Variable speed of light
Variable speed of light (VSL) refers to 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.
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 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.
- 1 Historical proposals
- 2 Modern VSL theories as an alternative to cosmic inflation
- 3 Various other VSL occurrences
- 4 Relation to other constants and their variation
- 5 Criticisms of various VSL concepts
- 6 References
- 7 External links
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 correct value later derived by his theory of general relativity. While the correct value was later measured by Eddington in 1919, Einstein never gave up his VSL theory but calculated it differently in his General Theory of 1915 and 5 editions of his book beginning with equation 107 and the several paragraphs that follow it. Notably, in 1911 he had considered variable time only, while in general relativity, albeit in another theoretical context, both space and time measurements are influenced by nearby masses. Many scientists including Peter Bergmann are in disagreement[clarification needed] with Einstein while other scientists most notably Max Born are in agreement.[clarification needed]
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 the velocity of light cannot be regarded as constant in a changing gravity field. 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 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/co where co is light speed in flat space.
α = ∫+∞-∞ (1/L)(∂L/∂x1)dx3
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.
dr/dt = ( 1- 2m/r)
where m is MG/c2 and where natural units are used such that co is equal to one.
The writing of Einstein, Born, and Tolman is generally ignored in favor of Peter Bergmann's interpretation.
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 does not contradict the SI definition of c= 299792458 m/s, since the time and length units second and meter can vary accordingly (p. 366).
Though Dicke's attempt presented an alternative to general relativity, the notion of a spatial variation of the speed of light as such does not contradict general relativity. Rather it is implicitly present in general relativity, occurring in the coordinate space description, as it is mentioned in several textbooks, e.g. Will, eqs. 6.14, 6.15, or Weinberg, eq. 9.2.5 ( denoting the gravitational potential −GM/r): "note that the photon speed is ... ." Based on this, variable speed of light models 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.
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 two-man team of Andreas Albrecht and João Magueijo in 1998 to explain the horizon problem of cosmology and propose an alternative to cosmic inflation. An alternative VSL model has also been proposed.
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. 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. Late-model[clarification needed] 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.
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 at the beginning of the universe. 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.[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. More modern formulations[by whom?] preserve local Lorentz invariance.
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 actually 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 . 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. 108
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 such a manner to have all dimensional quantities 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. Shorn 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.
- Albert Einstein (1907). Jahrbuch für Radioaktivität und Elektronik. 4: 411–462. Missing or empty
- A. Einstein (1911). "Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes" (PDF). Annalen der Physik. 35: 898–906. Bibcode:1911AnP...340..898E. doi:10.1002/andp.19113401005.
- A. Einstein (1912). "Lichtgeschwindigkeit und Statik des Gravitationsfeldes" (PDF). Annalen der Physik. 38: 355–369. Bibcode:1912AnP...343..355E. doi:10.1002/andp.19123430704.
- Einstein, Albert (1953). The Meaning of Relativity (5th ed.).
- Einstein, Albert (1961). Relativity - The Special And The General Theory (15th reprint from 1952 ed.). New York: Bonanza. p. 76. ISBN 0-517-029618.
- Einstein, Albert (1961). Relativity - The Special And The General Theory (15th reprint from 1952 ed.). New York: Bonanza. p. 145. ISBN 0-517-029618.
- Einstein, Albert (1984). The Meaning of Relativity (5th reprint from 1953 ed.). New York: Barns & Noble. p. 93. ISBN 1-56731-136-9.
- Bergmann, Peter (1976). Introduction to the Theory of Relativity (1st reprint from 1942 ed.). New York: Dover. p. 221. ISBN 0-486-63282-2.
- Bergmann, Peter (1992). The Riddle of Gravitation (1st reprint from 1968 ed.). New York: Dover. p. 66. ISBN 0-486-27378-4.
- Bergmann, Peter (1992). The Riddle of Gravitation (1st reprint from 1968 ed.). New York: Dover. p. 94. ISBN 0-486-27378-4.
- Born, Max (1965). Einstein’s Theory of Relativity (1st reprint from 1923 ed.). New York: Dover. p. 357. ISBN 0-486-60769-0.
- Tolman, Richard (1958). Relativity Cosmology and Thermodynamics (1st reprint from 1934 ed.). Oxford UK: Oxford. p. 212.
