Speed of light

The speed of light in vacuum, often called simply the speed of light and commonly denoted, is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time interval of second. The speed of light is the same for all observers, no matter their relative velocity. It is the upper limit for the speed at which information, matter, or energy can travel through space.
All forms of electromagnetic radiation, including visible light, travel in vacuum at the speed c. For many practical purposes, light and other electromagnetic waves will appear to propagate instantaneously, but for long distances and sensitive measurements, their finite speed has noticeable effects. Much starlight viewed on Earth is from the distant past, allowing humans to study the history of the universe by viewing distant objects. When communicating with distant space probes, it can take hours for signals to travel. In computing, the speed of light fixes the ultimate minimum communication delay. The speed of light can be used in time of flight measurements to measure large distances to extremely high precision.
Ole Rømer first demonstrated that light does not travel instantaneously by studying the apparent motion of Jupiter's moon Io. In an 1865 paper, James Clerk Maxwell proposed that light was an electromagnetic wave and, therefore, travelled at speed. Albert Einstein postulated that the speed of light with respect to any inertial frame of reference 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 so showed that the parameter had relevance outside of the context of light and electromagnetism.
Massless particles and field perturbations, such as gravitational waves, also travel at speed in vacuum. Such particles and waves travel at regardless of the motion of the source or the inertial reference frame of the observer. Particles with nonzero rest mass can be accelerated to approach but can never reach it, regardless of the frame of reference in which their speed is measured. In the theory of relativity, interrelates space and time and appears in the famous mass–energy equivalence,.
In some cases, objects or waves may appear to travel [|faster than light]. The expansion of the universe is understood to exceed the speed of light beyond a certain boundary.
The speed at which light propagates through transparent materials, such as glass or air, is less than ; similarly, the speed of electromagnetic waves around wire cables is slower than. The ratio between and the speed at which light travels in a material is called the refractive index of the material. For example, for visible light, the refractive index of glass is typically around 1.5, meaning that light in glass travels at ; the refractive index of air for visible light is about 1.0003, so the speed of light in air is about slower than.

Notation

The speed of light in vacuum is usually denoted by a lowercase. The origin of the letter choice is unclear, with guesses including "c" for "constant" or the Latin celeritas. The "c" was used for "celerity" meaning a velocity in books by Leonhard Euler and others, but this velocity was not specifically for light; Isaac Asimov wrote a popular science article, "C for Celeritas", but did not explain the origin. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used for a different constant that was later shown to equal times the speed of light in vacuum. Historically, the symbol was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1903, Max Abraham used with its modern meaning in a widely read textbook on electromagnetism. Einstein used in his original German-language papers on special relativity in 1905, but in 1907 he switched to, which by then had become the standard symbol for the speed of light.
Sometimes is used for the speed of waves in any material medium, and ₀ for the speed of light in vacuum. This subscripted notation, which is endorsed in official SI literature, has the same form as related electromagnetic constants: namely, for the vacuum permeability or magnetic constant, for the vacuum permittivity or electric constant, and for the impedance of free space. This article uses exclusively for the speed of light in vacuum.

In unit systems

Since 1983, the constant has been defined in the International System of Units as ; this relationship is used to define the metre as exactly the distance that light travels in vacuum in of a second. The second is, in turn, defined to be the length of time occupied by of the radiation emitted by a caesium-133 atom in a transition between two specified energy states. By using the value of, as well as an accurate measurement of the second, one can establish a standard for the metre.
The particular value chosen for the speed of light provided a more accurate definition of the metre that still agreed as much as possible with the definition used before 1983.
As a dimensional physical constant, the numerical value of is different for different unit systems. For example, in imperial units, the speed of light is approximately miles per second. This is value is less than 2% different from 1 billion feet per second or one foot per nanosecond. Naval officer and computer scientist Grace Murray Hopper distributed foot-long wires to colleagues in the late 1960 to visually illustrate the importance of designing smaller components to increase computing speed.
In branches of physics in which appears often, such as in relativity, it is common to use systems of natural units of measurement or the geometrized unit system where. Using these units, does not appear explicitly because multiplication or division by1 does not affect the result. Its unit of light-second per second is still relevant, even if omitted.

Fundamental role in physics

The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference 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 motion against the luminiferous aether. Experiments such as the Kennedy–Thorndike experiment and the Ives–Stilwell experiment have shown this postulate to match experimental observations.
The special theory of relativity explores the consequences of this invariance of with the assumption that the laws of physics are the same in all inertial frames of reference. One consequence is that is the speed at which all massless particles and waves, including light, must travel in vacuum.
Special relativity has many counterintuitive and experimentally verified implications. These include the equivalence of mass and energy, length contraction, Terrell rotation, and time dilation. The factor by which lengths contract and times dilate is known as the Lorentz factor and is given by, where is the speed of the object. The difference of from1 is negligible for speeds much slower than , such as most everyday speedsin which case special relativity is closely approximated by Galilean relativitybut it increases at relativistic speeds and diverges to infinity as approaches. For example, a time dilation factor of occurs at a relative velocity of 86.6% of the speed of light. Similarly, a time dilation factor of occurs at 99.5% the speed of light.
The results of special relativity can be summarized by treating space and time as a unified structure known as spacetime, and requiring that physical theories satisfy a special symmetry called Lorentz invariance, whose mathematical formulation contains the parameter . 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 is ubiquitous in modern physics, appearing in many contexts that are unrelated to light. For example, general relativity predicts that is also the speed of gravity and of gravitational waves, and observations of gravitational waves have been consistent with this prediction. In non-inertial frames of reference, the speed of light is constant and equal to , but the speed of light can differ from when measured from a remote frame of reference, depending on how measurements are extrapolated to the region.
It is generally assumed that fundamental constants such as 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 is generally assumed that the two-way 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, and of rotating optical resonators have put stringent limits on the possible two-way anisotropy.

Upper limit on speeds

An object with rest mass and speed relative to a laboratory has kinetic energy with respect to that lab, where is the Lorentz factor defined above. The factor approaches infinity as approaches , 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. Analysis of individual photons confirm that information cannot travel faster than the speed of light. This is experimentally established in many tests of relativistic energy and momentum.
More generally, it is impossible for signals or energy to travel faster than . One argument for this is known as causality. If the spatial distance between two events and is greater than the time interval between them multiplied by then there are frames of reference in which precedes , others in which precedes , and others in which they are simultaneous. As a result, if something were travelling faster than 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.
In some theoretical treatments, the Scharnhorst effect allows signals to travel faster than , by one part in 10³⁶. However other approaches to the same physical set up show no such effect. and it appears the special conditions in which this effect might occur would prevent one from using it to violate causality.