental to pseudorange calculations for positioning, but GPS accuracy also demands corrections for relativistic effects, including special relativistic time dilation from satellite velocities (reducing clock rates by about 7 μs/day) and general relativistic gravitational redshift (increasing rates by about 45 μs/day), ensuring positional errors remain below 10 meters. A common natural demonstration of light's finite speed occurs during thunderstorms, where the flash of lightning is visible almost instantaneously due to the near-instantaneous propagation of light over distances of several kilometers, while the accompanying thunder—sound waves traveling at roughly 343 m/s in air—arrives seconds later. For instance, a 5-second delay between seeing the flash and hearing the thunder indicates the strike was approximately 1.7 km away, calculated as the time for sound to cover that distance (using the rule of thumb: 5 seconds per mile or 3 seconds per kilometer). This disparity highlights light's vastly superior speed compared to sound, allowing observers to gauge storm proximity without specialized equipment. On microscopic scales within electronic devices, light's travel time becomes relevant in high-speed computing. In modern computer chips, where silicon dies measure on the order of 1–2 cm across, electromagnetic signals propagate at speeds close to that of light in vacuum, resulting in delays of approximately 30–70 picoseconds for signals to traverse the chip.[78] Similarly, in cathode-ray tube (CRT) displays, electron beams accelerate to velocities around 10^6 m/s (about 0.003c), taking microseconds to sweep across a 50 cm screen, whereas light from the phosphor would traverse the same distance in mere nanoseconds, underscoring why beam speed limits refresh rates in older televisions.[79] In particle accelerators, the approach to light speed introduces synchrotron radiation as a key terrestrial effect, where relativistic electrons (accelerated to 0.999c or higher) emit intense electromagnetic radiation when forced into curved paths by magnetic fields.[80] This energy loss imposes practical limits on accelerator design; for example, in electron synchrotrons like those at CERN, restoring the radiated energy requires powerful radio-frequency systems, constraining maximum energies to tens of GeV without prohibitively large or costly facilities, as the radiation power scales with the fourth power of the Lorentz factor γ.[81]

Astrophysical and Space Applications

In astrophysics, the finite speed of light, denoted as c≈3×108c \approx 3 \times 10^8 m/s, serves as a fundamental scale for measuring vast cosmic distances, most notably through the unit of the light-year. A light-year is defined as the distance light travels in one Julian year (365.25 days) in vacuum, equivalent to approximately 9.46×10129.46 \times 10^{12} km. This unit is essential for expressing scales in astronomy, such as the distance to the nearest star beyond the Sun, Proxima Centauri, which lies about 4.24 light-years away, allowing astronomers to conceptualize interstellar and intergalactic separations without cumbersome numerical values.[82] The speed of light imposes significant constraints on spaceflight operations, particularly communication delays due to signal propagation times. For missions to Mars, the round-trip light time varies with planetary positions but typically ranges from 6 to 40 minutes, with an average of about 20 minutes during closer approaches, precluding real-time control of rovers or spacecraft. This lag necessitates autonomous systems for immediate decision-making, as commands from Earth arrive long after events unfold, impacting mission design and safety protocols.[83] In observational astronomy, the finite cc enables insights into the universe's history via redshift and look-back time. Cosmological redshift occurs as light from distant galaxies stretches to longer wavelengths due to the expansion of space, quantified by the redshift parameter zz, which correlates with recession velocity and distance. Look-back time represents the duration light has traveled to reach Earth, meaning observations of remote objects reveal their past states; for instance, light from galaxies 10 billion light-years away shows the universe as it was 10 billion years ago, facilitating studies of cosmic evolution from the Big Bang onward.[84][85] The boundary of black holes, known as the event horizon, is defined in general relativity as the surface where the escape velocity equals cc, preventing any matter or radiation from escaping once crossed. For a non-rotating black hole, this horizon radius, the Schwarzschild radius rs=2GMc2r_s = \frac{2GM}{c^2}, marks the point of no return, with GG as the gravitational constant and MM the mass, underpinning phenomena like gravitational lensing and the information paradox in theoretical physics.[86]

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  54. Timing Measurements of the Relativistic Binary Pulsar PSR B1913+16
  55. Is Faster-Than-Light Travel or Communication Possible?
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  58. Superluminal signals: an engineer's perspective - ScienceDirect.com
  59. Clear message for causality - Physics World
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  65. July 1816: Fresnel's Evidence for the Wave Theory of Light
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  67. '…a paper …I hold to be great guns': a commentary on Maxwell ...
  68. The Site Where Albert Michelson Measured the Speed of Light 145 ...
  69. Speed of Light - Physclips.
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  72. Michelson Interferometer
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  82. Glossary term: Light Year - IAU Office of Astronomy for Education
  83. Key Differences in Operating a Rover on the Moon vs. Mars
  84. Hubble Cosmological Redshift - NASA Science
  85. Look Back in Time to the Early Universe! - NASA Science
  86. Black Hole Anatomy - NASA Science
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