Perturbations of Schwarzschild black holes in the Lorenz gauge: Formulation and numerical implementation

TL;DR

This paper reformulates Schwarzschild black hole perturbations in the Lorenz gauge, deriving a set of decoupled wave equations for scalar functions and implementing a numerical evolution scheme to analyze perturbations caused by orbiting particles.

Contribution

It introduces a new algebraic formulation of metric perturbations in the Lorenz gauge and develops a numerical method for evolving these perturbations in time.

Findings

Successfully solves wave equations for perturbations using numerical evolution.

Provides analytical solutions for axially-symmetric, odd-parity modes.

Demonstrates the framework's suitability for gravitational self-force calculations.

Abstract

We reformulate the theory of Schwarzschild black hole perturbations in terms of the metric perturbation in the Lorenz gauge. In this formulation, each tensor-harmonic mode of the perturbation is constructed algebraically from 10 scalar functions, satisfying a set of 10 wavelike equations, which are decoupled at their principal parts. We solve these equations using numerical evolution in the time domain, for the case of a pointlike test particle set in a circular geodesic orbit around the black hole. Our code uses characteristic coordinates, and incorporates a constraint damping scheme. The axially-symmetric, odd-parity modes of the perturbation are obtained analytically. The approach developed here is especially advantageous in applications requiring knowledge of the local metric perturbation near a point particle; in particular, it offers a useful framework for calculations of the gravitational self force.