Reconstruction of black hole metric perturbations from the Weyl curvature

Abstract

Perturbation theory of rotating black holes is usually described in terms of Weyl scalars ${\ensuremath{\psi}}_{4}$ and ${\ensuremath{\psi}}_{0},$ 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 $\ensuremath{\Psi}$ in outgoing or ingoing radiation gauges. In this paper we relate these potentials to what one actually computes in perturbation theory, i.e. ${\ensuremath{\psi}}_{4}$ and ${\ensuremath{\psi}}_{0}.$ 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.