speed of light

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The speed of light in
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Ph ...
, commonly denoted , is a universal
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant (mathematics), constant value in time. It is contrasted with a ...
that is important in many areas of
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
. The speed of light is exactly equal to ). According to the
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
, is the upper limit for the speed at which conventional
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
or
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
(and thus any
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
carrying
information Information is an Abstraction, abstract concept that refers to that which has the power to Communication, inform. At the most fundamental level information pertains to the Interpretation (logic), interpretation of that which may be sensed. ...
) can travel through
space Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...
. All forms of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
, including
visible light Light or visible light is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequency, fr ...
, travel at the speed of light. For many practical purposes, light and other electromagnetic waves will appear to propagate instantaneously, but for long distances and very sensitive measurements, their finite speed has noticeable effects.
Starlight Starlight is the light emitted by stars. It typically refers to visible spectrum, visible electromagnetic radiation from stars other than the Sun, observable from Earth at night, although a component of starlight is observable from Earth during ...
viewed on
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...
left the
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s many years ago, allowing humans to study the history of the universe by viewing distant objects. When
communicating Communication (from la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inquir ...
with distant
space probe A space probe is an Satellite, artificial satellite that travels through space to collect Data, scientific data. A space probe may orbit Earth; approach the Moon; travel through interplanetary space; planetary flyby, flyby, orbit, or landing, land ...
s, it can take minutes to hours for signals to travel from Earth to the spacecraft and vice versa. In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hardware and software. ...
, the speed of light fixes the ultimate minimum communication delay between computers, to
computer memory In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and Digital data, digital Electronics, electronic devices. The term ''memory'' is often synonymous with th ...
, and within a CPU. The speed of light can be used in
time of flight Time of flight (ToF) is the measurement of the time taken by an object, particle or wave (be it acoustic, electromagnetic, etc.) to travel a distance through a medium. This information can then be used to measure velocity or path length, or as a w ...
measurements to measure large distances to extremely high precision. Ole Rømer first demonstrated in 1676 that light travels at a finite speed (non-instantaneously) by studying the apparent motion of
Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
's moon Io. Progressively more accurate measurements of its speed came over the following centuries. In a
paper Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, Textile, rags, poaceae, grasses or other vegetable sources in water, draining the water through fine mesh leaving the fibre e ...
published in 1865,
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...
proposed that light was an
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
and, therefore, travelled at speed . In 1905,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
postulated that the speed of light with respect to any
inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
is a constant and is independent of the motion of the light source. He explored the consequences of that postulate by deriving the theory of relativity and, in doing so, showed that the parameter had relevance outside of the context of light and electromagnetism.
Massless particle In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, glu ...
s and field perturbations, such as
gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that Wave propagation, propagate as waves outward from their source at the speed of light. They were first proposed by Oliv ...
s, also travel at speed in a vacuum. Such particles and waves travel at regardless of the motion of the source or the inertial reference frame of the observer. Particles with nonzero
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system A system is a group of interacting or interrelated elements that act according ...
can be accelerated to approach but can never reach it, regardless of the frame of reference in which their speed is measured. In the special and general theories of relativity, interrelates
space and time Space and Time or Time and Space, or ''variation'', may refer to: * ''Space and time'' or ''time and space'' or ''spacetime'', any mathematical model that combines space and time into a single interwoven continuum * Philosophy of space and time Sp ...
and also appears in the famous equation of
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
, . In some cases, objects or waves may appear to travel faster than light (e.g., phase velocities of waves, the appearance of certain high-speed astronomical objects, and particular quantum effects). The
expansion of the universe The expansion of the universe is the increase in distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an est ...
is understood to exceed the speed of light beyond a certain boundary. The speed at which light propagates through transparent materials, such as glass or air, is less than ; similarly, the speed of
electromagnetic waves In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
in wire cables is slower than . The ratio between and the speed at which light travels in a material is called the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of the material (). 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 .

