Reconstruction of Black Hole Metric Perturbations from Weyl Curvature

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 $\mathrm{\Psi }$ 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.