Dynamical instabilities and quasi-normal modes, a spectral analysis with applications to black-hole physics

TL;DR

This paper analyzes the spectral properties of complex-frequency modes responsible for black hole dynamical instabilities, revealing their mathematical structure, how they emerge in spectra, and their evolution under parameter changes.

Contribution

It provides a general spectral analysis of unstable modes in black hole physics, including their square integrability, norm properties, and how they arise from quasi-normal modes or mode fusion.

Findings

Modes are square integrable with vanishing conserved norm.

Unstable modes appear from quasi-normal modes or mode fusion.

The analysis is illustrated with a generalized Friedrichs model.

Abstract

Black hole dynamical instabilities have been mostly studied in specific models. We here study the general properties of the complex-frequency modes responsible for such instabilities, guided by the example of a charged scalar field in an electrostatic potential. We show that these modes are square integrable, have a vanishing conserved norm, and appear in mode doublets or quartets. We also study how they appear in the spectrum and how their complex frequencies subsequently evolve when varying some external parameter. When working on an infinite domain, they appear from the reservoir of quasi-normal modes obeying outgoing boundary conditions. This is illustrated by generalizing, in a non-positive definite Krein space, a solvable model (Friedrichs model) which originally describes the appearance of a resonance when coupling an isolated system to a mode continuum. In a finite spatial domain instead, they arise from the fusion of two real frequency modes with opposite norms, through a process that closely resembles avoided crossing.