Tip:
Highlight text to annotate it
X
The speed of light in vacuum, commonly denoted c, is a universal physical
constant important in many areas of physics. Its value is exactly 299,792,458
metres per second, a figure that is exact because the length of the metre is
defined from this constant and the international standard for time. This is,
to three significant figures, 186,000 miles per second, or about 671 million
miles per hour. According to special relativity, c is the maximum speed at which
all energy, matter, and information in the universe can travel. It is the speed
at which all massless particles and associated fields (including electromagnetic
radiation such as light) travel in vacuum. It is also the speed of gravity (i.e.
of gravitational waves) predicted by current theories. Such particles and waves
travel at c regardless of the motion of the source or the inertial frame of
reference of the observer. In the theory of relativity, c interrelates space and
time, and also 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 n of the material (n = c / v).
For example, for visible light the refractive index of glass is typically around
1.5, meaning that light in glass travels at c / 1.5 ≈ 200,000 km/s; the
refractive index of air for visible light is 1.000293, so the speed of light in
air is 299,705 km/s or about 88 km/s slower than c.
In most practical cases, light and other electromagnetic waves can be thought of
as moving "instantaneously", but for long distances and very sensitive
measurements their finite speed has noticeable effects. For example, in videos
of an intense lightning storm on the Earth's surface taken from the
International Space Station, the expansion of light wavefronts from individual
flashes of lightning is clearly visible, and allows estimates of the speed of
light to be made from frame-to-frame analysis of the position of the light
wavefront. This is not surprising, as the time for light to propagate completely
around the Earth is of the order of 140 milliseconds. This transit time is what
causes the Schumann resonance. 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's moon Io.
In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave,
and therefore travelled at the speed c appearing in his theory of
electromagnetism. In 1905, Albert Einstein 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 299,792,458
m/s with a measurement uncertainty of 4 parts
per billion. In 1983, the metre was redefined in the International System of Units
(SI) as the distance travelled by light in vacuum in 1/299,792,458 of a second.
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 celeritas (meaning "swiftness"). (Capital C is the SI unit for coulomb of
electric charge.) Originally, the symbol V was used for the speed of light,
introduced by James Clerk Maxwell in 1865. In 1856, Wilhelm Eduard Weber and
Rudolf Kohlrausch had used c for a different constant later shown to equal √2
times the speed of light in vacuum. In 1894, Paul Drude redefined c with its
modern meaning. Einstein used V in his original German-language papers on
special relativity in 1905, but in 1907 he switched to c, which by then had
become the standard symbol.
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 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.
Since 1983, the metre has been defined in the International System of Units (SI)
as the distance light travels in vacuum in 1/299,792,458 of a second. This
definition fixes the speed of light in vacuum at exactly 299,792,458 m/s.
As a dimensional physical constant, the numerical value of c is different for
different unit systems.[Note 1] In branches of physics in which c appears often,
such as in relativity, it is common to use systems of natural units of
measurement or the geometrized unit system where c = 1. 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 of the observer.[Note 2]
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; 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 (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.
The Lorentz factor γ as a function of velocity. It starts at 1 and approaches
infinity as v approaches c.
Special relativity has many counterintuitive and experimentally verified
implications. These include the equivalence of mass and energy (E = mc2),
length contraction (moving objects shorten),[Note 3] and time dilation (moving
clocks run more slowly). The factor γ by which lengths contract and times dilate
is known as the Lorentz factor and is given by γ = (1 − v2/c2)−1/2, where v is
the speed of the object. The difference of γ from 1 is negligible for speeds
much slower than c, such as most everyday speeds—in which case special
relativity is closely approximated by Galilean relativity—but it increases at
relativistic speeds and diverges to infinity as v approaches c.
The results of special relativity can be summarized by treating space and time
as a unified structure known as spacetime (with c relating the units of space
and time), and requiring that physical theories satisfy a special symmetry
called Lorentz invariance, whose mathematical formulation contains the parameter
c. Lorentz invariance is an almost universal assumption for modern physical
theories, such as quantum electrodynamics, quantum chromodynamics, the Standard
Model of particle physics, and general relativity. As such, the parameter c is
ubiquitous in modern physics, appearing in many contexts that are unrelated to
light. For example, general relativity predicts that c is also the speed of
gravity and of gravitational waves. 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 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, meaning that
it has the same value regardless of the direction in which it is measured.