# Numerical value, notation, and units

The speed of light in vacuum is usually denoted by a lowercase , for "constant" or the Latin (meaning 'swiftness, celerity'). In 1856,
Wilhelm Eduard Weber Wilhelm Eduard Weber (; ; 24 October 1804 – 23 June 1891) was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph. Biography of Wilhelm Early years Weber was born in Schlossstrasse in ...
and Rudolf Kohlrausch had used for a different constant that was later shown to equal times the speed of light in vacuum. Historically, the symbol ''V'' was used as an alternative symbol for the speed of light, introduced by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...
in 1865. In 1894, Paul Drude redefined with its modern meaning.
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
used ''V'' in his original German-language papers on special relativity in 1905, but in 1907 he switched to , which by then had become the standard symbol for the speed of light. "The origins of the letter c being used for the speed of light can be traced back to a paper of 1856 by Weber and Kohlrausch ..Weber apparently meant c to stand for 'constant' in his force law, but there is evidence that physicists such as Lorentz and Einstein were accustomed to a common convention that c could be used as a variable for velocity. This usage can be traced back to the classic Latin texts in which c stood for 'celeritas', meaning 'speed'." Sometimes is used for the speed of waves in any material medium, and 0 for the speed of light in vacuum.See for example: * * * * This subscripted notation, which is endorsed in official SI literature, has the same form as related electromagnetic constants: namely, ''μ''0 for the
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
or magnetic constant, ''ε''0 for the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the Permittivity, absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the el ...
or electric constant, and ''Z''0 for the impedance of free space. This article uses exclusively for the speed of light in vacuum.

## Use in unit systems

Since 1983, the constant has been defined in the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
(SI) as ''exactly'' ; this relationship is used to define the metre as exactly the distance that light travels in a vacuum in of a
second The second (symbol: s) is the unit of Time in physics, time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally t ...
. By using the value of , as well as an accurate measurement of the
second The second (symbol: s) is the unit of Time in physics, time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally t ...
, one can thus establish a standard for the metre.See, for example: * * * As a dimensional physical constant, the numerical value of is different for different unit systems. For example, in
imperial units The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed thro ...
, the speed of light is approximately miles per second, or roughly 1
foot The foot (plural, : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb (anatomy), limb which bears weight and allows animal locomotion, locomotion. In many animals with feet, the foot is a separate o ...
per nanosecond. In branches of physics in which appears often, such as in relativity, it is common to use systems of
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
of measurement or the geometrized unit system where . Using these units, does not appear explicitly because multiplication or division by1 does not affect the result. Its unit of light-second per second is still relevant, even if omitted.

# 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 In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
of the observer.However, the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...
of light can depend on the motion of the source relative to the observer, due to the
Doppler effect The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who ...
.
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 or ether ("luminiferous", meaning "light-bearing") was the postulated Transmission medium, medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty ...
; it has since been consistently confirmed by many experiments.See
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 1887 ...
and Kennedy–Thorndike experiment, for example.
It is only possible to verify experimentally that the two-way speed of light (for example, from a source to a mirror and back again) is frame-independent, because it is impossible to measure the one-way speed of light (for example, from a source to a distant detector) without some convention as to how clocks at the source and at the detector should be synchronized. 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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
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 particle In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, glu ...
s 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 is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
(moving objects shorten), and
time dilation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
(moving clocks run more slowly). The factor ''γ'' by which lengths contract and times dilate is known as the
Lorentz factor The Lorentz factor or Lorentz term is a quantity (physics), quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations ...
and is given by , where ''v'' is the speed of the object. The difference of ''γ'' from1 is negligible for speeds much slower than ''c'', such as most everyday speedsin which case special relativity is closely approximated by
Galilean relativity Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, i ...
but it increases at relativistic speeds and diverges to infinity as ''v'' approaches ''c''. For example, a time dilation factor of ''γ'' = 2 occurs at a relative velocity of 86.6% of the speed of light (''v'' = 0.866 ''c''). Similarly, a time dilation factor of ''γ'' = 10 occurs at 99.5% the speed of light (''v'' = 0.995 ''c''). The results of special relativity can be summarized by treating space and time as a unified structure known as
spacetime In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...
(with ''c'' relating the units of space and time), and requiring that physical theories satisfy a special
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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 In particle physics, quantum electrodynamics (QED) is the relativity theory, 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 m ...
,
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
, the
Standard Model The Standard Model of particle physics Particle physics or high energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The fundamental particles ...
of
particle physics Particle physics or high energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standa ...
, and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
. As such, the parameter ''c'' is ubiquitous in modern physics, appearing in many contexts that are unrelated to light. For example, general relativity predicts that ''c'' is also the speed of gravity and of
gravitational waves Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that Wave propagation, propagate as waves outward from their source at the speed of light. They were first proposed by Oliv ...
, and observations of gravitational waves have been consistent with this prediction. In non-inertial frames of reference (gravitationally curved spacetime or accelerated reference frames), the ''local'' speed of light is constant and equal to ''c'', but the speed of light along a trajectory of finite length 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. 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 is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
, meaning that it has the same value regardless of the direction in which it is measured. Observations of the emissions from nuclear
energy level A Quantum mechanics, quantum mechanical system or particle that is Bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with Classical mechanics, classical pa ...
s as a function of the orientation of the emitting nuclei in a magnetic field (see Hughes–Drever experiment), and of rotating optical resonators (see Resonator experiments) have put stringent limits on the possible two-way
anisotropy Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on ...
.

