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 $r_{\rm isco}= 6M G/c^2$. 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 $\Delta r_{\rm isco}$ and $\Delta\Omega_{\rm isco}$ in the ISCO's radius and frequency, at leading order in the mass ratio $\mu/M$. We obtain, in the Lorenz gauge, $\Delta r_{\rm isco}=-3.269 (\pm 0.003)\mu G/c^2$ and $\Delta\Omega_{\rm isco}/\Omega_{\rm isco}=0.4870 (\pm 0.0006) \mu/M$. We discuss the implications of our result within the context of the extreme-mass-ratio binary inspiral problem.