Second-order Teukolsky formalism in Kerr spacetime: formulation and nonlinear source

TL;DR

This paper develops a second-order Teukolsky formalism for Kerr spacetime to improve gravitational wave modeling, providing a well-behaved nonlinear source term and addressing infrared divergences, aiding future gravitational self-force calculations.

Contribution

It formulates a second-order Teukolsky equation suitable for Kerr backgrounds, including a nonlinear source term and gauge choices to avoid divergences, advancing black hole perturbation theory.

Findings

Derived a nonlinear source term expression for second-order perturbations.

Identified a gauge (Bondi--Sachs) to evade infrared divergences.

Provided tools and a Mathematica notebook for practical calculations.

Abstract

To fully exploit the capabilities of next-generation gravitational wave detectors, we need to significantly improve the accuracy of our models of gravitational-wave-emitting systems. This paper focuses on one way of doing so: by taking black hole perturbation theory to second perturbative order. Such calculations are critical for the development of nonlinear ringdown models and of gravitational self-force models of extreme-mass-ratio inspirals. In the most astrophysically realistic case of a Kerr background, a second-order Teukolsky equation presents the most viable avenue for calculating second-order perturbations. Motivated by this, we analyse two second-order Teukolsky formalisms and advocate for the one that is well-behaved for gravitational self-force calculations and which meshes naturally with recent metric reconstruction methods due to Green, Hollands, and Zimmerman [CQG 37, 075001 (2020)] and others. Our main result is an expression for the nonlinear source term in the second-order field equation; we make this available, along with other useful tools, in an accompanying Mathematica notebook. Using our expression for the source, we also show that infrared divergences at second order can be evaded by adopting a Bondi--Sachs gauge.