Editing Variable speed of light (section)

=== Dimensionless and dimensionful quantities ===
To clarify 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 D. Barrow|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 the Planck 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 &ndash; 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 a form in which all dimensional quantities are normalized against like-dimensioned quantities (called ''[[nondimensionalization]]''), resulting in only [[dimensionless number|dimensionless quantities]] remaining. Physicists can ''choose'' their units so that the [[physical constants]] ''c'', [[gravitational constant|''G'']], [[Planck constant|''ħ''&nbsp;=&nbsp;''h''/(2π)]], [[vacuum permittivity|4π''ε''<sub>0</sub>]], and [[Boltzmann constant|''k''<sub>B</sub>]] 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.<ref name="uzan">{{Cite journal|arxiv=hep-ph/0205340|last1=Uzan|first1=Jean-Philippe|title=The fundamental constants and their variation: Observational status and theoretical motivations|journal=Reviews of Modern Physics|volume=75|issue=2|pages=403–455|year=2003|doi=10.1103/RevModPhys.75.403|bibcode=2003RvMP...75..403U|s2cid=118684485}}</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'', ''ħ'', ''ε''<sub>0</sub>, nor ''k''<sub>B</sub> remain, only dimensionless quantities, as predicted by the [[Buckingham π theorem]]. Short of their [[anthropometric]] unit dependence, there is no speed of light, gravitational constant, nor the [[Planck constant]], remaining in mathematical expressions of physical reality to be subject to such hypothetical variation.{{citation needed|date=February 2013}} 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'', ''ε''<sub>0</sub>), notably the fine-structure constant or the [[proton-to-electron mass ratio]], could in principle have meaningful variance and their possible variation continues to be studied.<ref name="uzan"/>