A Fast Frequency-Domain Algorithm for Gravitational Self-Force: I, Circular Orbits in Schwarzschild Spacetime

Fast, reliable orbital evolutions of compact objects around massive black holes will be needed as input for gravitational wave search algorithms in the data stream generated by the planned Laser Interferometer Space Antenna (LISA). Currently, the state of the art is a time-domain code by [Phys. Rev. D{\bf 81}, 084021, (2010)] that computes the gravitational self-force on a point-particle in an eccentric orbit around a Schwarzschild black hole. Currently, time-domain codes take up to a few days to compute just one point in parameter space. In a series of articles, we advocate the use of a frequency-domain approach to the problem of gravitational self-force (GSF) with the ultimate goal of orbital evolution in mind. Here, we compute the GSF for a particle in a circular orbit in Schwarzschild spacetime. We solve the linearized Einstein equations for the metric perturbation in Lorenz gauge. Our frequency-domain code reproduces the time-domain results for the GSF up to $\sim 1000$ times faster for small orbital radii. In forthcoming companion papers, we will generalize our frequency-domain methods to include bound (eccentric) orbits in Schwarzschild and (eventually) Kerr spacetimes for computing the GSF, where we will employ the method of extended homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)].