Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions

TL;DR

The paper addresses analytic solutions of the Teukolsky equation in Kerr spacetimes and their low-frequency behavior by constructing two complementary series: a hypergeometric-series near the horizon and a Coulomb-wave series away from it. It introduces a renormalized angular momentum $ν$ and derives three-term recurrence relations for the series coefficients, establishing their convergence and a precise matching between the two representations via a constant $K_{ν}$. A detailed low-frequency expansion in $ε=2Mω$ yields $ν = l + O(ε^2)$ with explicit coefficients, and provides analytic forms for the ingoing/outgoing amplitudes $A_{out}^{sν}$ and $A_{in}^{sν}$ up to $O(ε)$ (and $O(ε^2)$ in expanded form). The results enable controlled Post-Minkowskian/post-Newtonian analyses and practical numerical computation for gravitational-wave templates, enhancing template construction for LIGO/VIRGO.

Abstract

Analytic solutions of the Teukolsky equation in Kerr geometries are presented in the form of series of hypergeometric functions and Coulomb wave functions. Relations between these solutions are established. The solutions provide a very powerful method not only for examining the general properties of solutions and physical quantities when they are applied to, but also for numerical computations. The solutions are given in the expansion of a small parameter $ε\equiv 2Mω$, $M$ being the mass of black hole, which corresponds to Post-Minkowski expansion by $G$ and to post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. It is expected that these solutions will become a powerful weapon to construct the theoretical template towards LIGO and VIRGO projects.