On the massive wave equation on slowly rotating Kerr-AdS spacetimes
Gustav Holzegel
TL;DR
The paper proves uniform boundedness for the massive wave equation $\Box_g \psi - \alpha\frac{\Lambda}{3}\psi=0$ on Kerr–AdS spacetimes in the exterior region for $\alpha<\frac{9}{4}$, first in Schwarzschild–AdS and then for slowly rotating Kerr–AdS, using vector-field multipliers and Hardy inequalities to obtain positive energy despite the timelike AdS boundary.A redshift vector field near the horizon is constructed to overcome degeneracies in the radial derivative term, and a small correction $N=T+eY$ produces a robust energy identity that yields boundedness without requiring separability.The Kerr–AdS extension employs a causal Killing field $K=T+\lambda\Phi$, valid for small spin $|a|$, to maintain positivity of the energy; near-horizon estimates are combined with elliptic bounds on the horizon slices to control higher derivatives and obtain pointwise bounds.The results apply to spacetimes sufficiently close to Kerr–AdS that admit a causal Killing field null on the horizon, and the approach avoids separability techniques, suggesting avenues toward decay and nonlinear stability analyses.
Abstract
The massive wave equation $\Box_g ψ- α\fracΛ{3} ψ= 0$ is studied on a fixed Kerr-anti de Sitter background $(\mathcal{M},g_{M,a,Λ})$. We first prove that in the Schwarzschild case (a=0), $ψ$ remains uniformly bounded on the black hole exterior provided that $α< {9/4}$, 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 $K=\partial_t + λ\partial_φ$ for an appropriate $λ\sim a$, 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.