Gravitational self-force correction to the innermost stable circular orbit of a Schwarzschild black hole

TL;DR

The innermost stable circular orbit of a test particle around a Schwarzschild black hole of mass M has (areal) radius r_{isco}=6MG/c;{2} and if the particle is endowed with mass micro(<<M), it experiences a gravitational self-force whose conservative piece alters the location of the ISCO.

Abstract

The innermost stable circular orbit (ISCO) of a test particle around a Schwarzschild black hole of mass $M$ has (areal) radius $r_{\rm isco}= 6M G/c^2$. If the particle is endowed with mass $μ(\ll M)$, it experiences a gravitational self-force whose conservative piece alters the location of the ISCO. Here we calculate the resulting shifts $Δr_{\rm isco}$ and $ΔΩ_{\rm isco}$ in the ISCO's radius and frequency, at leading order in the mass ratio $μ/M$. We obtain, in the Lorenz gauge, $Δr_{\rm isco}=-3.269 (\pm 0.003)μG/c^2$ and $ΔΩ_{\rm isco}/Ω_{\rm isco}=0.4870 (\pm 0.0006) μ/M$. We discuss the implications of our result within the context of the extreme-mass-ratio binary inspiral problem.