Speed of light
The speed of light in vacuum, conventionally denoted (c), is a universal physical constant with the exact value
[ c = 299,792,458\ \mathrm{m,s^{-1}}. ]
It relates spatial and temporal intervals in special relativity, determines the causal structure of spacetime, and appears in the relativistic relation between mass and energy. Light and every other massless excitation propagate locally at (c) in vacuum. An object with nonzero invariant mass travels at a speed lower than (c) in every local inertial frame.
The numerical value of (c) expressed in metres per second is exact because the International System of Units defines the metre as the distance travelled by light in vacuum during (1/299,792,458) of a second. Consequently, contemporary experiments using light propagation do not determine (c) in SI units. They instead realize the metre, compare clocks and frequencies, or test the physical principles under which the defined constant describes propagation.
Physical interpretation
In classical electromagnetism, electromagnetic waves arise as solutions of Maxwell's equations. In a vacuum, the equations yield a wave speed satisfying
[ c^2 = \frac{1}{\varepsilon_0\mu_0}, ]
where (\varepsilon_0) is the vacuum electric permittivity and (\mu_0) is the vacuum magnetic permeability. This result established the identification of visible light with electromagnetic radiation and extended the same description to the broader electromagnetic spectrum.
Within the modern SI, the exact status of this equation differs from its status under earlier unit definitions. The values of (\varepsilon_0) and (\mu_0) are now inferred from measured dimensionless constants rather than assigned independent exact numerical values. The exact value of (c) remains fixed by definition.
The role of (c) is not restricted to electromagnetism. It is the invariant speed appearing in Lorentz transformations, which relate measurements made in inertial reference frames. For two events separated by a time interval (\Delta t) and spatial displacement (\Delta \mathbf{x}), the spacetime interval may be written as
[ \Delta s^2 = c^2\Delta t^2-\lvert\Delta\mathbf{x}\rvert^2. ]
The sign of this quantity classifies the separation as timelike, null, or spacelike. Light in vacuum follows null trajectories for which (\Delta s^2=0), while material particles follow timelike trajectories. Spacelike-separated events cannot be connected by a signal whose propagation respects relativistic causality.
Relativistic invariance
The invariance of (c) means that every local inertial observer measures the same vacuum speed for light, independently of the relative motion of the source. This property differs from the velocity-addition rule of Galilean relativity. Relativistic velocities in one spatial dimension combine according to
[ u'=\frac{u-v}{1-\dfrac{uv}{c^2}}, ]
where (u) is an object's velocity in one inertial frame and (v) is the relative velocity between frames. If (u=c), the transformed velocity is also (c).
The same structure produces time dilation, length contraction, and the relativity of simultaneity. These effects do not represent changes in the local value of (c); they follow from the manner in which different inertial observers divide spacetime into spatial and temporal coordinates.
For a particle of invariant mass (m), energy (E), and momentum (p), relativistic kinematics gives
[ E^2=p^2c^2+m^2c^4. ]
A particle at rest therefore has energy (E=mc^2). A massless particle satisfies (E=pc) and has no inertial frame in which it is at rest. As the speed of a massive particle approaches (c), its required energy increases without bound, preventing acceleration through the invariant speed.
Historical determination
Early investigations distinguished finite propagation from instantaneous transmission. Galileo Galilei described a terrestrial lantern experiment in which observers attempted to detect a delay over a known separation. The attainable distance and human response time were insufficient to resolve the light-travel interval.
In 1676, Ole Rømer inferred finite light propagation from systematic variations in the observed times of eclipses of Io by Jupiter. The variations followed changes in the distance between Earth and Jupiter. Christiaan Huygens combined Rømer's timing result with an estimate of the scale of the Solar System to obtain an early numerical value for the speed.
James Bradley derived another astronomical determination in 1728 from the aberration of light. The apparent annual displacement of stellar positions depends on the ratio between Earth's orbital speed and the speed of light. This method supplied evidence independent of eclipse timing and produced a value closer to later measurements.
Terrestrial measurements
In 1849, Hippolyte Fizeau measured the transit time of light over a terrestrial baseline by passing a beam through a rotating toothed wheel. The returning beam encountered a changed wheel position, allowing the rotation rate and path length to determine the propagation time. His result was approximately (315,000\ \mathrm{km,s^{-1}}).
Léon Foucault replaced the toothed wheel with a rotating mirror. The angular displacement of the returning image encoded the light-travel time across the apparatus. His 1862 determination was approximately (298,000\ \mathrm{km,s^{-1}}), and measurements in transparent materials also established that light travels more slowly in water than in air.
