|Einstein and Lorentz.|
(This question came up in the discussion of a recent proposal according to which photons with a tiny restmass might cause an effect similar to the cosmological constant.)
The short answer to your question is “No.” If photons had a restmass, special relativity would still be as valid as it’s always been.
The longer answer is that the invariance of the speed of light features prominently in the popular explanations of special relativity for historic reasons, not for technical reasons. Einstein was lead to special relativity contemplating what it would be like to travel with light, and then tried to find a way to accommodate an observer’s motion with the invariance of the speed of light. But the derivation of special relativity is much more general than that, and it is unnecessary to postulate that the speed of light is invariant.Special relativity is really just physics in Minkowski space, that is the 4-dimensional space-time you obtain after promoting time from a parameter to a coordinate. Einstein wanted the laws of physics to be the same for all inertial observers in Minkowski-space, ie observers moving at constant velocity. If you translate this requirement into mathematics, you are lead to ask for the symmetry transformations in Minkowski-space. These transformations form a group – the Poincaré-group – from which you can read off all the odd things you have heard of: time-dilatation, length-contraction, relativistic mass, and so on.
The Poincaré-group itself has two subgroups. One contains just translations in space and time. This tells you that if you have an infinitely extended and unchanging space then it doesn’t matter where or when you do your experiment, the outcome will be the same. The remaining part of the Poincaré-group is the Lorentz-group. The Lorentz-group contains rotations – this tells you it doesn’t matter in which direction you turn, the laws of nature will still be the same. Besides the rotations, the Lorentz-group contains boosts, that are basically rotations between space and time. Invariance under boosts tells you that it doesn’t matter at which velocity you move, the laws of nature will remain the same. It’s the boosts where all the special relativistic fun goes on.
Deriving the Lorentz-group, if you know how to do it, is a three-liner, and I assure you it has absolutely nothing to do with rocket ships and lasers and so on. It is merely based on the requirement that the metric of Minkowski-space has to remain invariant. Carry through with the math and you’ll find that the boosts depend on a free constant with the dimension of a speed. You can further show that this constant is the speed of massless particles.
Hence, if photons are massless, then the constant in the Lorentz-transformation is the speed of light. If photons are not massless, then the constant in the Lorentz-transformation is still there, but not identical to the speed of light. We already know however that these constants must be identical to very good precision, which is the same as saying the mass of photons must be very small.
Giving a mass to photons is unappealing not because it violates special relativity – it doesn’t – but because it violates gauge-invariance, the most cherished principle underlying the standard model. But that’s a different story and shall be told another time.
Thanks for an interesting question!