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Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes

Weihao Zheng

TL;DR

This work proves the existence of exponentially growing mode solutions for the Klein–Gordon equation on sub-extremal Reissner–Nordström–AdS black holes under reflecting boundary conditions, for a broad range of Klein–Gordon masses above the Breitenlohner–Freedman bound and including the conformal mass. The authors couple a variational construction of nontrivial stationary solutions with a robust perturbative argument (via the implicit function theorem) to produce growing modes, and they introduce a twisted-derivative framework to obtain coercivity and handle boundary issues. A novel near-extremal instability mechanism is uncovered for the uncharged case, while large scalar charge and near-extremality also yield growing modes in charged scenarios; these results distinguish RN–AdS from Schwarzschild–AdS/Kerr–AdS by showing linear instability in a spherically symmetric setting without relying on superradiance. The findings have implications for AdS/CFT holography and the existence of hairy black holes, indicating linear (and potentially nonlinear) instabilities for RN–AdS spacetimes under reflecting boundaries and enriching the landscape of AdS stability analyses.

Abstract

We construct growing mode solutions to the uncharged and charged Klein-Gordon equations on the sub-extremal Reissner-Nordström--anti-de-Sitter (AdS) spacetime under reflecting (Dirichlet or Neumann) boundary conditions. Our result applies to a range of Klein-Gordon masses above the so-called Breitenlohner-Freedman bound, notably including the conformal mass case. The mode instability of the Reissner-Nordström--AdS spacetime for some black hole parameters is in sharp contrast to the Schwarzschild-AdS spacetime, where the solution to the Klein-Gordon equation is known to decay in time. Contrary to other mode instability results on the Kerr and Kerr-AdS spacetimes, our growing mode solutions of the uncharged and weakly charged Klein-Gordon equation exist independently of the occurrence or absence of superradiance. We discover a novel mechanism leading to a growing mode solution, namely, a near-extremal instability for the Klein-Gordon equation. Our result seems to be the first rigorous mathematical realization of this instability.

Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes

TL;DR

This work proves the existence of exponentially growing mode solutions for the Klein–Gordon equation on sub-extremal Reissner–Nordström–AdS black holes under reflecting boundary conditions, for a broad range of Klein–Gordon masses above the Breitenlohner–Freedman bound and including the conformal mass. The authors couple a variational construction of nontrivial stationary solutions with a robust perturbative argument (via the implicit function theorem) to produce growing modes, and they introduce a twisted-derivative framework to obtain coercivity and handle boundary issues. A novel near-extremal instability mechanism is uncovered for the uncharged case, while large scalar charge and near-extremality also yield growing modes in charged scenarios; these results distinguish RN–AdS from Schwarzschild–AdS/Kerr–AdS by showing linear instability in a spherically symmetric setting without relying on superradiance. The findings have implications for AdS/CFT holography and the existence of hairy black holes, indicating linear (and potentially nonlinear) instabilities for RN–AdS spacetimes under reflecting boundaries and enriching the landscape of AdS stability analyses.

Abstract

We construct growing mode solutions to the uncharged and charged Klein-Gordon equations on the sub-extremal Reissner-Nordström--anti-de-Sitter (AdS) spacetime under reflecting (Dirichlet or Neumann) boundary conditions. Our result applies to a range of Klein-Gordon masses above the so-called Breitenlohner-Freedman bound, notably including the conformal mass case. The mode instability of the Reissner-Nordström--AdS spacetime for some black hole parameters is in sharp contrast to the Schwarzschild-AdS spacetime, where the solution to the Klein-Gordon equation is known to decay in time. Contrary to other mode instability results on the Kerr and Kerr-AdS spacetimes, our growing mode solutions of the uncharged and weakly charged Klein-Gordon equation exist independently of the occurrence or absence of superradiance. We discover a novel mechanism leading to a growing mode solution, namely, a near-extremal instability for the Klein-Gordon equation. Our result seems to be the first rigorous mathematical realization of this instability.
Paper Structure (43 sections, 21 theorems, 183 equations, 1 figure)

This paper contains 43 sections, 21 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The role of the black hole charge $e$ in the existence of growing mode solutions
  3. The role of the scalar field charge $q_{0}$ in the existence of growing mode solutions
  4. The Breitenlohner--Freedman bound and local well-posedness
  5. A toy model for the linearized Einstein--Maxwell equations
  6. Construction of hairy black holes
  7. Largely charged and weakly charged instability mechanism
  8. Outline of the rest of the introduction
  9. Previous results on stability/instability of the asymptotically AdS spacetime and decay of the field
  10. Local well-posedness results
  11. Results and conjectures on the stability of the pure AdS spacetime with dissipative boundary conditions
  12. Results and conjectures on the instability of the pure AdS spacetime with reflecting boundary conditions
  13. Linear (in)stability of AdS black hole
  14. Nonlinear (in)stability of the asymptotically AdS spacetime
  15. Comparing mechanisms for growing mode solutions to the Klein--Gordon equation on different spacetimes
  16. ...and 28 more sections

Key Result

Theorem 1.1

[Rough version of the main result] For the Klein--Gordon equation Klein-Gordon, let $C_{DN} = 0$ for Dirichlet boundary conditions and $C_{DN} = -\frac{5}{4}$ for Neumann boundary conditions respectively. Imposing reflecting boundary condition (Dirichlet or Neumann) for Klein-Gordon, we have the fol

Figures (1)

  • Figure 1: Penrose diagram for the asymptotically AdS spacetime.

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Conjecture 1.6: holzegel2020asymptotic
  • Conjecture 1.7: dafermos2006dynamicdafermos2006nonlinear
  • Conjecture 1.8: holzegel2013decay
  • Proposition 3.1
  • proof
  • ...and 45 more