Dynamical system approach to the spectral (in)stability of black holes under localised potential perturbations

TL;DR

This work analyzes how black hole resonance spectra, including quasinormal modes and Regge poles, deform under localized delta-function perturbations. It introduces a dynamical-systems framework where resonances migrate along trajectories in the complex plane, steered by attracting and repelling spectral points, first demonstrated in the analytically tractable Nariai spacetime and then applied to Schwarzschild black holes. Key contributions include deriving a perturbation condition $F( heta, u)+\\epsilon=0$, linking linear shifts to QNM excitation factors, and distinguishing linear, nonlinear, and anomalous (Elephant and Flea) instability with explicit thresholds. The findings clarify the global nature of spectral instability, showing that local perturbations can cause smooth global deformations governed by an attractor–repeller structure, with implications for BH spectroscopy and potential applications in analogue systems and photonics.

Abstract

The aim of this work is to improve understanding of the resonant spectra of black holes under perturbations arising from e.g. compact objects or accretion disks in their vicinity. It is known that adding a weak perturbation to the radial potential can strongly disrupt the spectrum of quasinormal modes and Regge poles of a black hole spacetime. Here we examine the effect of (weak or strong) localised delta-function perturbations on the resonant spectra of spherically-symmetric systems, to address fundamental questions around linear and non-linear spectral stability. We examine two cases: the Nariai spacetime with a Poschl-Teller potential and the Schwarzschild spacetime. We show that, in either case, the spectrum deforms in a smooth and continuous manner as the position and strength of the perturbation is varied. As the strength of the perturbation is increased, resonances migrate along trajectories in the complex plane which ultimately tend towards attracting points determined by a hard-wall scenario. However, for weak perturbations the trajectory near the unperturbed resonance is typically strongly influenced by a set of repelling points which, for perturbations far from the system, lie very close to the unperturbed resonances; hence there arises a non-linear instability (i.e. the failure of a linearised approximation). Taking a dynamical systems perspective, the sets of attracting and repelling spectral points follow their own trajectories as the position of the perturbation is varied, and these are tracked and understood.

Figures (9)

• Figure 1: The quasinormal mode spectrum for a Pöschl-Teller potential (\ref{['eq:nariai-radial']}) with a delta-perturbation at $\hat{x}_0 = 10 / \sqrt{27}$ of strength $\epsilon$ and angular momentum $\lambda = \ell + \tfrac{1}{2} = \tfrac{5}{2}$. The resonances migrate along trajectories shown as solid purple lines, ultimately towards the attracting points marked as open black circles. The unperturbed spectrum is shown with filled black circles and the perturbed spectrum for $\epsilon \in \{10^{-8}, 10^{-6}, 10^{-4}, 10^{-2} \}$ is shown with filled points (square, triangle, diamond and stars). The spectrum is symmetric about the imaginary axis. Trajectories that meet at $\text{Im} \ \omega = 0$ (from either side) then migrate along the imaginary axis to the attractors that lie (in alternating fashion) just above and below the points $\omega = -in$. The repelling points are marked with black crosses, and the integral curves of Eq. (\ref{['eq:ode']}) that move away from (towards) these points as $\epsilon$ increases are shown as dotted blue (red) lines.
• Figure 2: Example of the migration of the fundamental ($n=0$) quasinormal mode frequency in the perturbed Poschl-Teller potential for $\lambda_0 = 5/2$, $\hat{x}_0 = 10 / \sqrt{27}$. The QNF migrates from the unperturbed frequency $\lambda_0 - i /2$ (filled black circle) towards the frequency at which $u^{-}_{\lambda \omega}(x_0)$ is zero (open black circle) along the trajectory [blue solid]. The dotted lines show the linear and quadratic approximations. The points show the fundamental QNM frequency for perturbation strengths $\epsilon \in \{0.1,\ 1,\ 10,\ 50\}$.
• Figure 3: Examples of the switching of trajectories as the position of the perturbation $x$ is increased. The repelling point is marked with a black cross. The blue and red lines are the trajectories of the $n=0$ and $n=1$ QNFs for $\epsilon \in [0,30]$, with a perturbation at $x_0 = 12.1$ (solid) and $x_0=12.2$ (dashed).
• Figure 4: The attracting and repelling points of the QNM spectrum. The attracting (repelling) points migrate along dashed (solid) lines, as the position $x_0$ of the perturbation is increased from $x_0 = 0.4 / \sqrt{27}$ to $x_0 = 90 / \sqrt{27}$ (with $\lambda = 5/2$). The open blue circles show the positions of the unperturbed QNFs.
• Figure 5: The plot shows the Regge-pole spectrum for a Poschl-Teller potential (\ref{['eq:nariai-radial']}) with a perturbation at $x_0 = 10 / \sqrt{27}$ of strength $\epsilon$ and frequency $\omega = (3/2) \sqrt{27}$. As $\epsilon$ increases, most RPs move to the left (i.e. smaller real part), but certain poles ($n=4$, $9$ and $16$) migrate to the right, creating two branches. The attracting (open black circles) and repelling (black crosses) points defined in Sec. \ref{['subsec:flows']} are marked. The leftward-moving (rightward-moving) poles migrate towards the attractors with large (small) imaginary part. The blue (red) dotted lines show integral curves of Eq. (\ref{['eq:ode']}) that move away from the repellers as $\epsilon$ increases (decreases). As shown, the RP trajectories are attracted towards the blue dotted trajectories.