Analytic Solution of Regge-Wheeler Differential Equation for Black Hole Perturbations in Radial Coordinate and Time Domains

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 the time domain, and approximate and asymptotic for low or high frequencies in the Fourier domain. The former is of little 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 emitted gravitational radiation. Instead a time domain solution is essential for the determination of the radiation reaction of a particle falling into a black hole, i.e., the influence of 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 sourceterm (Regge-Wheeler-Zerilli equation) shall be attempted where a numerical contribution shall be present. Discussion on the features of the series shall be shown in further work.