Second order gauge invariant gravitational perturbations of a Kerr black hole

Abstract

We investigate higher than the first order gravitational perturbations in the Newman-Penrose formalism. Equations for the Weyl scalar $ψ_4,$ representing outgoing gravitational radiation, can be uncoupled into a single wave equation to any perturbative order. For second order perturbations about a Kerr black hole, we prove the existence of a first and second order gauge (coordinates) and tetrad invariant waveform, $ψ_I$, by explicit construction. This waveform is formed by the second order piece of $ψ_4$ plus a term, quadratic in first order perturbations, chosen to make $ψ_I$ totally invariant and to have the appropriate behavior in an asymptotically flat gauge. $ψ_I$ fulfills a single wave equation of the form ${\cal T}ψ_I=S,$ where ${\cal T}$ is the same wave operator as for first order perturbations and $S$ is a source term build up out of (known to this level) first order perturbations. We discuss the issues of imposition of initial data to this equation, computation of the energy and momentum radiated and wave extraction for direct comparison with full numerical approaches to solve Einstein equations.