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Gravitational energy and conserved currents in general relativity Keefer, Bowie Gordon

Abstract

The problem of the definition of gravitational energy is reconsidered. In the Einstein theory all matter and fields except gravity must have a well defined local distribution of energy that is described by the energy-momentum tensor. A gravitational "energy-momentum complex" may be defined in analogy to an energy-momentum tensor. However there is an infinite number of expressions for the gravitational complex, and each expression must depend explicitly upon the choice of reference system. Following a review of earlier works, a study is made of physical and geometric considerations which might select usefully distinguished gravitational complexes and reference frames. This investigation is conducted within the vierbein formulation of general relativity. Conserved currents corresponding to generalized energy, momentum and spin are derived from action principles. These currents transform as vector densities under general coordinate transformations, but depend on the vierbein frame chosen. The expressions for the energy and momentum currents are not unique, as their general expression contains three arbitrary constants. Physical examples are ised to test possible choices of these constants and possible vierbein frames. The generalized vierbein energy and momentum currents are calculated for asymptotically flat, radiative space-times. The physical requirements that the energy of an isolated system cannot increase when there is no incoming radiation, and that there, be invariance under vierbein transformations respecting boundary conditions appropriate to the asymptotic symmetry group, are imposed on the generalized energy integral. These requirements determine a unique expression for the energy current which contains no second order field derivatives. Since the boundary conditions do not specify the vierbein frame everywhere, the distribution of gravitational energy is not well defined even when the concept of total energy is made legitimate by asymptotic space-time symmetry. It has been conjectured repeatedly that a local density of gravitational energy could be defined even in the absence of space-time symmetries through a suitable choice of gravitational complex and of reference frame. This is certainly attainable in a formal sense, as invariant vierbein frames are defined by the principal directions of the curvature tensor and of the energy-momentum tensor-of matter. It is shown by the consideration of gravitational radiation fields that such definitions will not suffice to localize gravitational energy.

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