Speed of light

Here, laser light in air is traveling at 99.97% the speed of light in a vacuum.^[1]

The speed of light normally refers the speed of light in a vacuum, and is an important physical constant in modern physics. Light travels at different speeds through different materials, but it travels fastest in vacuum and its speed does not vary with the color, intensity, or direction of travel.^[2] Perhaps more surprisingly, the speed also does not depend on the motion of the light emitter or the observer. Therefore it makes sense to speak of the speed of light, which is the speed of light in vacuum and usually represented as c. The speed of light factors into much of modern physics, including special relativity, general relativity, and quantum mechanics.

By everyday standards, light travels very rapidly - approximately 300,000 km each second, in vacuum or air. This is roughly a million times faster than sound, and fast enough to circle the Earth more than 7 times in one second. Such a rapid speed is very hard to measure without specialized techniques, and in ancient times the speed of light was the subject of speculation. The first effective measurements of the speed of light were made in the seventeenth century, and were progressively refined. Today, time intervals can be measured extremely precisely, to the point where the metre is now defined officially as the distance light travels in "vacuum" in ¹⁄_299,792,458 of a second. As a consequence, according to NIST: "… the effect of this definition is to fix the speed of light in vacuum at exactly 299 792 458 m/s." (Further discussion is found later in the subsection Speed of light set by definition.)

The speed of light plays a crucial role in both classical and modern physics. Maxwell in the 19th century provided the first explanation of the speed of light in relation to other physical constants.^[3] In the early 20th century c assumed an even greater importance as a pivotal constant in Einstein's theory of special relativity, which holds that the speed of light has a special role connecting space and time in the structure of spacetime. As one result, the speed of light sets an absolute speed limit to how fast matter or information can move. As another result, energy and mass are connected by the speed of light in the famous mass-energy equation E = mc². This relationship is a foundation of quantum mechanics, including nuclear energy. General relativity further explains gravity as an effect of curved space time, and in this theory all distances must be computed using c. Today, the speed of light continues to be a subject of research, both in theory and experiment.

For many practical purposes, the speed of light is so great that it can be regarded to travel instantaneously. However, the finite speed of light becomes noticeable when very long distances, very short time intervals, or precise time measurements are involved. For instance, the speed of light is a critical factor in astronomy, modern electronics, and navigation systems such as global positioning systems.

Description

Notation and units

The symbol "c" for "constant" or the Latin celeritas (meaning “swiftness”)^[4] is used for the speed of light in free space, and in this article c is used exclusively this way. Some authors, however, use c for the speed of light in any material media.^[5] To avoid confusion, and for consistency with other constants of free space such as μ₀, ε₀ and Z₀, international bodies such as the International Bureau of Weights and Measures (BIPM) recommend using c₀ for the speed of light in free space.^{[6] [7]}

In branches of physics in which the speed of light plays an important part, such as in relativity, it is common to use a system of units known as natural units in which c is 1; thus no symbol for the speed of light is required.

When physicists refer to the speed of light in "vacuum", most commonly they mean free space, which means classically a region void of all matter and fields. Such an ideal vacuum is unrealizable in practice: for example, interstellar space and terrestrial vacuums are only approximations, and the notion of "empty space" is not compatible even in principle with today's notions of quantum vacuum or QCD vacuum. Therefore, measurements in real (imperfect) vacuums are related to those in "vacuum" using evolving corrections conforming to "best good practice" as defined by standards organizations and scientific and technical publications.^[8] In this article, "vacuum" refers to the hypothetically perfect vacuum of free space unless otherwise specified.

Description

Light is a type of electromagnetic radiation. According to electromagnetic theory, all electromagnetic radiation travels in free space at the speed of light in free space: 299,792,458 m/s. This definition applies regardless of the light's color, intensity, source, or direction. Examples of real media approximating ideal vacuum are: outer space, interstellar medium, interplanetary medium, quantum vacuum, QCD vacuum and ultra-high vacuum. Within experimental error, no variations have been found in the speed of light in these realizable media, so their behavior is well approximated as that of free space. Moreover, observations show the variation of the speed of light as the universe ages is less than two parts in 10¹⁶/year, for both microwaves and visible light.

