Speed of light

2008/9 Schools Wikipedia Selection. Related subjects: Physics

The speed of light in free space (a vacuum) is an important physical constant usually denoted by the letter c. It is the speed of all electromagnetic radiation, including visible light, in free space. More generally, it is the speed of anything having zero rest mass.

In metric units, the speed of light in vacuum is exactly 299,792,458 metres per second (1,079,252,849 km/h). The fundamental SI unit of length, the metre, has been defined since October 21, 1983, as the distance light travels in a vacuum in 1/299,792,458 of a second; any increase in the measurement precision of the speed of light would refine the definition of the metre, but not alter the numerical value of c. The approximate value of 3×10⁸ m/s is commonly used in rough estimates. In imperial units, the speed of light is about 670,616,630 miles per hour or 983,571,056 feet per second, which is about 186,282.397 miles per second, or roughly one foot per nanosecond. See also the later section of this article at "Speed of light set by definition".

The speed of light when it passes through a transparent or translucent material medium, like glass or air, is less than its speed in a vacuum. The ratio of the speed of light in the vacuum to the observed phase velocity is called the refractive index of the medium. See dispersion (optics). General relativity explains how a gravitational potential can affect the apparent speed of distant light in a vacuum, but locally light in a vacuum always passes an observer at a rate of c.

Overview

The speed of light in vacuum is now viewed as a fundamental physical constant. This postulate, together with the principle of relativity that all inertial frames are equivalent, forms the basis of Einstein's theory of special relativity. According to the currently prevailing definition, adopted in 1983, the speed of light is exactly 299,792,458 metres per second (approximately 3×10⁸ metres per second, or about 30  centimetres (1 foot) per nanosecond). See metre.

Experimental evidence has shown that the speed of light is independent of the motion of the source. It has also been confirmed experimentally that the two-way speed of light (for example from a source, to a mirror, and back again) is constant. It is not, however, possible 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 receiver should be synchronized. Einstein (who was aware of this fact) postulated that the speed of light should be taken as constant in all cases, one-way and two-way.

It is worth noting that it is the constant speed c, rather than light itself, that is fundamental to special relativity; thus if light is somehow manipulated to travel at less than c, this manipulation will not directly affect the theory of special relativity.

Observers traveling at large velocities will find that distances and times are distorted in accordance with the Lorentz transforms; however, the transformations distort times and distances in such a way that the speed of light remains constant. An observer moving with respect to a collection of light sources would find that light from the sources ahead was shifted toward the violet end of the spectrum while light from those behind was redshifted.

Use of the symbol 'c' for the speed of light

The symbol 'c' for the Latin celeritas ("swiftness"). is generally used for the speed of light. NIST and BIPM practice is to use c₀ for the speed of light in vacuum, in accord with ISO 31-5. However older publications use just c and many physicists may continue to do this in cases where there is no ambiguity. Occasionally some articles may use c for the speed of light in media other than vacuum. In accordance with common practice, c is used throughout this article exclusively to denote the speed of light in a vacuum.

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

Causality and information transfer

If information could travel faster than c in one reference frame, causality would be violated: 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.

To put it another way, information propagates to and from a point from 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.

Light years

Astronomical distances are sometimes measured in light years (the distance that light would travel in one Earth year, roughly 9.46×10¹² kilometres or about 5.88×10¹² miles). Because light travels at a large but finite speed, it takes time for light to cover large distances. Thus, the light we observe from distant objects in the universe was emitted from them long ago: in effect, we see their distant past.

Communications and GPS

The speed of light is of relevance to communications. For example, given the equatorial circumference of the Earth is about 40,100 km and c about  300,000 km/s, the theoretical shortest amount of time for a piece of information to travel half the globe along the surface is 0.0668 s.

The actual transit time is longer, in part because the speed of light is slower by about 30% in an optical fibre depending on its refractive index n, v = c / n and straight lines rarely occur in global communications situations, but also because delays are created when the signal passes through an electronic switch or signal regenerator. A typical time as of 2004 for a U.S. to Australia or Japan computer-to-computer ping is 0.18 s. The speed of light additionally affects wireless communications design.

