Energy-Momentum of Radiating Systems

Abstract

In this paper I shall discuss the null hypersurface treatment of asymptotically flat systems, with particular emphasis on the concept of energy. Before I get deeply involved in that subject, let me first recall the developments which made this approach attractive. At least in the far field region, where gravitational effects are weak, one would expect the linearized theory to give satisfactory results. These can be readily obtained through the standard introduction of harmonic coordinates and the consequent reduction of Einstein’s equations into flat space wave equations. However, one is also interested in the next approximation to the linearized theory, in which the gravitational field now serves as its own source, so that effects due to the non-linearity of general relativity can also be investigated. Fock,1,2 has carried out this next approximation in detail. When outgoing radiation is present in the linear approximation, he found that, aside from the Minkowski metric limit, an asymptotic log r/r behavior dominates the harmonic metric at large distances along the outward null cones. Although Fock was able to extract from this first approximation scheme finite values for physically relevant quantities such as energy flux, the appearance of these log r/r terms obscured the validity of the convergence of the approximation expansion and the consistency of any outgoing radiation conditions of the Sommerfeld type.3,4