According to special relativity, the speed of light has the same value in every inertial reference frame. The observed frequency may differ between two observers with different velocities, resulting in a phenomenon known as the Doppler effect.

As light propagates down the telescope, the telescope moves requiring a tilt to the telescope that depends on the speed of light. The apparent angle of the star φ differs from its true angle θ, a phenomenon called stellar aberration

Practical effect of the finite speed of light

The speed of light plays an important part in many modern sciences and technologies. Radar systems measure the distance to a target by measuring the time taken for an echo of the light pulse to return. Similarly, a global positioning system (GPS) receiver measures its distance to satellites based on how long it takes for a radio signal to arrive from the satellite. The distances to the moon, planets, and spacecraft are determined by measuring the round-trip travel time.

Another effect of the finite speed of light is stellar aberration. Suppose we look at a star with a telescope idealized as a narrow tube. The light enters the tube from a star at angle θ and travels at speed c taking a time h/c to reach the bottom of the tube, where our eye detects the light. Suppose observations are made from Earth, which is moving with a speed v. During the transit of the light, the tube moves a distance vh/c. Consequently, for the photon to reach the bottom of the tube, the tube must be inclined at an angle φ different from θ , resulting in an apparent position of the star at angle φ.

In astronomy beyond the solar system, distances are often measured in light-years, the distance light travels in a year.

In electronic systems, despite their small size, the speed of light can become a limiting factor in their maximum speed of operation.^[9][10]

Speed of light in various systems

Light in free space

In SI units the speed of all electromagnetic radiation in free space is related to the electric constant ε₀ (also called the permittivity of free space) and magnetic constant μ₀ (also called the permeability of free space) by the equation c₀²=1/(ε₀ μ₀)^[11] .

According to classical electromagnetism, the speed of electromagnetic radiation in free space is the same for all frequencies. That is, light in free space exhibits zero dispersion. The speed of light in free space is polarization independent, isotropic, and independent of field strength.

Free space is a reference state. Like absolute zero, it is an idealized state that only can be approximated in the physical world. Measurements in any real-world medium, such as air^[12][13] or a medium perturbed by gravity^[14], must account for the medium's refractive index to relate them to the reference standard of free space.

Light in realizable vacuum

Although free space is an idealized reference state unattainable in practice, some media accurately approximate free space, such as outer space and terrestrial ultra high vacuum.

A simple model often used to represent free-space-like vacuum is one where electric permittivity and magnetic permeability are constants with the values ε₀ and μ₀. In this classical model, the speed of light is the same for all wavelengths, and there exists perfect isotropy, zero dispersion, perfect linearity and zero dichroism. The refractive index of this classical model is unity.

Quantum electrodynamic theory predicts deviations from a unitary refractive index in the vacuum state for extremely strong electromagnetic fields.^[15] To date, there has been no experimental confirmation of that effect.

Measurements based on the arrival of electromagnetic radiation from distant astrophysical events put severe limits on the possible variation in the speed of light with frequency in outer space.^[16]

Light in transparent media

The refractive index of a transparent material indicates how much slower light propagates in that medium than in a vacuum. This slowing causes refraction. Classically, when an electromagnetic wave meets the surface of a dielectric material at an angle, the leading edge is slowed while the trailing edge continues normally. This causes the wave to change direction, as demonstrated by this prism (in the case of a prism splitting white light into a spectrum of colours, the refraction is known as dispersion).

When a light pulse comprised of multiple frequencies passes through transparent materials, the speed of light is characterized by two speeds: the phase velocity and the group velocity. The phase velocity of light may be found from knowledge of the frequency-dependent refractive index:

where ε_r is the material's relative permittivity, and μ_r is its relative permeability.

