Speed of light

# Speed of light

Overview
The speed of light in vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

, usually denoted by c, is a physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

important in many areas of physics. Its value is 299,792,458 metres per second
Metre per second
Metre per second is an SI derived unit of both speed and velocity , defined by distance in metres divided by time in seconds....

, a figure that is exact since the length of the metre is defined from this constant and the international standard for time. In imperial units this speed is approximately 186,282 miles per second.

According to special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, c is the maximum speed at which all energy, matter, and information
Physical information
In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics In physics,...

in the universe can travel.
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Encyclopedia
The speed of light in vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

, usually denoted by c, is a physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

important in many areas of physics. Its value is 299,792,458 metres per second
Metre per second
Metre per second is an SI derived unit of both speed and velocity , defined by distance in metres divided by time in seconds....

, a figure that is exact since the length of the metre is defined from this constant and the international standard for time. In imperial units this speed is approximately 186,282 miles per second.

According to special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, c is the maximum speed at which all energy, matter, and information
Physical information
In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics In physics, physical information refers generally to the information that is contained in a physical system. Its usage in quantum mechanics In physics,...

in the universe can travel. It is the speed of all massless particles and associated field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

such as light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

—in vacuum, and it is predicted by the current theory to be the speed of gravity
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...

(that is, gravitational wave
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...

s). Such particles and waves travel at c regardless of the motion of the source or the inertial frame of reference
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

of the observer. In the theory of relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

, c interrelates space and time
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

, and appears in the famous equation of mass–energy equivalence E = mc2.

The speed at which light propagates through transparent materials, such as glass or air, is less than c. The ratio between c and the speed v at which light travels in a material is called the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

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 ; the refractive index of air for visible light is about 1.0003, so the speed of light in air is about slower than c.

In most practical cases, light can be thought of as moving instantaneously, but for long distances and very sensitive measurements the finite speed of light has noticeable effects. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft or vice versa. The light we see from stars left them many years ago, allowing us to study the history of the universe by looking at distant objects. The finite speed of light also limits the theoretical maximum speed of computers, since information must be sent within the computer from chip to chip. Finally, the speed of light can be used with time of flight measurements to measure large distances to high precision.

Ole Rømer first demonstrated in 1676 that light travelled at a finite speed (as opposed to instantaneously) by studying the apparent motion of Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...

's moon Io
Io (moon)
Io ) is the innermost of the four Galilean moons of the planet Jupiter and, with a diameter of , the fourth-largest moon in the Solar System. It was named after the mythological character of Io, a priestess of Hera who became one of the lovers of Zeus....

. In 1865, James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

proposed that light was an electromagnetic wave, and therefore traveled at the speed c appearing in his theory of electromagnetism. In 1905, Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

postulated that the speed of light with respect to any inertial frame is independent of the motion of the light source, and explored the consequences of that postulate by deriving the special theory of relativity and showing that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of increasingly precise measurements, in 1975 the speed of light was known to be with a relative measurement uncertainty
Measurement uncertainty
In metrology, measurement uncertainty is a non-negative parameter characterizing the dispersion of the values attributed to a measured quantity. The uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and a measured...

of 4 parts per billion. In 1983, the metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

was redefined in the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI) as the distance travelled by light in vacuum in of a second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

. As a result, the numerical value of c in metres per second is now fixed exactly by the definition of the metre.

## Numerical value, notation, and units

The speed of light in vacuum is usually denoted by c, for "constant" or the Latin (meaning "swiftness"). Originally, the symbol V was used, introduced by James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

in 1865. In 1856, Wilhelm Eduard Weber
Wilhelm Eduard Weber
Wilhelm Eduard Weber was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph.-Early years:...

and Rudolf Kohlrausch
Rudolf Kohlrausch
Rudolf Hermann Arndt Kohlrausch was a German physicist.-Biography:He was a native of Göttingen, the son of educator Heinrich Friedrich Theodor Kohlrausch...

used c for a constant later shown to equal times the speed of light in vacuum. In 1894, Paul Drude redefined c with its modern meaning. Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

used V in his original German-language papers
Annus Mirabilis Papers
The Annus Mirabilis papers are the papers of Albert Einstein published in the Annalen der Physik scientific journal in 1905. These four articles contributed substantially to the foundation of modern physics and changed views on space, time, and matter...

on special relativity in 1905, but in 1907 he switched to c, which by then had become the standard symbol.

Sometimes c is used for the speed of waves in any material medium, and c0 for the speed of light in vacuum. This subscripted notation, which is endorsed in official SI literature, has the same form as other related constants: namely, μ0 for the vacuum permeability
Vacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...

