‘Quick and dirty’ methods for studying black-hole resonances

Abstract

We discuss simple integration methods for the calculation of black-hole scattering resonances both in the complex frequency plane (quasinormal modes) and the complex angular momentum plane (Regge poles). Our numerical schemes are based on variations of ‘phase-amplitude’ methods. In particular, we discuss the Prüfer transformation, where the original (frequency domain) Teukolsky wave equation is replaced by a pair of first-order nonlinear equations governing the introduced phase functions. Numerical integration of these equations, performed along the real r∗-axis (where r∗ denotes the usual tortoise radial coordinate), or along rotated contours in the complex r∗-plane, provides the required \U0001d4ae-matrix element (the ratio of amplitudes of the outgoing and ingoing waves at infinity). Müller's algorithm is then employed to conduct searches in the complex plane for the poles of this quantity (which are, by definition, the desired resonances). We have tested this method by verifying known results for Schwarzschild quasinormal modes and Regge poles, and provide new results for the Kerr black-hole problem. Results produced by our scheme prove to be accurate as long as the imaginary part of the resonance is not much larger than the real part. We also describe a new method for estimating the ‘excitation coefficients’ for quasinormal modes. The method is applied to scalar waves moving in the Kerr geometry, and the obtained results shed light on the long-lived quasinormal modes that exist for black holes rotating near the extreme Kerr limit.