Global Covariant Conservation Laws in Riemannian Spaces. I

Abstract

It is demonstrated that in the theory of general relativity integral conservation laws can be obtained from a vanishing covariant divergence of a tensor in a completely covariant form. This is achieved by introducing into tensor calculus a new operation, namely, tensor integration which is defined as an inverse of tensor differentiation. In particular, a one-dimensional absolute integration of a tensor along a curve is defined as an inverse operation to the same type of differentiation. A representation of absolute integration is developed by a perturbation method as an infinite series where each term consists of ordinary integrations only. As an example of absolute integration, a vector field is obtained whose components can be employed as a coordinate frame having a very close resemblance to the Riemannian coordinates. Covariant integration is then introduced as an inverse of covariant differentiation; however, its usefulness is severely limited by the conditions of integrability which have to be satisfied to make covariant integration possible. A set of unspecified covariantly constant base vectors is used to explain the idea of a covariantly constant tensor and to express symbolically absolute integration in terms of the ordinary integration. The one-dimensional absolute integration is then extended to higher dimensions in such a way that it is independent of the order of integrations. Finally, Gauss' theorem is proved for the absolute integration which enables one to convert a volume integral of a covariant divergence of a tensor into a corresponding surface integral of the same tensor.