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.

In the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI), the metre is defined as the distance light travels in vacuum in of a second. This definition fixes the speed of light in vacuum at exactly .
As a dimensional physical constant, the numerical value of c is different for different unit systems.
In branches of physics in which c appears often, such as in relativity, it is common to use systems of natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

of measurement in which . Using these units, c does not appear explicitly because multiplication or division by 1 does not affect the result.

## Fundamental role in physics

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
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

of the observer. However, the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

of light can depend on the motion of the source relative to the observer, due to the Doppler effect
Doppler effect
The Doppler effect , named after Austrian physicist Christian Doppler who proposed it in 1842 in Prague, is the change in frequency of a wave for an observer moving relative to the source of the wave. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from...

.
This invariance of the speed of light was postulated by Einstein in 1905, after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....

; it has since been consistently confirmed by many experiments. 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
One-way speed of light
The "one-way" speed of light from a source to a detector, cannot be measured independently of a convention as to how to synchronize the clocks at the source and the detector. What can however be experimentally measured is the round-trip speed from the source to the detector and back again...

(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. However, by adopting Einstein synchronization for the clocks, the one-way speed of light becomes equal to the two-way speed of light by definition. 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. One consequence is that c is the speed at which all massless particles and waves, including light, must travel in vacuum.

Special relativity has many counterintuitive and experimentally verified implications. These include the equivalence of mass and energy , length contraction
Length contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...

(moving objects shorten), and time dilation
Time dilation
In the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...

(moving clocks run slower). The factor γ by which lengths contract and times dilate, is known as the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...

and is given by , 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.

The results of special relativity can be summarized by treating space and time as a unified structure known as spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

(with c relating the units of space and time), and requiring that physical theories satisfy a special symmetry
Symmetry in physics
In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...

called Lorentz invariance, whose mathematical formulation contains the parameter c. Lorentz invariance is an almost universal assumption for modern physical theories, such as quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

, the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. 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
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...

and of gravitational waves. In non-inertial frames of reference (gravitationally curved space or accelerated reference frames), the local speed of light is constant and equal to c, but the speed of light along a trajectory of finite length
Propagation of light in non-inertial reference frames
The description of motion in relativity requires more than one concept of speed. Coordinate speed is the coordinate distance measured by the observer divided by the coordinate time of the observer. Proper speed is the local proper distance divided by the local proper time. For example, at the event...

can differ from c, depending on how distances and times are defined.

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
Variable speed of light
The variable speed of light concept states that the speed of light in a vacuum, usually denoted by c, may not be constant in most cases. In most situations in condensed matter physics when light is traveling through a medium, it effectively has a slower speed...

. No conclusive evidence for such changes has been found, but they remain the subject of ongoing research.

It also is generally assumed that the speed of light is isotropic
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...

, meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear energy level
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

s as a function of the orientation of the emitting nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

in a magnetic field (see Hughes–Drever experiment
Hughes–Drever experiment
Hughes–Drever experiments are testing the isotropy of mass and space. As in Michelson–Morley experiments, the existence of a preferred frame of reference, or deviations from Lorentz invariance can be tested, which also affects the validity of the equivalence principle...

), and of rotating optical resonators (see Resonator experiments) have put stringent limits on the possible two-way anisotropy.

### Upper limit on speeds

According to special relativity, the energy of an object with rest mass m and speed v is given by , where γ is the Lorentz factor defined above. When v is zero, γ is equal to one, giving rise to the famous formula for mass-energy equivalence
Mass-energy equivalence
In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant...

. Since the γ factor approaches infinity as v approaches c, it would take an infinite amount of energy to accelerate an object with mass to the speed of light. The speed of light is the upper limit for the speeds of objects with positive rest mass.

More generally, it is normally impossible for information or energy to travel faster than c. One argument for this follows from the counter-intuitive implication of special relativity known as the relativity of simultaneity
Relativity of simultaneity
In physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...

. 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
Causality (physics)
Causality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....

would be violated. In such a frame of reference, an "effect" could be observed before its "cause". Such a violation of causality has never been recorded, and would lead to paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

es such as the tachyonic antitelephone
Tachyonic antitelephone
A tachyonic antitelephone is a hypothetical device in theoretical physics that could be used to send signals into one's own past. Albert Einstein in 1907...

.

## Faster-than-light observations and experiments

There are situations in which it may seem that matter, energy, or information 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. For example, the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

of X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

s through most glasses can routinely exceed c, but such waves do not convey any information.

