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

The massive wave equation square g psi - alpha Lambda/3 psi = 0 is studied on a fixed Kerr-anti de Sitter background (M, gM, 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 total flux of 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 derivative(t) with K = partial derivative(t) + lambda partial derivative(phi) for an appropriate. lambda similar to 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.