Observations of the emissions from nuclear energy levels as a function of the
orientation of the emitting nuclei in a magnetic field (see Hughes–Drever
experiment), and of rotating optical resonators (see Resonator experiments) have
put stringent limits on the possible two-way anisotropy. [31]
Upper limit on speeds
According to special relativity, the energy of an object with rest mass m and
speed v is given by γmc2, where γ is the Lorentz factor defined above. When v is
zero, γ is equal to one, giving rise to the famous E = mc2 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. This is experimentally established in many tests of
relativistic energy and momentum.[32]
Event A precedes B in the red frame, is simultaneous with B in the green frame,
and follows B in the blue frame.
More generally, it is 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. If
the spatial distance between two events A and B is greater than the time
interval between them multiplied by c then there are frames of reference in
which A precedes B, others in which B precedes A, and others in which they are
simultaneous. As a result, if something were travelling faster than c relative
to an inertial frame of reference, it would be travelling backwards in time
relative to another frame, and causality would be violated.[Note 4][34] 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 paradoxes
such as the tachyonic antitelephone.[35]
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 of X-rays through most glasses can
routinely exceed c,[36] but phase velocity does not determine the velocity at
which waves convey information.[37]
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.[38] In neither case does any matter, energy, or
information travel faster than light.[39]
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.[39]
Certain quantum effects appear to be transmitted instantaneously and therefore
faster than c, as in the EPR paradox. An example involves the quantum states of
two particles that can be entangled. Until either of the particles is observed,
they exist in a superposition of two quantum states. If the particles are
separated and one particle's quantum state is observed, the other particle's
quantum state is determined instantaneously (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.[39][40]
Another quantum effect that predicts the occurrence of faster-than-light speeds
is called the Hartman effect; under certain conditions the time needed for a
virtual particle to tunnel through a barrier is constant, regardless of the
thickness of the barrier.[41][42] This could result in a virtual particle
crossing a large gap faster-than-light. However, no information can be sent
using this effect.[43]
So-called superluminal motion is seen in certain astronomical objects,[44] such
as the relativistic jets of radio galaxies and quasars. However, these jets are
not moving at speeds in excess of the speed of light: the apparent superluminal
motion is a projection effect caused by objects moving near the speed of light
and approaching Earth at a small angle to the line of sight: since the light
which was emitted when the jet was farther away took longer to reach the Earth,
the time between two successive observations corresponds to a longer time
between the instants at which the light rays were emitted.[45]
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.[39] 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.[46]
In September 2011, physicists working on the OPERA experiment published results
that suggested beams of neutrinos had travelled from CERN (in Geneva,
Switzerland) to LNGS (at the Gran Sasso, Italy) faster than the speed of light.[47]
These findings, sometimes referred to as the faster-than-light neutrino anomaly,
were subsequently determined—subject to further confirmation—to be the result of
a measurement error.[48]
Propagation of light
In classical physics, light is described as a type of electromagnetic wave. The
classical behaviour of the electromagnetic field is described by Maxwell's
equations, which predict that the speed c with which electromagnetic waves (such
as light) propagate through the vacuum is related to the electric constant ε0
and the magnetic constant μ0 by the equation c = 1/√ε0μ0.[49] In modern quantum
physics, the electromagnetic field is described by the theory of quantum
electrodynamics (QED). In this theory, light is described by the fundamental
excitations (or quanta) of the electromagnetic field, called photons. In QED,
photons are massless particles and thus, according to special relativity, they
travel at the speed of light in vacuum.
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,[50][51][52] 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,[53] the experimental upper bound for its mass is about 10−57 grams;[54]
if photon mass is generated by a Higgs mechanism, the experimental upper limit
is less sharp, m ≤ 10−14 eV/c2 [53] (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. 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 = √ħG/c3 ≈ 1.6163×10−35 m)
divided by 1.2.[55]
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 (a wave filling the
whole space, with only one frequency) propagate is called 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 vg,
and its earliest part travels at the front velocity vf.
