Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter 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 $\psi$ 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 $\psi$ 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.

46 pages, 3 figures; V2 significant update

References in corpus (4)

• On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions
• On quasinormal modes of asymptotically anti-de Sitter black holes
• A scalar field condensation instability of rotating anti-de Sitter black holes
• On the uniqueness and global dynamics of AdS spacetimes

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• A scattering theory construction of dynamical vacuum black holes
• A model problem for quasinormal ringdown on asymptotically flat or extremal black holes
• Unstable mode solutions to the Klein-Gordon equation in Kerr-anti-de Sitter spacetimes
• The holographic Hadamard condition on asymptotically Anti-de Sitter spacetimes
• Existence of quasinormal modes for Kerr-AdS black holes
• Vacuum polarization on topological black holes with Robin boundary conditions
• Instability of asymptotically anti de Sitter black holes under Robin conditions at the timelike boundary
• Determining an asymptotically AdS spacetime from data on its conformal boundary
• Scattering theory for the Dirac equation on the Schwarzschild-Anti-de Sitter spacetime
• The Klein-Gordon equation on the toric adS-Schwarzschild black hole
• Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis
• Dynamics of Nonlinear Waves on Bounded Domains
• Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes
• The Klein-Gordon equation on the hyperboloidal anti-de Sitter Schwarzschild black hole
• Non-linear instability of slowly rotating Kerr-AdS black holes
• Explicit formula and meromorphic extension of the resolvent for the massive Dirac operator in the Schwarzschild-Anti-de Sitter spacetime
• Instability of nonlinear scalar field on strongly charged asymptotically AdS black hole background
• Quasimodes and a lower bound for the local energy decay of the Dirac equation in Schwarzschild-Anti-de Sitter spacetime
• Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes
• Stability of the Toroidal AdS Schwarzschild Solution in the Einstein--Klein-Gordon System