Sourced metric perturbations of Kerr spacetime in Lorenz gauge

TL;DR

This paper develops a formalism to compute metric perturbations of Kerr spacetime in Lorenz gauge sourced by arbitrary stress-energy, enabling both linear and nonlinear analyses with a systematic scalar decomposition.

Contribution

It introduces a novel scalar-based formalism for solving Lorenz gauge metric perturbations of Kerr spacetime with arbitrary sources, including higher order nonlinearities.

Findings

Derivation of sourced Teukolsky equations for scalar components.

Explicit formalism for metric perturbations as sums of differential operators.

Applicable to both linear and nonlinear perturbation analyses.

Abstract

We derive a formalism for solving the Lorenz gauge equations for metric perturbations of Kerr spacetime sourced by an arbitrary stress-energy tensor. The metric perturbation is obtained as a sum of differential operators acting on a set of six scalars, with two of spin-weight $\pm2$, two of spin-weight $\pm1$, and two of spin-weight $0$. We derive the sourced Teukolsky equations satisfied by these scalars, with the sources given in terms of differential operators acting on the stress-energy tensor. The method can be used to obtain both linear and higher order nonlinear metric perturbations, and it fully determines the metric perturbation up to a time integral, omitting only static contributions which must be handled separately.