The variable speed of light (VSL) concept states that the speed of light in vacuum, usually denoted by c, may not be constant, for some reason. In most situations in condensed matter physics when light is traveling through a medium, it effectively has a slower speed. 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 is usually thought 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 speculative and controversial theories in cosmology, the speed of light also varies by changing the postulates of special relativity. A fundamental change to relativity is needed if c is changing because relativity shows that space and time are equivalent.

Photons move at a speed less than c, unless they are travelling in vacuum. This leads to several important effects, such as dispersion (see also refractive index). The slow-down in condensed matter, such as gases, liquids and solids, can be considerable. The group velocity of light can be lowered to arbitrary speeds, though only for an arbitrarily slow (low bandwidth) signal (see Slow light).

In certain highly unusual circumstances, it is also possible to prepare experiments in which the group or phase velocity of light exceeds c. Since these velocities are mathematical constructs, these faster than light observations do not indicate any contradiction with causality or special relativity, as no information or energy travels faster than c.

Varying c in classical physics

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.^{[1]} 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.^{[2]}

Varying c in quantum theory

Main article: Propagator (Section "Faster than light?")

See also: Scharnhorst effect

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."^{[3]} 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 interpreted not as a physical picture of what is actually taking place, but rather as a convenient calculation tool (which, in some cases, happen to involve faster-than-light velocity vectors).

Varying c in time

See also: Einstein equivalence principle

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 10^{11} 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 in his famous lectures^{[4]} 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.)

It is not clear 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:

"[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 m_{P}] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."^{[5]}

Any equation of physical law can be expressed in such a manner to have all dimensional quantities normalized against like dimensioned quantities (called nondimensionalization) resulting in only dimensionless quantities remaining. In fact, physicists often choose their units so that the physical constants c, G, h/(2π), and 4πε_{0} take the value one, resulting in every physical quantity being normalized against its corresponding Planck unit. As such, many physicists think that specifying the evolution of a dimensionful quantity is at best meaningless and at worst inconsistent.^{[6]} 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, or h remain, only dimensionless quantities. Shorn of their anthropometric unit dependence, there simply is no speed of light, gravitational constant, or Planck's constant, remaining in mathematical expressions of physical reality to be subject to such hypothetical variation. For example, in the case of the gravitational constant, G, the relevant dimensionless quantities that were assumed to vary ultimately became the ratios of the Planck mass to the masses of the fundamental particles. Some key dimensionless quantities (thought to be constant) depend on the speed of light, notably the fine-structure constant, would have meaningful variance and their possible variation continues to be studied^{[7]}.

In relativity, space-time is 4 dimensions of the same physical property of either space or time, depending on which perspective is chosen. The conversion factor of length=i*c*time is described in Appendix 2 of Einstein's Relativity. A changing c in relativity would mean the dimension of time is changing compared to the other three real-valued spacial dimensions of space-time.

Specifically regarding VSL, if the SImeter 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.

One group, studying distant quasars, has claimed to detect a variation of the fine structure constant ^{[8]} at the level in one part in 10^{5}. Other authors dispute these results. Other groups studying quasars claim no detectable variation at much higher sensitivities.^{[9]}^{[10]}^{[11]} Moreover, even more stringent constraints, placed by study of certain isotopic abundances in the Oklo natural nuclear fission reactor, seem to indicate no variation is present.^{[12]}^{[13]}

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.^{[14]} However, this has been disputed by others and is not generally accepted.^{[15]}^{[16]}

The varying speed of light cosmologies

A variable speed of light cosmology has been proposed independently by John Moffat and the two-man team of Andreas Albrecht and João Magueijo to explain the horizon problem of cosmology.^{[17]}^{[18]}^{[19]}^{[20]}^{[21]}^{[22]}^{[23]}
The idea is that light propagated as much as sixty times faster in the distant past, thus distant regions of the expanding universe have had time to interact since the beginning of the universe. As such, it was proposed as an alternative to cosmic inflation, although it is less clear how it reproduces the other successes of inflationary cosmology such as the monopole and isotropy of the universe, and the scale invariance of the spectrum of initial perturbations.

