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Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis

Oran Gannot

TL;DR

The paper analyzes quasinormal modes for massive scalar fields in Schwarzschild–AdS spacetimes by casting the radial equation as a semiclassical one-dimensional Schrödinger problem with an exponential barrier. It constructs exponentially accurate quasimodes and uses black box scattering techniques to infer the existence of resonances whose imaginary parts decay like e^{-Cℓ}; it also derives a detailed asymptotic expansion for the real parts of low-lying modes, with dimension-dependent vanishing of certain coefficients. The results provide rigorous insight into the long-lived nature of high-angular-momentum modes and connect to prior conjectures on coefficient vanishing, supported by numerical validation. Overall, the work bridges geometric black hole backgrounds, semiclassical analysis, and resonance theory to quantify the spectrum of perturbations in AdS spacetimes. The findings have implications for stability analyses and the late-time behavior of massive scalar fields in AdS geometries.

Abstract

We study quasinormal modes for massive scalar fields in Schwarzschild-anti-de Sitter black holes. When the mass-squared is above the Breitenlohner-Freedman bound we show that for large angular momenta, $\ell$, there exist quasinormal modes with imaginary parts of size $\exp(-\ell/C)$. We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.

Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis

TL;DR

The paper analyzes quasinormal modes for massive scalar fields in Schwarzschild–AdS spacetimes by casting the radial equation as a semiclassical one-dimensional Schrödinger problem with an exponential barrier. It constructs exponentially accurate quasimodes and uses black box scattering techniques to infer the existence of resonances whose imaginary parts decay like e^{-Cℓ}; it also derives a detailed asymptotic expansion for the real parts of low-lying modes, with dimension-dependent vanishing of certain coefficients. The results provide rigorous insight into the long-lived nature of high-angular-momentum modes and connect to prior conjectures on coefficient vanishing, supported by numerical validation. Overall, the work bridges geometric black hole backgrounds, semiclassical analysis, and resonance theory to quantify the spectrum of perturbations in AdS spacetimes. The findings have implications for stability analyses and the late-time behavior of massive scalar fields in AdS geometries.

Abstract

We study quasinormal modes for massive scalar fields in Schwarzschild-anti-de Sitter black holes. When the mass-squared is above the Breitenlohner-Freedman bound we show that for large angular momenta, , there exist quasinormal modes with imaginary parts of size . We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.

Paper Structure

This paper contains 15 sections, 31 theorems, 126 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Black holes in Anti-de Sitter spacetime
  3. Klein--Gordon equation
  4. Reduction to the Schrödinger equation
  5. Analysis of the effective potential
  6. Quasimodes
  7. Self-adjoint realizations
  8. Agmon estimates
  9. Asymptotic expansion for low lying quasimodes
  10. Construction of quasimodes
  11. Existence of resonances
  12. Black box model
  13. Schwarzschild--AdS problem in the black box framework
  14. Restoring the original parameters
  15. Numerical results

Key Result

Lemma 2.1

The inverse $z\mapsto r(z)$ satisfies $r(z) = \dfrac{1}{z} - \dfrac{z}{3} + O(z^2) \textrm{ as } z\rightarrow 0$ and $r(z) = r_+ + O(e^{-\gamma z}) \textrm{ as } z\rightarrow \infty$ for some $\gamma>0$. Both of these asymptotics are differentiable.

Figures (3)

  • Figure 1: Plots of $z$ versus $V(z;h)$ for different values of $d,\mu,\nu,h$. Left: $d=6,\, \mu =1/8,\, \nu = \sqrt{3/28}$. Right: $d=4,\,\mu=1/2,\,\nu = 1/2$. See Figure \ref{['f:2']} for a plot when $\nu > 1/2$.
  • Figure 2: A schematic plot of $V_0$ and $V$ for $\nu > 1/2$ illustrating the maximum and the left-most turning points.
  • Figure 3: A comparison between the asymptotic expansion for $E^\sharp_n(h)$ provided by Proposition \ref{['thm:expansion']} and $E^\sharp_n(h)$ as computed by SLEIGN2. Here the black hole parameters are $d=3, \mu = 1/10, \nu = 3/2$. Top: log-log plot of $h^{-1}$ against the difference between the SLEIGN2 value and the first two terms in the asymptotic expansion. Bottom: log-log plot of $h^{-1}$ against the difference between the SLEIGN2 value and the first three terms in the asymptotic expansion.

Theorems & Definitions (61)

  • Lemma 2.1
  • proof
  • Remark 1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 51 more