A quick snippet of the derivation of Einstein’s Mass-energy equivalence equation. This is always good for some good late night reading.
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In the particle’s rest frame, the momentum is (mc,0) and so for the force four-vector to be orthogonal, its time component must be zero in the rest frame as well, so F = (0,F). Applying a Lorentz transformation to an arbitrary frame, we find
Thus the time component of the relativistic version of Newton’s second law is
Recalling the definition of work done by the applied force as
and since the change in energy is given by the work done, we have
and so finally we see that, up to an additive constant,
from Wikipedia