If we have a mass and add linear momentum to it, we find that both the momentum and mass increases as per Einstein's work. Thus some photons increase the mass and other the velocity. If we add photons to a stationary object, the electrons will move to higher energy levels. this shows up as heat and the mass increases. If we add the same amount of photons to an object that can move, some of the energy will cause the object to move to a higher energy level or heat and the rest of the energy will become momentum. Is this correct?
If we bring a proton toward the speed of light in a cyclotron, what happens. The mass will increase greatly but how? Do the quarks move at higher speeds accounting for the energy increase? At the same time if we use Einstein's formula we have a large mass but what is the momentum? I would assume that it would be Mg C where Mg is the large Einsteinian mass. Thus at the same time that we have a huge mass we also have a huge momentum. Is this correct?