A consequence of the gravitational self-force for circular orbits of the Schwarzschild geometry

TL;DR

This paper numerically analyzes how the gravitational self-force affects the orbital frequency and proper time rate for a small mass in circular orbit around a Schwarzschild black hole, confirming results with post-Newtonian theory.

Contribution

It provides the first numerical calculation of the gravitational self-force effects on circular orbits in Schwarzschild spacetime, including gauge-invariant quantities.

Findings

Self-force causes measurable deviations in orbital frequency.

Proper time rate is affected by the self-force.

Results agree with post-Newtonian analysis for distant orbits.

Abstract

A small mass \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(\mu/M). This deviation is said to be caused by the gravitational self-force of the metric perturbation h_{ab} from \mu. For circular orbits about a non-rotating black hole we numerically calculate the O(\mu/M) effects upon the orbital frequency and upon the rate of passage of proper time on the worldline. 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.