Odd-Parity Negative Modes of Einstein-Yang-Mills Black Holes and Sphalerons

An analytical proof of the existence of negative modes in the odd–parity perturbation sector is given for all known non-Abelian Einstein–Yang–Mills black holes. The significance of the normalizability condition in the functional stability analysis is empha-sized. The role of the odd–parity negative modes in the sphaleron interpretation of the Bartnik–McKinnon solutions is discussed.