Schwarzschild perturbations in Lorenz gauge via elliptic differential equations
Accurate predictions of gravitational wave signals from asymmetric compact binaries are accessible through black hole perturbation theory and self-force calculations. Faithful waveform models will require contributions from first- and second-order terms in the small mass-ratio expansion. The problem of second-order Kerr perturbations is exacerbated by non-separability of the metric perturbation equations and non-linear mode coupling, which motivates this $m$-mode approach. This work moves towards the eventual goal of second-order Kerr perturbations by calculating first-order Schwarzschild metric perturbations via $m$-modes in the frequency domain for the first time. We solve the Lorenz gauge field equations as a system of coupled elliptic partial differential equations that govern each $m$-mode. Our Mathematica code implements a second-order finite difference representation of the field equations, which we solve as a sparse linear algebra problem. Regularization near the small body is achieved through the effective source method, and our presentation introduces a new puncture expansion of the singular field for a point mass in Kerr spacetime. Issues related to problematic near-horizon behavior are explored and then mitigated by applying sophisticated near-horizon boundary conditions. Our results illustrate the features of each component and $m$-mode of the metric perturbation, and we are able to calculate gravitational wave energy fluxes with sufficient accuracy to enable future second-order self-force calculations.