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. In neither case does any matter, energy, or information travel faster than light.

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.

Certain quantum effects appear to be transmitted instantaneously and therefore faster than c, as in the EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...

. An example involves the quantum states of two particles that can be entangled
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

. Until either of the particles is observed, they exist in a superposition
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

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 (i.e., faster than light could travel from one particle to the other). However, it is impossible to control which quantum state the first particle will take on when it is observed, so information cannot be transmitted in this manner.

Another quantum effect that predicts the occurrence of faster-than-light speeds is called the Hartman effect
Hartman effect
The delay time for a quantum tunneling particle is independent of the thickness of the opaque barrier. This is called the Hartman effect, after Thomas Hartman who discovered it in 1962...

; under certain conditions the time needed for a virtual particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

to tunnel
Quantum tunnelling
Quantum tunnelling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the sun, and has important...

through a barrier is constant, regardless of the thickness of the barrier. This could result in a virtual particle crossing a large gap faster-than-light. However, no information can be sent using this effect.

So-called superluminal motion
Superluminal motion
In astronomy, superluminal motion is the apparently faster-than-light motion seen in someradio galaxies, quasars and recently also in some galactic sources called microquasars...

is seen in certain astronomical objects, such as the relativistic jet
Relativistic jet
Relativistic jets are extremely powerful jets of plasma which emerge from presumed massive objects at the centers of some active galaxies, notably radio galaxies and quasars. Their lengths can reach several thousand or even hundreds of thousands of light years...

Radio galaxies and their relatives, radio-loud quasars and blazars, are types of active galaxy that are very luminous at radio wavelengths, with luminosities up to 1039 W between 10 MHz and 100 GHz. The radio emission is due to the synchrotron process...

and quasar
Quasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...

s. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection
Graphical projection
Graphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing.- Overview :...

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.

In models of the expanding universe, the farther galaxies are from each other, the faster they drift apart. This receding is not due to motion through space, but rather to the expansion of space
Metric expansion of space
The metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...

itself. For example, galaxies far away from Earth appear to be moving away from the Earth with a speed 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.

In September 2011, physicists working on the OPERA experiment
OPERA Experiment
The Oscillation Project with Emulsion-tRacking Apparatus is a scientific experiment for detecting tau neutrinos from muon neutrino oscillations. It is a collaboration between CERN in Geneva, Switzerland, and the Laboratori Nazionali del Gran Sasso in Gran Sasso, Italy and uses the CERN Neutrinos...

published results that suggest beams of neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

CERN
The European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...

(in Geneva, Switzerland) to LNGS
Laboratori Nazionali del Gran Sasso
Laboratori Nazionali del Gran Sasso is a particle physics laboratory of the INFN, situated near the Gran Sasso mountain in Italy, between the towns of L'Aquila and Teramo, about 120 km from Rome. In addition to a surface portion of the laboratory, there are extensive underground facilities...

(at the Gran Sasso, Italy) faster than the speed of light. These findings have yet to be independently verified. (See OPERA neutrino anomaly
OPERA neutrino anomaly
The OPERA neutrino anomaly is the detection of apparently faster-than-light neutrinos by the OPERA experiment as publicly announced in September 2011. The detection is anomalous because speeds exceeding that of light in a vacuum are generally thought to violate special relativity, a prevailing...

.)

## Propagation of light

In classical physics
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...

, light is described as a type of electromagnetic wave. The classical behaviour of the electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

is described by Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

ε0 and the magnetic constant μ0 by the equation . In modern quantum physics, the electromagnetic field is described by the theory of quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

(QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, called photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s. In QED, photons are massless particles and thus, according to special relativity, they travel at the speed of light in vacuum.

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. No variation of the speed of light with frequency has been observed in rigorous testing, putting stringent limits on the mass of the photon. The limit obtained depends on the model used: if the massive photon is described by Proca theory
Proca action
In physics, in the area of field theory, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The field involved is a real vector field A...

, the experimental upper bound for its mass is about 10−57 gram
Gram
The gram is a metric system unit of mass....

s; if photon mass is generated by a Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....

, the experimental upper limit is less sharp,   (roughly 2 × 10−47 g).

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
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

. In 2009, the observation of the spectrum of gamma-ray burst GRB 090510 did not find any difference in the speeds of photons of different energies, confirming that Lorentz invariance is verified at least down to the scale of the Planck length (lP =  ≈ ) divided by 1.2.

### In a medium

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
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....

