Sourced metric perturbations of Kerr spacetime in Lorenz gauge

TL;DR

This work develops a Lorenz-gauge formalism for Kerr perturbations sourced by arbitrary stress-energy, reducing metric reconstruction to six decoupled, separable Teukolsky equations for two spin-weight $\pm2$, two spin-weight $\pm1$, and two spin-weight $0$ scalars, and reconstructing the metric via explicit differential operators acting on these scalars. Central to the approach is the Aksteiner–Andersson–Bäckdahl construction, which provides a trace-free, time-derivative–driven perturbation whose Weyl-scalar content is linked to a gauge-corrected metric through Weyl- and stress-energy–sourced pieces; a trace piece and a trace-free piece are then transformed to Lorenz gauge using a carefully organized gauge-vector that satisfies coupled wave equations. The six sourced Teukolsky equations are solved with Green’s-function methods, including a circularity reformulation to handle the stress-energy contribution and to express the gauge-vector corrections in terms of adjoint operators, enabling explicit mode decompositions and radial solutions. The resulting framework fully determines the metric perturbation up to a time integral and is suitable for higher-order perturbations, with the main caveat that static modes require separate treatment, but a symmetric, zero-frequency variant ($h^+$) provides a path forward. Overall, the paper delivers a practical, general-purpose toolkit for Lorenz-gauge Kerr perturbations with arbitrary sources, advancing second-order and nonlinear perturbation theory for spinning black holes.

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.

Figures (1)

• Figure 1: Absolute value of the asymptotic amplitudes of the $(\ell,m)=(2,2)$ spin-weighted spheroidal harmonic mode of the solutions to Eqs. \ref{['eq:solutions']} for a point particle an a circular orbit of radius $r_0=p M$ in Kerr spacetime with $a=0.6M$. The coefficients of the dominant behaviour towards infinity are shown by solid lines and towards the horizon by dashed lines.