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

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.

References in corpus (6)

• On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions
• Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole
• Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes
• On the uniqueness and global dynamics of AdS spacetimes
• Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes
• The Dirac system on the Anti-de Sitter Universe

Cited by in corpus (32)

• Slowly Decaying Waves on Spherically Symmetric Spacetimes and an Instability of Ultracompact Neutron Stars
• Numerical Methods for Finding Stationary Gravitational Solutions
• Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes
• On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant. Part 1: Well posedness and breakdown criterion
• Quasinormal modes in extremal Reissner-Nordström spacetimes
• Echoes of Compact Objects in Scalar-Tensor Theories of Gravity
• Cavity effect in the quasinormal mode spectrum of topological stars
• A proof of the instability of AdS for the Einstein--null dust system with an inner mirror
• Quasinormal modes and stability of boosted Reissner-Nordström AdS black holes
• Unstable mode solutions to the Klein-Gordon equation in Kerr-anti-de Sitter spacetimes
• The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves
• Mathematical General Relativity
• The holographic Hadamard condition on asymptotically Anti-de Sitter spacetimes
• Long-term stable nonlinear evolutions of ultracompact black-hole mimickers
• Strong Cosmic Censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry
• From quasimodes to resonances: exponentially decaying perturbations
• Open problems in mathematical physics
• On turbulence for spacetimes with stable trapping
• Existence of quasinormal modes for Kerr-AdS black holes
• Evanescent ergosurface instability
• Assessing the stability of ultracompact spinning boson stars with nonlinear evolutions
• Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes
• Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere
• Instability of asymptotically anti de Sitter black holes under Robin conditions at the timelike boundary
• The Klein-Gordon equation on the toric adS-Schwarzschild black hole
• Gravitational Atoms from Topological Stars
• Strong cosmic censorship for the spherically symmetric Einstein-Maxwell-charged-Klein-Gordon system with positive $\Lambda$: stability of the Cauchy horizon and $H^1$ extensions
• Exponentially-growing Mode Instability on Reissner-Nordström--Anti-de-Sitter black holes
• Non-linear instability of slowly rotating Kerr-AdS black holes
• Instability of nonlinear scalar field on strongly charged asymptotically AdS black hole background
• Explicit formula and meromorphic extension of the resolvent for the massive Dirac operator in the Schwarzschild-Anti-de Sitter spacetime
• Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes