Imposition of Cauchy data to the Teukolsky equation. III. The rotating case

Abstract

Gravitational perturbations about a Kerr black hole in the Newman-Penrose formalism are concisely described by the Teukolsky equation. New numerical methods for studying the evolution of such perturbations require not only the construction of appropriate initial data to describe the collision of two orbiting black holes, but also to know how such new data must be imposed into the Teukolsky equation. In this paper we show how Cauchy data can be incorporated explicitly into the Teukolsky equation for nonrotating black holes. The Teukolsky function $\ensuremath{\Psi}$ and its first time derivative ${\ensuremath{\partial}}_{t}\ensuremath{\Psi}$ can be written in terms of only the three-geometry and the extrinsic curvature in a gauge-invariant way. Taking a Laplace transform of the Teukolsky equation incorporates initial data as a source term. We show that for astrophysical data the straightforward Green function method leads to divergent integrals that can be regularized like for the case of a source generated by a particle coming from infinity.