Stability analysis of Squashed Kaluza-Klein Black Holes with Charge

TL;DR

This paper investigates the stability of charged squashed Kaluza-Klein black holes under gravitational and electromagnetic perturbations, deriving master equations and analyzing the positivity of effective potentials for different modes.

Contribution

It provides a detailed stability analysis for specific perturbation modes of charged squashed Kaluza-Klein black holes using symmetry-based variable separation and master equations.

Findings

Modes with certain quantum numbers are stable due to positive effective potentials.

Stability of some modes remains uncertain due to negative potential regions.

Maximally charged case shows stability for all modes analyzed.

Abstract

We study gravitational and electromagnetic perturbation around the squashed Kaluza-Klein black holes with charge. Since the black hole spacetime focused on this paper have $SU(2) \times U(1) \simeq U(2)$ symmetry, we can separate the variables of the equations for perturbations by using Wigner function $D^{J}_{KM}$ which is the irreducible representation of the symmetry. In this paper, we mainly treat $J=0$ modes which preserve $SU(2)$ symmetry. We derive the master equations for the $J=0$ modes and discuss the stability of these modes. We show that the modes of $J = 0$ and $ K=0,\pm 2$ and the modes of $K = \pm (J + 2)$ are stable against small perturbations from the positivity of the effective potential. As for $J = 0, K=\pm 1$ modes, since there are domains where the effective potential is negative except for maximally charged case, it is hard to show the stability of these modes in general. To show stability for $J = 0, K=\pm 1$ modes in general is open issue. However, we can show the stability for $J = 0, K=\pm 1$ modes in maximally charged case where the effective potential are positive out side of the horizon.