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

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 nonrotating 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 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 $(v/c{)}^6$.

Figure 1

The quantity ${}_{1}u^t$, which is the gauge-independent $O(\mu )$ part of $u^t$ affected by the self-force, is given as a function of $R_{\Omega }$ for circular orbits in the Schwarzschild geometry. Also shown are ${}_{1}u^t$ as calculated with 1PN and 2PN analyses of Appendix  based upon Refs. 17, 18.

Figure 2

The same as Fig. 1, but including the 3PN analysis.