# New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation II. Solutions of the Continued Fraction Equation - General Relativity and Quantum Cosmology

New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation II. Solutions of the Continued Fraction Equation - General Relativity and Quantum Cosmology - Download this document for free, or read online. Document in PDF available to download.

Abstract: We investigate the solution of the continued fraction equation by which wedetermine -the renormalized angular momentum parameter-, $ u$, in theformalism developed by Leaver and Mano, Suzuki, and Takasugi. In thisformalism, we describe the homogeneous solutions of the radial Teukolskyequation, which is the basic equation of the black hole perturbation formalism.We find that, contrary to the assumption made in previous works, the solution,$ u$, becomes complex valued as $\omega$ the angular frequency becomes largefor each $l$ and $m$ the degree and order of the spin-weighted spheroidalharmonics. We compare the power radiated by gravitational waves from aparticle in a circular orbit in the equatorial plane around a Kerr black holein two ways, one using the Mano-Suzuki-Takasugi formalism with complex $ u$and the other using a direct numerical integration method. We find that the twomethods produce consistent results. These facts prove the validity of usingcomplex solutions to determine the homogeneous solutions of the Teukolskyequation.

Author: ** Ryuichi Fujita, Hideyuki Tagoshi**

Source: https://arxiv.org/