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

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.