The blue dot moves at the speed of the ripples, the phase velocity; the green
dot moves with the speed of the envelope, the group velocity; and the red dot
moves with the speed of the foremost part of the pulse, the front velocity
The phase velocity is important in determining how a light wave travels through
a material or from one material to another. It is often represented in terms of
a refractive index. The refractive index of a material is defined as the ratio
of c to the phase velocity vp in the material: larger indices of refraction
indicate lower speeds. The refractive index of a material may depend on the
light's frequency, intensity, polarization, or direction of propagation; in many
cases, though, it can be treated as a material-dependent constant. The
refractive index of air is approximately 1.0003.[56] Denser media, such as water,[57]
glass,[58] and diamond,[59] have refractive indexes of around 1.3, 1.5 and 2.4,
respectively, for visible light. In exotic materials like Bose–Einstein
condensates near absolute zero, the effective speed of light may be only a few
metres per second. However, this represents absorption and re-radiation delay
between atoms, as do all slower-than-c speeds in material substances. As an
extreme example of this, 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, one team at Harvard
University and the Rowland Institute for Science in Cambridge, Mass., and the
other at the Harvard–Smithsonian Center for Astrophysics, also in Cambridge.
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.[60]
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.[61] The requirement that causality is not violated implies that
the real and imaginary parts of the dielectric constant of any material,
corresponding respectively to the index of refraction and to the attenuation
coefficient, are linked by the Kramers–Kronig relations.[62] 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. Certain materials have an exceptionally low
(or even zero) group velocity for light waves, a phenomenon called slow light,
which has been confirmed in various experiments.[63][64][65][66] The opposite,
group velocities exceeding c, has also been shown in experiment.[67] It should
even be possible for the group velocity to become infinite or negative, with
pulses travelling instantaneously or backwards in time.[68]
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.[68]
It is possible for a particle to travel through a medium faster than the phase
velocity of light in that medium (but still slower than c). When a charged
particle does that in a dielectric material, the electromagnetic equivalent of a
shock wave, known as Cherenkov radiation, is emitted.[69]
Practical effects of finiteness
The speed of light is of relevance to communications: 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 supercomputers, the speed of light imposes a limit on how quickly data can be
sent between processors. If a processor operates at 1 gigahertz, a signal can
only travel a maximum of about 30 centimetres (1 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.[70]
Large distances on Earth
For example, given the equatorial circumference of the Earth is about 40,075 km
and c about 300,000 km/s, the theoretical shortest time for a piece of
information to travel half the globe along the surface is about 67 milliseconds.
When light is 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.[71] Furthermore,
straight lines rarely occur in global communications situations, and delays are
created when the signal passes through an electronic switch or signal
regenerator.[72]
Spaceflights and astronomy
A beam of light is depicted travelling between the Earth and the Moon in the
time it takes a light pulse to move between them: 1.255 seconds at their mean
orbital (surface-to-surface) distance. The relative sizes and separation of the
Earth–Moon system are shown to scale.
Similarly, communications between the Earth and spacecraft are not instantaneous.
There is a brief delay from the source to the receiver, which becomes more
noticeable as distances increase. This delay was significant for communications
between ground control and Apollo 8 when it became the first 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.[73] 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.
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.
Receiving light and other signals from distant astronomical sources can even
take much longer. For example, it has taken 13 billion (13×109) years for light
to travel to Earth from the faraway galaxies viewed in the Hubble Ultra Deep
Field images.[74][75] 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.[74] The fact that more distant objects appear to be younger,
due to the finite speed of light, allows astronomers to infer the evolution of
stars, of galaxies, and of the universe itself.
Astronomical distances are sometimes expressed in light-years, especially in
popular science publications and media.[76] A light-year is the distance light
travels in one 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.[77]
Distance measurement
Radar systems measure the distance to a target by the time it takes a radio-wave
pulse to return to the radar antenna after being reflected by the target: the
distance to the target is half the round-trip transit time multiplied by the
speed of light. A Global Positioning System (GPS) receiver measures its distance
to GPS satellites based on how long it takes for a radio signal to arrive from
each satellite, and from these distances calculates the receiver's position.
Because light travels about 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, radar astronomy and the Deep Space Network
determine distances to the Moon,[78] planets[79] and spacecraft,[80]
respectively, by measuring round-trip transit times.
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
equalling 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",[81] fixing the
value of the speed of light at 299,792,458 m/s by definition, as described below.
Consequently, accurate measurements of the speed of light yield an accurate
realization of the metre rather than an accurate value of c.
