### 6.2 Scalar wave equation in Kerr–Newman–AdS

The equations for probe scalars fields on the Kerr–Newman–AdS black hole can be obtained
straightforwardly. We consider only massless probes for simplicity. Using again the decomposition (197), the
Klein–Gordon equation is decoupled into an angular equation
and a radial equation
where is a separation constant and the various functions and parameters in the equations have been
defined in Section 2.4.4. In the flat limit, Eqs. (198) – (199) are recovered with .
The radial equation has a more involved form than the corresponding flat equation (199) due to the fact
that is a quartic instead of a quadratic polynomial in ; see (42). More precisely, the quartic
polynomial can be written as

where is a complex root. The radial equation is a general Heun’s equation due to the presence of two
conjugate complex poles in (203) in addition to the two real poles corresponding to the inner and outer
horizons and the pole at infinity.
It has been suggested that all these poles have a role to play in the microscopic description of the AdS
black hole [102]. It is an open problem to unravel the structure of the hidden symmetries,
if any, of the full non-extremal radial equation (203). It has been shown that in the context
of five-dimensional black holes, one can find hidden conformal symmetry in the near-horizon
region close to extremality [46]. It is expected that one could similarly neglect the two complex
poles in the near-horizon region of near-extremal black holes, but this remains to be checked in
detail.
Since much remains to be understood, we will not discuss AdS black holes further.