The group velocity of the wave is the speed at which the envelope of the pulse travels through the medium, and is dependent on the frequency content of the pulse as well as the properties of the medium. A wave with different group and phase velocities is said to undergo dispersion. If the light passing through the medium is monochromatic, the phase velocity is often referred to as the "speed of light".

When light enters materials its energy is absorbed. In the case of transparent materials (dielectrics) this energy is quickly re-radiated. However, this absorption and re-radiation introduces a delay. As light propagates through dielectric material it undergoes continuous absorption and re-radiation. Therefore when the speed of light in a medium is said to be less than c, this should be read as the speed of energy propagation at the macroscopic level. At an atomic level, electromagnetic waves always travel at c in the empty space between atoms. Two factors influence this slowing; stronger absorption leading to shorter path length between each re-radiation cycle and longer delays. The slowing is therefore the product of these two factors. This reduction in speed is also responsible for bending of light at an interface between two materials with different refractive indices, a phenomenon known as refraction.

The refractive index in air is only slightly larger than one ^[1]. Denser media, such as water and glass, have refractive indices of between 1.3 and 1.5 for visible light. Diamond has a refractive index of about 2.4.

History

Until relatively recent times, whether the speed of light was infinite was unknown. The debate began in ancient Greece. Empedocles maintained that light was something in motion, and therefore there had to be some time elapsed in traveling. Aristotle said that, on the contrary, "light is due to the presence of something, but it is not a movement".^[17] Euclid and Ptolemy advanced the emission theory of vision, which claimed light was emitted from the eye, thus enabling sight. Using that theory, Heron of Alexandria advanced the argument that the speed of light must be infinite, since distant objects such as stars appear immediately upon opening the eyes.

Early Muslim philosophers initially agreed with the Aristotelian view of the speed of light being infinite. In 1021, Iraqi physicist Alhazen (Ibn al-Haytham) published the Book of Optics, in which he used experiments related to the camera obscura to support the intromission theory of vision, which claimed that light moves from an object into the eye.^[18] This led to Alhazen proposing that light must therefore have a finite speed,^[17][19][20] and that the speed of light is variable, with its speed decreasing in denser bodies.^[20][21] He argued that light is a “substantial matter”, the propagation of which requires time "even if this is hidden to our senses".^[22] This debate continued in Europe and the Middle East throughout the Middle Ages.

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.^[23] In the 1270s, Witelo considered the possibility of light traveling at infinite speed in a vacuum but slowing down in denser bodies.^[24] A comment on a verse in the Rigveda by the 14th century Indian scholar Sayana^[25] may be interpreted as suggesting an estimate for the speed of light that is in good agreement with its actual speed. In 1574, the Ottoman astronomer and physicist Taqi al-Din concluded that the speed of light is constant, but variable in denser bodies, and suggested that it would take a long time for light from the stars which are millions of kilometres away to reach the Earth.^[26]

Early attempts to measure the speed of light

In the early 17th century, Johannes Kepler believed that the speed of light was infinite since empty space presents no obstacle to it. Francis Bacon argued that the speed of light was not necessarily infinite, since something can travel too fast to be perceived. 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 was found to be finite, his whole system of philosophy might be demolished.^[17]

In 1629, Isaac Beeckman proposed an experiment in which a person would observe 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 concluded that the speed of light is ten times faster than the speed of sound, a figure now known to be much too slow.^[20] Galileo's experiment was carried out by the Accademia del Cimento of Florence in 1667, with the lanterns separated by about one mile (1.6 km), but no delay was observed. Based on the modern value of the speed of light, the actual delay in this experiment would be about 11 microseconds. Robert Hooke explained the negative results as Galileo had by pointing out that such observations did not establish the infinite speed of light, but only that the speed must be very great.

Astronomical techniques

Rømer's observations of the occultations of Io from Earth.

