Speed of light

Light traveling through a medium such as air (for example, this laser) travels slower than light through a vacuum.

The speed of light in a vacuum (denoted as $c$ , reputedly from the Latin celeritas, "speed") is exactly equal to 299,792,458 metres per second, which is approximately 300,000 kilometres per second, or 186,000 miles per second. This exact speed is a definition, not a measurement, as the metre is defined in terms of the speed of light and not vice versa. The speed of light through a medium (that is, not in vacuum) is less than $c$ due to refraction.

Overview

According to standard modern physical theory, all electromagnetic radiation, including visible light, propagates (or moves) at a constant speed in vacuum, known as the speed of light, which is a physical constant denoted as $c$ .

According to the theory of special relativity, all observers will measure the speed of light as being the same, regardless of the reference frame of the observer or the velocity of the object emitting the light. A simple analysis can be used to suggest that this is the case:

the speed of light in a vacuum can be derived from Maxwell's equations, and
special relativity posits that the laws of physics (such as Maxwell's equations) are identical in all unaccelerated frames; therefore:
observers in all such frames must observe the same speed of light.

Observers travelling at large velocities will find that distances and times are distorted ("dilated") in accordance with the Lorentz transforms; however, the transforms distort times and distances in the same way so the speed of light remains constant. A person travelling near the speed of light would also find that colours of lights ahead were blue shifted and those of those behind were red shifted.

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 'cause' could be observed after the 'effect'. Due to special relativity's time dilation, the ratio between an external oberver's percieved time and the time percieved by an observer moving closer and closer to the speed of light approaches zero. If something could move faster than light, this ratio would not be a real number. Such a violation of causality has never been observed.

To put it another way, information propogates 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). However, there are also frames in which A precedes C (as shown) or in which C precedes A. Barring some way of travelling 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.

According to the currently prevailing definition, adopted in 1983, the speed of light is exactly 299,792,458 metres per second. This is approximately 3 × 10⁸ metres per second, that is, about thirty centimetres (12 inches) per nanosecond. The value of $c$ defines the permittivity of free space ( $\epsilon _{0}$ ) and the permeability of free space ( $\mu _{0}$ ) in SI units as:

\epsilon _{0}=10^{7}/4\pi c^{2}\quad \mathrm {(in~A^{2}\,s^{4}\,kg^{-1}\,m^{-3},\,or\,F\,m^{-1})}

\mu _{0}=4\,\pi \,10^{-7}\quad \mathrm {(in~kg\,m\,s^{-2}\,A^{-2},\,or\,N\,A^{-2})}

.

These constants appear in Maxwell's equations, which describe electromagnetism.

Astronomical distances are sometimes measured in light years (the distance that light would travel in one year, roughly 9.46 × 10¹² kilometres or about 5.88 × 10¹² miles) especially in popularized texts.

Definition of the metre

Historically, the metre has been defined as a fraction of the length of a meridian through Paris, by reference to the length of a standard bar, and by reference to the wavelength of a particular frequency of light. Since 1983, the metre has been defined by reference to the second and the speed of light.

In 1967, the Thirteenth General Conference on Weights and Measures defined the second of atomic time as 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, which remains the current definition of the second.

In 1983, the General Conference on Weights and Measures defined the metre as the length of the path travelled by light in absolute vacuum during a time interval of 1/299,792,458 of a second (i.e. one metre is $1/299792458\cdot c\cdot \mathrm {s} \,\!$ ). This relies on the constancy of the velocity of light for all observers.

Communications

The speed of a light is of relevance to communications. For example, given that the equatorial circumference of the Earth is 40,075 km and $c$ , the theoretical shortest amount of time for a piece of information to travel half the globe is 0.067 seconds.

As the speed of light is smaller in an optical fiber and straight lines rarely occur in communications situations, a typical time as of 2004 for an Australia or Japan to US computer-to-computer ping is 0.250 seconds. The speed of light additionally affects wireless communications design.

The finite speed of light became quite apparent to everybody following the communication of Houston ground control and Neil Armstrong when he set foot on the moon as first man ever: For every question, Houston had to wait nearly 3 seconds for the answer to arrive, and would have to do so even if the astronauts replied immediately.

The speed of light can also be of concern on short distances. In supercomputers, the speed of light imposes a limit on how quickly data can be sent between nodes. If a processor operates at 1 GHz, a signal can only travel a maximum of 300 mm in a single cycle. Nodes 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

Constant in all reference frames

It is important to realize that the speed of light is not a "speed limit" in the conventional sense. An observer chasing a beam of light will measure it moving away from him at the same speed as a stationary observer. This leads to some unusual consequences for velocities.

