Conservation Laws in the General Theory of Relativity with Electromagnetic Field

Abstract

The principal result of this paper is the reformulation of the so-called strong conservation laws of the general theory of relativity with electromagnetic field, resulting in relationships between closed two-dimensional surface integrals in ordinary three-dimensional space. This formulation is particularly useful if particles are represented as singularities of the field because it obviates the necessity for an examination of the detailed structure of the singularities themselves. This method has been used for determining the total energy (or mass) in a domain containing a Schwarzschild singularity. It turns out that the value of this energy depends on the choice of coordinate system unless the energy density is integrated over all of space, in which case the integral converges toward the expected value. In future papers the method of closed surface integrals will be applied to the rigorous treatment of the problem of motion.