Gauge invariant perturbations of black holes. II. Kerr space–time

We cast the perturbed Bianchi identities into a form involving only tetrad and coordinate-gauge invariant field quantities. From the perturbed Bianchi identities we then derive a system of gauge invariant wavelike gravitational perturbation equations. These are the Teukolsky equation, a gauge invariant analog of the Regge–Wheeler equation, and a new gravitational perturbation equation. The Bianchi identities also provide the transformations between the perturbation equations. In particular, the transformation between the Teukolsky equation and the analog of the Regge–Wheeler equation arises in this way. This work extends our previous analysis of the perturbations of the Schwarzschild black hole to the Kerr case.