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 $\hat{I}¼(\ll M)$, it experiences a gravitational self-force whose conservative piece alters the location of the ISCO. Here we calculate the resulting shifts $\hat{I}”r_{\mathrm{isco}}$ and $\hat{I}”\hat{I}{\text{c◯}}_{\mathrm{isco}}$ in the ISCO's radius and frequency, at leading order in the mass ratio $\hat{I}¼/M$. We obtain, in the Lorenz gauge, $\hat{I}”r_{\mathrm{isco}}=-3.269(\pm 0.003)\hat{I}¼G/c^2$ and $\hat{I}”\hat{I}{\text{c◯}}_{\mathrm{isco}}/\hat{I}{\text{c◯}}_{\mathrm{isco}}=0.4870(\pm 0.0006)\hat{I}¼/M$. We discuss the implications of our result within the context of the extreme-mass-ratio binary inspiral problem.

4 pages. v2: Added clarifications re. the definition of the conservative self-force and the gauge dependence of the frequency; some other minor changes. Accepted for publication in PRL