Evolution of the Carter constant for a resonant inspiral 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 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.
