Reconstruction of black hole metric perturbations from Weyl curvature: II. The Regge–Wheeler gauge

Perturbation theory of rotating black holes is described in terms of the Weyl scalars ψ 4 and ψ 0 , each satisfying Teukolsky's complex master wave equation with spin s = ∓2, and respectively representing outgoing and ingoing radiation. We explicitly construct the metric perturbations out of these Weyl scalars in the Regge–Wheeler gauge in the non-rotating limit. We propose a generalization of the Regge–Wheeler gauge for a Kerr background in Newman–Penrose language and discuss the approach for building up the perturbed spacetime of a rotating black hole. We also provide two-way relationships between waveforms defined in the metric and curvature approaches in the time domain, also known as (inverse) Chandrasekhar transformations, generalized to include matter.