On quasinormal modes of asymptotically anti-de Sitter black holes

TL;DR

This paper rigorously defines and proves properties of quasinormal modes for asymptotically anti-de Sitter black holes, showing they form a discrete set of eigenvalues without relying on separability or analyticity.

Contribution

It introduces a new eigenvalue-based framework for analyzing quasinormal modes, avoiding meromorphic extension and applicable to general linear fields.

Findings

Quasinormal frequencies form a discrete, countable set in the complex plane.

QNM are identified with eigenvalues of the solution generator on a Hilbert space.

The approach applies broadly to various linear fields on stationary black hole backgrounds.

Abstract

We consider the problem of quasinormal modes (QNM) for strongly hyperbolic systems on stationary, asymptotically anti-de Sitter black holes, with very general boundary conditions at infinity. We argue that for a time slicing regular at the horizon the QNM should be identified with certain H^k eigenvalues of the infinitesimal generator of the solution semigroup. Using this definition we are able to prove directly that the quasinormal frequencies form a discrete, countable subset of the complex plane, which in the globally stationary case accumulates only at infinity. We avoid any need for meromorphic extension, and the quasinormal modes are honest eigenfunctions of an operator on a Hilbert space. Our results apply to any of the linear fields usually considered (Klein-Gordon, Maxwell, Dirac etc.) on a stationary black hole background, and do not rely on any separability or analyticity properties of the metric. Our methods and results largely extend to the locally stationary case. We provide a counter-example to the conjecture that quasinormal modes are complete. We relate our approach directly to the approach via meromorphic continuation.