Reconstruction of Black Hole Metric Perturbations from Weyl Curvature II: The Regge-Wheeler gauge

TL;DR

This paper develops methods to reconstruct black hole metric perturbations from Weyl curvature scalars, extending the Regge-Wheeler gauge to Kerr black holes and relating waveform approaches in the time domain.

Contribution

It explicitly constructs metric perturbations from Weyl scalars in the Regge-Wheeler gauge for nonrotating black holes and proposes a generalized gauge for Kerr backgrounds.

Findings

Constructed metric perturbations from Weyl scalars in the Regge-Wheeler gauge.

Proposed a generalization of the Regge-Wheeler gauge for Kerr black holes.

Established relationships between waveform approaches in metric and curvature formalisms.

Abstract

Perturbation theory of rotating black holes is described in terms of the Weyl scalars $\psi_4$ and $\psi_0$; each satisfying the Teukolsky's complex master wave equation with spin $s=\mp2$, 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 nonrotating limit. We propose a generalization of the Regge-Wheeler gauge for Kerr background in the Newman-Penrose language, and discuss the approach for building up the perturbed spacetime of a rotating black hole. We also provide both-way relationships between waveforms defined in the metric and curvature approaches in the time domain, also known as the (inverse-) Chandrasekhar transformations, generalized to include matter.