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

For a self-gravitating particle of mass $\mu$ in orbit around a Kerr black hole of mass $M\gg \mu$, we compute the $\mathcal{O}(\mu /M)$ shift in the frequency of the innermost stable circular equatorial orbit due to the conservative piece of the gravitational self-force acting on the particle. Our treatment is based on a Hamiltonian formulation of the dynamics in terms of geodesic motion in a certain locally defined effective smooth spacetime. We recover the same result using the so-called first law of binary black-hole mechanics. We give numerical results for the innermost stable circular equatorial orbit frequency shift as a function of the black hole’s spin amplitude, and compare with predictions based on the post-Newtonian approximation and the effective one-body model. Our results provide an accurate strong-field benchmark for spin effects in the general-relativistic two-body problem.

Figure 1

Upper panel: Relative ISCO frequency shift $C_{\mathrm{\Omega }}$ [see Eq. (11)] as a function of the dimensionless black-hole spin, $q=S/M^2$. The red line interpolates the accurate numerical results from our method (i), while blue data points and error bars are from our method (ii). Lower panel: Relative difference $\delta C_{\mathrm{\Omega }}/C_{\mathrm{\Omega }}≔1-C_{\mathrm{\Omega }}^{\text{PN}/\mathrm{EOB}}/C_{\mathrm{\Omega }}$ between our “exact” GSF results and earlier predicted values for $C_{\mathrm{\Omega }}(q)$ from PN approximations [14, 16] and the EOB model of Ref. [54], the latter having been calibrated so as to reproduce the correct value for $C_{\mathrm{\Omega }}(0)$. The solid (dashed) lines indicate $\delta C_{\mathrm{\Omega }}>0$ ($\delta C_{\mathrm{\Omega }}<0$).