Astronomical measurements
Outer space is a natural setting for measuring the speed of light because of its
large scale and nearly perfect vacuum. Typically, one measures the time needed
for light to traverse some reference distance in the solar system, such as the
radius of the Earth's orbit. Historically, such measurements could be made
fairly accurately, compared to how accurately the length of the reference
distance is known in Earth-based units. It is customary to express the results
in astronomical units (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.[Note 5] 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.[84][85] 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 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 and deduced that light takes 22 minutes to cross the diameter
of the Earth's orbit.
Aberration of light: light from a distant source appears to be from a different
location for a moving telescope due to the finite speed of light.
Another method is to use the aberration of light, discovered and explained by
James Bradley in the 18th century.[86] 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)[87] 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 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.[86]
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. As of 2009, the best estimate, as approved by the
International Astronomical Union (IAU), is:[88][89]
light time for unit distance: 499.004783836(10) s
c = 0.00200398880410(4) AU/s = 173.144632674(3) AU/day.
The relative uncertainty in these measurements is 0.02 parts per billion (2×10−11),
equivalent to the uncertainty in Earth-based measurements of length by
interferometry.[90][Note 6] 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.[Note 7]
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.
Diagram of the Fizeau apparatus
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.[91]
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.[92]
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.[93][94][95]
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 μ0 established by Maxwell's
theory: c2 = 1/(ε0μ0). The vacuum permittivity may be determined by measuring
the capacitance and dimensions of a capacitor, whereas the value of the vacuum
permeability is fixed at exactly 4π×10−7 H⋅m−1 through the definition of the
ampere. Rosa and Dorsey used this method in 1907 to find a value of 299,710±22 km/s.[96][97]
Cavity resonance
Electromagnetic standing waves in a cavity.
Another way to measure the speed of light is to independently measure the
frequency f and wavelength λ of an electromagnetic wave in vacuum. The value of
c can then be found by using the relation c = 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 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.[96] 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.[96][98]
The Essen–Gordon-Smith result, 299,792±9 km/s, was substantially more precise
than those found by optical techniques.[96] By 1950, repeated measurements by
Essen established a result of 299,792.5±3.0 km/s.[99]
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".[100][101]
Interferometry
An interferometric determination of length. Left: constructive interference;
Right: destructive interference.
Interferometry is another method to find the wavelength of electromagnetic
radiation for determining the speed of light.[102] A coherent beam of light (e.g.
from a laser), with a known frequency (f), is split to follow two paths and then
recombined. By adjusting the path length while observing the interference
pattern and carefully measuring the change in path length, the wavelength of the
light (λ) can be determined. The speed of light is then calculated using the
equation c = λf.
Before the advent of laser technology, coherent radio sources were used for
interferometry measurements of the speed of light.[103] 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.[104] This technique was
due to a group at the National Bureau of Standards (NBS) (which later became
NIST). They used it in 1972 to measure the speed of light in vacuum with a
fractional uncertainty of 3.5×10−9.[104][105]
History History of measurements of c (in km/s) 1675
Rømer and Huygens, moons of Jupiter 220,000[85][106] 1729 James Bradley, aberration of light 301,000[91]
1849 Hippolyte Fizeau, toothed wheel 315,000[91] 1862 Léon Foucault, rotating mirror 298,000±500[91]
1907 Rosa and Dorsey, EM constants 299,710±30[96][97] 1926 Albert Michelson, rotating mirror 299,796±4[107]
1950 Essen and Gordon-Smith, cavity resonator 299,792.5±3.0[99]
1958 K.D. Froome, radio interferometry 299,792.50±0.10[103] 1972 Evenson et al., laser interferometry
299,792.4562±0.0011[105] 1983 17th CGPM, definition of the metre 299,792.458
(exact)[81]
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, 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 was the first to claim that light has a finite speed.[108] 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".[109] Euclid and Ptolemy advanced the
emission theory of vision, where light is emitted from the eye, thus enabling
sight. Based on that theory, Heron of Alexandria argued that the speed of light
must be infinite because distant objects such as stars appear immediately upon
opening the eyes.