The first quantitative estimate of the speed of light was made in 1676 by Ole Christensen Rømer, who was studying the motions of Jupiter's moon, Io, with a telescope. It is possible to time the orbital revolution of Io because it enters and exits Jupiter's shadow at regular intervals (at points C and D in the diagram at right). Rømer observed that Io revolved around Jupiter once every 42.5 hours when Earth was closest to Jupiter (at H). He also observed that, as Earth and Jupiter moved apart, (as from L to K), Io's exit from the shadow would begin progressively later than predicted. He realized that these exit "signals" took longer to reach Earth as a result of the extra time it took for light to cross the extra distance between the moving planets. The opposite is the case when they are approaching (as from F to G). Rømer observed 40 orbits of Io when Earth was approaching Jupiter to be 22 minutes shorter than 40 orbits of Io when Earth was moving away from Jupiter.^[27] On the basis of those observations, Rømer concluded that it took light 22 minutes to cross the distance the Earth traversed in 80 orbits of Io.^[27] This means that in travelling from L to K and F to G, whereas the earth took 80 periods of Io's orbits (42.5 hours), the light only took 22 minutes. This corresponds to a ratio between the speed of light and the speed at which the Earth travels in its orbit around the sun of 9,300. The modern value of the ratio is about 10,100.

Around the same time, the astronomical unit (roughly, the Earth-to-Sun distance) was estimated to be about 140 million kilometres (87 million miles). The astronomical unit and Rømer's time estimate were combined by Christiaan Huygens, who estimated the speed of light to be 1,000 Earth diameters per minute, based on having misinterpreted Rømer's value of 22 minutes to mean the time it would take light to cross the diameter of the orbit of the Earth.^[27] This is about 220,000 kilometres per second (136,000 miles per second), 26% lower than the currently accepted value, but still very much faster than any physical phenomenon then known.

Isaac Newton also accepted the finite speed. In his 1704 book Opticks he reports the value of 16.6 Earth diameters per second (210,000 kilometres per second. The same effect was subsequently observed by Rømer for a "spot" rotating with the surface of Jupiter. Later observations also showed the effect with the three other Galilean moons, where it was more difficult to observe, thus laying to rest some further objections that had been raised.

In 1728, James Bradley deduced that starlight falling on the Earth should appear to come from a slight angle, which could be calculated by comparing the speed of the Earth in its orbit to the speed of light. This "aberration of light", as it is called, was observed to be about 1/200 of a degree. Bradley calculated the speed of light as about 298,000 kilometres per second (186,000 miles per second).

Earth-bound techniques

Diagram of the Fizeau apparatus.

The first successful entirely earthbound measurement of the speed of light was carried out by Hippolyte Fizeau in 1849. Fizeau's experiment was conceptually similar to those proposed by Beeckman and Galileo. A beam of light was directed at a mirror 8 km away. On the way from the source to the mirror, the beam passed through a rotating cog wheel. At a certain rate of rotation, the beam could pass through one gap on the way out and another on the way back. But at slightly higher or lower rates, the beam would strike a tooth and not pass through the wheel. Knowing the distance to the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light could be calculated. Fizeau reported the speed of light as 313,000 kilometres per second. Leon Foucault improved on Fizeau's method by replacing the cogwheel with a rotating mirror. Foucault's estimate, published in 1862, was 298,000 kilometres per second.

In 1861 Maxwell proposed a theory which linked the speed of light to the electromagnetic field.^[3] In 1864 Maxwell compared known measurements of the speed of light with the ratio of electrostatic to electromagnetic units by Weber and Kohlrausch to support the connection between light and electromagnetic phenomena.^[3] Maxwell’s equations allow the speed of light to be calculated, in much the same way as the speed of sound can be calculated in normal matter.

A schematic representation of a Michelson interferometer, as used for the Michelson-Morley experiment. The point of reflection on the tilted beam splitter is shown as two separated points for clarity. Both beams travel equal length paths. If light speed is anisotropic, the interference pattern seen at the detector will vary with orientation of the apparatus.

