Computing radiation from Kerr black holes: Generalization of the Sasaki-Nakamura equation

As shown by Teukolsky, the master equation governing the propagation of weak radiation in a black hole spacetime can be separated into four ordinary differential equations, one for each spacetime coordinate. (“Weak” means the radiation’s amplitude is small enough that its own gravitation may be neglected.) Unfortunately, it is difficult to accurately compute solutions to the separated radial equation (the Teukolsky equation), particularly in a numerical implementation. The fundamental reason for this is that the Teukolsky equation’s potentials are long ranged. For nonspinning black holes, one can get around this difficulty by applying transformations which relate the Teukolsky solution to solutions of the Regge-Wheeler equation, which has a short-ranged potential. A particularly attractive generalization of this approach to spinning black holes for gravitational radiation (spin weight s=-2) was given by Sasaki and Nakamura. In this paper, I generalize the Sasaki-Nakamura results to encompass radiation fields of arbitrary integer spin weight, and give results directly applicable to scalar (s=0) and electromagnetic (s=-1) radiation. These results may be of interest for studies of astrophysical radiation processes near black holes, and of programs to compute radiation reaction forces in curved spacetime.