New perspectives on the irregular singular point of the wave equation for a massive scalar field in Schwarzschild spacetime

TL;DR

This paper analyzes the radial wave equation for a massive scalar field in Schwarzschild spacetime, focusing on the irregular singular point at infinity, and develops new solution techniques with applications to gravitational perturbations.

Contribution

It introduces a novel method for studying the irregular singular point of the wave equation and relates it to the Heun equation, providing detailed asymptotic expansions and integral representations.

Findings

Derived local solutions at singular points

Established connection to the Heun equation

Applied techniques to gravitational perturbations

Abstract

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background.