Quasinormal modes of nearly extreme Reissner-Nordström black holes.

We present detailed calculations of the quasinormal modes of ReissnerNordström black holes. While the first few, slowly damped, modes depend on the charge of the black hole in a relatively simple way, we find that the rapidly damped modes show several peculiar features. The higher modes generally spiral into the value for the extreme black hole as the charge increases. We also discuss the possible existence of a purely imaginary mode for the Schwarzschild black hole: Our data suggest that there is a quasinormal mode that limits to ωM = −2i as Q → 0. Typeset using REVTEX Preprint number: TIT/HEP-339/COSMO-76 and WUGRAV96-7 1 I. PERTURBING THE REISSNER-NORDSTRÖM BLACK HOLE Quasinormal modes of black holes have been studied ever since the seminal work of Vishveshwara [1] and Press [2] in the early 1970s. It soon became evident that exponentially damped mode-oscillations will dominate most processes involving perturbed black holes (see [3] for references). This means that the quasinormal modes provide a unique opportunity to identify a black hole, a possibility that hopefully will become reality when large-scale laserinterferometric detectors for gravitational waves come into operation in the near future [4]. In order to extract as much information as possible from a gravitational-wave signal it is important that we understand exactly how the quasinormal modes depend on the parameters of the black hole. The parameters of main astrophysical importance are the black holes mass and angular momentum. That is, the Kerr solution is the most relevant one from an astrophysical point of view. The solution that describes an electrically charged, nonrotating black hole — due to Reissner and Nordström — is of less direct importance because it seems unlikely that black holes with a considerable charge will exist in the Universe. Nevertheless, the Reissner-Nordström solution has several interesting features that warrant a closer inspection. The most intriguing one concerns the possible conversion of electromagnetic energy into gravitational energy and vice versa: In a charged environment an electromagnetic wave will inevitably give rise to gravitational waves. The Reissner-Nordström metric provides the simplest framework for studies of this effect. For this and several other reasons there has been a number of studies of perturbed Reissner-Nordström black holes. The equations governing a weak (massless) field in the geometry of an electrically charged black hole were first derived by Zerilli [5] and Moncrief [6]. Basically, these equations can be i) split into axial and polar perturbations (also known as odd and even parity, respectively) and then ii) reduced to two decoupled wave equations in each case. The final wave equations describe two variables Ψ1 and Ψ2, from which all components of the electromagnetic field and the perturbed metric can be reconstructed.