Light Energy

We know that the speed of light is the same for all observers in inertial frames. If so, are the momentum and energy of light as measured by all observers the same even when the light source is moving toward the observer?

Answer

A light source emitting light of frequency f approaching an observer at constant velocity v will appear blue shifted by an amount corresponding to the relativistic Doppler effect formula  because the clock ticking rates will be different for the source reference frame and the observer reference frame, and their separation distance is decreasing. When v << c, we can expand the formula in a Taylor series to obtain f ′ ~ f (1 + v/c – v²/2c² + . . . ), so the leading term in powers of v/c is positive, corresponding to the blueshift. We assume that an acceleration itself does not produce an additional fundamental frequency shift, although there will be acceleration effects because the source is changing instantaneous comoving inertial frames.

For a photon, its energy is E = hf and its momentum is p = E/c, so both energy and momentum are different in different reference frames because the observed frequencies are different. Notice that the recoil of the source on emission of light and of the observer on detection are not accounted for in the discussion and that the energy and momentum input necessary to keep the relative velocity of the source and observer fixed must be considered also. Of course, energy and momentum conservation laws are obeyed in this example.