Another consequence of the finite speed of light is that communications with spacecraft are not instantaneous, and the gap becomes more noticeable as distances increase. This delay was significant for communications between Houston ground control and Apollo 8 when it became the first spacecraft to orbit the Moon: for every question, Houston had to wait nearly 3 seconds for the answer to arrive, even when the astronauts replied immediately.

This effect forms the basis of the Global Positioning System (GPS) and similar navigation systems. One's position can be determined by means of the delays in radio signals received from a number of satellites, each carrying a very accurate atomic clock, and very carefully synchronized. It is remarkable that, to work properly, this method requires that (among many other effects) the relative motion of satellite and receiver be taken into effect, which was how (on an interplanetary scale) the finite speed of light was originally discovered (see the following section).

Similarly, instantaneous remote control of interplanetary spacecraft is impossible because it takes time for the Earth-based controllers to receive information from the craft, and an equal time for instructions to be received by the craft. It can take hours for controllers to become aware of a problem, respond with instructions, and have the spacecraft receive the instructions.

The speed of light can also be of concern on very short distances. In supercomputers, the speed of light imposes a limit on how quickly data can be sent between processors. If a processor operates at 1  GHz, a signal can only travel a maximum of 300 mm in a single cycle. Processors must therefore be placed close to each other to minimize communication latencies. If clock frequencies continue to increase, the speed of light will eventually become a limiting factor for the internal design of single chips.

Physics

Luminiferous aether (discredited)

Before the advent of special relativity, it was believed that light travels through a medium called the luminiferous aether. Maxwell’s equations predict a given speed of light, in much the same way as is the speed of sound in air. The speed of sound in air is relative to the movement of the air itself, and the speed of sound in air with respect to an observer may be changed if the observer is moving with respect to the air (or vice versa). The speed of light was believed to be relative to a medium of transmission for light that acted as air does for the transmission of sound—the luminiferous aether.

The Michelson–Morley experiment, arguably the most famous and useful failed experiment in the history of physics, was designed to detect the motion of the Earth through the luminiferous aether. It could not find any trace of this kind of motion, suggesting, as a result, that it is impossible to detect one's presumed absolute motion, that is, motion with respect to the hypothesized luminiferous aether. The Michelson–Morley experiment said little about the speed of light relative to the light’s source and observer’s velocity, as both the source and observer in this experiment were traveling at the same velocity together in space.

Interaction with transparent materials

In passing through materials, the observed speed of light can differ from c. The ratio of c to the phase velocity of light in the material is called the refractive index. The speed of light in air is only slightly less than c. Denser media, such as water and glass, can slow light much more, to fractions such as and of c. Through diamond, light is much slower—only about 124,000 kilometres per second, less than of c. This reduction in speed is also responsible for bending of light at an interface between two materials with different indices, a phenomenon known as refraction.

Since the speed of light in a material depends on the refractive index, and the refractive index may depend on the frequency of the light, light at different frequencies can travel at different speeds through the same material. This effect is called dispersion.

Classically, considering electromagnetic radiation to be a wave, the charges of each atom (primarily the electrons) interact with the electric and magnetic fields of the radiation, slowing its progress.

A more complete description of the passage of light through a medium is given by quantum electrodynamics.

Faster-than-light observations and experiments

It is generally considered that it is impossible for any information or matter to travel faster than c. The equations of relativity show that, for an object travelling faster than c, some physical quantities would be not represented by real numbers. However, there are many physical situations in which speeds greater than c are encountered.

