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Perturbation theory of rotating black holes is described in terms of the Weylscalars $\psi_4$ and $\psi_0$; each satisfying the Teukolsky's complex masterwave equation with spin $s=\mp2$, and respectively representing outgoing andingoing radiation. We explicitly construct the metric perturbations out ofthese Weyl scalars in the Regge-Wheeler gauge in the nonrotating limit. Wepropose a generalization of the Regge-Wheeler gauge for Kerr background in theNewman-Penrose language, and discuss the approach for building up the perturbedspacetime of a rotating black hole. We also provide both-way relationshipsbetween waveforms defined in the metric and curvature approaches in the timedomain, also known as the (inverse-) Chandrasekhar transformations, generalizedto include matter.Comment: 22 pages, no figure

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