An Analysis of the Wave Equation for the $U(1)^{2}$ Gauged Supergravity Black Hole

Abstract

We study the massless Klein-Gordon equation in the background of the most general rotating dyonic AdS black hole in gauged $\mathcal{N}=2$, $U(1)^{2}$ gauged supergravity in $D=4$, given by Chow and Comp\`{e}re [Phys. Rev. D\textbf{89} (2014) 065003]. The angular part of the separable wave equation is of the Heun type, while the radial part is a Fuchsian equation with five regular singularities. The radial equation is further analyzed and written in a specific form, that reveals the pole structure of the horizon equation, whose residua are expressed in terms of the surface gravities and angular velocities associated with respective horizons. The near-horizon (near-)extremal limits of the solution are also studied, where the expected hidden conformal symmetry is revealed. Furthermore, we present the retarded Green's functions for these limiting cases. We also comment on the generality of the charge-dependent parts of the metric parameters and address some further examples of limiting cases.