- R. Dicke (1957). "Gravitation without a Principle of Equivalence". Reviews of Modern Physics. 29: 363–376. Bibcode:1957RvMP...29..363D. doi:10.1103/RevModPhys.29.363.
- C.M. Will (1995). Theory and Experiment in Gravitational Physics. Cambridge University Press. p. 144.
- S. Weinberg (1972). Gravitation and Cosmology. London: Wiley. p. 222.
- J. Broekaert (2008). "A Spatially-VSL Gravity Model with 1-PN Limit of GRT". Foundations of Physics. 38: 409–435. arXiv:. Bibcode:2008FoPh...38..409B. doi:10.1007/s10701-008-9210-8.
- K. Krogh (1999). "Gravitation Without Curved Space-time". arXiv:. Bibcode:1999astro.ph.10325K.
- M. Arminjon (2006). "Space Isotropy and Weak Equivalence Principle in a Scalar Theory of Gravity". Brazilian Journal of Physics. 36: 177–189. arXiv:. Bibcode:2006BrJPh..36..177A. doi:10.1590/S0103-97332006000200010.
- A. Unzicker (2009). "A look at the abandoned contributions to cosmology of Dirac, Sciama, and Dicke". Annalen der Physik. 521: 57–70. arXiv:. Bibcode:2009AnP...521...57U. doi:10.1002/andp.200810335.
- J.P. Petit (1988). "An interpretation of cosmological model with variable light velocity" (PDF). Mod. Phys. Lett. A. 3 (16): 1527–1532. Bibcode:1988MPLA....3.1527P. doi:10.1142/S0217732388001823.
- J.P. Petit (1988). "Cosmological model with variable light velocity: the interpretation of red shifts" (PDF). Mod. Phys. Lett. A. 3 (18): 1733–1744. Bibcode:1988MPLA....3.1733P. doi:10.1142/S0217732388002099.
- J.P. Petit; M. Viton (1989). "Gauge cosmological model with variable light velocity. Comparizon with QSO observational data" (PDF). Mod. Phys. Lett. A. 4 (23): 2201–2210. Bibcode:1989MPLA....4.2201P. doi:10.1142/S0217732389002471.
- P. Midy; J.P. Petit (1989). "Scale invariant cosmology" (PDF). Int. J. Mod. Phys. D (8): 271–280.
- J. Moffat (1993). "Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology". Int. J. Mod. Phys. D. 2 (3): 351–366. arXiv:. Bibcode:1993IJMPD...2..351M. doi:10.1142/S0218271893000246.
- J.D. Barrow (1998). "Cosmologies with varying light-speed". Physical Review D. 59 (4). arXiv:. Bibcode:1999PhRvD..59d3515B. doi:10.1103/PhysRevD.59.043515.
- A. Albrecht; J. Magueijo (1999). "A time varying speed of light as a solution to cosmological puzzles". Phys. Rev. D59: 043516. arXiv:. Bibcode:1999PhRvD..59d3516A. doi:10.1103/PhysRevD.59.043516.
- J. Magueijo (2000). "Covariant and locally Lorentz-invariant varying speed of light theories". Phys. Rev. D62: 103521. arXiv:. Bibcode:2000PhRvD..62j3521M. doi:10.1103/PhysRevD.62.103521.
- J. Magueijo (2001). "Stars and black holes in varying speed of light theories". Phys. Rev. D63: 043502. arXiv:. Bibcode:2001PhRvD..63d3502M. doi:10.1103/PhysRevD.63.043502.
- J. Magueijo (2003). "New varying speed of light theories". Rept. Prog. Phys. 66 (11): 2025–2068. arXiv:. Bibcode:2003RPPh...66.2025M. doi:10.1088/0034-4885/66/11/R04.
- J. Magueijo (2003). Faster Than the Speed of Light: The Story of a Scientific Speculation. Massachusetts: Perseus Books Group. ISBN 0-7382-0525-7.
- J. Casado (2003). "A Simple Cosmological Model with Decreasing Light Speed". arXiv: [astro-ph].
- J.P. Petit; P. Midy; F. Landsheat (2001). "Twin matter against dark matter" (PDF). "Where is the matter?" (See sections 14 and 15 pp. 21–26). Int. Conf. on Astr. & Cosm.