The 'luminiferous aether'

After some proposals in the 17th and 18th century, Augustin Fresnel in 1818 argued that light moved through a rigid and stationary aether as a way to explain the existence of aberration and polarization. And because of James Clerk Maxwell's (1861) aether based concept of electromagnetism, the existence of the aether was widely accepted. In 1887, physicists Albert Michelson and Edward Morley designed an experiment to measure the velocity of the Earth through the aether.^[28] As the Earth is in orbit round the sun, and the aether was assumed to be fixed, the Earth would be expected to be in motion with respect to the aether for at least some of the time.^[29] As shown in the diagram of a Michelson interferometer, a half-silvered mirror was used to split a beam of monochromatic light into two beams traveling at right angles to one another. After leaving the splitter, each beam was reflected back and forth between mirrors several times then recombined to produce a pattern of constructive and destructive interference. Any slight change in speed of light along one arm of the interferometer compared with its speed along the other arm would then be observed as a change in the pattern of interference. The experiment gave a null result. Some interesting details of these experiments are found in Hollberg et al.^[30] Later experiments confirmed this result to a much higher accuracy.^[31][32]

At the beginning the experiment was not understood as disproving the existence of the luminiferous aether, and so some hypothesis (like the Aether drag hypothesis or the Lorentz–Fitzgerald contraction) were made to explain the result within the framework of an aether. So it lasted until the beginning of the 20th century, when some physicists cast doubt on the existence of the aether. Eventually, the concept was completely overthrown by Einstein. His theory of special relativity explains the null result of the Michelson–Morley experiment by postulating that the speed of light is always the same for all inertial observers. This means that the speed of light will always be the same in both arms of the interferometer, regardless of their orientation or state of inertial motion, thus no changes in the observed fringes would be expected when it was rotated. Einstein published his general theory of relativity, which extended his special theory to include gravitation, the concept of aether rapidly fell into disuse and it forms no part of physics today.

Modern methods

Electromagnetic standing waves in a cavity. Sinusoidal waves at the top have larger frequencies than below and are shifted upward for clarity. The conductive walls require the nodes of the standing wave to be at the wall surfaces, so the allowed wavelengths are: λ/2 = W, λ = W, 3λ/2 = W where W = width of cavity.

During World War II, the development of the cavity resonance wavemeter for use in radar, together with precision timing methods, opened the way to laboratory-based measurements of the speed of light. In 1946, Louis Essen and A.C. Gordon-Smith used a microwave cavity of precisely known dimensions to establish the frequency for a variety of normal modes of microwaves. 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, 299 792 ± 3 km/s, was substantially more precise than those found by optical techniques, and prompted much controversy. However, by 1950 repeated measurements by Essen established a result of 299 792.5 ± 1 km/s, which became the value adopted by the 12th General Assembly of the Radio-Scientific Union in 1957.

An idealized interferometric determination of wavelength obtained by looking at interference fringes between two coherent beams recombined after traveling different distances. Top: Constructive interference (in phase); If the difference in path length is a multiple of a wavelength, the recombined beams support one another and reconstitute the original beam. Bottom: Destructive interference (out of phase); If the two paths differ by half a wavelength, the recombined beams are out of phase and cancel each other. The bottom panel in the figure suggests the path length has been increased by half a wavelength by moving the right-hand point of reflection further out.

An alternative to the cavity resonator method to find the wavelength for determining the speed of light is to use a form of interferometer, indicated schematically in the figure.^[33] A coherent light beam with a known frequency, as from a laser, is split to follow two paths and then recombined. By carefully changing the path length and observing the interference pattern, the wavelength of the light can be determined, which can be related to the speed of light.

