Gravitating Non-Abelian Solitons and Black Holes with Yang-Mills Fields

TL;DR

This review synthesizes gravitating non-Abelian solitons and black holes in Einstein–Yang–Mills theory and its extensions, highlighting Bartnik–McKinnon solitons, non-Abelian black holes, and the rich interior dynamics. It links these solutions to flat-space soliton physics, notably sphalerons and topological vacua, and demonstrates violations of the traditional no-hair and uniqueness theorems in non-Abelian contexts. The work surveys generalizations with Λ, dilatons, higher gauge groups, and axial symmetry, and discusses stability, slow rotation, and self-gravitating lumps including monopoles and Skyrmions. A recurring theme is the emergence of discrete structures, bifurcations, and interior oscillations, governed by the interplay between massless gauge fields and gravity. The analysis emphasizes how gravity can stabilize or coexist with hair in some regimes, while interior dynamics often exhibit striking, highly nonlinear behavior near singularities or horizons.

Abstract

We present a review of gravitating particle-like and black hole solutions with non-Abelian gauge fields. The emphasis is given to the description of the structure of the solutions and to the connection with the results of flat space soliton physics. We describe the Bartnik-McKinnon solitons and the non-Abelian black holes arising in the Einstein-Yang-Mills theory, and their various generalizations. These include axially symmetric and slowly rotating configurations, solutions with higher gauge groups, $Λ$-term, dilaton, and higher curvature corrections. The stability issue is discussed as well. We also describe the gravitating generalizations for flat space monopoles, sphalerons, and Skyrmions.

Figures (14)

• Figure 1: $w(r)$ and the effective charge function $Q^2(r)$ for the lowest BK solutions.
• Figure 2: Metric functions $m(r)$ and $\sigma(r)$ for the lowest BK solutions.
• Figure 3: On the left: the radial energy density $r^2 T^0_0$ for the lowest BK solutions. On the right: the equations of state $T^r_r/T^{0}_{0}$ and $T^{\vartheta}_{\vartheta}/T^{0}_{0}$.
• Figure 4: On the left: The ADM mass along the field sequence (\ref{['5.5']})--(\ref{['5.7']}) through the $n=1$ BK solution. On the right: The potential energy (\ref{['5.18']}) in the vicinity of this solution.
• Figure 5: On the left: amplitudes $w$, $N$, $m$, and $\sigma$ for the asymptotically de Sitter solution with $n=3$ and $\Lambda=0.0003$. On the right, the conformal diagram for this solution (the black regions should not be confused with spacetime singularities). The definition of the null coordinates $U$ and $V$ is given in Volkov96a.