Variable speed of light
A variable speed of light (VSL) is a feature of a family of hypotheses stating that the speed of light in vacuum, usually denoted by c, may in some way not be constant, e.g. varying in space or time, or depending on frequency. A variable speed of light occurs in some situations of classical physics as equivalent formulations of accepted theories, but also in various alternative theories of gravitation and cosmology, many of them non-mainstream.
The speed of light in vacuum instead is considered a constant, and defined by the SI as 299,792,458 m/s. Variability of the speed of light is therefore equivalent with a variability of the SI meter and/or the SI second.
VSL should not be confused with faster than light theories; nor should it be confused with the fact that the speed of light in a medium is slower than the speed of light in vacuum depending on the medium's refractive index.
Einstein's early proposal (1911)
While Einstein first mentioned a variable speed of light in 1907, he reconsidered the idea more thoroughly in 1911. In analogy to the situation in a medium, where a shorter wavelength , by means of , leads to a lower speed of light, Einstein assumed that clocks in a gravitational field run slower, whereby the corresponding frequencies are influenced by the gravitational potential (eq.2, p. 903):
Einstein commented (pages 906–907):
"Aus dem soeben bewiesenen Satze, daß die Lichtgeschwindigkeit im Schwerefelde eine Funktion des Ortes ist, läßt sich leicht mittels des Huygensschen Prinzipes schließen, daß quer zum Schwerefeld sich fortpflanzende Lichtstrahlen eine Krümmung erfahren müssen." ("From the just proved assertion, that the speed of light in a gravity field is a function of position, it is easily deduced from Huygens's principle that light rays propagating at right angles to the gravity field must experience curvature.")
In a subsequent paper in 1912, he concluded that:
"Das Prinzip der Konstanz der Lichtgeschwindigkeit kann nur insofern aufrechterhalten werden, als man sich auf für Raum-Zeitliche-Gebiete mit konstantem Gravitationspotential beschränkt." ("The principle of the constancy of the speed of light can be kept only when one restricts oneself to space-time regions of constant gravitational potential.")
However, Einstein deduced a light deflection at the Sun of "almost one arcsecond" which is just one-half of the measured value later derived by his theory of general relativity. The measured value was provided by Eddington in 1919.
Einstein's updated proposals (1905–1915)
Albert Einstein went through several versions of light speed theory between 1905 and 1915, eventually concluding that light speed is constant when gravity does not have to be considered but that the speed of light cannot be constant in a gravitational field with variable strength. In the same book Einstein explained that he intended light speed to be a vector when it was described by coordinates in a reference frame.
Einstein was not explicit about whether the speed of light would change in a gravity field, or whether just the direction of the vector would change; this can be clarified from the text implicitly, however. A calculation of alpha (α) follows equation 107 and makes an unambiguous use of variable scalar light velocity (L) both as the argument of a partial differential function (proving a variable) and as the denominator in a fraction (proving not a vector) both in the same integrated quantity. Division of a scalar by a vector is not defined, so there is no other way to interpret the velocity of light in this usage except as a variable scalar speed.
In this calculation L = c/c0 where c0 is light speed in flat space.
Peter Bergmann did not agree with Einstein, but left the dispute out of his earlier book in 1942 to get Einstein's endorsement. After Einstein died Bergmann wrote a new book in 1968 claiming that vector light velocity could change direction but not speed. This has become a prevailing opinion in science, but not in agreement with Einstein's unambiguous math. Bergman did acknowledge that the apparent speed of light would change in a gravity field and go to zero at an event horizon as viewed by a distant observer.
Max Born agreed with Einstein and stated both speed and direction of light change in a gravity field first published 1923 in English, 1920 in German. Born expressed the magnitude of light speed as the square root of (−g44/g11) gravity components in general relativity to leave no doubt about his intent.
Richard Tolman also agreed with Einstein and expressed the radial speed of light as dr/dt in a gravity field.
where m is MG/c2 and where natural units are used such that c0 is equal to one.
Dicke's proposal (1957)
Robert Dicke, in 1957, developed a related VSL theory of gravity. In contrast to Einstein, Dicke assumed not only the frequencies to vary, but also the wavelengths. Since , this resulted in a relative change of c twice as much as considered by Einstein. Dicke assumed a refractive index (eqn.5) and proved it to be consistent with the observed value for light deflection. In a comment related to Mach's principle, Dicke suggested that, while the right part of the term in eq. 5 is small, the left part, 1, could have "its origin in the remainder of the matter in the universe".
