# Variable speed of light

Variable speed of light (VSL) is a hypothesis that states that the speed of light, usually denoted by c, may be a function of space and time. 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. In classical physics, the refractive index describes how light slows down when traveling through a medium. The speed of light in vacuum instead is considered a constant, and defined by the SI as 299792458 m/s. Alternative theories therefore usually modify the definitions of meter and seconds. VSL should not be confused with faster than light theories. Notable VSL attempts have been done by Einstein in 1911, by Robert Dicke in 1957, and by several researchers starting from the late 1980s. Since some of them contradict established concepts, VSL theories are a matter of debate. The VSL hypothesis have been used by young earth creationists (YEC) to explain why humans can apparently view astronomical objects millions and billions of lightyears away, which would normally imply that the age of these objects are much older than the 10,000 years that YECs believe the age of the Universe to be.

## Einstein's VSL attempt in 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 media, where a shorter wavelength $\lambda$ , by means of $c=\nu \lambda$ , leads to a lower speed of light, Einstein assumed that clocks in a gravitational field run slower, whereby the corresponding frequencies $\nu$ are influenced by the gravitational potential (eq.2, p. 903):

$\nu _{1}=\nu _{2}\left(1+{\frac {GM}{rc^{2}}}\right).$ 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 gave up his VSL theory for other reasons. 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.

## Dicke's 1957 attempt and Mach's principle

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 $c=\nu \lambda$ , this resulted in a relative change of c twice as much as considered by Einstein. Dicke assumed a refractive index $n={\frac {c}{c_{0}}}=1+{\frac {2GM}{rc^{2}}}$ (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).

## Other VSL attempts related to Einstein and Dicke

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 ($\phi$ denoting the gravitational potential −GM/r): "note that the photon speed is ... $|u|=1+2\phi +O(v^{3})$ ." Based on this, variable speed of light models have been developed which agree with all known tests of general relativity, but some distinguish 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 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 . 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. More modern formulations preserve local Lorentz invariance.

## Various other VSL occurrences

### Relation to relativity and definition of c

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

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.