Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

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>> m_1. More precisely, we construct the function h_{uu}(x) = h_{μν} u^μ u^ν (related to Detweiler's gauge-invariant "redshift" variable), where h_{μν} is the regularized metric perturbation in the Lorenz gauge, u^μ is the four-velocity of m_1, and x= [Gc^{-3}(m_1+m_2)Ω]^{2/3} is an invariant coordinate constructed from the orbital frequency Ω. 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 ν, of the main radial potential A(u,ν) of the Effective One Body (EOB) formalism, we compute a(u) over the entire domain 0<u<1/3. We find that a(u) diverges at the light-ring as ~0.25 (1-3u)^{-1/2}, explain the physical origin of this divergence, 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, as well as the O(ν) shift in the ISCO frequency. 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(ν) piece \bar d(u) of a second EOB radial potential {\bar D}(u,ν). 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.