Exponentially-Growing Mode Instability on the Reissner–Nordström-anti-de-Sitter Black Holes

We construct exponentially growing mode solutions to the uncharged and charged Klein–Gordon equations on the (3+1)-dimensional sub-extremal Reissner–Nordström-anti-de-Sitter (AdS) spacetime under reflecting (Dirichlet or Neumann) boundary conditions. Our result applies to a range of Klein–Gordon masses above the so-called Breitenlohner–Freedman bound, notably including the conformal mass case. The mode instability of the Reissner–Nordström-AdS spacetime for some black hole parameters is in sharp contrast to the Schwarzschild-AdS spacetime, where the solution to the Klein–Gordon equation is known to decay in time. Contrary to other mode instability results on the Kerr and Kerr-AdS spacetimes, our exponentially growing mode solutions of the uncharged and weakly charged Klein–Gordon equation exist independently of the occurrence or absence of superradiance. We discover a novel mechanism leading to an exponentially growing mode solution, namely, a near-extremal instability for the Klein–Gordon equation. Our result seems to be the first rigorous mathematical realization of this instability.