Besides traveling at the speed of light, how can we be sure that it is possible to have energy and momentum without mass? How can we be sure that it is possible to have energy and momentum without mass? If something were to continually lose energy, would it not also lose a corresponding amount of mass? I understand that photons are predicted to have no mass because they travel at the speed of light, but is there another reason this is believed?
 A: Photons are not " predicted" to have no mass, they have no rest mass, a photon with not light speed does not exist. You can measure its momentum $p=\frac{hf}{c}$ and thereby its mass . A particle with rest mass≠0 can never reach the speed c, approaching c its mass increases. If a particle losses energy it also looses mass, but you have to experiment with very fast particles to measure it, so you will not see it in cars which loose speed.
But to understand all this you have to study relativity.
A: You cannot separate the idea of having energy and momentum without mass from the idea of moving at the speed of light. I.e. there is no "besides". The basic equation that relates energy, momentum, and mass is: $$m^2 c^2 = E^2/c^2 -p^2$$ From this equation, in the special case that $p=0$ we get the famous $E=mc^2$. And in the special case that $m=0$ we get the usual photon relationship between momentum and energy $E=pc$. Since the velocity is always given by $$v=\frac{pc^2}{E}$$ for a massless particle we always get $v=c$. There is no way to separate $m=0$ from $v=c$, one implies the other.
A: How about this: even in classical electrodynamics (no quantum, no relativity) electromagnetic waves carry momentum. Waves can't have mass because they are not objects. And electromagnetic waves are disturbances in a medium that isn't made of stuff with mass.
And it has to be that way for conservation of momentum to be compatible with electrodynamics. Recall the basic picture of radiation: accelerating charges produce electromagnetic fields, which travel at a finite speed until they meet other charges and cause them to accelerate.
If the waves don't carry momentum then momentum appears to disappear from the first charge, then reappear in the second charge later. Conservation of momentum can only be true at every stage in this process if the wave itself has momentum.
So even with just classical electrodynamics you have to extend the concept of momentum to things beyond particles with mass. That means it should not be surprising to find other surprising kinds of particles with momentum.
