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Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes

Gustav Holzegel, Jacques Smulevici

TL;DR

This work proves that quasimodes exist for the Klein–Gordon equation on Kerr–AdS exteriors, derived from a semiclassical nonlinear eigenproblem with semiclassical parameter $h=1/\sqrt{\ell(\ell+1)}$, and with exponentially small residuals. The axisymmetric reduction reduces the problem to a one-dimensional radial operator with a well-structured potential, enabling Weyl-law counting for bound states and Agmon-type exponential decay in the classically forbidden region. By cutting off these bound states, the authors construct quasimodes whose time evolution is nearly exact for times up to $t\sim e^{C\ell}$, and they show a positive limsup for a suitably defined energy functional, implying that uniform decay cannot occur faster than a logarithmic rate (matching previous upper bounds). The results sharpen the understanding of slow energy decay in Kerr–AdS spacetimes and connect to broader questions about universal minimal decay rates in black-hole geometries with AdS asymptotics. The methodology combines separation of variables, semiclassical spectral theory, Agmon estimates, and Duhamel-type arguments to establish sharp, frequency-resolved lower bounds on decay rates.

Abstract

We construct quasimodes for the Klein-Gordon equation on the black hole exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are associated with time-periodic approximate solutions of the Klein Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semi-classical non-linear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semi-classical parameter. Our construction results in exponentially small errors in the semi-classical parameter. This implies that general solutions to the Klein Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes

TL;DR

This work proves that quasimodes exist for the Klein–Gordon equation on Kerr–AdS exteriors, derived from a semiclassical nonlinear eigenproblem with semiclassical parameter , and with exponentially small residuals. The axisymmetric reduction reduces the problem to a one-dimensional radial operator with a well-structured potential, enabling Weyl-law counting for bound states and Agmon-type exponential decay in the classically forbidden region. By cutting off these bound states, the authors construct quasimodes whose time evolution is nearly exact for times up to , and they show a positive limsup for a suitably defined energy functional, implying that uniform decay cannot occur faster than a logarithmic rate (matching previous upper bounds). The results sharpen the understanding of slow energy decay in Kerr–AdS spacetimes and connect to broader questions about universal minimal decay rates in black-hole geometries with AdS asymptotics. The methodology combines separation of variables, semiclassical spectral theory, Agmon estimates, and Duhamel-type arguments to establish sharp, frequency-resolved lower bounds on decay rates.

Abstract

We construct quasimodes for the Klein-Gordon equation on the black hole exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are associated with time-periodic approximate solutions of the Klein Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semi-classical non-linear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semi-classical parameter. Our construction results in exponentially small errors in the semi-classical parameter. This implies that general solutions to the Klein Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

Paper Structure

This paper contains 31 sections, 16 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Related works and discussion
  4. Non-linear analysis on asymptotically AdS spacetimes
  5. Quasi-normal modes of the asymptotically AdS black holes
  6. Universal minimal decay rates of waves outside stationary black holes
  7. Outline and overview of the proof
  8. Acknowledgments
  9. Preliminaries
  10. The Kerr-AdS family of spacetimes
  11. The fixed manifold with boundary $\mathcal{R}$
  12. The parameter space and the radial function
  13. More coordinates: $r^\star$, $t$, $\tilde{\phi}$
  14. The Kerr-AdS metric for fixed $(a,M,l)$
  15. The norms
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $(g,\mathcal{R})$ denote the black hole exterior of a Kerr-AdS spacetime with mass $M>0$, angular momentum per unit mass $a$ and cosmological constant $\Lambda=-\frac{3}{l^2}$. Assume that the parameters satisfy $\alpha < \frac{9}{4}$, $|a| < l$. Fix a spacelike slice $\Sigma_{t^\star_0}$ inters

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2: Quasimodes for Kerr-AdS
  • Corollary 1.1
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 28 more