Recipes for computing radiation from a Kerr black hole using a generalized Sasaki-Nakamura formalism: Homogeneous solutions

Central to black hole perturbation theory calculations is the Teukolsky equation that governs the propagation and the generation of radiation emitted by Kerr black holes. However, it is plagued by a long-ranged potential associated with the perturbation equation and hence a direct numerical integration of the equation is challenging. Sasaki and Nakamura devised a formulation that transforms the equation into a new equation that is free from the issue for the case of outgoing gravitational radiation. The formulation was later generalized by Hughes to work for any type of radiation. In this work, we revamp the generalized Sasaki-Nakamura (GSN) formalism and explicitly show the transformations that convert solutions between the Teukolsky and the GSN formalism for both in- and outgoing radiation of scalar, electromagnetic, and gravitational types. We derive all necessary ingredients for the GSN formalism to be used in numerical computations. In particular, we present a new numerical implementation of the formalism, GeneralizedSasakiNakamura.jl, that computes homogeneous solutions to both perturbation equations in the Teukolsky and the GSN formalism. The code works well at low frequencies and is even better at high frequencies by leveraging the fact that black holes are highly permeable to waves at high frequencies. This work lays the foundation for an efficient scheme to compute gravitational radiation from Kerr black holes and an alternative way to compute quasinormal modes of Kerr black holes.