(a wave filling the whole space, with only one frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

) propagate is called the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

vp. An actual physical signal with a finite extent (a pulse of light) travels at a different speed. The largest part of the pulse travels at the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

vg, and its earliest part travels at the front velocity
Front velocity
In physics, front velocity is the speed at which the first rise of a pulse above zero moves forward.In mathematics, it is also used to describe the velocity of a possibly propagating front in the solution of hyperbolic partial differential equation....

vf.
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. Denser media, such as water
Optical properties of water and ice
The refractive index of water at 20°C is 1.332986. The refractive index of normal ice is 1.31. In general, an index of refraction is a complex number with both a real and imaginary part, where the latter indicates the strength of absorption loss at a particular wavelength...

, glass, and diamond, have refractive indexes of around 1.3, 1.5 and 2.4 respectively for visible light.

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. The requirement that causality is not violated implies that the real and imaginary parts
Real and imaginary parts
In mathematics, a hypercomplex number has a real part and an imaginary part associated with it. This is most familiar in the context of complex numbers, but extends to the other hypercomplex algebras such as split-complex numbers and quaternions....

of the dielectric constant
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...

of any material, corresponding respectively to the index of refraction and to the attenuation coefficient
Attenuation coefficient
The attenuation coefficient is a quantity that characterizes how easily a material or medium can be penetrated by a beam of light, sound, particles, or other energy or matter. A large attenuation coefficient means that the beam is quickly "attenuated" as it passes through the medium, and a small...

, are linked by the Kramers–Kronig relations. In practical terms, this means that in a material with refractive index less than 1, the absorption of the wave is so quick that no signal can be sent faster than c.

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
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called slow light
Slow light
Slow light is the propagation of an optical pulse or other modulation of an optical carrier at a very low group velocity. Slow light occurs when a propagating pulse is substantially slowed down by the interaction with the medium in which the propagation take place.Researchers at the Rowland...

, which has been confirmed in various experiments.
The opposite, group velocities exceeding c, has also been shown in experiment. It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.

None of these options, however, 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
Front velocity
In physics, front velocity is the speed at which the first rise of a pulse above zero moves forward.In mathematics, it is also used to describe the velocity of a possibly propagating front in the solution of hyperbolic partial differential equation....

). It can be shown that this is (under certain assumptions) always equal to c.

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
Charged particle
In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a plasma, which is called the fourth state of matter because its...

does that in an electrical insulator, the electromagnetic equivalent of a shock wave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

Cherenkov radiation is electromagnetic radiation emitted when a charged particle passes through a dielectric medium at a speed greater than the phase velocity of light in that medium...

, is emitted.

## Practical effects of finiteness

The finiteness of the speed of light has implications for various sciences and technologies. In some cases, it is a hindrance: for example, c, being the upper limit of the speed with which signals can be sent, provides a theoretical upper limit for the operating speed of microprocessors. On the other hand, some techniques depend on it, for example in distance measurements. Also, earth-based controllers must wait for round-trip communication lag that increases as spacecraft get farther away; NASA must wait several hours for information from a probe orbiting Jupiter, and if it needs to correct a navigation error, the fix will not arrive at the spacecraft for an equal amount of time, creating a risk of the correction not arriving in time.

The speed of light is of relevance to communications
Telecommunication
Telecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...

. For example, given the equatorial circumference of the Earth is about and c about , the theoretical shortest time for a piece of information to travel half the globe along the surface is about 67 milliseconds. When light is travelling around the globe in an optical fibre, the actual transit time is longer, in part because the speed of light is slower by about 35% in an optical fibre, depending on its refractive index n. Furthermore, straight lines rarely occur in global communications situations, and delays are created when the signal passes through an electronic switch or signal regenerator.

Another consequence of the finite speed of light is that 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
Mission Control Center
A mission control center is an entity that manages aerospace vehicle flights, usually from the point of lift-off until the landing or the end of the mission. A staff of flight controllers and other support personnel monitor all aspects of the mission using telemetry, and send commands to the...

and Apollo 8
Apollo 8
Apollo 8, the second manned mission in the American Apollo space program, was the first human spaceflight to leave Earth orbit; the first to be captured by and escape from the gravitational field of another celestial body; and the first crewed voyage to return to Earth from another celestial...

when it became the first manned spacecraft to orbit the Moon: for every question, the ground control station had to wait at least three seconds for the answer to arrive. 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 at least five minutes later, and possibly up to twenty minutes later; it would then take a further five to twenty minutes for instructions to travel from Earth to Mars.

