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

Sarp Akcay, Leor Barack, Thibault Damour, Norichika Sago

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107 Citations (Scopus)


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 orbit and the light ring of a Schwarzschild black hole of mass m 2m 1. More precisely, we construct the function huuR,L(x)hμνR,Luμuν (related to Detweiler's gauge-invariant "redshift" variable), where hμνR,L( 1) is the regularized metric perturbation in the Lorenz gauge, uμ is the four-velocity of m 1 in the background Schwarzschild metric of m 2, and [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 huuR,L just outside the "light ring" at x=13 (i.e., r=3Gm 2/c2), where the circular orbit becomes null. Using the recently discovered link between huuR,L and the piece a(u), linear in the symmetric mass ratio νm 1m 2/(m 1+m 2) 2, of the main radial potential A(u,ν)=1-2u+νa(u)+O(ν2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0<u<13 (thereby extending previous results limited to u≤15). We find that a(u) diverges like a(u)0.25(1-3u) -1 /2 at the light-ring limit, u→(13) -, 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<13 (and possibly beyond), and give accurate numerical estimates of the values of a(u) and its first three derivatives at the innermost stable circular orbit u=16, as well as the associated O(ν) shift in the frequency of that orbit. 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 of a second EOB radial potential D̄(u)=1+νd̄(u)+O(ν2). Combining these results with our present global analytic representation of a(u), we numerically compute d̄(u) on the interval 0<u≤16.

Original languageEnglish
Article number104041
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Issue number10
Publication statusPublished - Nov 16 2012

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics
  • Physics and Astronomy (miscellaneous)


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