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 Ė, ${\dot{L}}_z$, and $\dot{Q}$. Then, we compare our results for Ė and ${\dot{L}}_z$ 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.