Given that in a universe with an increasing horizon more and more masses contribute to the above refractive index, Dicke considered a cosmology where c decreased in time, providing an alternative explanation to the cosmological redshift  (p. 374). Dicke's theory is compatible with the SI definition of c since the time and length units second and meter can vary in concert, keeping the value of c fixed (p. 366).
"note that the photon speed is ... ." ( denoting the gravitational potential −GM/r)
Based on this observation, variable speed of light models, including Dicke's, have been developed which agree with all known tests of general relativity, but some are distinct for higher-order tests.
Several hypotheses for varying speed of light, seemingly in contradiction to general relativity theory, have been published, including those of Giere and Tan (1986) and Sanejouand (2005). In 2003 Magueijo gave a review of such hypotheses.
Modern VSL theories as an alternative to cosmic inflation
The varying speed of light cosmology[clarification needed] has been proposed independently by Jean-Pierre Petit in 1988, John Moffat in 1992, and the team of Andreas Albrecht and João Magueijo in 1998 to explain the horizon problem of cosmology and propose an alternative to cosmic inflation.
In Petit's VSL model, the variation of c accompanies the joint variations of all physical constants combined to space and time scale factors changes, so that all equations and measurements of these constants remain unchanged through the evolution of the universe. This provides a universal gauge relationship and the secular variation of the parameters usually taken to be constants:
see figure "secular variation of constant". The Einstein field equations remain invariant through convenient joint variations of c and G in the Einstein gravitational constant. According to this model, the cosmological horizon grows like R, the space scale, which ensures the homogeneity of the primeval universe, which fits the observational data. Subsequent developments of the model restricts the variation of constants to the higher energy density of the early universe, at the very beginning of the radiation-dominated era where spacetime is identified to space-entropy with a metric conformally flat.
Moffat and the team Albrecht–Magueijo present the idea that the speed of light had a very large value of about 1035 km/s in the early universe, thus their models show distant regions of the expanding universe interacting at the beginning of the universe[clarification needed]. There is no known way to solve the horizon problem with a variation of the fine-structure constant[clarification needed], because its variation does not change the causal structure of spacetime. To do so would require[why?] modifying gravity by varying Newton's constant or redefining special relativity.[clarification needed]
Classically,[clarification needed] varying speed of light cosmologies propose to circumvent this by varying the dimensionful quantity c by breaking the Lorentz invariance of Einstein's theories of general and special relativity in a particular way. Magueijo has also proposed a formulation that preserves local Lorentz invariance.
C. Köhn actually took over the idea by Albrecht and Magueijo  and investigated the motion of a relativistic particle in two time dimensions . He showed that the speed of light depends on time as . However, this time dependency is negligible for the current state/size of the Universe which is the reason why the speed of light appears to be constant at present times. However, two time dimensions explain why the speed of light was not constant in the Early Universe as proposed by Albrecht and Magueijo.
Various other VSL occurrences
Virtual photons in some calculations in quantum field theory may also travel at a different speed for short distances; however, this doesn't imply that anything can travel faster than light. While it has been claimed (see VSL criticism below) that no meaning can be ascribed to a dimensional quantity such as the speed of light varying in time (as opposed to a dimensionless number such as the fine structure constant), in some controversial theories in cosmology, the speed of light also varies by changing the postulates of special relativity.
Varying photon speed
The photon, the particle of light which mediates the electromagnetic force is believed to be massless. The so-called Proca action describes a theory of a massive photon. Classically, it is possible to have a photon which is extremely light but nonetheless has a tiny mass, like the neutrino. These photons would propagate at less than the speed of light defined by special relativity and have three directions of polarization. However, in quantum field theory, the photon mass is not consistent with gauge invariance or renormalizability and so is usually ignored. However, a quantum theory of the massive photon can be considered in the Wilsonian effective field theory approach to quantum field theory, where, depending on whether the photon mass is generated by a Higgs mechanism or is inserted in an ad hoc way in the Proca Lagrangian, the limits implied by various observations/experiments may be different. So, therefore, the speed of light is not constant.
Varying c in quantum theory
In quantum field theory the Heisenberg uncertainty relations indicate that photons can travel at any speed for short periods. In the Feynman diagram interpretation of the theory, these are known as "virtual photons", and are distinguished by propagating off the mass shell. These photons may have any velocity, including velocities greater than the speed of light. To quote Richard Feynman "...there is also an amplitude for light to go faster (or slower) than the conventional speed of light. You found out in the last lecture that light doesn't go only in straight lines; now, you find out that it doesn't go only at the speed of light! It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c." These virtual photons, however, do not violate causality or special relativity, as they are not directly observable and information cannot be transmitted acausally in the theory. Feynman diagrams and virtual photons are usually interpreted not as a physical picture of what is taking place, but rather as a convenient calculation tool (which, in some cases, happen to involve faster-than-light velocity vectors).
