Gravitational self-force on a particle in circular orbit around a Schwarzschild black hole

Abstract

We calculate the gravitational self-force acting on a pointlike particle of mass $\ensuremath{\mu}$, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; we then compute the (finite) backreaction force from each of the multipole modes of the perturbation; finally, we apply the ``mode-sum'' method to obtain the total, physical self-force. The temporal component of the self-force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The radial component of the self-force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the $O(\ensuremath{\mu})$ correction to the specific energy and angular momentum parameters (in the Lorenz gauge), as well as the $O(\ensuremath{\mu})$ shift in the orbital frequency (which is gauge invariant).