Unstable Mode Solutions to the Klein–Gordon Equation in Kerr-anti-de Sitter Spacetimes

Abstract

For any cosmological constant Λ = - 3 / ℓ2< 0 and any α< 9 / 4 , we find a Kerr-AdS spacetime (M, gKAdS) , in which the Klein–Gordon equation □gKAdSψ+α/ℓ2ψ=0 has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking–Reall bound r+2>|a|ℓ. We obtain an analogous result for Neumann boundary conditions if 5 / 4 < α< 9 / 4. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking–Reall bound, there exists an open family of masses α such that the corresponding Klein–Gordon equation permits exponentially growing mode solutions. Our result adopts methods of Shlapentokh-Rothman developed in (Commun. Math. Phys. 329:859–891, 2014) and provides the first rigorous construction of a superradiant instability for negative cosmological constant.