**Authors**: Sarp Akcay, Leor Barack, Thibault Damour, Norichika Sago

**Date**: 5 Sep 2012

**Abstract**: We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass $m_1$ as it moves along an (unstable) circular geodesic orbit between the innermost stable circular orbit (ISCO) and the light ring of a Schwarzschild black hole of mass $m_2\gg m_1$. More precisely, we construct the function $h_{uu}(x) \equiv h_{\mu\nu} uˆ{\mu} uˆ{\nu}$ (related to Detweiler's gauge-invariant "redshift" variable), where $h_{\mu\nu}$ is the regularized metric perturbation in the Lorenz gauge, $uˆ{\mu}$ is the four-velocity of $m_1$, and $x\equiv [Gcˆ{-3}(m_1+m_2)\Omega]ˆ{2/3}$ is an invariant coordinate constructed from the orbital frequency $\Omega$. In particular, we explore the behavior of $h_{uu}$ just outside the "light ring" at $x=1/3$, where the circular orbit becomes null. Using the recently discovered link between $h_{uu}$ and the piece $a(u)$, linear in the symmetric mass ratio $nu$, of the main radial potential $A(u,\nu)$ of the Effective One Body (EOB) formalism, we compute $a(u)$ over the entire domain $0<u<1/3$ (extending previous results for $u\leq 1/5$). We find that $a(u)$ {\it diverges} like $\approx 0.25 (1-3u)ˆ{-½}$ at the light-ring limit, explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for $a(u)$, valid on the entire domain $0<u<1/3$ (and possibly beyond), and give accurate numerical estimates of the values of $a(u)$ and its first 3 derivatives at the ISCO. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of $a(u)$ and its first two derivatives, involving also the $O(\nu)$ piece $\bar d(u)$ of a second EOB radial potential ${\bar D}(u,\nu)$. Combining these results with our present global analytic representation of $a(u)$, we numerically compute $\bar d(u)$ on the interval $0<u\leq 1/6$.

1209.0964
(/preprints)

2012-09-21, 10:52
**[edit]**

© M. Vallisneri 2012 — last modified on 2010/01/29

*Tantum in modicis, quantum in maximis*