Self-force on a scalar charge in Kerr spacetime: Circular equatorial orbits

Abstract

We present a calculation of the scalar field self-force (SSF) acting on a scalar-charge particle in a strong-field orbit around a Kerr black hole. Our calculation specializes to circular and equatorial geodesic orbits. The analysis is an implementation of the standard mode-sum regularization scheme: We first calculate the multipole modes of the scalar-field perturbation using numerical integration in the frequency domain, and then apply a certain regularization procedure to each of the modes. The dissipative piece of the SSF is found to be consistent with the flux of energy and angular momentum carried by the scalar waves through the event horizon and out to infinity. The conservative (radial) component of the SSF is found to be attractive (inward pointing) for $r_0>r_{\rm c}(a)$ and repulsive (outward pointing) for $r_0<r_{\rm c}(a)$, where $a$ is the Kerr spin parameter, $r_0$ is the Boyer-Lindquist orbital radius, and $r_{\rm c}$ is a critical $a$-dependent radius at which the conservative SSF vanishes. When the motion is retrograde the conservative SSF is repulsive for all $r_0$ (as in the Schwarzschild case). The dominant conservative effect of the SSF in Schwarzschild spacetime is known to be of 3rd post-Newtonian (PN) order (with a logarithmic running). Our numerical results suggest that the leading-order PN correction due to the black hole's spin arises from spin-orbit coupling at 3PN, which dominates the overall SSF effect at large $r_0$. In PN language, the change-of-sign of the radial SSF is attributed to an interplay between the spin-orbit term ($\propto -ar_0^{-4.5}$) and the "Schwarzschild" term ($\propto r_0^{-5}\log r_0$).