## Upper limit on speeds

According to special relativity, the energy of an object with
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system A system is a group of interacting or interrelated elements that act according ...
''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. The ''γ'' factor approaches infinity as ''v'' approaches ''c'', and it would take an infinite amount of energy to accelerate an object with mass to the speed of light. The speed of light is the upper limit for the speeds of objects with positive rest mass, and individual photons cannot travel faster than the speed of light. This is experimentally established in many tests of relativistic energy and momentum. More generally, it is impossible for signals 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 In physics, the relativity of simultaneity is the concept that ''distant wikt:simultaneity, simultaneity'' – whether two spatially separated events occur at the same Time in physics, time – is not absolute time and space, absolute ...
. 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 (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
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 A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
es such as the
tachyonic antitelephone A tachyonic antitelephone is a hypothetical device in theoretical physics that could be used to send Signal (electronics), signals into one's own past. Albert Einstein in 1907 presented a thought experiment of how faster-than-light signals can lead ...
.

# Faster-than-light observations and experiments

There are situations in which it may seem that matter, energy, or information-carrying signal travels at speeds greater than ''c'', but they do not. For example, as is discussed in the propagation of light in a medium section below, many wave velocities can exceed ''c''. The
phase velocity The phase velocity of a wave is the rate at which the wave Wave propagation, propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wa ...
of
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
s through most glasses can routinely exceed ''c'', but phase velocity does not determine the velocity at which waves convey 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 EPR may refer to: Science and technology * EPR (nuclear reactor), European Pressurised-Water Reactor * EPR paradox (Einstein–Podolsky–Rosen paradox), in physics * Earth potential rise, in electrical engineering * East Pacific Rise, a mid-ocean ...
. An example involves the
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
s of two particles that can be entangled. Until either of the particles is observed, they exist in a superposition of two quantum states. If the particles are separated and one particle's quantum state is observed, the other particle's quantum state is determined instantaneously. However, it is impossible to control which quantum state the first particle will take on when it is observed, so information cannot be transmitted in this manner. Another quantum effect that predicts the occurrence of faster-than-light speeds is called the Hartman effect: under certain conditions the time needed for a
virtual particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle In quantum mechanics, the uncertainty principle (als ...
to
tunnel A tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A Pipeline transport, pipeline is not a tunnel, though some recent tunnels have used ...
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.
archive
/ref> So-called superluminal motion is seen in certain astronomical objects, such as the
relativistic jet An astrophysical jet is an astronomy, astronomical phenomenon where outflows of Ionization, ionised matter are emitted as an extended beam along the rotation, axis of rotation. When this greatly accelerated matter in the beam approaches the speed ...
quasar A quasar is an extremely Luminosity, luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a m ...
s. However, these jets are not moving at speeds in excess of the speed of light: the apparent superluminal motion is a projection effect caused by objects moving near the speed of light and approaching Earth at a small angle to the line of sight: since the light which was emitted when the jet was farther away took longer to reach the Earth, the time between two successive observations corresponds to a longer time between the instants at which the light rays were emitted. A 2011 experiment where neutrinos were observed to travel faster than light turned out to be due to experimental error. 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 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.