Most individuals are accustomed to the additive rule of velocities: if two cars approach each other, each travelling at a speed of 50 kilometres per hour (31 miles per hour), one expects that each car will perceive the other as approaching at a combined speed of 50 + 50 = 100 km/h (62 km/h) to a very high degree of accuracy.

At velocities approaching or at the speed of light, however, it becomes clear from experimental results that this additive rule does not apply. Two spaceships approaching each other, each travelling at 90% the speed of light relative to some third observer between them, do not perceive each other as approaching at 90 + 90 = 180% the speed of light; instead they each perceive the other as approaching at slightly less than 99.5% the speed of light.

This last result is given by the Einstein velocity addition formula:

u={v+w \over 1+vw/c^{2}}

where $v$ and $w$ are the speeds of the spaceships relative to the observer, and $u$ is the speed perceived by the observer.

Contrary to one's usual intuitions, regardless of the speed at which one observer is moving relative to another observer, both will measure the speed of an incoming light beam as the same constant value, the speed of light.

The above equation was derived by Albert Einstein from his theory of special relativity, which takes the principle of relativity as a main premise. This principle (originally proposed by Galileo Galilei) requires physical laws to act in the same way in all reference frames. As Maxwell's equations directly give a speed of light, it should be the same for every observer—a consequence which sounded obviously wrong to the 19th century physicists, who assumed that the speed of light given by Maxwell's theory is valid relative to the luminiferous aether. But the Michelson-Morley experiment, arguably the most famous and useful failed experiment in the history of physics, could not find this aether, suggesting instead that the speed of light is constant in all frames of reference.

Although it is uncertain whether Einstein knew the results of the Michelson-Morley experiment, he took the speed of light being constant as a given fact, understood it as reaffirming Galilei's principle of relativity, and deduced the consequences, now known as the theory of special relativity which includes the counter-intuitive addition formula above.

Refraction

In passing through materials, light is slowed to less than $c$ by the ratio called the refractive index of the material. 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 3/4 and 2/3 of $c$ .

On the microscopic scale, considering electromagnetic radiation to be like a particle, refraction is caused by continual absorption and re-emission of the photons that compose the light by the atoms or molecules through which it is passing. In some sense, the light itself travels only through the vacuum existing between these atoms, and is impeded by the atoms. Alternatively, considering electromagnetic radiation to be like a wave, the charges of each atom (primarily the electrons) interfere with the electric and magnetic fields of the radiation, slowing its progress.

"Faster-than-light" observations and experiments

Recent experimental evidence shows that it is possible for the group velocity of light to exceed c. One experiment made the group velocity of laser beams travel for extremely short distances through caesium atoms at 300 times $c$ . 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 in a distantly spaced line of people, 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 last person shouting.

The speed of light may also appear to be exceeded in some phenomena involving evanescent waves, such as tunneling. Experiments indicate that the phase velocity of evanescent waves may exceed $c$ ; however, it would appear that neither the group velocity nor the front velocity exceed $c$ , so, again, it is not possible for information to be transmitted faster than $c$ .

Certain quantum effects may be transmitted at speeds greater than $c$ (indeed, action at a distance has long been perceived as a problem with quantum mechanics: see EPR paradox). 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. Experiments show that the 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 partcle will take on when it is observed, so no information can be transmitted in this manner.

So-called superluminal motion is also seen in certain astronomical objects, such as the jets of radio galaxies and quasars. However, these jets are not actually 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.

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 polarized 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 (material), the disruption 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, the photons constructively interfere and intensify the observed radiation. The result (analogous to a sonic boom) is known as Cherenkov radiation.

The ability to communicate or travel faster-than-light is a popular topic in science fiction. Particles that travel faster than light, dubbed tachyons, have been proposed by particle physicists but have yet to be observed.

Light slowing experiments

In a sense, any light travelling through a medium other than a vacuum travels below $c$ as a result of refraction. However, certain materials have an exceptionally high refractive index: in particular, the optical density of a Bose-Einstein condensate can be very high. In 1999, a team of scientists led by Lene Hau were able to slow the speed of a light beam to about 17 meters per second, and, 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 mass of hot rubidium gas, the atoms of which, in Lukin's words, "[behaved] like tiny mirrors", due to an interference pattern in two "control" beams.