Things which can travel faster than c

Wave velocities and synchronized events

It has long been known theoretically that it is possible for the " group velocity" of light to exceed c. One recent experiment made the group velocity of laser beams travel for extremely short distances through caesium atoms at 300 times c. In 2002, at the Université de Moncton, physicist Alain Haché made history by sending pulses at a group velocity of three times light speed over a long distance for the first time, transmitted through a 120-metre cable made from a coaxial photonic crystal. However, it is not possible to use this technique to transfer information faster than c: the velocity of information transfer depends on the front velocity (the speed at which the first rise of a pulse above zero moves forward) and the product of the group velocity and the front velocity is equal to the square of the normal speed of light in the material.

Exceeding the group velocity of light in this manner is comparable to exceeding the speed of sound by arranging people distantly spaced in a line, and asking them all to shout "I'm here!", one after another with short intervals, each one timing it by looking at their own wristwatch so they don't have to wait until they hear the previous person shouting. Another example can be seen when watching ocean waves washing up on shore. With a narrow enough angle between the wave and the shoreline, the breakers travel along the waves' length much faster than the waves' movement inland.

Light spots and shadows

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

Quantum mechanics

The speed of light may also appear to be exceeded in some phenomena involving evanescent waves, such as tunnelling. Experiments indicate that the phase velocity and the group velocity of evanescent waves may exceed c; however, it would appear that the front velocity does not exceed c, so, again, it is not possible for information to be transmitted faster than c.

In quantum mechanics, certain quantum effects may be transmitted at speeds greater than c (indeed, action at a distance has long been perceived by some as a problem with quantum mechanics: see EPR paradox, interpretations of quantum mechanics). For example, the quantum states of two particles can be entangled, so the state of one particle fixes the state of the other particle (say, one must have spin +½ and the other must have spin −½). Until the particles are observed, they exist in a superposition of two quantum states, (+½, −½) and (−½, +½). 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. If, as in some interpretations of quantum mechanics, one presumes that the information about the quantum state is local to one particle, then one must conclude that second particle takes up its quantum state instantaneously, as soon as the first observation is carried out. 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. The laws of physics also appear to prevent information from being transferred through more clever ways and this has led to the formulation of rules such as the no-cloning theorem and the no-communication theorem.

Closing speeds

If two objects are travelling towards one another, each at at 0.8c as measured in a particular inertial frame of reference, then they are getting closer together at 1.6c as measured in that frame. This is called a closing speed. Note that a closing speed does not represent the speed of any object in an inertial frame.

Proper Speeds

If an spaceship travels to a planet 1 light year (as measured in the Earth's rest frame) away from Earth at high speed, the time taken to reach that planet could be less than one year as measured by the traveller's clock (although it will always be more than one year as measured by a clock on Earth). The value obtained by dividing the distance travelled, as determined in the Earth's frame, by the time taken, measured by the traveller's clock, is known a proper speed or a proper velocity. There is no limit on the value of a proper speed as a proper speed does not represent a speed measured in a single inertial frame. Note that a light signal that left the Earth at the same time as the traveller would always get to the destination before the traveller.

Things which only appear to travel faster than c

So-called superluminal motion is seen in certain astronomical objects, 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.

Travel faster than the speed of light in a medium

Although it may sound paradoxical, it is possible for shock waves to be formed with electromagnetic radiation. As a charged particle travels through an insulating medium, it disrupts the local electromagnetic field in the medium. Electrons in the atoms of the medium will be displaced and polarised by the passing field of the charged particle, and photons are emitted as the electrons in the medium restore themselves to equilibrium after the disruption has passed. (In a conductor, the equilibrium can be restored without emitting a photon.) In normal circumstances, these photons destructively interfere with each other and no radiation is detected. However, if the disruption travels faster than the photons themselves travel, as when a charged particle exceeds the speed of light in that medium, the photons constructively interfere and intensify the observed radiation. The result (analogous to a sonic boom) is known as Čerenkov radiation.

General relativity

Some topics (such as the expansion of the universe, and wormholes) require the application of general relativity and are covered in the main faster than light article.

Other theories concerning the speed of light

Particles that travel faster than light, dubbed tachyons, have been proposed by particle physicists but have yet to be observed, and would potentially violate causality if they were.

