Variable speed of light
A variable speed of light 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.
Notable attempts to incorporate a variable speed of light into physics have been made by Einstein in 1911, by Robert Dicke in 1957, and by several researchers starting from the late 1980s.
The speed of light in vacuum instead is considered a constant, and defined by the SI as 299792458 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.
Historical proposals
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 :Einstein commented :
In a subsequent paper in 1912, he concluded that:
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 follows equation 107 and makes an unambiguous use of variable scalar light velocity both as the argument of a partial differential function and as the denominator in a fraction 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 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)
, 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 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 . 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.
Subsequent proposals
Spatial variation of the speed of light is implicitly present in general relativity, occurring in the coordinate space description.:"note that the photon speed is...."
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.
Other models claim to shed light on the equivalence principle or make a link to Dirac's large numbers hypothesis.
Several hypotheses for varying speed of light, seemingly in contradiction to general relativity theory, have been published, including those of Giere and Tan and Sanejouand. In 2003 Magueijo gave a review of such hypotheses.
Modern VSL theories as an alternative to cosmic inflation
The varying speed of light cosmology 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.
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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. There is no known way to solve the horizon problem with a variation of the fine-structure constant, because its variation does not change the causal structure of spacetime. To do so would require modifying gravity by varying Newton's constant or redefining special relativity.
Classically, 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
s 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 that no meaning can be ascribed to a dimensional quantity such as the speed of light varying in time, 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 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.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.
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 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. 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.
Paul Davies and collaborators have suggested that it is in principle possible to disentangle which of the dimensionful constants 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
Dimensionless and dimensionful quantities
It has to be clarified what a variation in a dimensionful quantity actually means, since any such quantity can be changed merely by changing one's choice of units. John Barrow wrote:Any equation of physical law can be expressed in a form in which all dimensional quantities are normalized against like-dimensioned quantities, resulting in only dimensionless quantities remaining. In fact, physicists can choose their units so that the physical constants c, G, ħ = h/, 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 that are related to the speed of light, notably the fine-structure constant or the proton-to-electron mass ratio, does have meaningful variance and their possible variation continues to be studied.