On the stability of soliton and hairy black hole solutions of ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant

TL;DR

This paper analyzes the linear stability of spherically symmetric soliton and black hole solutions in ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with negative cosmological constant, identifying conditions for stability in different perturbation sectors.

Contribution

It provides a detailed stability analysis of ${\mathfrak {su}}(N)$ solutions, including conditions for stability in the sphaleronic and gravitational sectors, extending previous results to higher N and large negative Lambda.

Findings

No instabilities in sphaleronic sector if gauge functions have no zeros and satisfy inequalities.

Existence of stable solutions near embedded ${\mathfrak {su}}(2)$ solutions for large |Lambda|.

Stability conditions depend on the properties of gauge functions and the magnitude of the cosmological constant.

Abstract

We investigate the stability of spherically symmetric, purely magnetic, soliton and black hole solutions of four-dimensional ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant $\Lambda $. These solutions are described by $N-1$ magnetic gauge field functions $\omega _{j}$. We consider linear, spherically symmetric, perturbations of these solutions. The perturbations decouple into two sectors, known as the sphaleronic and gravitational sectors. For any $N$, there are no instabilities in the sphaleronic sector if all the magnetic gauge field functions $\omega _{j}$ have no zeros, and satisfy a set of $N-1$ inequalities. In the gravitational sector, we are able to prove that there are solutions which have no instabilities in a neighbourhood of stable embedded ${\mathfrak {su}}(2)$ solutions, provided the magnitude of the cosmological constant $\left| \Lambda \right| $ is sufficiently large.