Reconstruction of Black Hole Metric Perturbations from Weyl Curvature

TL;DR

This paper develops a method to reconstruct black hole metric perturbations from Weyl curvature scalars, facilitating better understanding of gravitational radiation and perturbation effects in black hole spacetimes.

Contribution

It explicitly relates metric perturbation potentials to Weyl scalars in the nonrotating limit, aiding future analysis of rotating black holes and second order perturbations.

Findings

Constructed relations between potentials and Weyl scalars in the nonrotating limit

Facilitates second order perturbation analysis and radiation reaction studies

Provides groundwork for extending to rotating black holes

Abstract

Perturbation theory of rotating black holes is usually described in terms of Weyl scalars $\psi_4$ and $\psi_0$, which each satisfy Teukolsky's complex master wave equation and respectively represent outgoing and ingoing radiation. On the other hand metric perturbations of a Kerr hole can be described in terms of (Hertz-like) potentials $\Psi$ in outgoing or ingoing {\it radiation gauges}. In this paper we relate these potentials to what one actually computes in perturbation theory, i.e $\psi_4$ and $\psi_0$. We explicitly construct these relations in the nonrotating limit, preparatory to devising a corresponding approach for building up the perturbed spacetime of a rotating black hole. We discuss the application of our procedure to second order perturbation theory and to the study of radiation reaction effects for a particle orbiting a massive black hole.