E = mc^2 Without Relativity

Abstract

The equivalence of mass and energy is indelibly linked with relativity, both by scientists and in the popular mind. I prove that E = mc^2 by demanding momentum conservation of an object that emits two equal electromagnetic wave packets in opposite directions in its own frame. In contrast to Einstein's derivation of this equation, which applies energy conservation to a similar thought experiment, the new derivation employs no effects that are greater than first order in v/c and therefore does not rely on results from Special Relativity. In addition to momentum conservation, it uses only aberration of starlight and the electromagnetic-wave momentum-energy relation p=E/c, both of which were established by 1884. In particular, no assumption is made about the constancy of the speed of light, and the derivation proceeds equally well if one assumes that light is governed by a Galilean transformation. In view of this, it is somewhat puzzling that the equivalence of mass and energy was not derived well before the advent of Special Relativity. The new derivation is simpler and more transparent than Einstein's and is therefore pedagogically useful.