On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole

TL;DR

This paper employs Kovacic's algorithm to classify all Liouvillian solutions of the perturbation equations for Schwarzschild black holes, revealing their structure in terms of special functions and polynomial solutions, and extending previous results.

Contribution

It provides a complete classification of Liouvillian solutions for Schwarzschild perturbation equations, including explicit polynomial solutions and their expansions, extending prior work by Hautot and Chandrasekhar.

Findings

Liouvillian solutions contain polynomial solutions to confluent Heun equations.

Explicit polynomial solutions can be expanded in confluent hypergeometric functions and Laguerre polynomials.

The paper identifies only two Liouvillian solutions, confirming previous results with improved proofs.

Abstract

We use Kovacic's algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of the master equation which governs the evolution of first order perturbations of the Schwarzschild geometry. We show that all solutions in quadratures of this equation contain a polynomial solution to an associated ordinary differential equation (ODE). This ODE, apart from a few trivial cases, falls into the confluent Heun class. In the case of the gravitational perturbations, for the Liouvillian solution $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$, we find in "closed form" the polynomial solution P to the associated confluent Heun ODE. We prove that the Liouvillian solution $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$ is a product of elementary functions, one of them being the polynomial P. We extend previous results by Hautot and use the extended results we derive in order to prove that P admits a finite expansion in terms of truncated confluent hypergeometric functions of the first kind. We also prove, by using the extended results we derive, that P admits also a finite expansion in terms of associated Laguerre polynomials. We prove, save for two unresolved cases, that the Liouvillian solutions $\chi$ and $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$, initially found by Chandrasekhar, are the only Liouvillian solutions to the master equation. We improve previous results in the literature on this problem and compare our results with theirs. Comments are made for a more efficient implementation of Kovacic's algorithm to any second order ODE with rational function coefficients. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher.