Electromagnetic quasinormal modes of Schwarzschild-anti-de Sitter black holes: bifurcations, spectral similarity, and exact solutions in the large black hole limit

Abstract

We revisit the peculiar electromagnetic quasinormal mode spectrum of an asymptotically anti-de Sitter Schwarzschild black hole. Recent numerical calculations have shown that this quasinormal mode spectrum becomes purely overdamped at some critical black hole sizes, where the spectrum also bifurcates. In this paper, we shed light on unnoticed and unexplained properties of this spectrum, by exploiting some novel analytic results for the large black hole limit. In particular, we demonstrate that the quasinormal mode spectrum is self-similar in this limit. We take advantage of this self-similarity to derive a precise analytic expression for the locations of the bifurcations, in which a surprising Feigenbaum-like constant appears. Also in this large black hole limit, we show that the differential operator is of Sturm-Liouville type and thus self-adjoint. We derive an exact solution for its spectrum and eigenfunctions, and find that large black holes cannot be made to vibrate with electromagnetic perturbations, independently of the boundary conditions imposed at spatial infinity. Finally, we characterize the insensitivity of the spectrum to different boundary conditions by analyzing the expansion of the quasinormal mode spectrum around the large black hole limit.