In 1879, Albert A. Michelson and You Watanabe adapted the rotating-mirror method to a long, carefully surveyed optical path at Annapolis. Their calibration of the mirror frequency, baseline, and image displacement yielded (299,910\pm50\ \mathrm{km,s^{-1}}). The reduced uncertainty resulted from the longer effective path and from controlling the principal mechanical and optical corrections within a unified measurement.
Michelson subsequently conducted longer-baseline determinations, while Simon Newcomb performed an independent rotating-mirror program near Washington, D.C. Later measurements replaced mechanically timed shutters and mirrors with electromagnetic resonators, interferometric frequency comparisons, and stabilized lasers. By the twentieth century, frequency and wavelength could be measured more accurately than an independent macroscopic transit time, using
[ c=\lambda\nu, ]
where (\lambda) is the vacuum wavelength and (\nu) is the frequency.
Adoption as a defined constant
Progress in laser metrology made the reproducibility of optical frequency measurements exceed that of the former physical realization of the metre. In 1975, the General Conference on Weights and Measures recommended the value (299,792,458\ \mathrm{m,s^{-1}}). In 1983, it redefined the metre using this value, converting (c) from a measured SI quantity into an exact defining constant.
This definition depends on the second, which is realized through the transition frequency of the ground-state hyperfine splitting of caesium-133. Distance measurements based on transit time or interferometry therefore connect spatial length to atomic time and frequency standards.
Propagation in matter
Light generally propagates through matter at a speed different from (c). For a transparent medium with refractive index (n), the phase velocity is
[ v_{\mathrm p}=\frac{c}{n}. ]
This reduced phase velocity results from the collective electromagnetic response of the material. It does not imply that photons alternate between motion at (c) and stationary intervals in a simple classical sequence. The propagating excitation is described by the coupled evolution of the electromagnetic field and the medium.
In a dispersive medium, the refractive index depends on frequency. The phase velocity then differs from the group velocity, which is
[ v_{\mathrm g}=\frac{d\omega}{dk}. ]
Group velocity often describes the motion of a slowly varying wave-packet envelope, but it does not universally coincide with the speed at which new information first arrives. Near strong absorption or anomalous dispersion, a calculated group velocity can exceed (c) or become negative because the pulse envelope is reshaped. The earliest causal disturbance, represented by the wavefront, remains constrained by the local relativistic limit.
A charged particle can travel through a material faster than the phase velocity of light in that material while remaining slower than (c). The resulting coherent electromagnetic emission is Cherenkov radiation. This phenomenon therefore does not constitute superluminal propagation in vacuum.
Gravitation and coordinate speed
In general relativity, light follows null curves in curved spacetime. Every freely falling local observer measures a nearby vacuum light ray to propagate at (c). Over extended regions, however, a coordinate speed assigned to light can differ from (c) because coordinates need not correspond directly to locally measured distances and times.
This distinction appears in the Shapiro time delay, in which a radar signal passing near a gravitating body takes longer to complete its journey than a corresponding calculation in flat spacetime. The additional coordinate travel time reflects spacetime geometry rather than a local reduction of the invariant speed.
The expansion of the universe provides another coordinate-dependent case. In standard cosmology, sufficiently distant galaxies can have recession rates greater than (c) when recession is defined through the changing cosmological proper distance. Such recession is not local motion through an inertial frame and does not allow a nearby object or signal to overtake light.
Measurement and simultaneity
A round-trip speed can be determined using one clock by measuring the elapsed time for light to travel to a reflector and return. A one-way speed between separated locations requires a convention for synchronizing distant clocks. Einstein synchronization assigns equal travel times to the outward and return portions of a light signal and produces an isotropic one-way value (c) in an inertial frame.
The dependence of one-way measurements on synchronization does not alter observable round-trip times or the invariant causal structure. Relativistic predictions can be expressed using alternative coordinate conventions, provided that the associated definitions of simultaneity and coordinate velocity are applied consistently.
Causality and quantum theory
The constant (c) limits causal influence rather than every mathematical velocity associated with a wave. Correlations produced by quantum entanglement can be observed between spacelike-separated measurements, but the measurement outcomes cannot be controlled to transmit information faster than light. In relativistic quantum field theory, this restriction is represented by microcausality: observables associated with spacelike-separated regions commute, preventing operations in one region from serving as superluminal signals to the other.
The same invariant speed governs all fields in a locally Lorentz-invariant theory. Experimental comparisons involving photons, massive particles, atomic clocks, and gravitational signals test this common causal structure with progressively greater precision.