Analytic Solutions of the Teukolsky Equation and their Properties

TL;DR

This paper presents analytical solutions to the Teukolsky equation using hypergeometric and Coulomb wave functions, explores their properties, and applies them to black hole absorption and evaporation rates.

Contribution

It provides a detailed analysis of the properties of Teukolsky solutions, including normalization and identities, enhancing computational accuracy and understanding.

Findings

Derived analytical normalization relations using T-S identities.

Discovered nontrivial identities between series coefficients.

Obtained analytic results for black hole absorption and evaporation rates.

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.