The speed of light can also be of concern over very short distances. In supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s, the speed of light imposes a limit on how quickly data can be sent between processor
Central processing unit
The central processing unit is the portion of a computer system that carries out the instructions of a computer program, to perform the basic arithmetical, logical, and input/output operations of the system. The CPU plays a role somewhat analogous to the brain in the computer. The term has been in...

s. If a processor operates at 1 gigahertz, a signal can only travel a maximum of about 30 centimetre (0.984251968503937 ft) in a single cycle. Processors must therefore be placed close to each other to minimize communication latencies; this can cause difficulty with cooling. If clock frequencies continue to increase, the speed of light will eventually become a limiting factor for the internal design of single chips
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...

.

### Distance measurement

Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

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
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...

(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 300,000 kilometres (186,000 miles) in one second, these measurements of small fractions of a second must be very precise. The Lunar Laser Ranging Experiment
Lunar laser ranging experiment
The ongoing Lunar Laser Ranging Experiment measures the distance between the Earth and the Moon using laser ranging. Lasers on Earth are aimed at retroreflectors planted on the moon during the Apollo program, and the time for the reflected light to return is determined...

Radar astronomy is a technique of observing nearby astronomical objects by reflecting microwaves off target objects and analyzing the echoes. This research has been conducted for six decades. Radar astronomy differs from radio astronomy in that the latter is a passive observation and the former an...

and the Deep Space Network
Deep Space Network
The Deep Space Network, or DSN, is a world-wide network of large antennas and communication facilities that supports interplanetary spacecraft missions. It also performs radio and radar astronomy observations for the exploration of the solar system and the universe, and supports selected...

determine distances to the Moon, planets and spacecraft, respectively, by measuring round-trip transit times.

### Astronomy

The finite speed of light is important in astronomy. Due to the vast distances involved, it can take a very long time for light to travel from its source to Earth. For example, it has taken 13 billion (13) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field
Hubble Ultra Deep Field
The Hubble Ultra-Deep Field is an image of a small region of space in the constellation Fornax, composited from Hubble Space Telescope data accumulated over a period from September 24, 2003, through to January 16, 2004...

images. Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when the universe was less than a billion years old. The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars, of galaxies
Galaxy formation and evolution
The study of galaxy formation and evolution is concerned with the processes that formed a heterogeneous universe from a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time, and the processes that have generated the variety of structures observed in nearby...

, and of the universe itself.

Astronomical distances are sometimes expressed in light-year
Light-year
A light-year, also light year or lightyear is a unit of length, equal to just under 10 trillion kilometres...

s, especially in popular science
Popular science
Popular science, sometimes called literature of science, is interpretation of science intended for a general audience. While science journalism focuses on recent scientific developments, popular science is broad-ranging, often written by scientists as well as journalists, and is presented in many...

publications and media. A light-year is the distance light travels in one year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsec
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....

s. Proxima Centauri
Proxima Centauri
Proxima Centauri is a red dwarf star about 4.2 light-years distant in the constellation of Centaurus. It was discovered in 1915 by Robert Innes, the Director of the Union Observatory in South Africa, and is the nearest known star to the Sun, although it is too faint to be seen with the naked eye...

, the closest star to Earth after the Sun, is around 4.2 light-years away.

## Measurement

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. However, 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 equaling c.

In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of 1⁄299,792,458 of a second", fixing the value of the speed of light at 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

Outer space
Outer space
Outer space is the void that exists between celestial bodies, including the Earth. It is not completely empty, but consists of a hard vacuum containing a low density of particles: predominantly a plasma of hydrogen and helium, as well as electromagnetic radiation, magnetic fields, and neutrinos....

is a natural setting for measuring the speed of light because of its large scale and nearly perfect vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

. Typically, one measures the time needed for light to traverse some reference distance in the solar system
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

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. It is customary to express the results in astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

s (AU) per day. An astronomical unit is approximately the average distance between the Earth and Sun; it is not based on the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

. Because the AU determines an actual length, and is not based upon time-of-flight like the SI units, modern measurements of the speed of light in astronomical units per day can be compared with the defined value of c in the International System of Units.

Ole Christensen Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light
Rømer's determination of the speed of light
Rømer's determination of the speed of light was the demonstration in 1676 that light has a finite speed, and so doesn't travel instantaneously. The discovery is usually attributed to Danish astronomer Ole Rømer ,There are several alternative spellings of Rømer's surname: Roemer, Rœmer, Römer etc....