With modern electronics, particularly oscilloscopes with time resolutions of less than one nanosecond, the speed of light can now be directly measured by timing the delay of a light pulse from a laser or a LED in reflecting from a mirror, although this method is less precise than either the cavity resonator or the interferometric methods.^[34][35][36]

Speed of light set by definition

It is well-known that for light of a given single frequency, the wavelength of light and its frequency are related via the speed of light by the relation:^[37]

with f = frequency and λ = wavelength. The frequency of the light is related to the period of oscillation of the light source as:

where T = period of oscillation in s, so a known frequency in turn implies a known time interval.

The wavelength is the spacing, along the direction of propagation, between successive crests or troughs, and can be observed directly using an interferometer. In 1960 the meter was defined as a specific number of wavelengths of a particular light source:^[38]

"realization being achieved using an interferometer with a travelling microscope to measure the optical path difference as the fringes were counted"^[39]

Of course, such a definition coupled with an accurate frequency allows determination of the speed of light as c = λ f.

Because of advances in producing light sources with very precise frequencies, it became possible to measure time intervals very accurately. For reasons stated in Resolution 1, ^[40] in 1983 the 17th Conférence Générale des Poids et Mesures defined the metre in terms of the distance traveled by light in a given amount of time in perfect vacuum:^[6]

"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."

A consequence of this definition is that the speed of light is a defined numerical value c = 299 792 458 m/s in "vacuum".

Rather than measure a time-of-flight, one implementation of this definition is to use a recommended source with established frequency in conjunction with the wavelength relation pointed out above, and determine wavelength in terms of the metre using the defined numerical value of c as λ = c / f.^[41]

Because the second is precisely known, the new definition allows for a more precise definition of the metre, assuming an adequate realization of "vacuum" is available. The practical realization (or embodiment) of vacuum is obtained by measurement in a real medium (which may be simply a controlled volume of air^[42]) and employing various corrections to reduce the measurements to what they are expected to be in ideal vacuum. Practical realizations of the metre use recommended wavelengths of visible light in a laboratory vacuum with corrections being applied to take account of actual conditions such as diffraction, gravitation or imperfection in the vacuum.^[43] With regard to the light source, as suggested by the BIPM: "it is not sufficient just to meet the specifications for the listed parameters. In addition, it is necessary to follow the best good practice concerning methods of stabilization as described in numerous scientific and technical publications."^[41] Naturally, what constitutes "best good practice" evolves as measurement accuracy and reproducibility improve with advances in technique.

Speed of light in scientific fields

In Gaussian units, the speed of light fixes the ratio between electrostatic units and electromagnetic units.

Speed of light in special relativity

The special theory of relativity says that the speed of light is the same for all inertial observers regardless of the movement of the light source.^[44] Experimental evidence is in agreement.^[45][46]

The finite speed of light in relativity leads to some counter-intuitive consequences, which include length contraction, time dilation and the relativity of simultaneity, this last item contradicting the classical notion that the duration of the time interval between two events is equal for all observers.

Speed of light in astronomy

The relative sizes and separation of the Earth–Moon system are shown to scale above. The beam of light is depicted traveling between the Earth and the Moon in the same time it actually takes light to scale the real distance between them: 1.255 seconds at its mean orbital distance (surface to surface). The light beam helps provide the sense of scale of the Earth-Moon system relative to the Sun, which is 8.28 light-minutes away (photosphere to Earth surface).

The speed of light is particularly important in astronomy. Due to the vast distances involved it can take a very long time for light to travel from its source to Earth. For example, it takes 13 billion 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 (near the beginning of the universe). The fact that farther-away objects appear younger (due to the finite speed of light) is crucial in astronomy, allowing astronomers to infer the evolution of stars, galaxies, and the universe itself.

