Does Einstein's relationE=mc ²arise from the metric?

Abstract

In this Letter we propose to consider the ‘four-energy-space’ whose ‘coordinates’ are composed as follows: (i) the ‘coordinate’ ε⁰ refers to the internal energy of the body (it is involved as an unknown function of the rest-energy and the kinetic energy of the body), and (ii) the ‘coordinates’ ε¹, ε², ε³ relate to the presence of gravitational, electromagnetic, and thermal energy at the location of the body respectively. We involve yet the proper energy interval dε² by analogy to the four-interval ds ² in general relativity. From such ‘metric field’ we calculate the ‘Ricci tensor’ in the simplest case. In addition, we require its form to be the same one as that considered by Schwarzschild. Comparing both solutions we obtain Einstein's relationE=mc ².