.Translated in (As reproduced in )

The account published in Journal des sçavans was based on a report that Rømer read to the French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

in November 1676 (Cohen, 1940, p. 346). 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 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
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

of the Earth's orbit around the Sun. The observed change in the moon's orbital period is actually the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect for Jupiter's innermost moon Io
Io (moon)
Io ) is the innermost of the four Galilean moons of the planet Jupiter and, with a diameter of , the fourth-largest moon in the Solar System. It was named after the mythological character of Io, a priestess of Hera who became one of the lovers of Zeus....

and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.
Another method is to use the aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

, discovered and explained by James Bradley
James Bradley FRS was an English astronomer and served as Astronomer Royal from 1742, succeeding Edmund Halley. He is best known for two fundamental discoveries in astronomy, the aberration of light , and the nutation of the Earth's axis...

in the 18th century. 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) 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 easily converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled 10,210 times faster than the Earth in its orbit (the modern figure is 10,066 times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.

Nowadays, the "light time for unit distance"—the inverse of c, expressed in seconds per astronomical unit—is measured by comparing the time for radio signals to reach different spacecraft in the Solar System, with their position calculated from the gravitational effects of the Sun and various planets. By combining many such measurements, a best fit value for the light time per unit distance is obtained. , the best estimate, as approved by the International Astronomical Union
International Astronomical Union
The International Astronomical Union IAU is a collection of professional astronomers, at the Ph.D. level and beyond, active in professional research and education in astronomy...

(IAU), is:
light time for unit distance:
c =  =

The relative uncertainty in these measurements is 0.02 parts per billion (2), equivalent to the uncertainty in Earth-based measurements of length by interferometry. Since the metre is defined to be the length travelled by light in a certain time interval, the measurement of the light time for unit distance can also be interpreted as measuring the length of an AU in metres.

### Time of flight techniques

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 the Fizeau–Foucault apparatus developed by Hippolyte Fizeau
Hippolyte Fizeau
Armand Hippolyte Louis Fizeau was a French physicist.-Biography:Fizeau was born in Paris. His earliest work was concerned with improvements in photographic processes. Following suggestions by François Arago, Léon Foucault and Fizeau collaborated in a series of investigations on the interference of...

and Léon Foucault
Léon Foucault
Jean Bernard Léon Foucault was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation...

.

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.

The method of Foucault replaces the cogwheel by 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.

Nowadays, 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.

### Electromagnetic constants

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
Vacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...

μ0 established by Maxwell's theory: c2 = 1/(ε0μ0). The vacuum permittivity may be determined by measuring the capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

and dimensions of a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

, whereas the value of the vacuum permeability
Vacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...

is fixed at exactly through the definition of the ampere. Rosa and Dorsey used this method in 1907 to find a value of .

### Cavity resonance

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 = . 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 determine the wavelength of the wave. In 1946, Louis Essen
Louis Essen
Louis Essen FRS O.B.E. was an English physicist whose most notable achievements were in the precise measurement of time and the determination of the speed of light...

and A.C. Gordon-Smith establish the frequency for a variety of normal mode
Normal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...

s of microwaves of a microwave cavity
Microwave cavity
A microwave cavity is a closed metal structure that confines electromagnetic fields in the microwave region of the spectrum. Such cavities act as resonant circuits with extremely low loss at their frequency of operation...

of precisely known dimensions. The dimensions were established to an accuracy of about ±0.8 μm using gauges calibrated by interferometry. As the wavelength of the modes was known from the geometry of the cavity and from electromagnetic theory, knowledge of the associated frequencies enabled a calculation of the speed of light.

The Essen–Gordon-Smith result, , was substantially more precise than those found by optical techniques. By 1950, repeated measurements by Essen established a result of .

A household demonstration of this technique is possible, using a microwave oven
Microwave oven
A microwave oven is a kitchen appliance that heats food by dielectric heating, using microwave radiation to heat polarized molecules within the food...

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".

### Interferometry

Interferometry
Interferometry
Interferometry refers to a family of techniques in which electromagnetic waves are superimposed in order to extract information about the waves. An instrument used to interfere waves is called an interferometer. Interferometry is an important investigative technique in the fields of astronomy,...

is another method to find the wavelength of electromagnetic radiation for determining the speed of light. A coherent
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

beam of light (e.g. from a laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

), 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. However interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~0.4 cm) 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. 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. This technique was due to a group at the National Bureau of Standards (NBS) (which later became NIST
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

). They used it in 1972 to measure the speed of light in vacuum with a fractional uncertainty
Measurement uncertainty
In metrology, measurement uncertainty is a non-negative parameter characterizing the dispersion of the values attributed to a measured quantity. The uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and a measured...

of .

