Addentum to: Conformal Invariance and Near-extreme Rotating AdS Black Holes

TL;DR

This paper completes the derivation of the wave equation for massless scalar fields in five-dimensional rotating charged AdS black holes, revealing how horizon properties influence the Klein-Gordon equation's residues.

Contribution

It provides a complete derivation of the radial Klein-Gordon equation with nonzero azimuthal eigenvalues, including horizon angular velocities, for these black holes.

Findings

Radial equation is a general Heun's equation with singularities at horizons and infinity.

Residues at horizons relate to surface gravity and angular velocities.

Full wave equation analysis now includes nonzero azimuthal eigenvalues.

Abstract

We obtained retarded Green's functions for massless scalar fields in the background of near-extreme, near-horizon rotating charged black hole of five-dimensional minimal gauged supergravity in Phys. Rev. D84, 044018 (2011). For general nonextreme black holes, we also derived the radial part of the massless Klein-Gordon equation with zero azimuthal-angle eigenvalues, and showed that it is a general Heun's equation with a regular singularity at each horizon $u_k$ ($k=1,2,3$) and at infinity. We derived explicitly that the residuum of a pole at each $u_k$ is associated with the surface gravity there. In this addendum, probing regular singularities at each $u_k$ we complete the derivation of the full radial equation with nonzero azimuthal-angle eigenvalues. The residua now include modifications by the angular velocities at respective horizons. This result completes the analysis of the wave equation for the massless Klein-Gordon equation for the general rotating charged black hole of five-dimensional minimal gauged supergravity.