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Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

Gustav H. Holzegel, Claude M. Warnick

TL;DR

This work presents a spectral stability framework for the linear Klein–Gordon equation with mass parameter $\alpha<\frac{9}{4}$ on general stationary, asymptotically anti-de Sitter black holes under Dirichlet, Neumann, or Robin boundary conditions. Central to the approach is a degenerate elliptic operator $L$ on a Cauchy slice, whose lowest eigenvalue $\omega_1$ determines boundedness or growth of solutions after twisting the energy via an appropriately chosen function $f$ (the twisted energy). The authors develop both untwisted and twisted energy methods, construct a renormalized twisted energy momentum tensor, and prove a Rellich–Kondrachov type compactness for twisted Sobolev spaces to establish a discrete spectrum for $L$. They then apply the formalism to AdS–Schwarzschild and AdS–Kerr spacetimes, showing Dirichlet BCs yield $\omega_1>0$ (stability) under the Hawking–Reall bound, while Robin data can lead to linear hair or instability at critical parameters. This provides a practical, boundary-condition–dependent criterion for linear stability and clarifies the threshold between stability and scalar hair in rotating AdS black holes.

Abstract

We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet-, Neumann- or Robin- boundary conditions imposed on $ψ$ at infinity. This class includes the regular Kerr-AdS black holes satisfying the Hawking Reall bound $r_+^2 > |a|l$. We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of $ψ$ and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr-AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions.

Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

TL;DR

This work presents a spectral stability framework for the linear Klein–Gordon equation with mass parameter on general stationary, asymptotically anti-de Sitter black holes under Dirichlet, Neumann, or Robin boundary conditions. Central to the approach is a degenerate elliptic operator on a Cauchy slice, whose lowest eigenvalue determines boundedness or growth of solutions after twisting the energy via an appropriately chosen function (the twisted energy). The authors develop both untwisted and twisted energy methods, construct a renormalized twisted energy momentum tensor, and prove a Rellich–Kondrachov type compactness for twisted Sobolev spaces to establish a discrete spectrum for . They then apply the formalism to AdS–Schwarzschild and AdS–Kerr spacetimes, showing Dirichlet BCs yield (stability) under the Hawking–Reall bound, while Robin data can lead to linear hair or instability at critical parameters. This provides a practical, boundary-condition–dependent criterion for linear stability and clarifies the threshold between stability and scalar hair in rotating AdS black holes.

Abstract

We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet-, Neumann- or Robin- boundary conditions imposed on at infinity. This class includes the regular Kerr-AdS black holes satisfying the Hawking Reall bound . We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr-AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions.

Paper Structure

This paper contains 17 sections, 28 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The AdS-Schwarzschild Black Hole
  3. The untwisted energies
  4. The fluxes
  5. The twisted energies
  6. The case of negative Robin function
  7. The twisted energy momentum tensor
  8. Boundedness for general stationary black holes
  9. Decomposing the metric
  10. Decomposing the wave operator
  11. The structure of $L$ at the horizon and infinity
  12. Boundedness of solutions to the Klein-Gordon equation
  13. The AdS-Kerr Black Hole
  14. The Dirichlet case
  15. Proof of Theorem \ref{['RKthm']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

If the initial conditions satisfy stronger regularity and asymptotic conditionsthese conditions are of the form of those appearing in Theorem schbdn, then in fact $\psi|_\mathcal{S} \in H^k_{\textrm{loc.}}(\mathcal{S})$, $\hat{n}_\mathcal{S}\psi|_\mathcal{S} \in H^{k-1}_{\textrm{loc.}}(\mathcal{S} The functions $\psi_i^\pm\in H^{k-1-i}(\mathcal{I})$ satisfy: Thus for sufficiently regular initia

Figures (3)

  • Figure 1: The region of interest
  • Figure 2: The critical value of $\beta/l$ as a function of $r_{+}/l$ for the conformally coupled case, $\alpha = 2$ .
  • Figure 3: A schematic Penrose diagram for a stationary, asymptotically anti-de Sitter, black hole space time. Each point represents a compact $2$-surface diffeomorphic to $\mathcal{K}$.

Theorems & Definitions (53)

  • Definition 1
  • Theorem 1.1: Well Posedness
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1
  • Theorem 2.1
  • Proposition 2
  • Definition 2
  • Proposition 3: Properties of $\tilde{T}_{\mu\nu}$
  • Definition 3
  • ...and 43 more