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

TL;DR

This work addresses analytic solutions of the Regge-Wheeler equation by formulating two convergent series, one in hypergeometric functions and one in Coulomb wave functions, both governed by a renormalized angular momentum $\nu$. The authors derive three-term recurrence relations, determine $\nu$ via continued fractions, and establish precise inter-relations between the two series, enabling a consistent description across convergence regions. They develop Post-Minkowskian expansions in $\epsilon=2M\omega$ up to $O(\epsilon^2)$, give explicit coefficients $a_n^{\nu}$, and compute asymptotic amplitudes $A_{out}^{\nu}$, $A_{in}^{\nu}$ and the absorption coefficient $\Gamma^{\nu}$, providing practical tools for gravitational-wave calculations and numerical checks. The results connect RW and Teukolsky solutions in Schwarzschild geometry and offer high-accuracy analytic templates for testing PN/PM predictions and aiding LIGO/VIRGO template construction.

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 $ε\equiv 2Mω$, $M$ 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.