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Speed of light

From Academic Kids

 in a "swimming pool" . The effect is due to  moving faster than the speed at which light moves in water.
Cherenkov effect in a "swimming pool" nuclear reactor. The effect is due to electrons moving faster than the speed at which light moves in water.

The speed of light in a vacuum is exactly equal to 299,792,458 metres per second (approximately 186,282.4 miles per second). The speed of light is denoted by the letter c, reputedly from the Latin celeritas, "speed", and also known as Einstein's constant. This exact speed is a definition, not a measurement, as the metre itself is defined in terms of the speed of light and the second. The speed of light through a medium (that is, not in vacuum) is less than c (defining the refractive index of the medium). "Speed of light" is sometimes abbreviated SOL.

Contents

Overview

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

One consequence of the laws of electromagnetism (such as Maxwell's equations) is that the speed c of electromagnetic radiation does not depend on the velocity of the object emitting the radiation; thus for instance the light emitted from a rapidly moving light source would travel at the same speed as the light coming from a stationary light source (although the colour, frequency, energy, and momentum of the light will be shifted, thanks to the Doppler effect). If one combines this observation with the principle of relativity, one concludes that all observers will measure the speed of light in vacuum as being the same, regardless of the reference frame of the observer or the velocity of the object emitting the light. Because of this, one can view c as a fundamental physical constant. This fact can then be used as a basis for the theory of special relativity. It is worth noting that it is the constant speed c, rather than light itself, which is fundamental to special relativity; thus if light is somehow slowed to travel at less than c, this will not directly affect the theory of special relativity.

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 such a way that 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 observer's perceived time and the time perceived 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.

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

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 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 (approximately 3 × 108 metres per second, or about thirty centimetres (one foot) per nanosecond). The value of <math>c<math> defines the permittivity of free space (<math>\epsilon_0<math>) in SI units as:

<math> \varepsilon_0 = 10^{7}/4\pi c^2 \quad \mathrm{(in~ A^2\, s^4\, kg^{-1}\, m^{-3}, \, or \, F \, m^{-1})}<math>

The permeability of free space (<math>\mu_0<math>) is not dependant on <math>c<math> and is defined in SI units as:

<math> \mu_0 = 4\,\pi\, 10^{-7} \quad \mathrm{(in~ kg\, m\, s^{-2}\, A^{-2}, \, or \, N \, A^{-2})}<math>.

These constants appear in Maxwell's equations, which describe electromagnetism, and are related by:

<math>c= \frac {1} {\sqrt{\varepsilon_0\mu_0}}<math>

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 popularised texts.

Definition of the metre

Historically, the metre had 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 1/299,792,458 light second). This relies on the constancy of the velocity of light for all observers. "What, then, does it mean to measure the speed of light?", one may ask. The answer is that finding any measured difference from the defined value means that one's length or time standard is wrong, or is exhibiting a change since it was calibrated. If such change is true physics, and not error or ascribable to a perturbation (such as temperature change or mechanical shock), one has made an important discovery.

The motivation in changing the definition of the metre, as with all unit definition changes, was to provide a very precise definition of the unit that could easily be used to calibrate equipment throughout the world. The standard bar was impractical in this sense, since it could not be removed from its vault or used by two scientists at once. It was also prone to massive length changes (compared to the accuracy required) due to temperature variations, so it required a long settling time. Wear of the ends, oxidation, etc. became major issues in the quest for ultimate accuracy.

Communications

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

The actual transit time is longer, in part because the speed of light is slower by about 30% in an optical fibre 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 an Australia or Japan to US computer-to-computer ping is 0.18 second. 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 became the first man to set foot on the Moon: 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. (See animation.)

Similarly, instantaneous remote control of an interplanetary spacecraft is impossible, in the sense that the time it takes, for example, for the earth-based controllers to become aware of a problem, plus the time it takes for the spacecraft to receive their response, can be some hours.

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 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 minimise 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 velocity from all reference frames

It is important to realise 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 addition rule of velocities: if two cars approach each other from opposite directions, 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 mph) to a very high degree of accuracy.

At velocities at or approaching the speed of light, however, it becomes clear from experimental results that this 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:

<math>u = {v + w \over 1 + v w / c^2} \,\!<math>

where v and w are the speeds of the spaceships as observed by the third observer, and u is the speed of either space ship as observed by the other.

