Schwarzschild radial perturbations in Eddington-Finkelstein and Painlev\'e-Gullstrand coordinates

TL;DR

This paper extends previous work on Schwarzschild perturbations by constructing exact solutions for massive scalar fields in Eddington-Finkelstein coordinates and analyzing massless fields in Painlevé-Gullstrand coordinates, relevant for black hole physics.

Contribution

It introduces solutions for massive scalar fields in Eddington-Finkelstein coordinates and extends the analysis of massless fields to Painlevé-Gullstrand coordinates, enhancing understanding of horizon-regular perturbations.

Findings

Exact solutions for massive scalar fields in Eddington-Finkelstein coordinates.

Extension of massless field analysis to Painlevé-Gullstrand coordinates.

Regular solutions valid from the singularity to infinity.

Abstract

In a previous paper we have considered the Regge-Wheeler equation for fields of spin $s=0$, $1$ or $2$ on the Schwarzschild spacetime in coordinates that are regular at the horizon. In particular, we have constructed in Eddington-Finkelstein coordinates exact solutions in terms of series that are regular at the horizon and converge on the entire open domain from the central singularity to infinity. Here we extend this earlier work in two different directions. Firstly, we consider in Eddington-Finkelstein coordinates a massive scalar field that can serve as a dark matter candidate. Secondly, we extend the treatment of the massless case to Painlev\'e-Gullstrand coordinates, which are associated with radially infalling observers.