The speed of light in vacuum, commonly denoted
c, is a universal physical constant important
in many areas of physics. Its exact value
is 299,792,458 metres per second (approximately
300,000 km/s (186,000 mi/s)). It is exact
because by international agreement a metre
is defined to be the length of the path travelled
by light in vacuum during a time interval
of 1/299792458 second. According to special
relativity, c is the maximum speed at which
all conventional matter and hence all known
forms of information in the universe can travel.
Though this speed is most commonly associated
with light, it is in fact the speed at which
all massless particles and changes of the
associated fields travel in vacuum (including
electromagnetic radiation and gravitational
waves). Such particles and waves travel at
c regardless of the motion of the source or
the inertial reference frame of the observer.
In the special and general theories 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; similarly, the speed
of electromagnetic waves in wire cables is
slower than c. The ratio between c and the
speed v at which light travels in a material
is called the refractive index n of the material
(n = c / v). For example, for visible light
the refractive index of glass is typically
around 1.5, meaning that light in glass travels
at c / 1.5 ≈ 200,000 km/s (124,000 mi/s);
the refractive index of air for visible light
is about 1.0003, so the speed of light in
air is about 299,700 km/s (186,220 mi/s),
which is about 90 km/s (56 mi/s) slower than
c.
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. 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 seen
from stars left them many years ago, allowing
the study of 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.
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 travels 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 c with respect to any inertial
frame 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 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
299792458 m/s (983571056 ft/s; 186282.397
mi/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 1/299792458 of a second.
== Numerical value, notation, and units ==
The speed of light in vacuum is usually denoted
by a lowercase c, for "constant" or the Latin
celeritas (meaning "swiftness, celerity").
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. Historically, the symbol V was
used as an alternative symbol for the speed
of light, introduced by James Clerk Maxwell
in 1865. 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 for the speed of light.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 ​1⁄299792458
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.
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. 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.
Special relativity has many counterintuitive
and experimentally verified implications.
These include the equivalence of mass and
energy (E = mc2), length contraction (moving
objects shorten), and time dilation (moving
clocks run more slowly). The factor γ by
which lengths contract and times dilate is
known as the Lorentz factor and is given by
γ = (1 − v2/c2)−1/2, where v is the speed
of the object. The difference of γ from 1
is negligible for speeds much slower than
c, such as most everyday speeds—in which
case special relativity is closely approximated
by Galilean relativity—but it increases
at relativistic speeds and diverges to infinity
as v approaches c. For example, a time dilation
factor of γ = 2 occurs at a relative velocity
of 86.6% of the speed of light (v = .866c).
Similarly, a time dilation factor of γ = 10
occurs at v = 99.5% 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 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, 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.
=== 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, 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 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.
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.
== 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, 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. 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.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.
This could result in a virtual particle crossing
a large gap faster-than-light. However, no
information can be sent using this effect.So-called
superluminal motion is seen in certain astronomical
objects, 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.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, 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 distributed capacitance
and inductance of the vacuum, otherwise respectively
known as the electric constant ε0 and the
magnetic constant μ0, by the equation
c
=
1
ε
0
μ
0
.
{\displaystyle c={\frac {1}{\sqrt {\varepsilon
_{0}\mu _{0}}}}\ .}
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, 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
grams; if photon mass is generated by a Higgs
mechanism, the experimental upper limit is
less sharp, m ≤ 10−14 eV/c2 (roughly 2
× 10−47 g).
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.
=== 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 phase velocity is important in determining
how a light wave travels through a material
or from one material to another. It is often
represented in terms of a refractive index.
The refractive index of a material is defined
as the ratio of c to the phase velocity vp
in the material: larger indices of refraction
indicate lower speeds. The refractive index
of a material may depend on the light's frequency,
intensity, polarization, or direction of propagation;
in many cases, though, it can be treated as
a material-dependent constant. The refractive
index of air is approximately 1.0003. Denser
media, such as water, glass, and diamond,
have refractive indexes of around 1.3, 1.5
and 2.4, respectively, for visible light.
In exotic materials like Bose–Einstein condensates
near absolute zero, the effective speed of
light may be only a few metres per second.
However, this represents absorption and re-radiation
delay between atoms, as do all slower-than-c
speeds in material substances. As an extreme
example of light "slowing" in matter, two
independent teams of physicists claimed to
bring light to a "complete standstill" by
passing it through a Bose–Einstein condensate
of the element rubidium, 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.In transparent
materials, the refractive index generally
is greater than 1, meaning that the phase
velocity is less than c. In other materials,
it is possible for the refractive index to
become smaller than 1 for some frequencies;
in some exotic materials it is even possible
for the index of refraction to become negative.
The requirement that causality is not violated
implies that the real and imaginary parts
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.
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.
The opposite, group velocities exceeding c,
has also been shown in experiment. It should
even be possible for the group velocity to
become infinite or negative, with pulses travelling
instantaneously or backwards in time.