Early Islamic philosophers initially agreed with the Aristotelian view that
light had no speed of travel. In 1021, Alhazen (Ibn al-Haytham) published the
Book of Optics, in which he presented a series of arguments dismissing the
emission theory in favour of the now accepted intromission theory of vision, in
which light moves from an object into the eye.[110] This led Alhazen to propose
that light must have a finite speed,[109][111][112] and that the speed of light
is variable, decreasing in denser bodies.[112][113] He argued that light is
substantial matter, the propagation of which requires time, even if this is
hidden from our senses.[114] 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.[115]
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.[116][117] In the 1270s, Witelo considered the possibility of light
travelling at infinite speed in vacuum, but slowing down in denser bodies.[118]
In the early 17th century, Johannes Kepler believed that the speed of light was
infinite, since empty space presents no obstacle to it. René Descartes argued
that if the speed of light were finite, the Sun, Earth, and Moon would be
noticeably out of alignment during a lunar eclipse. Since such misalignment had
not been observed, Descartes concluded the speed of light was infinite.
Descartes speculated that if the speed of light were found to be finite, his
whole system of philosophy might be demolished.[109] In Descartes' derivation of
Snell's law, he assumed that even though the speed of light was instantaneous,
the more dense the medium, the faster was light's speed.[119] Pierre de Fermat
derived Snell's law using the opposing assumption, the more dense the medium the
slower light traveled. Fermat also argued in support of a finite speed of light.[120]
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.[121][122]
Galileo's experiment was carried out by the Accademia del Cimento 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
microseconds.
Rømer's observations of the occultations of Io from Earth
The first quantitative estimate of the speed of light was made in 1676 by Rømer
(see Rømer's determination of the speed of light).[84][85] 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 220,000 km/s, 26% lower than the actual value.[106]
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).[123] 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 the aberration of
light.[86] 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.[86]
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 315,000 km/s.
His method was improved upon by Léon Foucault who obtained a value of 298,000 km/s
in 1862.[91] In the year 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch
measured the ratio of the electromagnetic and electrostatic units of charge, 1/√ε0μ0,
by discharging a Leyden jar, and found that its numerical value was very close
to the speed of light as measured directly by Fizeau. The following year Gustav
Kirchhoff calculated that an electric signal in a resistanceless wire travels
along the wire at this speed.[124] In the early 1860s, Maxwell showed that,
according to the theory of electromagnetism he was working on, electromagnetic
waves propagate in empty space[125][126][127] 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.[128]
"Luminiferous aether"
Hendrik Lorentz (right) with Albert Einstein.
It was thought at the time that empty space was filled with a background medium
called the luminiferous aether 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 the experiment performed
by Albert Michelson and Edward Morley in 1887.[129] 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.[130] Because of this experiment Hendrik Lorentz proposed
that the motion of the apparatus through the aether may cause the apparatus to
contract along its length in the direction of motion, and he further assumed,
that the time variable for moving systems must also be changed accordingly ("local
time"), which led to the formulation of the Lorentz transformation. Based on
Lorentz's aether theory, Henri Poincaré (1900) showed that this local time (to
first order in v/c) is indicated by clocks moving in the aether, which are
synchronized under the assumption of constant light speed. In 1904, he
speculated that the speed of light could be a limiting velocity in dynamics,
provided that the assumptions of Lorentz's theory are all confirmed. In 1905,
Poincaré brought Lorentz's aether theory into full observational agreement with
the principle of relativity.[131][132]
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 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.[133][134]
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 1960, the metre was
redefined in terms of the wavelength of a particular spectral line of krypton-86,
and, in 1967, the second was redefined in terms of the hyperfine transition
frequency of the ground state of caesium-133.
In 1972, using the laser interferometer method and the new definitions, a group
at NBS in Boulder, Colorado determined the speed of light in vacuum to be c = 299,792,456.2±1.1
m/s. This was 100 times less uncertain than the
previously accepted value. The remaining uncertainty was mainly related to
the definition of the metre.[Note 8][105] As similar experiments found comparable results
for c, the 15th Conférence Générale des Poids et Mesures (CGPM) in
1975 recommended using the value 299,792,458 m/s
for the speed of light.[137]
Defining the speed of light as an explicit constant
In 1983 the 17th CGPM found that wavelengths from frequency measurements and a
given value for the speed of light are more reproducible than the previous
standard. They kept the 1967 standard for time, so the Caesium hyperfine
frequency would now determine both the second and the metre. To do this, they
redefined the metre thus, "The metre is the length of the path travelled by
light in vacuum during a time interval of 1/299 792 458 of a second."[81] As a
result of this definition, the value of the speed of light in vacuum is exactly 299,792,458
m/s[138][139] 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 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.[140][141]
In 2011, the CGPM said it intends to define 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 299 792 458 when it is
expressed in the SI unit m s–1." [142]