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

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 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 inspiraling 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 amethod of computing the averaged rate of change of the Carter constant in a relatively simple manner at the resonance point.