Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions

TL;DR

This paper derives analytic solutions to the Teukolsky equation in Kerr geometries using hypergeometric and Coulomb functions, providing a basis for gravitational wave modeling relevant to LIGO and VIRGO.

Contribution

It introduces series solutions of the Teukolsky equation in terms of hypergeometric and Coulomb functions, with low-frequency expansions applicable to gravitational wave analysis.

Findings

Solutions facilitate numerical computations and analysis of physical properties.

Expansion in small parameter $$ links to post-Newtonian approximations.

Potential to improve gravitational wave templates for LIGO and 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 $\epsilon \equiv 2M\omega$, $M$ 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.