# Propagation of light

In
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, light is described as a type of
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
. The classical behaviour of the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field (physics), field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is ...
is described by
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
, which predict that the speed ''c'' with which electromagnetic waves (such as light) propagate in vacuum is related to the distributed capacitance and inductance of vacuum, otherwise respectively known as the
electric constant Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the Permittivity, absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the el ...
''ε''0 and the
magnetic constant The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant ...
''μ''0, by the equation :$c =\frac \ .$ In modern
quantum physics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
, the electromagnetic field is described by the theory of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativity theory, 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 m ...
(QED). In this theory, light is described by the fundamental excitations (or quanta) of the electromagnetic field, called
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
s. In QED, photons are
massless particle In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, glu ...
s 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, the experimental upper bound for its mass is about 10−57
gram The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram. Originally defined as of 1795 as "the absolute weight of a volume Volume is a measure ...
s; if photon mass is generated by a Higgs mechanism, 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 (QG) is a field of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, ...
. In 2009, the observation of
gamma-ray burst In gamma-ray astronomy, gamma-ray bursts (GRBs) are immensely energetic explosions that have been observed in distant Galaxy, galaxies. They are the most energetic and luminous Electromagnetic pulse, electromagnetic events since the Big Bang. Bu ...
GRB 090510 found no evidence for a dependence of photon speed on energy, supporting tight constraints in specific models of spacetime quantization on how this speed is affected by photon energy for energies approaching the Planck scale.

## 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
(a wave filling the whole space, with only one
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...
) propagate is called the
phase velocity The phase velocity of a wave is the rate at which the wave Wave propagation, propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wa ...
''v''p. A physical signal with a finite extent (a pulse of light) travels at a different speed. The overall
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...
of the pulse travels at the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thr ...
''v''g, and its earliest part travels at the front velocity ''v''f. 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 ''v''p 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 Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
, 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 Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
,
glass Glass is a non-Crystallinity, crystalline, often transparency and translucency, transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most ...
, and
diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material ...
, have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light. In exotic materials like
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low Density, densities is cooled to temperatures very close to absolute zero#Relation with Bose–Einste ...
s near absolute zero, the effective speed of light may be only a few metres per second. However, this represents absorption and re-radiation delay between atoms, as do all slower-than-''c'' speeds in material substances. As an extreme example of light "slowing" in matter, two independent teams of physicists claimed to bring light to a "complete standstill" by passing it through a Bose–Einstein condensate of the element
rubidium Rubidium is the chemical element with the Symbol (chemistry), symbol Rb and atomic number 37. It is a very soft, whitish-grey solid in the alkali metal group, similar to potassium and caesium. Rubidium is the first alkali metal in the group to ha ...
. However, the popular description of light being "stopped" in these experiments refers only to light being stored in the excited states of atoms, then re-emitted at an arbitrarily later time, as stimulated by a second laser pulse. During the time it had "stopped", it had ceased to be light. This type of behaviour is generally microscopically true of all transparent media which "slow" the speed of light. 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 than1 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 of the
dielectric constant The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the vacuum permittivity, electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric co ...
of any material, corresponding respectively to the index of refraction and to the
attenuation coefficient The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
, are linked by the Kramers–Kronig relations. In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly. A pulse with different group and phase velocities (which occurs if the phase velocity is not the same for all the frequencies of the pulse) smears out over time, a process known as dispersion. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called
slow light 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 by the interaction with the medium in which the propagati ...
. The opposite, group velocities exceeding ''c'', was proposed theoretically in 1993 and achieved experimentally in 2000. 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). 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 In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
does that in a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electr ...
material, the electromagnetic equivalent of a
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propaga ...
, known as Cherenkov radiation, is emitted.

# Practical effects of finiteness

The speed of light is of relevance to
communications Communication (from la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inquir ...
: the one-way and round-trip delay time are greater than zero. This applies from small to astronomical scales. On the other hand, some techniques depend on the finite speed of light, for example in distance measurements.

