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

Abstract

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 $\ensuremath{\mu}(\ensuremath{\ll}M)$, it experiences a gravitational self-force whose conservative piece alters the location of the ISCO. Here we calculate the resulting shifts $\ensuremath{\Delta}{r}_{\mathrm{isco}}$ and $\ensuremath{\Delta}{\ensuremath{\Omega}}_{\mathrm{isco}}$ in the ISCO's radius and frequency, at leading order in the mass ratio $\ensuremath{\mu}/M$. We obtain, in the Lorenz gauge, $\ensuremath{\Delta}{r}_{\mathrm{isco}}=\ensuremath{-}3.269(\ifmmode\pm\else\textpm\fi{}0.003)\ensuremath{\mu}G/{c}^{2}$ and $\ensuremath{\Delta}{\ensuremath{\Omega}}_{\mathrm{isco}}/{\ensuremath{\Omega}}_{\mathrm{isco}}=0.4870(\ifmmode\pm\else\textpm\fi{}0.0006)\ensuremath{\mu}/M$. We discuss the implications of our result within the context of the extreme-mass-ratio binary inspiral problem.