Breit-Wigner resonances and the quasinormal modes of anti-de Sitter black holes

TL;DR

This work tackles efficient computation of quasinormal modes for Schwarzschild-anti-de Sitter black holes using Breit-Wigner resonances. The authors show that the resonance method yields accurate quasi-bound states and damping rates, especially for small BHs, with $\omega_I L \propto (r_+/L)^{2l+2}$ and real parts approaching AdS values $\omega_R L \to l+3+2n$ as $r_+\to0$, in agreement with analytic predictions. In the eikonal limit, they confirm the existence of very long-lived modes and provide a WKB description that matches prior predictions by Festuccia and Liu. The results have implications for AdS/CFT, as these long-lived modes dictate the perturbation decay and thermalization timescales in the dual field theory, and establish the resonance method as a powerful tool for spectral studies in AdS spacetimes.

Abstract

The purpose of this short communication is to show that the theory of Breit-Wigner resonances can be used as an efficient numerical tool to compute black hole quasinormal modes. For illustration we focus on the Schwarzschild anti-de Sitter (SAdS) spacetime. The resonance method is better suited to small SAdS black holes than the traditional series expansion method, allowing us to confirm that the damping timescale of small SAdS black holes for scalar and gravitational fields is proportional to r_+^(-2l-2), where r_+ is the horizon radius. The proportionality coefficients are in good agreement with analytic calculations. We also examine the eikonal limit of SAdS quasinormal modes, confirming quantitatively Festuccia and Liu's prediction of the existence of very long-lived modes in asymptotically AdS spacetimes. Our results are particularly relevant for the AdS/CFT correspondence, since long-lived modes presumably dominate the decay timescale of the perturbations.

Figures (3)

• Figure 1: Potential for scalar field ($s=0$) perturbations of a SAdS background with $l=0$. Different lines refer to different values of $r_+/L$. A potential well develops for small BHs ($r_+/L<1$).
• Figure 2: A plot of $\alpha^2+\beta^2$ for scalar field SAdS perturbations with $l=0$, $r_+/L=10^{-2}$. Resonances are seen when $\omega_R\simeq 3+2n$, i.e. close to the resonant frequencies of the pure AdS spacetime. In the inset we show the behavior near the minimum, which allows us to extract the decay time by a parabolic fit.
• Figure 3: Track traced in the complex plane ($\omega_R L,\omega_I L$) by the fundamental $l=0,1$ scalar field QNM frequencies as we vary the BH size $r_+/L$. Counterclockwise along these tracks we mark by circles and diamonds the frequencies corresponding to decreasing decades in $r_+/L$ ($r_+/L=10^2,10^1,10^0,10^{-1},...$).