Dirty black holes: Quasinormal modes

TL;DR

The paper analyzes the asymptotic spectrum of quasinormal modes for static, spherically symmetric (and hence potentially 'dirty') black holes. It uses the first Born approximation to locate poles of the scattering amplitude, linking those poles to the quasinormal-mode frequencies and showing that the leading imaginary spacing is $\kappa$, where $\kappa$ is the horizon surface gravity. This result holds for Schwarzschild and generic single-horizon geometries and extends to dual-horizon spacetimes, demonstrating a universal, geometry-driven mechanism for the mode spacing independent of surrounding matter. The offset term remains outside the reach of this approximation, and extremal/degenerate-horizon cases reveal limitations of the method, highlighting that the full spectrum requires beyond-Born analysis.

Abstract

In this paper, we investigate the asymptotic nature of the quasinormal modes for "dirty" black holes -- generic static and spherically symmetric spacetimes for which a central black hole is surrounded by arbitrary "matter" fields. We demonstrate that, to the leading asymptotic order, the [imaginary] spacing between modes is precisely equal to the surface gravity, independent of the specifics of the black hole system. Our analytical method is based on locating the complex poles in the first Born approximation for the scattering amplitude. We first verify that our formalism agrees, asymptotically, with previous studies on the Schwarzschild black hole. The analysis is then generalized to more exotic black hole geometries. We also extend considerations to spacetimes with two horizons and briefly discuss the degenerate-horizon scenario.