Exact Solutions of Teukolsky Master Equation with Continuous Spectrum

Abstract

Weak gravitational, electromagnetic, neutrino and scalar fields, considered as perturbations on Kerr background satisfy Teukolsky Master Equation. The two non-trivial equations obtained after separating the variables are the polar angle equation and the radial equation. We solve them by transforming each one into the form of a confluent Heun equation. The transformation depends on a set of parameters, which can be chosen in a such a way, so the resulting equations have simple polynomial solutions for neutrino, electromagnetic, and gravitational perturbations, provided some additional conditions are satisfied. Remarkably there exists a class of solutions for which these additional conditions are the same for the two different equations for $|s|=1/2$ and $|s|=1$. As a result the additional conditions fix the dependence of the separation constant on the angular frequency but the frequency itself remains unconstrained and belongs to a continuous spectrum.