There is no known way to solve the horizon problem with 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. (See equivalence principle for further details.) 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.^{[24]} However, it has been pointed out by Ellis and Uzan^{[25]} that the VSL cosmology is an ad hoc modification of various equations of physics without a consistent underlying scheme, such as a Lagrangian from which the equations of motion can be derived. It has been suggested^{[26]} that a modification of the Einstein-Maxwell action can cause light to propagate at a speed faster than the speed of light defined by the metric, but this necessarily causes problems with causality and quantum mechanics.^{[27]}

References

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^ R. Feynman (1988). QED: the strange theory of light and matter. Princeton University Press, 89.

^ R. P. Feynman (1970). Lectures on Physics. Addison Wesley Longman.

^ John D. Barrow, The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, New York, 2002, ISBN 0-375-42221-8.

^ J. P. Uzan, "The fundamental constants and their variation: Observational status and theoretical motivations," Rev. Mod. Phys.75, 403 (2003). arXiv:hep-ph/0205340

^ J.K. Webb, M.T. Murphy, V.V. Flambaum, V.A. Dzuba, J.D. Barrow, C.W. Churchill, J.X. Prochaska and A.M. Wolfe (2001). "Further Evidence for Cosmological Evolution of the Fine Structure Constant". Phys.Rev.Lett.87: 091301. arXiv:astro-ph/0012539

^ H. Chand, R. Srianand, P. Petitjean and B. Aracil (2004). "Probing the cosmological variation of the fine-structure constant: results based on VLT-UVES sample". Astron. Astrophys.417. arXiv:astro-ph/0401094

^ R. Srianand, H. Chand, P. Petitjean and B. Aracil (2004). "Limits on the time variation of the electromagnetic ne-structure constant in the low energy limit from absorption lines in the spectra of distant quasars". Phys. Rev. Lett.92. arXiv:astro-ph/0402177

^ S. A. Levshakov, M. Centurion, P. Molaro and S. D’Odorico. "VLT/UVES constraints on
the cosmological variability of the fine-structure constant". Astron. Astrophys.. arXiv:astro-ph/0408188

^ A. I. Shlyakhter (1976). "Direct test of the constancy of fundamental nuclear constants". Nature264.

^ T. Damour and F. Dyson (1996). "The Oklo bound on the time variation of the fine-structure constant revisited". Nucl. Phys.B480. arXiv:hep-ph/9606486

^ P.C.W. Davies, Tamara M. Davis, Charles H. Lineweaver (2002). "Cosmology: Black holes constrain varying constants". Nature418.

^ M. J. Duff, "Comment on time-variation of fundamental constants", arXiv:hep-th/0208093.

^ S. Carlip and S. Vaidya (2003). "Black holes may not constrain varying constants". Nature421. arXiv:hep-th/0209249

^ Moffat, John; "Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology" arXiv:gr-qc/9211020

^ João Magueijo, Faster Than the Speed of Light: The Story of a Scientific Speculation, Perseus Books Group , Massachusetts , 2003, ISBN 0-7382-0525-7.

^ Andreas Albrecht and João Magueijo (1999). "A time varying speed of light as a solution to cosmological puzzles". Phys. Rev.D59. arXiv:astro-ph/9811018

^ J. D. Barrow. "Cosmologies with varying light-speed". arXiv:astro-ph/9811022

^ J. Magueijo (2000). "Covariant and locally Lorentz-invariant varying speed of light theories". Phys. Rev.D62. arXiv:gr-qc/0007036

^ J. Magueijo (2001). "Stars and black holes in varying speed of light theories". Phys. Rev.D63. arXiv:astro-ph/0010591

^ J. Magueijo (2003). "New varying speed of light theories". Rept. Prog. Phys.66. arXiv:astro-ph/0305457

^ Bruce A. Bassett, Stefano Liberati, Carmen Molina-Paris and Matt Visser (2000). "Geometrodynamics of variable-speed-of-light cosmologies". Phys. Rev.D62. arXiv:astro-ph/0001441

^ George F. R. Ellis and J.-P. Uzan (2005). "'c' is the speed of light, isn't it?". Am. J. Phys.73: 240–7.

^ P. Teyssandier, "Variation of the speed of light due to non-minimal coupling between electromagnetism and gravity", Ann. de la Fondation Louis de Broglie, 29 173–186, 2004 [1].

^ A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and Riccardo Rattazzi, "Causality, Analyticity and an IR Obstruction to UV Completion", arXiv:hep-th/0602178