Variable speed of light
Encyclopedia
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 most 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
. However, this would require a rewrite of much of modern physics to replace the current system which depends on a constant c.
, 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.
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).
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 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:
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. 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.
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 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. 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.
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.
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 constant
s 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
. 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 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.
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in a vacuum, usually denoted by c, may not be constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...
in most cases. In most situations in condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
when light is traveling through a medium, it effectively has a slower speed. Virtual photons in some calculations in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
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
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. However, this would require a rewrite of much of modern physics to replace the current system which depends on a constant c.
Varying c in classical physics
The photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
, the particle of light which mediates the electromagnetic force is believed to be massless. The so-called Proca action
Proca action
In physics, in the area of field theory, the Proca action describes a massive spin-1 field of mass m in Minkowski spacetime. The field involved is a real vector field A...
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
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
. These photons would propagate at less than the speed of light defined by special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and have three directions of polarization. However, in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the photon mass is not consistent with gauge invariance or renormalizability
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
and so is usually ignored. However, a quantum theory of the massive photon can be considered in the Wilsonian effective field theory
Effective field theory
In physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...
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 theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
the Heisenberg uncertainty relations indicate that photons can travel at any speed for short periods. In the Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
interpretation of the theory, these are known as "virtual photons
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
", 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
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
"...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).
Varying c in time
In 1937, Paul DiracPaul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
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
Dirac large numbers hypothesis
The Dirac large numbers hypothesis is an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of magnitude in the present cosmological epoch...
. However, Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
showed in his famous lectures that the gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
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
John D. Barrow
-External links:****** The Forum-Publications available on the Internet:************...
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 lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, 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
Physical law
A physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...
can be expressed in such a manner to have all dimensional quantities normalized against like dimensioned quantities (called nondimensionalization
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...
) 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. 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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
, 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
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...
, would have meaningful variance and their possible variation continues to be studied.
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
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
meter definition was reverted to its pre-1960 definition as a length on a prototype
Prototype
A prototype is an early sample or model built to test a concept or process or to act as a thing to be replicated or learned from.The word prototype derives from the Greek πρωτότυπον , "primitive form", neutral of πρωτότυπος , "original, primitive", from πρῶτος , "first" and τύπος ,...
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
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
to the Planck time
Planck time
In physics, the Planck time, , is the unit of time in the system of natural units known as Planck units. It is the time required for light to travel, in a vacuum, a distance of 1 Planck length...
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
Bohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...
or, alternatively, as the dimensionless ratio of the Planck time to the period
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of a particular caesium
Caesium
Caesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-gold alkali metal with a melting point of 28 °C , which makes it one of only five elemental metals that are liquid at room temperature...
-133 radiation
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...
or both.
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. Moreover, even more stringent constraints, placed by study of certain isotopic
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...
abundances in the Oklo
Oklo
Oklo is a region near the town of Franceville, in the Haut-Ogooué province of the Central African state of Gabon. Several natural nuclear fission reactors were discovered in the uranium mines in the region in 1972.-History:...
natural nuclear fission reactor
Natural nuclear fission reactor
A natural nuclear fission reactor is a uranium deposit where analysis of isotope ratios has shown that self-sustaining nuclear chain reactions have occurred. The existence of this phenomenon was discovered in 1972 at Oklo in Gabon, Africa, by French physicist Francis Perrin. The conditions under...
, seem to indicate no variation is present.
Paul Davies
Paul Davies
Paul Charles William Davies, AM is an English physicist, writer and broadcaster, currently a professor at Arizona State University as well as the Director of BEYOND: Center for Fundamental Concepts in Science...
and collaborators have suggested that it is in principle possible to disentangle which of the dimensionful constants (the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
, Planck's constant, and the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
) 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.
The varying speed of light cosmology
The varying speed of light cosmology has been proposed independently by Jean-Pierre PetitJean-Pierre Petit
Jean-Pierre Petit is a French scientist, senior researcher at National Center for Scientific Research as an astrophysicist in Marseille Observatory, now retired...
in 1988, John Moffat in 1992,
and the two-man team of Andreas Albrecht
Andreas Albrecht
Andreas Albrecht is a cosmologist involved in research and teaching in the University of California, Davis. Along with João Magueijo, Andreas Albrecht independently proposed a model of varying speed of light cosmology...
and João Magueijo
João Magueijo
João Magueijo is a Portuguese cosmologist and professor in Theoretical Physics at Imperial College London. He is a pioneer of the varying speed of light theory.- Career :...
in 1998
to explain the horizon problem
Horizon problem
The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the 1970s. It points out that different regions of the universe have not "contacted" each other because of the great distances between them, but nevertheless they have the same temperature...
of cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
and propose an alternative to cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
. An alternative VSL model has also been proposed.
In Petit's VSL model, the variation of c accompanies the joint variations of all physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...
s combined to space and time scale factors
Scale factor (Universe)
The scale factor or cosmic scale factor parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. It is sometimes called the Robertson-Walker scale factor...
changes, so that all equations and measurements of these constants remain unchanged through the evolution of the universe. The Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
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
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...
of the early universe, at the very beginning of the Radiation-Dominated Era
Radiation-Dominated Era
The radiation-dominated era refers to one of the three phases of the known universe, the other two being the matter-dominated era and the dark-energy-dominated era. During this era, the dynamics of the universe were set by radiation, which refers generally to the constituents of the universe which...
where spacetime is identified to space-entropy with a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
conformally flat
Conformally flat
A Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.More formally, let be a pseudo-Riemannian manifold...
. 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 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
Causality (physics)
Causality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....
of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. To do so would require modifying gravity by varying Newton's constant or redefining special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. Classically, varying speed of light cosmologies propose to circumvent this by varying the dimensionful quantity c by breaking the Lorentz invariance of Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theories of general and special relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
in a particular way. More modern formulations preserve local Lorentz invariance.