Analytic solution of the Regge-Wheeler differential equation for black hole pertubations in radial coordinate and time domains

TL;DR

This paper presents an analytic series solution to the Regge-Wheeler equation in the time and radial domains, enhancing understanding of black hole perturbations and aiding in gravitational wave modeling.

Contribution

It provides the first analytic Frobenius series solution of the Regge-Wheeler equation at the horizon's singularity, moving beyond numerical and approximate methods.

Findings

Analytic series solution at the horizon's singularity.

Improved understanding of black hole perturbations.

Foundation for semi-analytic solutions with source terms.

Abstract

An analytic solution of the Regge-Wheeler (RW) equation has been found via the Frobenius method at the regular singularity of the horizon 2M, in the form of a time and radial coordinate dependent series. The RW partial differential equation, derived from the Einstein field equations, represents the first order perturbations of the Schwarzschild metric. The known solutions are numerical in time domain or approximate and asymptotic for low or high frequencies in Fourier domain. The former is of scarce relevance for comprehension of the geodesic equations for a body in the black hole field, while the latter is mainly useful for the description of the emitted gravitational radiation. Instead a time domain solution is essential for the determination of radiation reaction of the falling particle into the black hole, i.e. the influence of the emitted radiation on the motion of the perturbing mass in the black hole field. To this end, a semi-analytic solution of the inhomogeneous RW equation with the source term (Regge-Wheeler-Zerilli equation) shall be the next development.