Simplified derivation of the gravitational wave stress tensor from the linearized Einstein field equations

TL;DR

This paper presents a simplified, physically motivated derivation of the gravitational wave stress tensor directly from the linearized Einstein equations, clarifying its properties and gauge invariance.

Contribution

It introduces a shorter, more straightforward method for deriving the gravitational wave stress tensor from the linearized wave equation, avoiding complex second-order calculations.

Findings

The stress tensor is explicitly derived in any gauge, with a symmetric form in harmonic gauges.

The method directly links energy flux to work done by gravitational waves on sources.

Angular momentum conservation is naturally obtained from the derived tensor.

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.