## Small scales

In
computers A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
, the speed of light imposes a limit on how quickly data can be sent between
processors A central processing unit (CPU), also called a central processor, main processor or just Processor (computing), processor, is the electronic circuitry that executes Instruction (computing), instructions comprising a computer program. The CPU per ...
. If a processor operates at 1
gigahertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, me ...
, a signal can travel only a maximum of about in a single clock cycle — in practice, this distance is even shorter since the
printed circuit board A printed circuit board (PCB; also printed wiring board or PWB) is a medium used in Electrical engineering, electrical and electronic engineering to connect electronic components to one another in a controlled manner. It takes the form of a L ...
itself has a refractive index and slows down signals. Processors must therefore be placed close to each other, as well as
memory Memory is the faculty of the mind by which data or information is Encoding (memory), encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If Foresight (psycholo ...
chips, to minimize communication latencies, and care must be exercised when routing wires between them to ensure
signal integrity Signal integrity or SI is a set of measures of the quality of an electrical signal. In digital electronics, a stream of binary values is represented by a voltage (or current) waveform. However, digital signals are fundamentally analog signal, ana ...
. If clock frequencies continue to increase, the speed of light may eventually become a limiting factor for the internal design of single
chips ''CHiPs'' is an American crime film, crime drama television series created by Rick Rosner and originally aired on NBC from September 15, 1977, to May 1, 1983. It follows the lives of two motorcycle police officer, officers of the California Hig ...
.

## Large distances on Earth

Given that the equatorial circumference of the Earth is about and that ''c'' is 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 traveling in
optical fibre An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparency and translucency, transparent fiber made by Drawing (manufacturing), drawing glass (silica) or plastic to a diameter slightly thicker than that of a Hair ...
(a transparent material) the actual transit time is longer, in part because the speed of light is slower by about 35% in optical fibre, depending on its refractive index ''n''. Furthermore, straight lines are rare in global communications and the travel time increases when signals pass through electronic switches or signal regenerators. Although this distance is largely irrelevant for most applications, latency becomes important in fields such as high-frequency trading, where traders seek to gain minute advantages by delivering their trades to exchanges fractions of a second ahead of other traders. For example, traders have been switching to
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequency, frequencies between 300 MHz and 300 GHz respectively. Different sources define different fre ...
communications between trading hubs, because of the advantage which radio waves travelling at near to the speed of light through air have over comparatively slower fibre optic signals.

## Spaceflight and astronomy

Similarly, communications between the Earth and spacecraft are not instantaneous. There is a brief delay from the source to the receiver, which becomes more noticeable as distances increase. This delay was significant for communications between ground control and
Apollo 8 Apollo 8 (December 21–27, 1968) was the first crewed spacecraft to leave low Earth orbit and the first human spaceflight to reach the Moon. The crew orbited the Moon ten times without landing, and then departed safely back to Earth. These t ...
when it became the first crewed spacecraft to orbit the Moon: for every question, the ground control station had to wait at least three seconds for the answer to arrive. 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 later. It would then take a further for commands to travel from Earth to Mars. Receiving light and other signals from distant astronomical sources takes much longer. For example, it takes 13 billion (13) years for light to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep Field images. Those photographs, taken today, capture images of the galaxies as they appeared 13 billion years ago, when the universe was less than a billion years old. The fact that more distant objects appear to be younger, due to the finite speed of light, allows astronomers to infer the evolution of stars, Galaxy formation and evolution, of galaxies, and history of the universe, of the universe itself. Astronomical distances are sometimes expressed in light-years, especially in popular science publications and media. A light-year is the distance light travels in one Julian year (astronomy), Julian year, around 9461 billion kilometres, 5879 billion miles, or 0.3066 parsecs. In round figures, a light year is nearly 10 trillion kilometres or nearly 6 trillion miles. Proxima Centauri, the closest star to Earth after the Sun, is around 4.2 light-years away.Further discussion can be found at

# 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 relative permittivity, ''ε''0 and permeability (electromagnetism), ''μ''0 and using their relation to ''c''. Historically, the most accurate results have been obtained by separately determining the frequency and wavelength of a light beam, with their product equalling ''c''. This is described in more detail in the Speed of light#Interferometry, "Interferometry" section below. In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of of a second", fixing the value of the speed of light at by definition, as #Increased accuracy of c and redefinition of the metre and second, 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 is a convenient setting for measuring the speed of light because of its large scale and nearly perfect
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Ph ...
. Typically, one measures the time needed for light to traverse some reference distance in the Solar System, such as the radius of the Earth's orbit. Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units. Ole Christensen Rømer used an astronomical measurement to make Rømer's determination of the speed of light, the first quantitative estimate of the speed of light in the year 1676.
Translated in
Reproduced in
The account published in ''Journal des sçavans'' was based on a report that Rømer read to the French Academy of Sciences in November 1676 #cohen-1940, (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 of the Earth's orbit around the Sun. The observed change in the moon's orbital period is caused by the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect for
Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
's innermost major moon Io 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, discovered and explained by James Bradley 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 converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled times faster than the Earth in its orbit (the modern figure is times faster) or, equivalently, that it would take light 8 minutes 12 seconds to travel from the Sun to the Earth.