Astronomical distances are sometimes measured in light-years, the distance light travels in one year. A light‑year is around 9 trillion km, 6 trillion miles, or 0.3 parsecs. Next to the Sun, the closest star to Earth, Proxima Centauri, is around 4.2 light‑years away.^[47]

Speed of light in cosmology

Weyl, Eddington and Dirac suggested the questions of just why the fundamental constants of nature have the values they do, and whether they are changing as the universe evolves.^{[48][49][50][51]} A change with time of the speed of light also affects the fine structure constant:

so theories describing an evolution of α have much in common with theories involving the evolution of c.^[52][53][54] Kafatos et al. have explored the possibility that the speed of light is identical to the rate of change of the scale of the universe, and summarize some recent work of this type.^[55] Experiment and theory continue to explore these ideas. ^[56][57]

Quantum gravity models suggest that the speed of light exhibits dispersion.^[58] While being smooth at large distances, space-time might show a complex, foamy, structure due to quantum fluctuations at short distances on the order of the Planck length ℓ_P  :

^[59][60] where:

• G is the gravitational constant;
• (pronounced "h-bar") is the reduced Planck constant.

An energy dependence of the speed of light in vacuum may arise from photon propagation through such a gravitational medium.^[61] Lehnert and Roy^[62] also discussed as a possible effect of fluctuations of permittivity and permeability in vacuum that photons may be gaining mass, if indeed photons have non-zero masses. Recently, Rañada proposed that due to variation of physical constants, there will be change of permittivity and permeability of quantum vacuum causing a change of refractive index of the vacuum. There should be an effect upon the rest mass of a photon as well, because such a vacuum can shift the frequency of a photon propagating through it.^[63]

Observations of astrophysical events at high redshifts can be used to place severe limits on the variation of the speed of light itself Δc/c, as well as on the photon mass m_γ. Schaefer presented limits on Δc/c < 6.3 × 10⁻²¹ and a limit on m_γ < 4.2 × 10⁻⁴⁴g based upon the difference in arrival times on Earth of distant, explosive events that simultaneously emit radiation at multiple frequencies.^[64] A different experimental approach is to compare the energy level separations of atomic transitions in distant objects from those near at hand. At higher redshifts, a possible time dependence of α will be registered in the form of small shifts in the absorption line spectra seen in distant quasars because the energy of the atomic transitions depend on α. Interesting experimental observations using absorption systems in the spectra of distant quasars may suggest time evolution of the fine structure constant.^[65][66] An overview of time variation of fundamental constants is provided by Landau et al.^[67] Recent analysis of experimental data suggests −0.050 ≤ Δα/α ≤ 0.042,^[68] an experimental uncertainty that includes the possibility that α is constant. This limit applies to a change occurring in the time between today and the so-called recombination epoch when the Universe became cool enough for protons to capture electrons and form neutral hydrogen (5–10 billion years, or a redshift of z_* = 1078 ± 11).^[69][70] Laboratory measurements based upon precision comparisons of different atomic frequency standards over a period of a few years set a rate of variation as dℓn α /dt = (−0.26 ± 0.39) × 10⁻¹⁵ /yr.^[71]

Speed of light as the maximum speed of information transfer

Causality and information transfer

A light cone defines locations that are in causal contact and those that are not.

According to the theory of special relativity, causality would be violated if information could travel faster than c in some reference frame, which frame would then become a frame privileged above all others.^[72][73][74] In some other reference frames, the information would be received before it had been sent, so the "effect" could be observed before the "cause". Such a violation of causality has never been recorded.^[75]

Information propagates to and from a point forming regions defined by a light cone. The interval AB in the diagram to the right is "time-like" (that is, there is a frame of reference in which event A and event B occur at the same location in space, separated only by their occurring at different times, and if A precedes B in that frame then A precedes B in all frames: there is no frame of reference in which event A and event B occur simultaneously). Thus, it is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the "cause" and B the "effect").

On the other hand, the interval AC in the diagram to the right is "space-like" (that is, there is a frame of reference in which event A and event C occur simultaneously, separated only in space; see simultaneity). However, there are also frames in which A precedes C (as shown) or in which C precedes A. Barring some way of traveling faster than light, it is not possible for any matter (or information) to travel from A to C or from C to A. Thus there is no causal connection between A and C.

