Kerr metric perturbations in Lorenz gauge via numerical relativity methods

Gravitational self-force calculations are needed to accurately describe the motion of a small body in a spinning (Kerr) black hole spacetime, which is vital to predict gravitational wave signals from asymmetric-mass compact binaries. Accurate calculations for the upcoming LISA mission require a minimum of second perturbative order in the mass ratio, which has thus far only been achieved for nonspinning (Schwarzschild) black holes due to nonlinear complexities of the Einstein equations. In this work, we calculate Kerr metric perturbations with a new computational approach based on numerical relativity methods, extending recent scalar self-force results to full general relativity. We formulate the perturbative Lorenz-gauge Einstein equations in terms of azimuthal $m$-modes in the frequency domain and solve the resulting elliptic partial differential equations using a spectral discontinuous Galerkin discretization, adaptive mesh refinement, and preconditioned iterative solvers. With our new method, we achieve exponentially convergent first-order gravitational self-force calculations for circular equatorial orbits in Kerr spacetime up to spin $|a|/M=0.998$ (the Thorne limit). Our framework is designed to carry over to second order by delivering accurate and well-behaved first-order data for nonlinear source construction, and as a solver for the second-order metric perturbations themselves.