Localizing the Energy and Momentum of Linear Gravity

TL;DR

This paper develops a gauge-specific framework to quantify local energy and momentum exchange in linear gravity, deriving a unique energy-momentum tensor with desirable physical properties and applying it to various gravitational scenarios.

Contribution

It introduces a gauge-fixed, unambiguous gravitational energy-momentum tensor for linearized gravity, ensuring positive energy and subluminal energy flow.

Findings

The tensor is gauge-invariant for gravitational plane-waves.

Energy density is non-negative for transverse-traceless fields.

Applied to gravitational waves and static spacetimes, it quantifies energy-momentum content.

Abstract

A framework is developed which quantifies the local exchange of energy and momentum between matter and the linearized gravitational field. We derive the unique gravitational energy-momentum tensor consistent with this description, and find that this tensor only exists in the harmonic gauge. Consequently, nearly all the gauge freedom of our framework is naturally and unavoidably removed. The gravitational energy-momentum tensor is then shown to have two exceptional properties: (a) it is gauge-invariant for gravitational plane-waves, (b) for arbitrary transverse-traceless fields, the energy-density is never negative, and the energy-flux is never spacelike. We analyse in detail the local gauge invariant energy-momentum transferred between the gravitational field and an infinitesimal point-source, and show that these invariants depend only on the transverse-traceless components of the field. As a result, we are led to a natural gauge-fixing program which at last renders the energy-momentum of the linear gravitational field completely unambiguous, and additionally ensures that gravitational energy is never negative nor flows faster than light. Finally, we calculate the energy-momentum content of gravitational plane-waves, the linearized Schwarzschild spacetime (extending to arbitrary static linear spacetimes) and the gravitational radiation outside two compact sources: a vibrating rod, and an equal-mass binary.