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 (197View Equation), the Klein–Gordon equation is decoupled into an angular equation
[ 1 m2 Ξ2 2ma Ξ ω − a2ω2 sin2 πœƒ ] -----∂πœƒ(sin πœƒΔ πœƒ∂πœƒ) − --2-----+ -------------------- + Cl S (πœƒ) = 0, (202 ) sinπœƒ sin πœƒΔ πœƒ Δ πœƒ
and a radial equation
[ 2 2 2 ] ∂ (Δ ∂ ) + [ω(r-+--a-) −-ma-Ξ-−-qeQer]- − C R (r) = 0, (203 ) r r Δr l
where Cl 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. (198View Equation) – (199View Equation) are recovered with 2 2 Kl = Cl + 2ma ω − a ω.

The radial equation has a more involved form than the corresponding flat equation (199View Equation) due to the fact that Δr is a quartic instead of a quadratic polynomial in r; see (42View Equation). More precisely, the quartic polynomial Δr can be written as

−2 ∗ Δr = l (r − r+)(r − r− )(r − rc)(r − rc), (204 )
where rc is a complex root. The radial equation is a general Heun’s equation due to the presence of two conjugate complex poles in (203View Equation) 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 [102Jump To The Next Citation Point]. It is an open problem to unravel the structure of the hidden symmetries, if any, of the full non-extremal radial equation (203View Equation). 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.17 Since much remains to be understood, we will not discuss AdS black holes further.

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