Analytic Solutions of the Teukolsky Equation and their Properties

TL;DR

This work advances analytic solutions of the Teukolsky equation for Kerr black holes by formulating radial functions as both hypergeometric and Coulomb series, then matching them to obtain globally convergent solutions for finite $\epsilon$. It fixes the relative normalization between spin weights $s$ and $-s$ via Teukolsky-Starobinsky identities and the Wronskian, and derives nontrivial identities for sums of series coefficients that yield compact, testable expressions for asymptotic amplitudes. The authors extract compact relations among horizon and infinity amplitudes, enabling robust checks and efficient computation of Teukolsky functions. As a key application, they obtain analytic forms for the black hole absorption and evaporation rates, reproducing Page-type results in the small-$\epsilon$ limit and providing a practical framework for gravitational-wave and Hawking-evaporation studies in Kerr spacetimes.

Abstract

The analytical solutions reported in our previous paper are given as series of hypergeometric or Coulomb wave functions. By using them, we can get the Teukolsky functions analytically in a desired accuracy. For the computation, the deep understanding of their properties is necessary. We summarize the main result: The relative normalization between the solutions with a spin weight s and -s is given analytically by using the Teukolsky-Starobinsky (T-S) identities. By examining the asymptotic behaviors of our solution and combined with the T-S identities and the Wronskian, we found nontrivial identities between the sums of coefficients of the series. These identities will serve to make various expression in simpler forms and also become a powerful tool to test the accuracy of the computation. As an application, we investigated the absorption rate and the evaporation rate of black hole and obtain interesting analytic results.