History

The Ancients

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 traveling. 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 is that light is emitted from the eye, instead of being reflected into the eye from another source. On this theory, Heron of Alexandria advanced the argument that the speed of light must be infinite, since distant objects such as stars appear immediately when one opens one's eyes.

Whether the speed of light is infinite

The Islamic philosophers Avicenna and Alhazen believed that light has a finite speed, although most philosophers agreed with Aristotle on this point. 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. Rene 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. In fact, Descartes was convinced that if the speed of light were finite, his whole system of philosophy would be demolished.

Measurement of the speed of light

Isaac Beeckman, a friend of Descartes, proposed an experiment (1629) in which one 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 have 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. Descartes criticized this experiment as superfluous, in that the observation of eclipses, which had more power to detect a finite speed, gave a negative result. 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.

The first quantitative estimate of the speed of light was made in 1676 by Ole Rømer, who was studying the motions of Jupiter's satellite Io. Rømer observed that eclipses of Io by Jupiter appeared sooner when Earth was approaching Jupiter and later when Earth was moving farther away. Rømer correctly deduced that this discrepancy was due to the time it took for light to cross the lesser or greater distance between the planets. 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 closer to 16 minutes and 40 seconds. Around the same time, the astronomical unit was estimated to be about 140 million kilometres. The two results were combined by Christiaan Huygens, who estimated the speed of light to be 16 2/3 Earth diameters per second. This is about 220,000 kilometres per second (136,000 miles per second), well below the currently accepted value, but still very much faster than any physical phenomenon then known.

The finite speed of light was not conclusively established by these observations, as it could be argued the differences in the times of eclipses were due to perturbations of the orbits of the satellites. However, 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 185,000 miles per second (298,000 km 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 Magnus Nyren.

The first successful measurement of the speed of light using an earthbound apparatus 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 several thousand meters 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.

Relativity

Due to the works of James Clerk Maxwell, it was known that the speed of electromagnetic radiation was a constant defined by the electromagnetic properties of the vacuum (permittivity and permeability).

A schematic representation of a Michelson interferometer, as used for the Michelson-Morley experiment.

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", the medium that was then thought to be necessary for the tramsission of light. 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 travelling at right angles to one other. 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 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 actually amounted to a disproof of the aether theory. Developments in theoretical physics had already begun to provide an alternate 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. Einstein's theory did not require an aether but was entirely consistent with the null result of the experiment: aether did not exist and the speed of light was the same in each direction. The constant speed of the speed of light is one of the fundamental Postulates (together with causality and the equivalence of inertial frames) of special relativity.

References

Historical references

Edmund Halley. "Monsieur Cassini, his New and Exact Tables for the Eclipses of the First Satellite of Jupiter, reduced to the Julian Stile and Meridian of London", Philosophical Transactions XVIII, No. 214, pp 237-256, Nov.-Dec., 1694.
H.L. Fizeau. "Sur une experience relative a la vitesse de propogation de la lumiere", Comptes Rendus 29, 90-92, 132, 1849.
J.L. Foucault. "Determination experimentale de la vitesse de la lumiere: parallaxe du Soleil", Comptes Rendus 55, 501-503, 792-796, 1862.
A.A. Michelson. "Experimental Determination of the Velocity of Light", Proceedings of the American Association for the Advancement of Science 27, 71-77, 1878.
Simon Newcomb. "The Velocity of Light", Nature, pp 29-32, May 13, 1886.
Joseph Perrotin. "Sur la vitesse de la lumiere", Comptes Rendus 131, 731-734, 1900.
A.A. Michelson, F.G. Pease, and F. Pearson. "Measurement Of The Velocity Of Light In A Partial Vacuum", Astrophysical Journal 82, 26-61, 1935.

Modern references

John David Jackson. Classical electrodynamics. John Wiley & Sons, 2nd edition, 1975; 3rd edition, 1998. ISBN 047130932X
R.J. MacKay and R.W. Oldford. "Scientific Method, Statistical Method and the Speed of Light", Statistical Science 15(3):254–278, 2000. (Also available on line: [1])

External links

speed of light in vacuum (at NIST)
A Brief History of c
Data Gallery: Michelson Speed of Light (Univariate Location Estimation) (download data gathered by A.A. Michelson)
Switching light on and off (news article on stopping light)
Beam smashes light barrier (news article on group velocity experiment)
Subluminal (Java applet demonstrating group velocity information limits)