Second order gauge invariant gravitational perturbations of a Kerr black hole

TL;DR

The work extends gravitational perturbation theory to second order on a Kerr background by deriving a Teukolsky-like master equation with a source term built from first-order perturbations and by constructing a second-order gauge- and tetrad-invariant radiation waveform ψ_I^{(2)}. This invariant is formed by combining the second-order Weyl scalar ψ_4^{(2)} with carefully chosen quadratic corrections in first-order quantities, yielding an evolution equation with the same operator as at first order and a physically meaningful source. The authors provide explicit prescriptions for achieving tetrad and coordinate invariance, discuss initial data construction, and give formulas for computing radiated energy and momentum, enabling direct comparison with full numerical relativity in a time-domain setting. The framework lays the groundwork for accurate second-order waveforms in Kerr spacetimes and can be extended to Schwarzschild and other Petrov type D backgrounds, offering a robust tool for gravitational wave modeling and error estimation in perturbative regimes.

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.