### Astronomical unit

An astronomical unit (AU) is approximately the average distance between the Earth and Sun. It was redefined in 2012 as exactly . Previously the AU was not based on the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
but in terms of the gravitational force exerted by the Sun in the framework of classical mechanics. The current definition uses the recommended value in metres for the previous definition of the astronomical unit, which was determined by measurement. This redefinition is analogous to that of the metre and likewise has the effect of fixing the speed of light to an exact value in astronomical units per second (via the exact speed of light in metres per second). Previously, the inverse of  expressed in seconds per astronomical unit was 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 could be obtained. For example, in 2009, the best estimate, as approved by the International Astronomical Union (IAU), was: :light time for unit distance: ''t''au =  :''c'' =  =  The relative uncertainty in these measurements is 0.02 parts per billion (), 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 in terms of the previous definition of the astronomical unit can also be interpreted as measuring the length of an AU (old definition) 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 and Léon Foucault, based on a suggestion by François Arago. The setup as used by Fizeau consists of a beam of light directed at a mirror 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 with a rotating mirror. Because the mirror keeps rotating while the light travels to the distant mirror and back, the light is reflected from the rotating mirror at a different angle on its way out than it is on its way back. From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated. Today, using oscilloscopes with time resolutions of less than one nanosecond, the speed of light can be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror. This method is less precise (with errors of the order of 1%) than other modern techniques, but it is sometimes used as a laboratory experiment in college physics classes.

## 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 Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the Permittivity, absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the el ...
''ε''0 and
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
''μ''0 established by Maxwell's theory: ''c''2 = 1/(''ε''0''μ''0). The vacuum permittivity may be determined by measuring the capacitance and dimensions of a capacitor, whereas the value of the vacuum permeability was historically fixed at exactly through the definition of the ampere (unit), ampere. Edward Bennett Rosa, Rosa and Noah Ernest Dorsey, Dorsey used this method in 1907 to find a value of . Their method depended upon having a standard unit of electrical resistance, the "international ohm", and so its accuracy was limited by how this standard was defined.

## 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'' = ''fλ''. One option is to measure the resonance frequency of a cavity resonator. If the dimensions of the resonance cavity are also known, these can be used to determine the wavelength of the wave. In 1946, Louis Essen and A.C. Gordon-Smith established the frequency for a variety of normal modes of microwaves of a microwave cavity of precisely known dimensions. The dimensions were established to an accuracy of about ±0.8 μm using gauges calibrated by interferometry. 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 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 is another method to find the wavelength of electromagnetic radiation for determining the speed of light. A Coherence (physics), coherent beam of light (e.g. from a laser), with a known frequency (''f''), is split to follow two paths and then recombined. By adjusting the path length while observing the interference (wave propagation), 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 radiowave, 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 (~) 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 (which later became the National Institute of Standards and Technology). They used it in 1972 to measure the speed of light in vacuum with a Measurement uncertainty, fractional uncertainty of .

