New Numerical Methods to Evaluate Homogeneous Solutions of the Teukolsky Equation

TL;DR

This paper introduces a numerical method based on the Mano, Suzuki, and Takasugi formalism for accurately computing homogeneous solutions of the Teukolsky equation, crucial for black hole perturbation analysis, and demonstrates its effectiveness in gravitational wave flux calculations.

Contribution

The paper extends the application of the Mano, Suzuki, and Takasugi formalism from analytical to numerical evaluation of Teukolsky solutions, achieving high precision in gravitational wave flux computations.

Findings

Numerical method achieves energy flux accuracy of about 10^{-14}.

Method is effective for computing homogeneous solutions of the Teukolsky equation.

The renormalized angular momentum parameter $ u$ is limited to real values in certain frequency regions.

Abstract

We discuss a numerical method to compute the homogeneous solutions of the Teukolsky equation which is the basic equation of the black hole perturbation method. We use the formalism developed by Mano, Suzuki and Takasugi, in which the homogeneous solutions of the radial Teukolsky equation are expressed in terms of two kinds of series of special functions, and the formulas for the asymptotic amplitudes are derived explicitly.Although the application of this method was previously limited to the analytical evaluation of the homogeneous solutions, we find that it is also useful for numerical computation. We also find that so-called "renormalized angular momentum parameter", $\nu$, can be found only in the limited region of $\omega$ for each $l,m$ if we assume $\nu$ is real (here, $\omega$ is the angular frequency, and $l$ and $m$ are degree and order of the spin-weighted spheroidal harmonics respectively). We also compute the flux of the gravitational waves induced by a compact star in a circular orbit on the equatorial plane around a rotating black hole. We find that the relative error of the energy flux is about $10^{-14}$ which is much smaller than the one obtained by usual numerical integration methods.