None of these options, however, allow information
to be transmitted faster than c. It is impossible
to transmit information with a light pulse
any faster than the speed of the earliest
part of the pulse (the front velocity). 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 does that in a
dielectric material, the electromagnetic equivalent
of a shock wave, known as Cherenkov radiation,
is emitted.
== 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.
=== Large distances on Earth ===
Given that the equatorial circumference of
the Earth is about 40075 km and that c is
about 300000 km/s, the theoretical shortest
time for a piece of information to travel
half the globe along the surface is about
67 milliseconds. When light is travelling
around the globe in an optical fibre, the
actual transit time is longer, in part because
the speed of light is slower by about 35%
in an optical fibre, depending on its refractive
index n. Furthermore, straight lines rarely
occur in global communications situations,
and delays are created when the signal passes
through an electronic switch or signal regenerator.
=== Spaceflights 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 when it
became the first manned spacecraft to orbit
the Moon: for every question, the ground control
station had to wait at least three seconds
for the answer to arrive. The communications
delay between Earth and Mars can vary between
five and twenty minutes depending upon the
relative positions of the two planets. As
a consequence of this, if a robot on the surface
of Mars were to encounter a problem, its human
controllers would not be aware of it until
at least five minutes later, and possibly
up to twenty minutes later; it would then
take a further five to twenty minutes for
instructions to travel from Earth to Mars.
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. Those photographs, taken
today, capture images of the galaxies as they
appeared 13 billion years ago, when the universe
was less than a billion years old. The fact
that more distant objects appear to be younger,
due to the finite speed of light, allows astronomers
to infer the evolution of stars, of galaxies,
and 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 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.
=== Distance measurement ===
Radar systems measure the distance to a target
by the time it takes a radio-wave pulse to
return to the radar antenna after being reflected
by the target: the distance to the target
is half the round-trip transit time multiplied
by the speed of light. A Global Positioning
System (GPS) receiver measures its distance
to GPS satellites based on how long it takes
for a radio signal to arrive from each satellite,
and from these distances calculates the receiver's
position. Because light travels about 300000
kilometres (186000 mi) 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, planets
and spacecraft, respectively, by measuring
round-trip transit times.
=== High-frequency trading ===
The speed of light has become important in
high-frequency trading, where traders seek
to gain minute advantages by delivering their
trades to exchanges fractions of a second
ahead of other traders. For example, traders
have been switching to microwave communications
between trading hubs, because of the advantage
which microwaves travelling at near to the
speed of light in air, have over fibre optic
signals which travel 30–40% slower at the
speed of light through glass.
== 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 ​1⁄299792458 of a second",
fixing the value of the speed of light at
299792458 m/s by definition, as described
below. Consequently, accurate measurements
of the speed of light yield an accurate realization
of the metre rather than an accurate value
of c.
=== Astronomical measurements ===
Outer space is a convenient setting for measuring
the speed of light because of its large scale
and nearly perfect vacuum. Typically, one
measures the time needed for light to traverse
some reference distance in the solar system,
such as the radius of the Earth's orbit. Historically,
such measurements could be made fairly accurately,
compared to how accurately the length of the
reference distance is known in Earth-based
units. It is customary to express the results
in astronomical units (AU) per day.
Ole Christensen Rømer used an astronomical
measurement to make the first quantitative
estimate of the speed of light. 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's innermost 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 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.
==== Astronomical unit ====
An astronomical unit (AU) is approximately
the average distance between the Earth and
Sun. It was redefined in 2012 as exactly 149597870700
m. Previously the AU was not based on the
International System of Units 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 c 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: tau = 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. 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.
The setup as used by Fizeau consists of a
beam of light directed at a mirror 8 kilometres
(5 mi) away. On the way from the source to
the mirror, the beam passes through a rotating
cogwheel. At a certain rate of rotation, the
beam passes through one gap on the way out
and another on the way back, but at slightly
higher or lower rates, the beam strikes a
tooth and does not pass through the wheel.
Knowing the distance between the wheel and
the mirror, the number of teeth on the wheel,
and the rate of rotation, the speed of light
can be calculated.The method of Foucault replaces
the cogwheel by a rotating mirror. Because
the mirror keeps rotating while the light
travels to the distant mirror and back, the
light is reflected from the rotating mirror
at a different angle on its way out than it
is on its way back. From this difference in
angle, the known speed of rotation and the
distance to the distant mirror the speed of
light may be calculated.
Nowadays, using oscilloscopes with time resolutions
of less than one nanosecond, the speed of
light can be directly measured by timing the
delay of a light pulse from a laser or an
LED reflected from a mirror. This method is
less precise (with errors of the order of
1%) than other modern techniques, but it is
sometimes used as a laboratory experiment
in college physics classes.
=== Electromagnetic constants ===
An option for deriving c that does not directly
depend on a measurement of the propagation
of electromagnetic waves is to use the relation
between c and the vacuum permittivity ε0
and vacuum permeability 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 299710±22 km/s.
=== 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, 299792±9 km/s, was substantially
more precise than those found by optical techniques.
