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

Abstract

We solve the problem of expressing the Weyl scalars $ψ$ 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 $ψ$ and $\partial _tψ$ 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 $ψ$ 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.