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

The massive wave equation ${\square_{g}\psi - \alpha \frac{\Lambda}{3}\psi = 0}The massive wave equation \square_gy- a\fracL3y = 0{\square_{g}\psi - \alpha \frac{\Lambda}{3}\psi = 0} is studied on a fixed Kerr-anti de Sitter background (M,g_M,a,L){\left(\mathcal{M},g_{M,a,\Lambda}\right)}. We first prove that in the Schwarzschild case (a = 0), ψ remains uniformly bounded on the black hole exterior provided that a < \frac94{\alpha < \frac{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 = ∂ _t with K=?_t + l?_f{K=\partial_t + \lambda \partial_\phi} for an appropriate λ ~ 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.