Faster-than-light observations and experiments

The blue glow in this "swimming pool" nuclear reactor is Čerenkov radiation, emitted as a result of electrons traveling faster than the speed of light in water.

Only zero-rest mass particles can travel at the speed of light.^[76] It is generally considered that it is impossible for any information or matter to travel faster than c, because it would travel backwards in time relative to some observers.^[77] However, there are many physical situations in which speeds greater than c are encountered.

Some of these situations involve entities that actually travel faster than c in a particular reference frame but none involves either matter, energy, or information traveling faster than the speed of light in vacuum.

It is possible for the group velocity of light to exceed c^[78][79] and in an experiment in 2000 laser beams traveled for extremely short distances through caesium atoms with a group velocity of 300 times c.^[80] It is not, however, possible to use this technique to transfer information faster than c since the velocity of information transfer depends on the front velocity, which is always less than c.^[81]

If a laser is swept across a distant object, the spot of light can easily be made to move at a speed greater than c.^[82] Similarly, a shadow projected onto a distant object can be made to move faster than c.^[83] In neither case does any matter or information travel faster than light.

In some interpretations of quantum mechanics, certain quantum effects may be transmitted at speeds greater than c. For example, the quantum states of two particles can be entangled. Until the particles are observed, they exist in a superposition of two quantum states. If the particles are separated and one of them is observed to determine its quantum state then the quantum state of the second particle is determined automatically and faster than a light signal could travel between the two particles. However, it is impossible to control which quantum state the first particle will take on when it is observed, so no information can be transmitted in this manner.

Another prediction of faster-than-light speeds occurs for tunneling and is called the Hartman effect.^{[84] [85]} However, no information can be sent using these effects.^[86]

Quantum field theory predicts an apparent superluminal propagation of photons due to vacuum polarization. This prediction raises the question of whether causality may be violated by quantum effects in curved spacetime.^[87][88][89]

Closing speeds and proper speeds are examples of calculated speeds that may have value in excess of c but that do not represent the speed of an object as measured in a single inertial frame.

So-called superluminal motion is seen in certain astronomical objects,^[90] such as the 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 at a small angle to the line of sight.

Čerenkov radiation

It is possible for shock waves to be formed with electromagnetic radiation.^[91][92] If a charged particle travels through an insulating medium faster than the speed of light in that medium then radiation is emitted which is analogous to a sonic boom and is known as Čerenkov radiation.

Galaxies moving faster than light

In models of the expanding universe, the further things are from Earth, the faster they move away from us. This movement is not considered to be a straightforward travel, like a rocket for example, but a movement due to the expansion of space itself. This expansion moves distant objects away from us faster and faster the further away they are. At a boundary called the Hubble sphere, the recessional velocity is the speed of light.

At distances beyond the Hubble sphere, objects move away faster than the speed of light. One view is that this speed does not contradict special relativity because each observer is the center of their own Hubble sphere, so the motion occurs outside any particular observer's inertial frame.^[93] A different explanation is that the "velocity" calculated this way does not correspond to a velocity seen in any single inertial frame, but is concatenated from distances observed in an infinite sequence of local inertial frames between the observer and the object (there are no global inertial frames), and special relativity refers to observations made in a single inertial frame, not an assembly of such frames.^[94]

So it happens that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light. The most distant objects that we can see now were outside the Hubble sphere when they emitted the photons we see now. The current recession velocity of the points from which the cosmic microwave background was emitted is v = 3.2c. We routinely see radiation from objects that lie outside the Hubble sphere.^[93]

See also

References

Footnotes

Historical references

Modern references

External links

Wikibooks has a book on the topic of Special Relativity

Wikinews has related news: Two German scientists claim to have broken the speed of light

• Speed of light in vacuum (at NIST)
• Definition of the metre (BIPM)
• Data Gallery: Michelson Speed of Light (Univariate Location Estimation) (download data gathered by A.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)
• How is the speed of light measured?
• The Fizeau "Rapidly Rotating Toothed Wheel" Method