# History

Until the early modern period, it was not known whether light travelled instantaneously or at a very fast finite speed. The first extant recorded examination of this subject was in ancient Greece. The ancient Greeks, Arabic scholars, and classical European scientists long debated this until Rømer provided the first calculation of the speed of light. Einstein's Theory of Special Relativity 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 (c. 490–430 BCE) was the first to propose a theory of light and claimed that light has a finite speed. He maintained that light was something in motion, and therefore must take some time to travel. Aristotle argued, to the contrary, that "light is due to the presence of something, but it is not a movement". (click on "Historical background" in the table of contents) Euclid and Ptolemy advanced Empedocles' Emission theory (vision), emission theory of vision, where light is emitted from the eye, thus enabling sight. Based on that theory, Heron of Alexandria argued that the speed of light must be Infinity, infinite because distant objects such as stars appear immediately upon opening the eyes. Early Islamic philosophy, Early Islamic philosophers initially agreed with the Aristotelian physics, Aristotelian view that light had no speed of travel. In 1021, Alhazen (Ibn al-Haytham) published the ''Book of Optics'', in which he presented a series of arguments dismissing the emission theory of Visual perception, vision in favour of the now accepted intromission theory, 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 the 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. In the 13th century, Roger Bacon argued that the speed of light in air was not infinite, using philosophical arguments backed by the writing of Alhazen and Aristotle. In the 1270s, Witelo considered the possibility of light travelling at infinite speed in vacuum, but slowing down in denser bodies. In the early 17th century, Johannes Kepler believed that the speed of light was infinite since empty space presents no obstacle to it. René Descartes argued that if the speed of light were to be finite, the Sun, Earth, and Moon would be noticeably out of alignment during a lunar eclipse. (Although this argument fails when aberration of light is taken into account, the latter was not recognized until the following century.) 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. Despite this, in his derivation of Snell's law, Descartes assumed that some kind of motion associated with light was faster in denser media. Pierre de Fermat derived Snell's law using the opposing assumption, the denser the medium the slower light travelled. Fermat also argued in support of a finite speed of light.

## First measurement attempts

In 1629, Isaac Beeckman proposed an experiment in which a person observes the flash of a cannon reflecting off a mirror about one mile (1.6 km) away. In 1638, Galileo Galilei proposed an experiment, with an apparent claim to having performed it some years earlier, to measure the speed of light by observing the delay between uncovering a lantern and its perception some distance away. He was unable to distinguish whether light travel was instantaneous or not, but concluded that if it were not, it must nevertheless be extraordinarily rapid. In 1667, the Accademia del Cimento of Florence reported that it had performed Galileo's experiment, with the lanterns separated by about one mile, but no delay was observed. The actual delay in this experiment would have been about 11 microseconds. Rømer's determination of the speed of light, The first quantitative estimate of the speed of light was made in 1676 by Ole Rømer. From the observation that the periods of Jupiter's innermost moon Io appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth's orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth's orbit to obtain an estimate of speed of light of , which is 27% lower than the actual value. In his 1704 book ''Opticks'', Isaac Newton reported Rømer's calculations of the finite speed of light and gave a value of "seven or eight minutes" for the time taken for light to travel from the Sun to the Earth (the modern value is 8 minutes 19 seconds). Newton queried whether Rømer's eclipse shadows were coloured; hearing that they were not, he concluded the different colours travelled at the same speed. In 1729, James Bradley discovered aberration of light, stellar aberration. 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.

## Connections with electromagnetism

In the 19th century Hippolyte Fizeau developed a method to determine the speed of light based on time-of-flight measurements on Earth and reported a value of . His method was improved upon by Léon Foucault who obtained a value of in 1862. In the year 1856,
Wilhelm Eduard Weber Wilhelm Eduard Weber (; ; 24 October 1804 – 23 June 1891) was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph. Biography of Wilhelm Early years Weber was born in Schlossstrasse in ...
and Rudolf Kohlrausch measured the ratio of the electromagnetic and electrostatic units of charge, 1/, by discharging a Leyden jar, and found that its numerical value was very close to the speed of light as measured directly by Fizeau. The following year Gustav Kirchhoff calculated that an electric signal in a electrical resistance, resistanceless wire travels along the wire at this speed. In the early 1860s, Maxwell showed that, according to the theory of electromagnetism he was working on, electromagnetic waves propagate in empty space at a speed equal to the above Weber/Kohlrausch ratio, and drawing attention to the numerical proximity of this value to the speed of light as measured by Fizeau, he proposed that light is in fact an electromagnetic wave.

