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The '''Count's of Falkenstein''', were German nobility named after the [[Burg Falkenstein (Donnersberg)|'''Burg Falkenstein''']] at [[Donnersbergkreis|Donnersberg]] in what is now the [[Rhineland-Palatinate]].
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 photon]]s 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]].
In 1255 the family inherited by marriage land from the Count's of '''Hagen-Münzenberg'''. In this manner they came to be established in the [[Frankfurt Rhein-Main Region]]. At '''[[Königstein im Taunus]]''' they built their new Castle '''[[Burg Falkenstein (Taunus)|Burg (Neu-)Falkenstein]]'''.
==Varying ''c'' in condensed matter physics==
The Falkenstein's als inherited the town of [[Offenbach am Main]] from the Count of Münzenberg. They leased the town to the neighbouring town of [[Frankfurt am Main]] for 1,000 [[Gulden]].
Photons [[Speed of light#Interaction with transparent materials|move at a speed less than ''c'']], unless they are travelling in vacuum. This leads to several important effects, such as [[dispersion (optics)|dispersion]] (''see also [[refractive index]]''). The slow-down in [[condensed matter physics|condensed matter]], such as [[gas|gases]], [[liquid]]s and [[Solid state physics|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]]).
Count [[Werner von Falkenstein|Werner III. von Falkenstein]] († 1418) was [[Archbishop of Trier]] from 1388 until his death in 1418. He is remembered for his provokation against the people of Frankfurt by developing rivalry with that town and Offenbach.
In certain highly unusual circumstances, it is also possible to prepare experiments in which the [[group velocity|group]] or [[phase velocity]] of light exceeds ''c''. Since these velocities are mathematical constructs, these [[Speed of light#.22Faster-than-light.22 observations and experiments|faster than light observations]] do not indicate any contradiction with [[causality (physics)|causality]] or [[special relativity]], as no information or energy travels faster than ''c''.
The Falkenstein male line died out with the death of Werner III. in 1418. Their landed estates were inherited by '''[[Herren von Eppstein]]''' family as well as the Counts von Solms.
==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.<ref>{{cite book|author=J. D. Jackson|title=Classical Electrodynamics|year=1998|publisher=Wiley|edition=3rd ed.}}</ref> 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.<ref name="adel">E. Adelberger, G. Dvali and A. Gruzinov, ''Photon Mass Bound Destroyed by Vortices'' [http://arxiv.org/abs/hep-ph/0306245 preprint].</ref>
==Varying ''c'' in quantum theory==
{{main|Propagator#Faster than light?}}
{{seealso|Scharnhorst effect}}
[[Category: Units of velocityGerman_Nobility]] ▼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 particle|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''."<ref>{{cite book|author=R. Feynman|title=QED: the strange theory of light and matter|publisher=Princeton University Press|page=89|year=1988}}</ref> 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).
[[Category: Special relativityTaunus]] ▼
==Varying ''c'' in time==
{{seealso|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<sup>11</sup> 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<ref>{{cite book|author=R. P. Feynman|title=Lectures on Physics|year=1970|publisher=Addison Wesley Longman|Volume=1|Chapter=7}}</ref> 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<sub>P</sub>'' ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."<ref>[[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.</ref>
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 number|dimensionless quantities]] remaining. In fact, physicists often ''choose'' their units so that the [[physical constants]] ''c'', ''G'', ''h''/(2π), and 4πε<sub>0</sub> 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.<ref>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}}</ref> 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<ref>''ibid''</ref>.
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 [[frequency|period]] of a particular [[caesium]]-133 [[atomic clock|radiation]] or both.