Some physicists, notably João Magueijo and John Moffat, have proposed that in the past light traveled much faster than the current speed of light. This theory is called variable speed of light (VSL) and its supporters claim that it has the ability to explain many cosmological puzzles better than its rival, the inflation model of the universe. However, it has not gained wide acceptance.

Science fiction

The ability to communicate or travel faster than light is a popular topic in science fiction.

Slow light

Light traveling through a medium other than a vacuum travels below c as a result of the time lag between the polarization response of the medium and the incident light. However, certain materials have an exceptionally high group index and a correspondingly low group velocity. In 1999, a team of scientists led by Lene Hau were able to slow the speed of a light pulse to about 17 metres per second; in 2001, they were able to momentarily stop a beam.

In 2003, Mikhail Lukin, with scientists at Harvard University and the Lebedev Institute in Moscow, succeeded in completely halting light by directing it into a Bose–Einstein condensate of the element rubidium, the atoms of which, in Lukin's words, behaved "like tiny mirrors" due to an interference pattern in two "control" beams.

History

Until relatively recent times, the speed of light was largely a matter of conjecture. Empedocles maintained that light was something in motion, and therefore there had to be some time elapsed in travelling. Aristotle said that, on the contrary, "light is due to the presence of something, but it is not a movement". Furthermore, if light had a finite speed, it would have to be very great; Aristotle asserted "the strain upon our powers of belief is too great" to believe this.

One of the ancient theories of vision was that light was emitted from the eye, instead of entering the eye from another source. Using this 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.

Medieval and early modern theories

Early Muslim philosophers initially agreed with Aristotle's view that light has an infinite speed. In the 11th century, however, Muslim scientists realized that light has a finite speed. The Iraqi scientist Ibn al-Haytham (Alhacen), the father of optics, using an early experimental scientific method in his Book of Optics, discovered that light has a finite speed. Some of his contemporaries, notably the Persian scientists Avicenna and al-Biruni, also agreed with Alhacen that light has a finite speed. Avicenna "observed that if the perception of light is due to the emission of some sort of particles by a luminous source, the speed of light must be finite". Al-Biruni further discovered that the speed of light is much faster than the speed of sound.

Commenting on a verse in the Rigveda ("Swift and all beautiful art thou, O Surya, maker of the light; illuminating all the radiant realm."), the 14th century Indian scholar Sayana wrote "Thus it is remembered: [O Sun] you who traverse 2202 yojanas [1 yojana is about 9 miles] [ca. 14,000 to 30,000 km] in half a nimesa [ca. 0.1 to 0.2 s]", corresponding to between 65,000 and 300,000 km/s, for high values of yojana and low values of nimesa consistent with the actual speed of light.

Johannes Kepler believed that the speed of light is infinite since empty space presents no obstacle to it. Francis Bacon argued that the speed of light is 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 is infinite. Descartes speculated that if the speed of light was found to be finite, his whole system of philosophy might be demolished.

Measurement of the speed of light

Early attempts

Isaac Beeckman proposed an experiment (1629) in which a person would observe the flash of a cannon reflecting off a mirror about one mile away. Galileo proposed an experiment (1638), 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. This experiment was carried out by the Accademia del Cimento of Florence in 1667, with the lanterns separated by about one mile. No delay was observed. 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

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 C or D). 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. It was clear that these exit "signals" took longer to reach Earth, as Earth and Jupiter moved further apart. This was as a result of the extra time it took for light to cross the extra distance between the planets, time which had accumulated in the interval between one signal and the next. The opposite is the case when they are approaching (as from F to G). On the basis of his observations, Rømer estimated that it would take light 22 minutes to cross the diameter of the orbit of the Earth (that is, twice the astronomical unit); the modern estimate is about 16 minutes and 40 seconds.