## "Luminiferous aether"

It was thought at the time that empty space was filled with a background medium called the
luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated Transmission medium, medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty ...
in which the electromagnetic field existed. Some physicists thought that this aether acted as a preferred frame of reference for the propagation of light and therefore it should be possible to measure the motion of the Earth with respect to this medium, by measuring the isotropy of the speed of light. Beginning in the 1880s several experiments were performed to try to detect this motion, the most famous of which is Michelson–Morley experiment, the experiment performed by Albert A. Michelson and Edward W. Morley 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 proposed that the motion of the apparatus through the aether may cause the apparatus to Lorentz contraction, contract along its length in the direction of motion, and he further assumed that the time variable for moving systems must also be changed accordingly ("local time"), which led to the formulation of the Lorentz transformation. Based on Lorentz ether theory, Lorentz's aether theory, Henri Poincaré (1900) showed that this local time (to first order in ''v''/''c'') is indicated by clocks moving in the aether, which are synchronized under the assumption of constant light speed. In 1904, he speculated that the speed of light could be a limiting velocity in dynamics, provided that the assumptions of Lorentz's theory are all confirmed. In 1905, Poincaré brought Lorentz's aether theory into full observational agreement with the principle of relativity.

## 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 physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
, in which the speed of light in vacuum ''c'' featured as a fundamental constant, also appearing in contexts unrelated to light. This made the concept of the stationary aether (to which Lorentz and Poincaré still adhered) useless and revolutionized the concepts of space and time.

## Increased accuracy of ''c'' and redefinition of the metre and second

In the second half of the 20th century, much progress was made in increasing the accuracy of measurements of the speed of light, first by cavity resonance techniques and later by laser interferometer techniques. These were aided by new, more precise, definitions of the metre and second. In 1950, Louis Essen determined the speed as , using cavity resonance. This value was adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. In 1960, the history of the metre#Krypton standard, metre was redefined in terms of the wavelength of a particular spectral line of krypton-86, and, in 1967, the
second The second (symbol: s) is the unit of Time in physics, time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally t ...
was redefined in terms of the hyperfine transition frequency of the ground state of caesium-133. In 1972, using the laser interferometer method and the new definitions, a group at the US National Institute of Standards and Technology, National Bureau of Standards in Boulder, Colorado determined the speed of light in vacuum to be ''c'' = . This was 100 times less uncertain than the previously accepted value. The remaining uncertainty was mainly related to the definition of the metre. As similar experiments found comparable results for ''c'', the 15th General Conference on Weights and Measures in 1975 recommended using the value for the speed of light.

## Defined as an explicit constant

In 1983 the 17th meeting of the General Conference on Weights and Measures (CGPM) found that wavelengths from frequency measurements and a given value for the speed of light are more reproducibility, reproducible than the previous standard. They kept the 1967 definition of second, so the caesium Hyperfine structure#Use in defining the SI second and meter, hyperfine frequency would now determine both the second and the metre. To do this, they redefined the metre as "the length of the path traveled by light in vacuum during a time interval of 1/ of a second." As a result of this definition, the value of the speed of light in vacuum is exactly and has become a defined constant in the SI system of units. Improved experimental techniques that, prior to 1983, would have measured the speed of light no longer affect the known value of the speed of light in SI units, but instead allow a more precise realization of the metre by more accurately measuring the wavelength of krypton-86 and other light sources. In 2011, the CGPM stated its intention to redefine all seven SI base units using what it calls "the explicit-constant formulation", where each "unit is defined indirectly by specifying explicitly an exact value for a well-recognized fundamental constant", as was done for the speed of light. It proposed a new, but completely equivalent, wording of the metre's definition: "The metre, symbol m, is the unit of length; its magnitude is set by fixing the numerical value of the speed of light in vacuum to be equal to exactly when it is expressed in the SI unit ." This was one of the changes that was incorporated in the 2019 redefinition of the SI base units, also termed the ''New SI''.See, for example: * * *

* Light-second * Speed of electricity * Speed of gravity * Speed of sound * Velocity factor * Warp drive, Warp factor (fictional)

# References

## Historical references

* ** Translated as * * * * * * *

## Modern references

* * * * * *

"Test Light Speed in Mile Long Vacuum Tube."
''Popular Science Monthly'', September 1930, pp. 17–18.

(International Bureau of Weights and Measures, BIPM)
Speed of light in vacuum
(National Institute of Standards and Technology, NIST)

Subluminal
(Java applet by Greg Egan demonstrating group velocity information limits)