Radiative evolution of the Carter constant for generic orbits around a Kerr black hole

Recently, we proposed a new approach for the radiative evolution of generic orbits around a Kerr black hole. This involves the calculation of the evolution rate of all three constants of motion: The energy E, the azimuthal angular momentum L(z), and the Carter constant Q. In a previous paper we presented the general approach, and outlined its application to a scalar-field radiation reaction. In the present paper, we give the full details of the scalar-field calculations: We first derive the explicit expressions for E, L(z), and Q. Then, we compare our results for E and L(z) to those obtained from the standard method, and show that they are in full agreement. In addition, we outline the application of our method to the electromagnetic and gravitational radiation-reaction problem.