Classical super-radiance in Kerr–Newman–anti-de Sitter black holes

We study in detail the modes of a classical scalar field on a Kerr--Newman--anti-de Sitter (KN-AdS) black hole. We construct sets of basis modes appropriate to the two possible boundary conditions (``reflective'' and ``transparent'') at timelike infinity, and consider whether super-radiance is possible. If we employ ``reflective'' boundary conditions, all modes are non-super-radiant. On the other hand, for ``transparent'' boundary conditions, the presence of super-radiance depends on our definition of positive frequency. For those KN-AdS black holes having a globally time-like Killing vector, the natural choice of positive frequency leads to no super-radiance. For other KN-AdS black holes, there is a choice of positive frequency which gives no super-radiance, but for other choices there will, in general, be super-radiance.