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.

Interaction with transparent materials

Missing image
PrismAndLight.jpg
The refractive index of a material indicates how much slower the speed of light is in that medium than in a vacuum. The slower speed of light in materials can cause refraction, 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).

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 <math>c<math>. Denser media, such as water and glass, can slow light much more, to fractions such as 3/4 and 2/3 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 depends on the frequency of the light, light at different frequencies travels at different speeds through the same material. This can cause distortion of electromagnetic waves that consist of multiple frequencies, called dispersion.

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, 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.

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 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.

In some interpretations of quantum mechanics, 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. 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.

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 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 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.

Some physicists, notably Joao Magueijo and John Moffat, have proposed that in the past light travelled 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 yet to gain wide acceptance.

Light-slowing experiments

 phenomena, such as this , are due to the slower speed of light in a medium (water, in this case).
Enlarge
Refractive phenomena, such as this rainbow, are due to the slower speed of light in a medium (water, in this case).

In a sense, any light travelling through a medium other than a vacuum travels below <math>c<math> 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 metres 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

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 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.

Medieval and early modern theories

The Islamic philosophers Avicenna and Alhazen believed that light has a finite speed, although most philosophers agreed with Aristotle on this point. The Aryan school of philosophy in ancient India also held the speed of light to be finite. The 14th century philosopher Sayana wrote the following comment on verse 1.50 of the Rig Veda:

"Thus it is remembered: [O Sun] you who traverse 2202 yojanas in half a nimesa."

According to some, this refers to the speed of light. It is not known exactly how long a yojana and a nimesa is, but this value is possibly accurate to within 1% (Kak, 1998), though by adopting other possible values of these units the accuracy of the statement can be reduced to a factor of 4.

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. Ren 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 criticised 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 Rmer, who was studying the motions of Jupiter's satellite Io with a telescope. It is possible to time the revolution of Io because it is entering/exiting Jupiter's shadow at regular intervals. Rmer observed that Io revolved around Jupiter once every 42.5 hours when Earth was closest to Jupiter. He also observed that, as Earth and Jupiter moved apart, 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, as a result of the extra time it took for light to cross the extra distance between the planets, which had accumulated in the interval between one signal and the next. Similarly, about half a year later, Io's entries into the shadow happened more frequently, as Earth and Jupiter were now drawing closer together. On the basis of his observations, Rmer 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 astronomical unit and Rmer's time estimate were combined by Christiaan Huygens, who estimated the speed of light to be 1000 Earth diameters per minute. 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.

Isaac Newton also accepted the finite speed. In his book "Opticks" he, in fact, reports the more accurate value of 16 minutes per diameter, which it seems he inferred for himself (whether from Rmer's data, or otherwise, is not known). The same effect was subsequently observed by Rmer 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 185,000 miles per second (298,000 kilometres 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.

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Speed_of_light_(Fizeau).PNG
Diagram of the Fizeau-Foucault apparatus.

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 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 rotating mirrors to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California. The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second).

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).

Missing image
Interferometer.png
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 transmission 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 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 (due to the fact that 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 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 light is one of the fundamental Postulates (together with causality and the equivalence of inertial frames) of special relativity.

See also

References

Historical references

  • Ole Rmer. "Dmonstration touchant le mouvement de la lumire", Journal des Savans, 7 Dcembre 1676, pp. 223-236. Translated as "A Demonstration concerning the Motion of Light", Philosophical Transactions of the Royal Society no. 136, pp. 893-894; June 25, 1677. (Rmer's 1676 paper, in English and French, as bitmap images: [1] (http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html), and in French as plain text: [2] (http://astro.campus.ecp.fr/histoire/roemer.html))
  • 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
  • Subhash Kak. The Speed of Light and Purāṇic Cosmology. In T.R.N. Rao and S. Kak, Computing Science in Ancient India, pages 80–90. USL Press, Lafayette, 1998. Available as e-print physics/9804020 (http://uk.arxiv.org/abs/physics/9804020) on the arXiv.
  • 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: [3] (http://www.stats.uwaterloo.ca/~rwoldfor/papers/sci-method/paperrev))

External links

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