One group, studying distant quasars, has claimed to detect a variation of the fine structure constant <ref>{{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}} {{arxiv|archive=astro-ph|id=0012539}}</ref> at the level in one part in 10<sup>5</sup>. Other authors dispute these results. Other groups studying quasars claim no detectable variation at much higher sensitivities.<ref>{{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|page=853|year=2004}} {{arxiv|archive=astro-ph|id=0401094}}</ref><ref>{{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|page=121302|year=2004}} {{arxiv|archive=astro-ph|id=0402177}}</ref><ref>{{cite journal|author=S. A. Levshakov, M. Centurion, P. Molaro and S. D’Odorico|title=VLT/UVES constraints on
the cosmological variability of the fine-structure constant|journal=Astron. Astrophys.}} {{arxiv|archive=astro-ph|id=0408188}}</ref> Moreover, even more stringent constraints, placed by study of certain [[isotope|isotopic]] abundances in the [[Oklo]] [[natural nuclear fission reactor]], seem to indicate no variation is present.<ref>{{cite journal|author=A. I. Shlyakhter|title=Direct test of the constancy of fundamental nuclear constants|journal=Nature|volume=264|page=340|year=1976}}</ref><ref>{{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|page=37|year=1996}} {{arxiv|archive=hep-ph|id=9606486}}</ref>
[[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.<ref>{{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|page=602–603}}</ref> However, this has been disputed by others and is not generally accepted.<ref>M. J. Duff, "Comment on time-variation of fundamental constants", {{arxiv|archive=hep-th|id=0208093}}.</ref><ref>{{cite journal|title=Black holes may not constrain varying constants|author=S. Carlip and S. Vaidya|year=2003|journal=Nature|volume=421|page=498}} {{arxiv|archive=hep-th|id=0209249}}</ref>
Some [[Young Earth Creationism|Young Earth Creationists]] have investigated the idea of a changing ''c'' in 1987<ref>[http://www.ldolphin.org/constc.shtml A collection of links]</ref> as an explanation for the discrepancy between the biblical and measured ages of the universe. This idea was mentioned by [[Marilyn vos Savant]] in Parade magazine.<ref>M. vos Savant, ''Parade'', May 22, 1988</ref> This idea has been refuted.<ref>See, ''e.g'', the talkorigins c-decay FAQ (also contains links) [http://talkorigins.org/faqs/c-decay.html]</ref>
==The varying speed of light cosmology==
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 [[physical cosmology|cosmology]].<ref>Moffat, John; "Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology" {{arxiv|archive=gr-qc|id=9211020}}</ref><ref>João Magueijo, ''Faster Than the Speed of Light: The Story of a Scientific Speculation,'' Perseus Books Group , Massachusetts , 2003, ISBN 0-7382-0525-7.</ref><ref>{{cite journal|title=A time varying speed of light as a solution to cosmological puzzles|author=Andreas Albrecht and João Magueijo|journal=Phys. Rev.|volume=D59|year=1999|page=043516}} {{arxiv|archive=astro-ph|id=9811018}}</ref>
<ref>{{cite journal|author=J. D. Barrow|title=Cosmologies with varying light-speed}} {{arxiv|archive=astro-ph|id=9811022}}</ref>
<ref>{{cite journal|author=J. Magueijo|title=Covariant and locally Lorentz-invariant varying speed of light theories|journal=Phys. Rev.|volume=D62|year=2000|page=103521}} {{arxiv|archive=gr-qc|id=0007036}}</ref>
<ref>{{cite journal|author=J. Magueijo|title=Stars and black holes in varying speed of light theories|journal=Phys. Rev.|volume=D63|year=2001|page=043502}} {{arxiv|archive=astro-ph|id=0010591}}</ref>
<ref>{{cite journal|author=J. Magueijo|title=New varying speed of light theories|journal=Rept. Prog. Phys.|volume=66|year=2003|page=2025}} {{arxiv|archive=astro-ph|id=0305457}}</ref>
The idea is that light propagated as much as sixty times faster in the distant past, and 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 problem|monopole]] and [[flatness problem]]s and how it reproduces the observed homogeneity 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 [[causality (physics)|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 [[Albert Einstein|Einstein]]'s [[theory of relativity|theories of general and special relativity]] in a particular way.<ref>{{cite journal|title=Geometrodynamics of variable-speed-of-light cosmologies|author=Bruce A. Bassett, Stefano Liberati, Carmen Molina-Paris and Matt Visser|journal=Phys. Rev.|volume=D62|year=2000|page=103518}} {{arxiv|archive=astro-ph|id=0001441}}</ref> However, it has been pointed out by [[George Ellis|Ellis]] and Uzan<ref>{{Cite journal | title = 'c' is the speed of light, isn't it? | author = George F. R. Ellis and J.-P. Uzan | journal = Am. J. Phys. | volume = 73 | year = 2005 | pages = 240–7 | url = http://arxiv.org/abs/gr-qc/0305099}}</ref> 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<ref>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 [http://www.ensmp.fr/aflb/AFLB-291/aflb291p173.pdf].</ref> 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 (mathematics)|metric]], but this necessarily causes problems with causality and [[quantum mechanics]].<ref>A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and Riccardo Rattazzi, "Causality, Analyticity and an IR Obstruction to UV Completion", {{arxiv|archive=hep-th|id=0602178}}</ref>
==References==
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[[Category:Electromagnetic radiation]]
▲[[Category:Units of velocity]]
▲[[Category:Special relativity]]
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