Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion

TL;DR

This paper derives analytic solutions to the Regge-Wheeler equation using hypergeometric and Coulomb functions, and introduces a Post-Minkowskian expansion relevant for gravitational wave analysis around black holes.

Contribution

It provides new series solutions to the Regge-Wheeler equation and connects them with the Post-Minkowskian expansion, enhancing analytical and numerical approaches in black hole perturbation theory.

Findings

Series solutions expressed in hypergeometric and Coulomb functions.

Established relations between different solution regions.

Post-Minkowskian expansion applicable to gravitational radiation calculations.

Abstract

Analytic solutions of the Regge-Wheeler equation are presented in the form of series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations between these solutions are established. The series solutions are given as the Post-Minkowskian expansion with respect to a parameter $\epsilon \equiv 2M\omega$, $M$ being the mass of black hole. This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. These solutions can also be useful for numerical computations.