Wave Equation for the Wu Black Hole

TL;DR

This paper analyzes the wave equation in the Wu black hole, revealing that both radial and angular parts are Heun's equations with four singularities, and links singularity residues to surface gravity and horizon properties.

Contribution

It demonstrates the separability of the Klein-Gordon equation in the Wu black hole background and relates singularity residues to physical horizon parameters, extending understanding of wave behavior in complex black hole spacetimes.

Findings

Radial and angular equations are Heun's equations with four singularities.

Residues of singularities relate to surface gravity and horizon angular velocities.

Separability holds for general stationary, axisymmetric metrics with orthogonal transitivity.

Abstract

Wu black hole is the most general solution of maximally supersymmetric gauged supergravity in D=5, containing $U(1)^{3}$ gauge symmetry. We study the separability of the massless Klein-Gordon equation and probe its singularities for a general stationary, axisymmetric metric with orthogonal transitivity, and apply the results to the Wu black hole solution. We start with the zero azimuthal-angle eigenvalues in the scalar field Ansatz and find that the residuum of a pole in the radial equation is associated with the surface gravity calculated at this horizon. We then generalize our calculations to nonzero azimuthal eigenvalues and probing each horizon singularity, we show that the residua of the singularities for each horizon are in general associated with a specific combination of the surface gravity and the angular velocities at the associated horizon. It turns out that for the Wu black hole both the radial and angular equations are general Heun's equations with four regular singularities.