## History

 1675 Rømer and Huygens, moons of Jupiter 1729 James Bradley, aberration of light 1849 Hippolyte Fizeau, toothed wheel 1862 Léon Foucault, rotating mirror 1907 Rosa and Dorsey, EM constants 1926 Albert Michelson, rotating mirror 1950 , cavity resonator 1958 K.D. Froome, radio interferometry 1972 Evenson et al., laser interferometry 1983 17th CGPM, definition of the metre (exact)

Until the early modern period
Early modern period
In history, the early modern period of modern history follows the late Middle Ages. Although the chronological limits of the period are open to debate, the timeframe spans the period after the late portion of the Middle Ages through the beginning of the Age of Revolutions...

, 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
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

. The ancient Greeks, Muslim 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 concluded that the speed of light is constant regardless of one's frame of reference. Since then, scientists have provided increasingly accurate measurements.

### Early history

Empedocles
Empedocles
Empedocles was a Greek pre-Socratic philosopher and a citizen of Agrigentum, a Greek city in Sicily. Empedocles' philosophy is best known for being the originator of the cosmogenic theory of the four Classical elements...

was the first to claim that the light has a finite speed. He maintained that light was something in motion, and therefore must take some time to travel. Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

argued, to the contrary, that "light is due to the presence of something, but it is not a movement". Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

and Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

Emission theory (vision)
Emission theory or extramission theory is the proposal that visual perception is accomplished by rays of light emitted by the eyes. This theory has been replaced by intromission theory, which states that visual perception comes from something representative of the object entering the eyes...

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.

Early Islamic philosophers
Early Islamic philosophy
Early Islamic philosophy or classical Islamic philosophy is a period of intense philosophical development beginning in the 2nd century AH of the Islamic calendar and lasting until the 6th century AH...

initially agreed with the Aristotelian view
Aristotelian physics
Aristotelian Physics the natural sciences, are described in the works of the Greek philosopher Aristotle . In the Physics, Aristotle established general principles of change that govern all natural bodies; both living and inanimate, celestial and terrestrial—including all motion, change in respect...

that light had no speed of travel. In 1021, Alhazen (Ibn al-Haytham) published the Book of Optics
Book of Optics
The Book of Optics ; ; Latin: De Aspectibus or Opticae Thesaurus: Alhazeni Arabis; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Muslim scholar Alhazen .-See also:* Science in medieval Islam...

, in which he presented a series of arguments dismissing the emission theory in favour of the now accepted intromission theory of vision
Visual perception
Visual perception is the ability to interpret information and surroundings from the effects of visible light reaching the eye. The resulting perception is also known as eyesight, sight, or vision...

, in which light moves from an object into the eye. This led Alhazen to propose that light must have a finite speed, and that the speed of light is variable, decreasing in denser bodies. He argued that light is substantial matter, the propagation of which requires time, even if this is hidden from our senses.

Also in the 11th century, Abū Rayhān al-Bīrūnī agreed that light has a finite speed, and observed that the speed of light is much faster than the speed of sound. Roger Bacon
Roger Bacon
Roger Bacon, O.F.M. , also known as Doctor Mirabilis , was an English philosopher and Franciscan friar who placed considerable emphasis on the study of nature through empirical methods...

argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle. In the 1270s, Witelo
Witelo
Witelo was a friar, theologian and scientist: a physicist, natural philosopher, mathematician. He is an important figure in the history of philosophy in Poland...

considered the possibility of light travelling at infinite speed in vacuum, but slowing down in denser bodies.

In the early 17th century, Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

believed that the speed of light was infinite, since empty space presents no obstacle to it. René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

argued that if the speed of light were finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse
Lunar eclipse
A lunar eclipse occurs when the Moon passes behind the Earth so that the Earth blocks the Sun's rays from striking the Moon. This can occur only when the Sun, Earth, and Moon are aligned exactly, or very closely so, with the Earth in the middle. Hence, a lunar eclipse can only occur the night of a...

. Since such misalignment had not been observed, Descartes concluded the speed of light was infinite. Descartes speculated that if the speed of light were found to be finite, his whole system of philosophy might be demolished.

### First measurement attempts

In 1629, Isaac Beeckman
Isaac Beeckman
Isaac Beeckman was a Dutch philosopher and scientist, who, through his studies and contact with leading natural philosophers, may have "virtually given birth to modern atomism".-Biography:...

