The imposition of Cauchy data to the Teukolsky equation I: The nonrotating case

TL;DR

This paper develops a method to explicitly incorporate Cauchy initial data into the Teukolsky equation for non-rotating black holes, facilitating numerical studies of gravitational perturbations relevant to black hole collisions.

Contribution

It provides a gauge-invariant formulation of initial data incorporation into the Teukolsky equation and addresses regularization of divergent integrals in the Green function approach.

Findings

Explicit expression of Teukolsky function and its derivative in terms of 3-geometry and extrinsic curvature.

Regularization method for divergent integrals in Green function approach.

Framework applicable to astrophysical initial data for black hole perturbations.

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 non-rotating black holes. The Teukolsky function $% \Psi $ and its first time derivative $\partial_t \Psi $ can be written in terms of only the 3-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.