Mass Energy Example

Consider a particle, that is stationary relative to the Earth, emitting photons in opposite directions each with energy -ΔE / 2 where ΔE is the energy change of the particle:

The photons have the following momenta:

(ΔE / (2c), 0, 0)

(-ΔE / (2c), 0, 0)

where c is the speed of light in a vacuum.  Relative to a reference frame moving to the right with a speed v relative to the Earth, the photons have the following momenta:

((ΔE / (2c))√(1 + v / c) / (1 - v / c), 0, 0)

((-ΔE / (2c))√(1 - v / c) / (1 + v / c), 0, 0).

This implies the total photon momentum is -γΔEv / c² and that the particle must have lost the same amount.  However, the particles speed is constant! Therefore, the change in momentum must have come from a change in mass:

γΔmv = γΔEv / c²

Δm = ΔE / c²

ΔE = Δmc².