Gravitational-wave dynamics and black-hole dynamics: second quasi-spherical approximation

TL;DR

This paper introduces a second quasi-spherical approximation for gravitational waves and black-hole dynamics, incorporating non-linear effects, energy conservation, and a local wave equation, advancing understanding of gravitational radiation reaction.

Contribution

It generalizes the first approximation to include non-linear radiation effects, defining a local energy tensor and a non-linear wave equation for gravitational waves.

Findings

Energy tensor satisfies local energy conditions

Formulation of a local first law of black-hole dynamics

Derivation of a non-linear wave equation for gravitational radiation

Abstract

Gravitational radiation with roughly spherical wavefronts, produced by roughly spherical black holes or other astrophysical objects, is described by an approximation scheme. The first quasi-spherical approximation, describing radiation propagation on a background, is generalized to include additional non-linear effects, due to the radiation itself. The gravitational radiation is locally defined and admits an energy tensor, satisfying all standard local energy conditions and entering the truncated Einstein equations as an effective energy tensor. This second quasi-spherical approximation thereby includes gravitational radiation reaction, such as the back-reaction on the black hole. With respect to a canonical flow of time, the combined energy-momentum of the matter and gravitational radiation is covariantly conserved. The corresponding Noether charge is a local gravitational mass-energy. Energy conservation is formulated as a local first law relating the gradient of the gravitational mass to work and energy-supply terms, including the energy flux of the gravitational radiation. Zeroth, first and second laws of black-hole dynamics are given, involving a dynamic surface gravity. Local gravitational-wave dynamics is described by a non-linear wave equation. In terms of a complex gravitational- radiation potential, the energy tensor has a scalar-field form and the wave equation is an Ernst equation, holding independently at each spherical angle. The strain to be measured by a distant detector is simply defined.