Abstract

We discuss the inspiral of a small body in a background Kerr spacetime. When the time scale of the radiation reaction is sufficiently longer than its orbital period, the inspiral is in the adiabatic regime. In this case, the orbital evolution is rather accurately 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 the 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 resonant point, one cannot find a simultaneous turning point in general. Hence, even for the averaged rate of the change of the Carter constant, a direct computation of the “self-forces”, which is quite challenging especially in the case of the Kerr background, seems to be necessary.

In this talk, however, we show that it is possible to compute the averaged rate of the change of the Carter constant in a relatively simple manner even at a “resonance” point.