Gravitational Self-Force Correction to the Innermost Stable Circular Orbit of a Schwarzschild Black Hole

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