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
Galileo Galilei
Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

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. Galileo's experiment was carried out by the Accademia del Cimento
The Accademia del Cimento , an early scientific society, was founded in Florence 1657 by students of Galileo, Giovanni Alfonso Borelli and Vincenzo Viviani. The foundation of Academy was funded by Prince Leopoldo and Grand Duke Ferdinando II de' Medici...

of Florence, Italy, in 1667, with the lanterns separated by about one mile, but no delay was observed. The actual delay in this experiment would have been about 11 microsecond
Microsecond
A microsecond is an SI unit of time equal to one millionth of a second. Its symbol is µs.A microsecond is equal to 1000 nanoseconds or 1/1000 millisecond...

s.

The first quantitative estimate of the speed of light was made in 1676 by Rømer (see Rømer's determination of the speed of light
Rømer's determination of the speed of light
Rømer's determination of the speed of light was the demonstration in 1676 that light has a finite speed, and so doesn't travel instantaneously. The discovery is usually attributed to Danish astronomer Ole Rømer ,There are several alternative spellings of Rømer's surname: Roemer, Rœmer, Römer etc....

). From the observation that the periods of Jupiter's innermost moon Io
Io (moon)
Io ) is the innermost of the four Galilean moons of the planet Jupiter and, with a diameter of , the fourth-largest moon in the Solar System. It was named after the mythological character of Io, a priestess of Hera who became one of the lovers of Zeus....

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 , 26% lower than the actual value.

In his 1704 book Opticks
Opticks
Opticks is a book written by English physicist Isaac Newton that was released to the public in 1704. It is about optics and the refraction of light, and is considered one of the great works of science in history...

, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

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). 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
James Bradley FRS was an English astronomer and served as Astronomer Royal from 1742, succeeding Edmund Halley. He is best known for two fundamental discoveries in astronomy, the aberration of light , and the nutation of the Earth's axis...

discovered the aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

. 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.

### The speed of light and electromagnetism

In the 19th century Hippolyte Fizeau
Hippolyte Fizeau
Armand Hippolyte Louis Fizeau was a French physicist.-Biography:Fizeau was born in Paris. His earliest work was concerned with improvements in photographic processes. Following suggestions by François Arago, Léon Foucault and Fizeau collaborated in a series of investigations on the interference of...

developed a method to determine the speed of light based on time-of-flight measurements on Earth and reported a value of . His method was improved upon by Léon Foucault
Léon Foucault
Jean Bernard Léon Foucault was a French physicist best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation...

who obtained a value of in 1862. In the year 1856, Wilhelm Eduard Weber
Wilhelm Eduard Weber
Wilhelm Eduard Weber was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph.-Early years:...

and Rudolf Kohlrausch
Rudolf Kohlrausch
Rudolf Hermann Arndt Kohlrausch was a German physicist.-Biography:He was a native of Göttingen, the son of educator Heinrich Friedrich Theodor Kohlrausch...

measured the ratio of the electromagnetic and electrostatic units of charge, 1/√ε0μ0, by discharging a Leyden jar
Leyden jar
A Leyden jar, or Leiden jar, is a device that "stores" static electricity between two electrodes on the inside and outside of a jar. It was invented independently by German cleric Ewald Georg von Kleist on 11 October 1745 and by Dutch scientist Pieter van Musschenbroek of Leiden in 1745–1746. The...

, and found that its numerical value was very close to the speed of light as measured directly by Fizeau. The following year Gustav Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

calculated that an electric signal in a resistanceless
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

wire travels along the wire at this speed. In the early 1860s, Maxwell showed that according to the theory of electromagnetism which he was working on, that electromagnetic waves propagate in empty space at a speed equal to the above Weber/Kohrausch 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.

### "Luminiferous aether"

It was thought at the time that empty space was filled with a background medium called the luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....

in which the electromagnetic field existed. Some physicists thought that this aether acted as a preferred frame
Preferred frame
In theoretical physics, a preferred or privileged frame is usually a special hypothetical frame of reference in which the laws of physics might appear to be identifiably different from those in other frames....

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 Michelson and Edward Morley
Edward Morley
Edward Williams Morley was an American scientist famous for the Michelson–Morley experiment.-Biography:...

in 1887. The detected motion was always less than the 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.
Because of this experiment Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...

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
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

. Based on Lorentz's aether theory
Lorentz ether theory
What is now often called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....

, Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

(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
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....

.

### Special relativity

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 parameter, 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.

### Increased accuracy of c and redefinition 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. In 1972, using the latter method and the 1960 definition of the metre in terms of a particular spectral line of krypton-86, a group at NBS
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...