Relation to other constants and their variation
Gravitational constant G
In 1937, Paul Dirac and others began investigating the consequences of natural constants changing with time. For example, Dirac proposed a change of only 5 parts in 1011 per year of Newton's constant G to explain the relative weakness of the gravitational force compared to other fundamental forces. This has become known as the Dirac large numbers hypothesis.
However, Richard Feynman showed that the gravitational constant most likely could not have changed this much in the past 4 billion years based on geological and solar system observations (although this may depend on assumptions about the constant not changing other constants). (See also strong equivalence principle.)
Fine structure constant α
One group, studying distant quasars, has claimed to detect a variation of the fine structure constant  at the level in one part in 105. Other authors dispute these results. Other groups studying quasars claim no detectable variation at much higher sensitivities.
For over three decades since the discovery of the Oklo natural nuclear fission reactor in 1972, even more stringent constraints, placed by the study of certain isotopic abundances determined to be the products of a (estimated) 2 billion year-old fission reaction, seemed to indicate no variation was present. However, Lamoreaux and Torgerson of the Los Alamos National Laboratory conducted a new analysis of the data from Oklo in 2004, and concluded that α has changed in the past 2 billion years by 4.5 parts in 108. They claimed that this finding was "probably accurate to within 20%." Accuracy is dependent on estimates of impurities and temperature in the natural reactor. These conclusions have yet to be verified by other researchers.[needs update]
Paul Davies and collaborators have suggested that it is in principle possible to disentangle which of the dimensionful constants (the elementary charge, Planck's constant, and the speed of light) of which the fine-structure constant is composed is responsible for the variation. However, this has been disputed by others and is not generally accepted.
Criticisms of various VSL concepts
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Dimensionless and dimensionful quantities
- "[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass mP] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."
Any equation of physical law can be expressed in a form in which all dimensional quantities are normalized against like-dimensioned quantities (called nondimensionalization), resulting in only dimensionless quantities remaining. In fact, physicists can choose their units so that the physical constants c, G, ħ = h/(2π), 4πε0, and kB take the value one, resulting in every physical quantity being normalized against its corresponding Planck unit. For that, it has been claimed that specifying the evolution of a dimensional quantity is meaningless and does not make sense. When Planck units are used and such equations of physical law are expressed in this nondimensionalized form, no dimensional physical constants such as c, G, ħ, ε0, nor kB remain, only dimensionless quantities. Short of their anthropometric unit dependence, there simply is no speed of light, gravitational constant, nor Planck's constant, remaining in mathematical expressions of physical reality to be subject to such hypothetical variation. For example, in the case of a hypothetically varying gravitational constant, G, the relevant dimensionless quantities that potentially vary ultimately become the ratios of the Planck mass to the masses of the fundamental particles. Some key dimensionless quantities (thought to be constant) that are related to the speed of light (among other dimensional quantities such as ħ, e, ε0), notably the fine-structure constant or the proton-to-electron mass ratio, does have meaningful variance and their possible variation continues to be studied.
Relation to relativity and definition of c
Specifically regarding VSL, if the SI meter definition was reverted to its pre-1960 definition as a length on a prototype bar (making it possible for the measure of c to change), then a conceivable change in c (the reciprocal of the amount of time taken for light to travel this prototype length) could be more fundamentally interpreted as a change in the dimensionless ratio of the meter prototype to the Planck length or as the dimensionless ratio of the SI second to the Planck time or a change in both. If the number of atoms making up the meter prototype remains unchanged (as it should for a stable prototype), then a perceived change in the value of c would be the consequence of the more fundamental change in the dimensionless ratio of the Planck length to the sizes of atoms or to the Bohr radius or, alternatively, as the dimensionless ratio of the Planck time to the period of a particular caesium-133 radiation or both.
General critique of varying c cosmologies
From a very general point of view, G. Ellis expressed concerns that a varying c would require a rewrite of much of modern physics to replace the current system which depends on a constant c. Ellis claimed that any varying c theory (1) must redefine distance measurements (2) must provide an alternative expression for the metric tensor in general relativity (3) might contradict Lorentz invariance (4) must modify Maxwell's equations (5) must be done consistently with respect to all other physical theories. Whether these concerns apply to the proposals of Einstein (1911) and Dicke (1957) is a matter of debate, though VSL cosmologies remain out of mainstream physics.
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