Around the same time, the astronomical unit was estimated to be about 140 million kilometres. 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. 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, 30% less than the actual value), which it seems he inferred for himself (whether from Rømer's data, or otherwise, is not known). The same effect was subsequently observed by Rømer for a "spot" rotating with the surface of Jupiter. And 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.

Even if, by these observations, the finite speed of light may not have been established to everyone's satisfaction (notably Jean-Dominique Cassini's), after the observations of James Bradley (1728), the hypothesis of infinite speed was considered discredited. 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 (185,000 miles per second). This is only slightly less than the currently accepted value. The aberration effect has been studied extensively over the succeeding centuries, notably by Friedrich Georg Wilhelm Struve and de:Magnus Nyrén.

Earth-bound techniques

The first successful measurement of the speed of light using an earthbound apparatus was carried out by Hippolyte Fizeau in 1849. (This measures the speed of light in air, which is slower than the speed of light in vacuum by a factor of the refractive index of air, about 1.0003.) Fizeau's experiment was conceptually similar to those proposed by Beeckman and Galileo. A beam of light was directed at a mirror several thousand metres 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. Fizeau's method was later refined by Marie Alfred Cornu (1872) and Joseph Perrotin (1900).

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. Foucault's method was also used by Simon Newcomb and Albert A. Michelson. Michelson began his lengthy career by replicating and improving on Foucault's method.

In 1926, Michelson used a rotating prism to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California, a distance of about 22 miles (36 km). The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second).

Laboratory-based methods

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 in collaboration with A.C. Gordon-Smith used a microwave cavity of precisely known dimensions to establish the frequency for a variety of normal modes of microwaves—which, in common with all electromagnetic radiation, travels at the speed of light in vacuum. 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. Their result, 299,792±3 km/s, was substantially greater 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; this became the value adopted by the 12th General Assembly of the Radio-Scientific Union in 1957. Most subsequent measurements have been consistent with this value.

Speed of light set by definition

In 1983, the 17th Conférence Générale des Poids et Mesures defined the meter in terms of the distance traveled by light in a given amount of time, which amounts to adopting a standard value for the speed of light in vacuum:

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.

Here, the term vacuum is meant in the technical sense of free space. This definition of the metre relies on the definition of the second, which is:

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

The consequence of this definition is that further refinements in the current experimental value of the speed of light would only refine the definition of a metre. This point is made explicit by nondimensionalization in the article on Maxwell's equations. The value of c, or c₀, namely:

combined with the definition of magnetic constant μ₀, also defines the electric constant ε₀ in SI units. The magnetic constant μ₀ is not dependent on c and as a result of the definition of the ampere, has a standard value in SI units of:

.

The electric constant has then the exact value

These constants appear in Maxwell's equations.

Special relativity

After the work of James Clerk Maxwell, it was believed that light travelled at a constant speed relative to the "luminiferous aether", the medium that was then thought to be necessary for the transmission of light. This speed was determined by the ( permittivity and permeability) of the aether.

In 1887, the physicists Albert Michelson and Edward Morley performed the influential Michelson-Morley experiment to measure the speed of light relative to the motion of the earth, the goal being to measure the velocity of the Earth through the aether. 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 (the same number for each beam to give a long but equal path length; the actual Michelson-Morley experiment used more mirrors than shown) then recombined to produce a pattern of constructive and destructive interference. Any slight change in speed of light along each arm of the interferometer (because the apparatus was moving with the Earth through the proposed "aether") would change the amount of time that the beam spent in transit, which would then be observed as a change in the pattern of interference. In the event, the experiment gave a null result.

Ernst Mach was among the first physicists to suggest that the experiment amounted to a disproof of the aether theory. Developments in theoretical physics had already begun to provide an alternative theory, Fitzgerald-Lorentz contraction, which explained the null result of the experiment.

It is uncertain whether Albert Einstein knew the results of the Michelson-Morley experiment, but the null result of the experiment greatly assisted the acceptance of his theory of relativity. The constant speed of light is one of the fundamental Postulates (together with causality and the equivalence of inertial frames) of special relativity.