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 ${\tau }_R^2\le 0$. However, we also notice that the APD for large angular momentum $l$ always moves towards the north and south poles , regardless the choice of ${\tau }^2$.

6 pages, 2 tables and 5 figures