Negative modes around Einstein-Yang-Mills sphalerons and black holes

Abstract

The dynamics of small perturbations around sphaleron and black hole solutions in the Einstein-Yang-Mills theory for the gauge group $SU(2)$ is investigated. The perturbations can be split into the two independent sectors in accordance with their parity; each sector contains negative modes. The even-parity negative modes are shown to correspond to the negative second variations of the height of the potential energy barrier near the sphaleron. For the odd-parity sector, the existence of precisely $n$ (the number of nodes of the sphaleron solution gauge field function) negative modes is rigorously proven. The same results hold for the Einstein-Yang-Mills black holes as well.