A Remark on the Instability of the Bartnik-McKinnon Solutions

TL;DR

This paper critically reviews the stability of Bartnik-McKinnon solutions in Einstein-Yang-Mills theory, clarifying the number and nature of their unstable modes and distinguishing two types of instabilities.

Contribution

It introduces a clear distinction between gravitational and sphaleron instabilities and precisely counts the unstable modes for each solution.

Findings

The nth solution has exactly 2n unstable modes.

There are two types of instabilities: gravitational and sphaleron.

Clarifies previous confusion about the stability characteristics.

Abstract

The aim of the present letter is to critically review the stability of the Bartnik-McKinnon solutions of the Einstein-Yang-Mills theory. The stability question was already studied by several authors, but there seems to be some confusion about the nature and the number of unstable modes. We suggest to distinguish two different kind of instabilities, which we call `gravitational' respectively `sphaleron' instabilities. We claim that the $n^{\rm th}$ Bartnik-McKinnon solution has exactly $2n$ unstable modes, $n$ of either type.