This is interesting actually. I think it is a shame that 'c' has the name 'speed of light'. Let me explain.
Relativity theory, although Einstein rambled on about light a lot when he was formulating it, actually has nothing to do with light directly. It is a theory of spacetime. 'c' is just a parameter which characterises the structure of spacetime. The fact that light travels at c is a consequence of this structure (and the fact that light is massless), not the other way around.
So it is possible that statements like "photons travel at the speed of light" might become really confusing one day, if we find out that photons have a really tiny mass. 'c' would remain the same number, but the speed of light would actually be frame dependent, you would just need amazingly high energies to notice (i.e. higher than the energies the protons at the LHC are circulating...). At least that is one possible scenario
. I don't actually think that will happen, at least not in my lifetime, because the theory that says photons are massless is very convincing and well supported, but I like to acknowledge the possibility
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