Consequence of the gravitational self-force for circular orbits of the Schwarzschild geometry

Abstract

A small mass $\ensuremath{\mu}$ in orbit about a much more massive black hole $m$ moves along a world line that deviates from a geodesic of the black hole geometry by $O(\ensuremath{\mu}/m)$. This deviation is said to be caused by the gravitational self-force of the metric perturbation ${h}_{ab}$ from $\ensuremath{\mu}$. For circular orbits about a nonrotating black hole we numerically calculate the $O(\ensuremath{\mu}/m)$ effects upon the orbital frequency and upon the rate of passage of proper time on the world line. These two effects are independent of the choice of gauge for ${h}_{ab}$ and are observable in principle. For distant orbits, our numerical results agree with a post-Newtonian analysis including terms of order $(v/c{)}^{6}$.