Revamping the Generalized Sasaki-Nakamura formalism for efficient computations of radiation from black holes

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-range potential and a divergent source term (at infinity for fields with spin weight s = -2 and at the horizon for fields with s = +2) associated with the equation. Sasaki and Nakamura proposed a formulation that is free from the issues above for s = -2, relevant for extracting gravitational radiation at infinity. The formulation was later generalized by Hughes for any integer s. In this work, we revisit the Generalized Sasaki-Nakamura (GSN) formalism and derive expressions for the higher-order corrections to the asymptotic boundary conditions of the GSN equation. In addition, we derive the frequency-dependent factors for converting the limiting behaviors of a solution to the GSN equation at both infinity and the horizon to that of a solution to the Teukolsky equation, which are both essential ingredients in using the GSN formalism for numerical analyses. We also extend the construction of the source term for the GSN equation to work for both s = ±2, which allows for numerically stable computations of gravitational radiation at both infinity and the horizon for Kerr black holes.