Quasinormal modes of black holes. II. Pad\'e summation of the higher-order WKB terms

TL;DR

This paper enhances the WKB method for calculating black hole quasinormal modes by using high-order Padé summation, achieving extremely accurate results that match numerical methods to many decimal places.

Contribution

It extends previous work by demonstrating that high-order Padé approximants from very high-order series can reproduce known numerical results with exceptional accuracy.

Findings

Padé approximants from series up to order 700 yield highly accurate quasinormal mode frequencies.

The method achieves agreement with numerical results to 24 decimal places for certain modes.

High-order series are necessary for the stabilization of quasinormal frequency calculations.

Abstract

In previous work [1] we proposed an improvement of the WKB-based semianalytic technique of Iyer and Will for calculation of the quasiormal modes of black holes by constructing the Pad\'e approximants of the formal series for $\omega^{2}.$ It has been demonstrated that (within the domain of applicability) the diagonal Pad\'e transforms $\mathcal{P}_{6}^{6}$ and $\mathcal{P}_{7}^{6}$ are always in a very good agreement with the numerical results. In this paper we present a further extension of the method. We show that it is possible to reproduce many known numerical results with a great accuracy (or even exactly) if the Pad\'e transforms are constructed from the perturbative series of a really high order. In our calculations the order depends on the problem but it never exceeds 700. For example, the frequencies of the gravitational mode $l=2,$ $n=0$ calculated with the aid of the Pad\'e approximants and within the framework of the continued fractions method agree to 24 decimal places. The use of such a large number of terms is necessary as the stabilization of the quasinormal frequencies can be slow. Our results reveal some unexpected features of the WKB-based approximations and may shed some fresh light on the problem of overtones.