Kerr self-force via elliptic PDEs

Our long-term goal is to calculate the Lorenz gauge gravitational self-force for an extreme mass-ratio binary system in Kerr spacetime. Prior time domain efforts encountered catastrophic instabilities, which we overcome by entering the frequency domain and solving elliptic PDEs. To develop techniques, we first consider the self-force exerted on a scalar charge in Kerr spacetime via the effective source method. Challenges include handling a large sparse linear system and how imperfect boundary conditions can introduce errors that pollute the entire domain. We have applied various techniques to overcome these challenges, such as analyzing the boundary behavior to impose more sophisticated boundary conditions with improved accuracy. Various preliminary self-force results will be presented and discussed.