Exact solutions of the angular Teukolsky equation in particular cases

Abstract

In this work, we propose a new scheme to solve the angular Teukolsky equation for the particular case: $m=0, s=0$. We first transform this equation to a confluent Heun differential equation and then construct the Wronskian determinant to calculate the eigenvalues and normalized eigenfunctions. We find that the eigenvalues for larger $l$ are approximately given by $_{0}{A_{l0}} \approx [l(l + 1) - \tau_{R}^2/2] - i\;\tau_{I}^2/2$ with an arbitrary $\tau^2=\tau_R^2 + i\,\tau_{I}^2$. The angular probability distribution (APD) for the ground state moves towards the north and south poles for $\tau_R^2>0$, but aggregates to the equator for $\tau_R^2\leq0$. However, we also notice that the APD for large angular momentum $l$ always moves towards the north and south poles , regardless the choice of $\tau^2$.