Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes

Abstract

This paper investigates the decay properties of solutions to the massive linear wave equation ${\square }_g\psi +\frac{\alpha }{l^2}\psi =0$ for $g$ the metric of a Kerr-AdS spacetime satisfying $|a|l<r_{+}^2$ and $\alpha <9/4$ satisfying the Breitenlohner Freedman bound. We prove that the non-degenerate energy of $\psi$ with respect to an appropriate foliation of spacelike slices decays like $(\log t^{\star }{)}^{-2}$. Our estimates are expected to be sharp from heuristic and numerical arguments in the physics literature suggesting that general solutions will only decay logarithmically. The underlying reason for the slow decay rate can be traced back to a stable trapping phenomenon for asymptotically anti de Sitter black holes which is in turn a consequence of the reflecting boundary conditions for $\psi$ at null-infinity.

47 pages, 1 figure; typos corrected, minor improvement of the results

References in corpus (8)

• On weakly turbulent instability of anti-de Sitter space
• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• On the uniqueness and global dynamics of AdS spacetimes
• Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes
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Cited by in corpus (5)

• On quasinormal modes of asymptotically anti-de Sitter black holes
• A scattering theory construction of dynamical vacuum black holes
• Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis
• A spacetime characterization of the Kerr-NUT-(A)de Sitter and related metrics
• Stability of the Toroidal AdS Schwarzschild Solution in the Einstein--Klein-Gordon System