- J.P Petit; G. d'Agostini (2007). "Bigravity: a bimetric model of the Universe with variable constants, including VSL (variable speed of light)". arXiv: [physics.gen-ph].
- M. A. Clayton; J. W. Moffat (1999). "Dynamical Mechanism for Varying Light Velocity as a Solution to Cosmological Problems". Phys. Lett. B460: 263–270. arXiv:. Bibcode:1999PhLB..460..263C. doi:10.1016/S0370-2693(99)00774-1.
- B.A. Bassett; S. Liberati; C. Molina-Paris; M. Visser (2000). "Geometrodynamics of variable-speed-of-light cosmologies". Phys. Rev. D62: 103518. arXiv:. Bibcode:2000PhRvD..62j3518B. doi:10.1103/PhysRevD.62.103518.
- J. D. Jackson (1998). Classical Electrodynamics (3rd ed.). Wiley.
- Eric Adelberger; Gia Dvali; Andrei Gruzinov (2007). "Photon Mass Bound Destroyed by Vortices". Physical Review Letters. 98 (2): 010402. arXiv:. Bibcode:2007PhRvL..98a0402A. doi:10.1103/PhysRevLett.98.010402. PMID 17358459.
- R. Feynman (1988). QED: the strange theory of light and matter. Princeton University Press. p. 89.
- P.A.M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society A. 165 (921): 199–208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
- R. P. Feynman (1970). "7". Lectures on Physics. 1. Addison Wesley Longman.
- 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 (9): 091301. arXiv:. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.
- 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 (3): 853–871. arXiv:. Bibcode:2004A&A...417..853C. doi:10.1051/0004-6361:20035701.
- 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 (12): 121302. arXiv:. Bibcode:2004PhRvL..92l1302S. doi:10.1103/PhysRevLett.92.121302. PMID 15089663.
- 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. 434: 827–838. arXiv:. Bibcode:2005A&A...434..827L. doi:10.1051/0004-6361:20041827.
- A. I. Shlyakhter (1976). "Direct test of the constancy of fundamental nuclear constants". Nature. 264 (5584): 340. Bibcode:1976Natur.264..340S. doi:10.1038/264340a0.
- T. Damour & F. Dyson (1996). "The Oklo bound on the time variation of the fine-structure constant revisited". Nucl. Phys. B480 (1–2): 37–54. arXiv:. Bibcode:1996NuPhB.480...37D. doi:10.1016/S0550-3213(96)00467-1.
- S.K. Lamoreaux; J.R. Torgerson (2004). "Neutron Moderation in the Oklo Natural Reactor and the Time Variation of Alpha". Physical Review D. 69 (12). arXiv:. Bibcode:2004PhRvD..69l1701L. doi:10.1103/PhysRevD.69.121701.
- E.S. Reich (30 June 2004). "Speed of Light May Have Changed Recently". New Scientist. Retrieved 30 January 2009.
- "Scientists Discover One Of The Constants Of The Universe Might Not Be Constant". ScienceDaily. 12 May 2005. Retrieved 30 January 2009.
- P.C.W. Davies; Tamara M. Davis; Charles H. Lineweaver (2002). "Cosmology: Black holes constrain varying constants". Nature. 418 (6898): 602–603. Bibcode:2002Natur.418..602D. doi:10.1038/418602a. PMID 12167848.
- M. J. Duff, "Comment on time-variation of fundamental constants", arXiv:hep-th/0208093.
- S. Carlip & S. Vaidya (2003). "Black holes may not constrain varying constants". Nature. 421 (6922): 498. arXiv:. Bibcode:2003Natur.421..498C. doi:10.1038/421498a. PMID 12556883.
- 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
- George F R Ellis (April 2007). "Note on Varying Speed of Light Cosmologies". General Relativity and Gravitation. 39 (4): 511–520. arXiv:. Bibcode:2007GReGr..39..511E. doi:10.1007/s10714-007-0396-4.
- A. Unzicker. "The VSL Discussion: What Does Variable Speed of Light Mean and Should we be Allowed to Think About ?". arXiv:. Bibcode:2007arXiv0708.2927U.