On energy conservation and the question of its localization in static Riemannian space-times

Abstract

I t is well known that the total energy of static Riemannian space-times, reprcscnting physically ch)sed systems in general relativity (e.g., the Schwarzschild solution), is a well-defined conserved quanti ty that has the constant value me c 2. Furthermore, it has been demonstrated that it is possible to make total-energy calculations of this type in any physically well-defined co-ordinate frame (1). Previously, the total energy of such gravitational systems h~ts been generally regarded as well defined only asymptotically at large spatial distances from the gravitational mass. This requirement, is, of course, in accord with the generaUy accepted notion of gravitational energy being a nonlocalizable quanti ty where only the global or total energy of a static physically closed system is well defined (~). General]y, no successful at tempt has been made to at tr ibute a welldefined meaning to similar calculations made in some subdomain of the given physically closed system. The purpose of this note is to show that the above considerations for the energy of certain static, Riemannian space-times are unnecessarily severe. We will show that the energy of a static, spherically symmetric mass distribution is localized within a given and well-defined (, rad ius , where R,j r 0. The usual asymptotic limits are therefore not necessary in these calculations as there are no contributions to the total energy outside the physically defined mass boundary for these special static space-times. Indeed, it will be shown that these particular Riemannian spaces admit a type of ~, Gauss law ,> in direct analogy to Newtonian gravitationM theory. We will further be able to draw some additional conclusions and analogies for the case of static, spherically symmetric, closed gravitational systems including charge. Also, results relating to the axially symmetric spinning solution of Einstein's field equations (Kerr metric) are briefly mentioned.