On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes

Abstract

The massive wave equation $\Box_g \psi - \alpha\frac{\Lambda}{3} \psi = 0$ 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 $K=\partial_t + \lambda \partial_\phi$ 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.