Stability analysis of Squashed Kaluza-Klein Black Holes with Charge

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.