By 1950, repeated measurements by Essen established
a result 
of 299792.5±3.0 km/s.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 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. 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 (0.16 in))
of the radiowaves. The precision can be improved
by using light with a shorter wavelength,
but then it becomes difficult to directly
measure the frequency of the light. One way
around this problem is to start with a low
frequency signal of which the frequency can
be precisely measured, and from this signal
progressively synthesize higher frequency
signals whose frequency can then be linked
to the original signal. A laser can then be
locked to the frequency, and its wavelength
can be determined using interferometry. This
technique was due to a group at the National
Bureau of Standards (NBS) (which later became
NIST). They used it in 1972 to measure the
speed of light in vacuum with a fractional
uncertainty of 3.5×10−9.
== 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,
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 (c. 490–430 BC) 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".
Euclid and Ptolemy advanced Empedocles' emission
theory of vision, where light is emitted from
the eye, thus enabling sight. Based on that
theory, Heron of Alexandria argued that the
speed of light must be infinite because distant
objects such as stars appear immediately upon
opening the eyes.
Early Islamic philosophers initially agreed
with the Aristotelian view that light had
no speed of travel. In 1021, Alhazen (Ibn
al-Haytham) published the Book of Optics,
in which he presented a series of arguments
dismissing the emission theory of vision in
favour of the now accepted intromission theory,
in which light moves from an object into the
eye. This led Alhazen to propose that light
must have a finite speed, and that the speed
of light is variable, decreasing in denser
bodies. He argued that light is substantial
matter, the propagation of which requires
time, even if this is hidden from our senses.
Also in the 11th century, Abū Rayhān al-Bīrūnī
agreed that light has a finite speed, and
observed that the speed of light is much faster
than the speed of sound.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. 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. In Descartes'
derivation of Snell's law, he assumed that
even though the speed of light was instantaneous,
the denser the medium, the faster was light's
speed. Pierre de Fermat derived Snell's law
using the opposing assumption, the denser
the medium the slower light traveled. 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.
The first quantitative estimate of the speed
of light was made in 1676 by Rømer (see Rømer's
determination of the speed of light). From
the observation that the periods of Jupiter's
innermost moon Io appeared to be shorter when
the Earth was approaching Jupiter than when
receding from it, he concluded that light
travels at a finite speed, and estimated that
it takes light 22 minutes to cross the diameter
of Earth's orbit. Christiaan Huygens combined
this estimate with an estimate for the diameter
of the Earth's orbit to obtain an estimate
of speed of light of 220000 km/s, 26% 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 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 315000 km/s. His method
was improved upon by Léon Foucault who obtained
a value of 298000 km/s in 1862. 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.
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 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 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 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.
=== 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.
=== 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 299792.5±1 km/s,
using cavity resonance. This value was adopted
by the 12th General Assembly of the Radio-Scientific
Union in 1957. In 1960, the metre was redefined
in terms of the wavelength of a particular
spectral line of krypton-86, and, in 1967,
the second was redefined in terms of the hyperfine
transition frequency of the ground state of
caesium-133.
In 1972, using the laser interferometer method
and the new definitions, a group at the US
National Bureau of Standards in Boulder, Colorado
determined the speed of light in vacuum to
be 
c = 299792456.2±1.1 m/s. This was 100 times
less uncertain than the previously accepted
value. The remaining uncertainty was mainly
related to the definition of the metre. As
similar experiments found comparable results
for c, the 15th General Conference on Weights
and Measures in 1975 recommended using the
value 299792458 m/s for the speed of light.
=== 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 definition of second, so the caesium
hyperfine frequency would now determine both
the second and the metre. To do this, they
redefined the metre as: "The metre is the
length of the path travelled by light in vacuum
during a time interval of 1/299792458 of a
second." As a result of this definition, the
value of the speed of light in vacuum is exactly
299792458 m/s 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 299792458 when it is expressed
in the SI unit m s−1." This is one of the
proposed changes to be incorporated in the
next revision of the SI also termed the New
SI.
== See also ==
Light-second
Speed of electricity
Speed of gravity
Speed of sound
Velocity factor
Warp factor (fictional)
== Notes ==
== References ==
== Further reading ==
=== Historical references ===
=== Modern references ===
== External links ==
"Test Light Speed in Mile Long Vacuum Tube."
Popular Science Monthly, September 1930, p.
17–18.
Definition of the metre (International Bureau
of Weights and Measures, BIPM)
Speed of light in vacuum (National Institute
of Standards and Technology, NIST)
Data Gallery: Michelson Speed of Light (Univariate
Location Estimation) (download data gathered
by Albert A. Michelson)
Subluminal (Java applet demonstrating group
velocity information limits)
De Mora Luminis at MathPages
Light discussion on adding velocities
Speed of Light (University of Colorado Department
of Physics)
c: Speed of Light (Sixty Symbols, University
of Nottingham Department of Physics [video])
Usenet Physics FAQ
Speed of light illustration (Speed of light
as Live-Counter)
The Speed of Light, BBC Radio 4 discussion
with John Barrow, Iwan Morus & Jocelyn Bell
Burnell (In Our Time, 30 Nov. 2006)
