The imposition of Cauchy data to the Teukolsky equation III: The rotating case

TL;DR

This paper develops a method to express gravitational perturbation data of a Kerr black hole in terms of initial Cauchy data, facilitating the computation of gravitational waves in numerical relativity.

Contribution

It explicitly relates Weyl scalars to initial geometric data on a spacelike slice for the rotating Kerr black hole, bridging initial conditions and Teukolsky evolution.

Findings

Provides explicit formulas for  and _t in terms of initial data

Enables computation of gravitational radiation from black hole binaries

Facilitates waveform extraction in numerical simulations

Abstract

We solve the problem of expressing the Weyl scalars $\psi $ that describe gravitational perturbations of a Kerr black hole in terms of Cauchy data. To do so we use geometrical identities (like the Gauss-Codazzi relations) as well as Einstein equations. We are able to explicitly express $\psi $ and $\partial _t\psi $ as functions only of the extrinsic curvature and the three-metric (and geometrical objects built out of it) of a generic spacelike slice of the spacetime. These results provide the link between initial data and $\psi $ to be evolved by the Teukolsky equation, and can be used to compute the gravitational radiation generated by two orbiting black holes in the close limit approximation. They can also be used to extract waveforms from spacetimes completely generated by numerical methods.