Variable speed of light

The variable speed of light (VSL) concept states that the speed of light in a vacuum, usually denoted by "c", may not be constant in some cases. 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 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.

Varying "c" in condensed matter physics

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. [cite book|author=J. D. Jackson|title=Classical Electrodynamics|year=1998|publisher=Wiley|edition=3rd ed.] 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.E. Adelberger, G. Dvali and A. Gruzinov, "Photon Mass Bound Destroyed by Vortices" [http://arxiv.org/abs/hep-ph/0306245 preprint] .]

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"." [cite book|author=R. Feynman|title=QED: the strange theory of light and matter|publisher=Princeton University Press|page=89|year=1988] 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

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 in his famous lectures [cite book|author=R. P. Feynman|title=Lectures on Physics|year=1970|publisher=Addison Wesley Longman|Volume=1|Chapter=7] 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." [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.]

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 4πε₀ 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. [J. P. Uzan, "The fundamental constants and their variation: Observational status and theoretical motivations," "Rev. Mod. Phys." 75, 403 (2003). arxiv|archive=hep-ph|id=0205340] 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 ["ibid"] .

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 imaginary dimension of time is changing compared to the other three real-valued spacial dimensions of space-time.

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.

One group, studying distant quasars, has claimed to detect a variation of the fine structure constant [cite journal|author=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|title=Further Evidence for Cosmological Evolution of the Fine Structure Constant|journal=Phys. Rev.Lett.|volume=87|year=2001|pages=091301|doi=10.1103/PhysRevLett.87.091301 arxiv|archive=astro-ph|id=0012539] 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. [cite journal|author=H. Chand, R. Srianand, P. Petitjean and B. Aracil|title=Probing the cosmological variation of the fine-structure constant: results based on VLT-UVES sample|journal=Astron. Astrophys.|volume=417|year=2004|pages=853|doi=10.1051/0004-6361:20035701 arxiv|archive=astro-ph|id=0401094] [cite journal|author=R. Srianand, H. Chand, P. Petitjean and B. Aracil|title=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|journal=Phys. Rev. Lett.|volume=92|year=2004|pages=121302|doi=10.1103/PhysRevLett.92.121302 arxiv|archive=astro-ph|id=0402177] [cite journal|author=S. A. Levshakov, M. Centurion, P. Molaro and S. D’Odorico|title=VLT/UVES constraints onthe cosmological variability of the fine-structure constant|journal=Astron. Astrophys. arxiv|archive=astro-ph|id=0408188] 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. [cite journal|author=A. I. Shlyakhter|title=Direct test of the constancy of fundamental nuclear constants|journal=Nature|volume=264|year=1976|pages=340|doi=10.1038/264340a0] [cite journal|author=T. Damour and F. Dyson|title=The Oklo bound on the time variation of the fine-structure constant revisited|journal=Nucl. Phys.|volume=B480|year=1996|pages=37|doi=10.1016/S0550-3213(96)00467-1 arxiv|archive=hep-ph|id=9606486]

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. [cite journal|title = Cosmology: Black holes constrain varying constants|author=P.C.W. Davies, Tamara M. Davis, Charles H. Lineweaver|year=2002|journal=Nature|volume=418|pages=602–603|doi = 10.1038/418602a] However, this has been disputed by others and is not generally accepted. [M. J. Duff, "Comment on time-variation of fundamental constants", arxiv|archive=hep-th|id=0208093.] [cite journal|title=Black holes may not constrain varying constants|author=S. Carlip and S. Vaidya|year=2003|journal=Nature|volume=421|pages=498|doi=10.1038/421498a arxiv|archive=hep-th|id=0209249]

