Energy and momentum in general relativity. I The 4-momentum expressed in terms of four invariants when space-time is asymptotically flat

Abstract

A simple derivation is given of a class of pseudo-tensor conservation equations which includes those due to Einstein and Landau & Lifshitz. When space-time is asymptotically flat the asymptotic form of the metric is taken as the metric of a flat background space-time and all events are mapped on to this background. In the map the conservation equations may be written as tensor equations and from them invariant integral conservation laws are derived. The conservation laws are used to suggest possible invariant expressions for the components of the 4-momentum on a 3-flat in the map and to distinguish between radiating and non-radiating systems. When certain conditions are satisfied it is shown that the total 4-momentum on a 3-flat is the same whichever conservation equation is taken as the basis of the theory. The theory based on the Landau-Lifshitz conservation equation is regarded as physically more useful than the other forms as it alone leads easily to six further invariants which are related to the angular momentum.