On the massive wave equation on slowly rotating Kerr-AdS spacetimes

Abstract

The massive wave equation ${\square }_g\psi -\alpha \frac{\Lambda }{3}\psi =0$ is studied on a fixed Kerr-anti de Sitter background $(\mathcal{M},g_{M,a,\Lambda })$. We first prove that in the Schwarzschild case (a=0), $\psi$ remains uniformly bounded on the black hole exterior provided that $\alpha <9/4$, i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The usual energy current arising from the timelike Killing vector field $T$ (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to $T$, we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield $T={\partial }_t$ with $K={\partial }_t+\lambda {\partial }_{\varphi }$ for an appropriate $\lambda \sim a$, which is also Killing and--in contrast to the asymptotically flat case--everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field $K$ which is null on the horizon.

1 figure; typos corrected, references added, introduction revised; to appear in CMP

References in corpus (4)

• Breaking an Abelian gauge symmetry near a black hole horizon
• Gravitational Perturbations of Higher Dimensional Rotating Black Holes: Tensor Perturbations
• A Rotating Holographic Superconductor
• The Dirac system on the Anti-de Sitter Universe

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