The varying speed of light cosmology

The varying speed of light cosmology has been proposed independently by Jean-Pierre Petit from 1988 [cite journal |title= An interpretation of cosmological model with variable light velocity |author= J.P. Petit |journal= Mod. Phys. Lett. A |volume= 3 |issue= 16 |year= 1988 |pages= 1527–1532 |doi= 10.1142/S0217732388001823 [http://www.bigravitytheory.com/pdf/modern_physics_letters_a1.pdf] ] [Cite journal |title= Cosmological model with variable light velocity: the interpretation of red shifts |author= J.P. Petit |journal= Mod. Phys. Lett. A |volume= 3 |issue= 18 |year= 1988 |pages= 1733–1744 |doi= 10.1142/S0217732388002099 [http://www.bigravitytheory.com/pdf/modern_physics_letters_a2.pdf] ] [cite journal |title= Gauge cosmological model with variable light velocity. Comparizon with QSO observational data |author= J.P. Petit, M. Viton |journal= Mod. Phys. Lett. A |volume= 4 |issue= 23 |year= 1989 |pages= 2201–2210 |doi= 10.1142/S0217732389002471 [http://www.bigravitytheory.com/pdf/modern_physics_letters_a3.pdf] ] [ cite journal |title= Scale invariant cosmology |author= P. Midy, J.P. Petit |journal= Int. J. Mod. Phys. D |issue= 8 |year= 1989 |pages= 271–280 [http://www.bigravitytheory.com/pdf/scale_inv.pdf] ] and then John Moffat in 1992 [cite journal | title= Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology | author= J. Moffat | journal= Int. J. Mod. Phys. D |volume= 2 |issue= 3 |year=1993 | pages=351–366 | doi= 10.1142/S0218271893000246arxiv|archive=gr-qc|id=9211020] and the two-man team of Andreas Albrecht and João Magueijo in 1998 [cite journal |author= J.D. Barrow |title= Cosmologies with varying light-speed |year= 1998arxiv|archive=astro-ph|id=9811022] [cite journal |title= A time varying speed of light as a solution to cosmological puzzles |author= A. Albrecht, J. Magueijo |journal= Phys. Rev. |volume= D59 |pages= 043516 |year= 1999arxiv|archive=astro-ph|id=9811018] [cite journal |author= J. Magueijo |title= Covariant and locally Lorentz-invariant varying speed of light theories |journal=Phys. Rev. |volume= D62 |year= 2000 |pages= 103521arxiv|archive=gr-qc|id=0007036] [cite journal |author= J. Magueijo |title= Stars and black holes in varying speed of light theories |journal= Phys. Rev. |volume= D63 |year= 2001 |pages= 043502arxiv|archive=astro-ph|id=0010591] [cite journal |author= J. Magueijo |title= New varying speed of light theories |journal= Rept. Prog. Phys. |volume= 66 |year= 2003 |pages= 2025 |doi= 10.1088/0034-4885/66/11/R04arxiv|archive=astro-ph|id=0305457] [cite book |title= Faster Than the Speed of Light: The Story of a Scientific Speculation |author= J. Magueijo |publisher= Perseus Books Group |location= Massachusetts |year= 2003 |isbn= 0-7382-0525-7] 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. The Einstein field equations remain invariant through convenient joint variations of "c" and "G" in Einstein's constant. Late-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. [cite conference |title= Twin matter against dark matter |author= J.P. Petit, P. Midy, F. Landsheat |conference= Int. Conf. on Astr. & Cosm |booktitle= "Where is the matter?" (See sections 14 and 15 pp. 21–26) |date= 2001 [http://www.bigravitytheory.com/pdf/where_is_the_matter_2001.pdf] ] [cite paper |title= Bigravity: a bimetric model of the Universe with variable constants, including VSL (variable speed of light) |author= J.P Petit, G. d'Agostini |publisher= Int. Meet. Var. Tech. CITV |date = 2007arxiv|id=0803.1362] However it should be noted that while this was the first VSL model to be published, and the sole to date where an evolution law is given relating the joint variations of constants through time while leaving the physics unchanged, these papers received few citations in the later VSL literature.

The idea from Moffat and the team Albrecht-Magueijo is that light propagated as much as 60 orders of magnitude faster in the early universe, thus distant regions of the expanding universe have had time to interact since the beginning of the universe. But 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). 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. [cite journal |title= Geometrodynamics of variable-speed-of-light cosmologies |author= B.A. Bassett, S. Liberati, C. Molina-Paris, M. Visser |journal= Phys. Rev. |volume= D62 |year= 2000 |pages= 103518arxiv|archive=astro-ph|id=0001441] Alternative VSL models have also been proposed [cite journal |author= J. Casado |title= A Simple Cosmological Model with Decreasing Light Speed |journal= |volume= |year= 2003 |pages= |doi=arxiv|archive=astro-ph|id=0310178] .

References

External links

* [http://arxiv.org/abs/0708.2927 A. Unzicker: The VSL Discussion: What does variable speed of light mean and should we be allowed to think about ? (arXiv:0708.2927)]