Evolution of the Carter constant for a resonant inspiral into a Kerr black hole: I. The scalar case

TL;DR

This paper introduces a method to compute the averaged rate of change of the Carter constant during a resonant inspiral into a Kerr black hole, addressing challenges posed by orbital resonances.

Contribution

It provides a novel approach for calculating the Carter constant's evolution at resonance points, where traditional methods face difficulties.

Findings

Derived a simple formula for the Carter constant's rate of change at resonance

Addressed the challenge of orbital resonances in Kerr black hole inspirals

Proposed a practical computational method for the scalar field case

Abstract

We discuss the inspiral of a small body around a Kerr black hole. When the time scale of the radiation reaction is sufficiently longer than its orbital period, the leading order orbital evolution is described only by the knowledge of the averaged evolution of the constants of motion, i.e., the energy, azimuthal angular momentum and the Carter constant. Although there is no conserved current composed of the perturbation field corresponding to the Carter constant, it has been shown that the averaged rate of change of the Carter constant can be given by a simple formula, when there exists a simultaneous turning point of the radial and polar oscillations. However, an inspiralling orbit may cross a "resonance" point, where the frequencies of the radial and polar orbital oscillations are in a rational ratio. At the resonance point, one cannot find a simultaneous turning point in general. Hence, even for the averaged rate of change of the Carter constant, a direct computation of the 'self-force", which is still challenging especially in the case of the Kerr background, seems to be necessary. In this paper, we present a method of computing the averaged rate of change of the Carter constant in a relatively simple manner at the "resonance" point.