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

Abstract

It is well known that the equations governing the evolution of scalar, electromagnetic, and gravitational perturbations of the background geometry of a Schwarzschild black hole can be reduced to a single master equation. We use Kovacic’s algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of this master equation. We prove that the algebraically special Liouvillian solutions χ and χ∫dr *χ2, initially found by Chandrasekhar in the gravitational case, are the only Liouvillian solutions to the master equation. We show that the Liouvillian solution χ∫dr *χ2 is a product of elementary functions, one of them being a polynomial solution P to an associated confluent Heun equation. P admits a finite expansion both in terms of truncated confluent hypergeometric functions of the first kind, and also in terms of associated Laguerre polynomials. Remarkably both expansions entail not constant coefficients but appropriate function coefficients instead. We highlight the relation of these results with inspiring new developments. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher.