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All of chemistry depends on their values, and significant changes would alter the chemical and mechanical properties of all substances.Furthermore, the speed of light itself would change by different amounts according to which definition of units was used.
Then, the metre was defined as 1,650,763.73 wavelengths of the reddish-orange light from a krypton-86 source, and the second was defined (then as now) as 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of caesium-133.In that case, it would make more sense to attribute the changes to variations in the charge on the electron or the particle masses than to changes in the speed of light.In any case, there is good observational evidence to indicate that those parameters have not changed over most of the lifetime of the universe.Obviously it would be more natural to attribute those changes to variations in the units of measurement than to changes in the speed of light itself, but by the same token it's nonsense to say that the speed of light is now constant just because the SI definitions of units define its numerical value to be constant.But the SI definition highlights the point that we need first to be very clear about what we mean by constancy of the speed of light, before we answer our question.See the FAQ article Have physical constants changed with time?(Note that the fine-structure constant does change with energy scale, but I am referring to the constancy of its low-energy limit.) Another assumption on the laws of physics made by the SI definition of the metre is that the theory of relativity is correct.The short answer is that it depends on who is doing the measuring: the speed of light is only guaranteed to have a value of 299,792,458 m/s in a vacuum when measured by someone situated right next to it.But let's approach the question by considering its various meanings. Light is slowed down in transparent media such as air, water and glass.Experiments have shown that the mass of the photon must be very small if it is not zero (see the FAQ entry What is the mass of the photon? Any such possible photon rest mass is certainly too small to have any practical significance for the definition of the metre in the foreseeable future, but it cannot be shown to be exactly zero—even though currently accepted theories indicate that it is.If the mass weren't zero, the speed of light would not be constant; but from a theoretical point of view we would then take is constant.