Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric

TL;DR

This paper develops a method to reconstruct the metric perturbations of Kerr black holes at second order, based on first order solutions, aiding gravitational wave analysis of black hole mergers.

Contribution

It introduces a novel procedure to reconstruct the first order metric perturbation from Teukolsky solutions without Hertz potentials, specifically for Kerr black holes.

Findings

Reconstruction method demonstrated in the asymptotic limit for Kerr quasi-normal modes.

Provides a foundation for numerical implementation of second order perturbations.

Enhances understanding of nonlinear gravitational wave interactions around Kerr black holes.

Abstract

Motivated by gravitational wave observations of binary black hole mergers, we present a procedure to compute the leading order nonlinear gravitational wave interactions around a Kerr black hole. We describe the formalism used to derive the equations for second order perturbations. We develop a procedure that allows us to reconstruct the first order metric perturbation solely from knowledge of the solution to the first order Teukolsky equation, without the need of Hertz potentials. Finally, we illustrate this metric reconstruction procedure in the asymptotic limit for the first order quasi-normal modes of Kerr. In a companion paper, we present a numerical implementation of these ideas.