Conservation Laws in General Relativity as the Generators of Coordinate Transformations

Abstract

The components of the so-called canonical energy-stress pseudotensor in general relativity may be thought of as the generators of infinitesimal coordinate transformations corresponding to a rigid parallel displacement of the coordinate origin, just as in Lorentz-covariant theories. In this paper it is shown that the canonical expressions, as well as the expressions proposed by Landau and Lifshitz and the expressions for the angular momentum density, are all special cases of an infinity of conservation laws whose pseudovectors generate arbitrary curvilinear coordinate transformations. This approach enables us to construct the transform of every one of these conservation laws under an arbitrary (finite) coordinate transformation. Finally it is shown that every one of these conservation laws may be used to obtain a surface integral relationship that describes the motion of singularities in a general-relativistic theory. It is concluded that there is an infinite number of parameters that describes a singularity of the field, a fact that had previously been in doubt.