Simplified derivation of the gravitational wave stress tensor from the linearized Einstein field equations
arXiv:1604.05974 · doi:10.1073/pnas.1614681113
Abstract
A conserved stress energy tensor for weak field gravitational waves propagating in vacuum is derived directly from the linearized wave equation alone, for an arbitrary gauge using standard general relativity. In any harmonic gauge, the form of the tensor leads directly to the classical expression for the outgoing wave energy. The method described here, however, is a much simpler, shorter, and more physically motivated approach than is the customary procedure, which involves a lengthy and cumbersome second-order (in wave-amplitude) calculation starting with the Einstein tensor. Our method has the added advantage of exhibiting the direct coupling between the outgoing energy flux in gravitational waves and the work done by the gravitational field on the sources. For nonharmonic gauges, the directly derived wave stress tensor has an apparent index asymmetry. This coordinate artefact may be removed, and the symmetrized (still gauge-invariant) tensor then takes on its widely-used form. Angular momentum conservation follows immediately. For any harmonic gauge, however, the stress tensor is manifestly symmetric from the start and its derivation depends, in its entirety, on the structure of the linearized wave equation.
11 pages, no figures. Published in PNAS
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