A Proposal for Determining the Energy Content of Gravitational Waves by Using Approximate Symmetries of Differential Equations

TL;DR

This paper proposes a novel method to define the energy content of gravitational waves using approximate symmetries of differential equations, addressing the challenge of energy localization in time-varying vacuum solutions.

Contribution

It introduces a new approach employing 'slightly broken' Noether symmetries and approximate Lie symmetries to evaluate gravitational wave energy content.

Findings

The proposed definition is physically acceptable and consistent.

Application to various gravitational wave solutions demonstrates its usefulness.

The method provides insights into whether gravitational waves experience self-damping.

Abstract

Since gravitational wave spacetimes are time-varying vacuum solutions of Einstein's field equations, there is no unambiguous means to define their energy content. However, Weber and Wheeler had demonstrated that they do impart energy to test particles. There have been various proposals to define the energy content but they have not met with great success. Here we propose a definition using "slightly broken" Noether symmetries. We check whether this definition is physically acceptable. The procedure adopted is to appeal to "approximate symmetries" as defined in Lie analysis and use them in the limit of the exact symmetry holding. A problem is noted with the use of the proposal for plane-fronted gravitational waves. To attain a better understanding of the implications of this proposal we also use an artificially constructed time-varying non-vacuum metric and evaluate its Weyl and stress-energy tensors so as to obtain the gravitational and matter components separately and compare them with the energy content obtained by our proposal. The procedure is also used for cylindrical gravitational wave solutions. The usefulness of the definition is demonstrated by the fact that it leads to a result on whether gravitational waves suffer self-damping.