Classical Yang-Mills black hole hair in anti-de Sitter space

TL;DR

This work investigates hairy black holes in Einstein–Yang–Mills theory with anti-de Sitter boundary conditions. It develops the ${\\mathfrak{su}}(N)$ EYM framework, derives the static, spherically symmetric ansatz with $N-1$ magnetic gauge functions $\\omega_j(r)$, and analyzes both field equations and linear perturbations that split into sphaleronic and gravitational sectors. The key contribution is showing the existence of stable AdS black holes with nontrivial gauge hair for arbitrarily large $N$, with analytic existence proofs near embedded ${\\mathfrak{su}}(2)$ solutions and stability results in the spherically symmetric sector for large $|\\Lambda|$. This implies a much richer landscape of “furry” black holes in AdS and prompts exploration of their holographic implications within the AdS/CFT correspondence.

Abstract

The properties of hairy black holes in Einstein-Yang-Mills (EYM) theory are reviewed, focusing on spherically symmetric solutions. In particular, in asymptotically anti-de Sitter space (adS) stable black hole hair is known to exist for su(2) EYM. We review recent work in which it is shown that stable hair also exists in su(N) EYM for arbitrary N, so that there is no upper limit on how much stable hair a black hole in adS can possess.

Figures (14)

• Figure 1: An example of an ${\mathfrak {su}}(2)$ EYM black hole in adS in which the gauge field function $\omega (r)$ has no zeros. Here, $\Lambda = -1$, $r_{h}=1$ and $\omega (r_{h})=0.7$.
• Figure 2: The space of ${\mathfrak {su}}(2)$ black hole solutions when $\Lambda = -0.01$, for varying $r_{h}$. The shaded region indicates values of the gauge field function $\omega (r_{h})$ at the event horizon for which the constraint (\ref{['win:eq:su2bhconstraint']}) is satisfied, but for which we find no well-behaved black hole solution. The number of zeros $n$ of the gauge field function $\omega$ are indicated in those regions of the phase space where we find black hole solutions. Elsewhere on the diagram, the constraint (\ref{['win:eq:su2bhconstraint']}) is not satisfied. Between the region where $n=2$ and the shaded region we find black hole solutions with $n=3$, $4$ and $5$, but these regions are too small to indicate on the graph. Taken from win:Baxter3.
• Figure 3: Phase space of ${\mathfrak {su}}(2)$ black holes with $r_{h}=1$ and varying $\Lambda$. The shaded region indicates values of the gauge field function $\omega (r_{h})$ at the event horizon for which the constraint (\ref{['win:eq:su2bhconstraint']}) is satisfied, but for which we find no well-behaved black hole solution. The number of zeros $n$ of the gauge field function $\omega$ are indicated in those regions of the phase space where we find black hole solutions. Elsewhere on the diagram, the constraint (\ref{['win:eq:su2bhconstraint']}) is not satisfied. As well as the regions where $n=0,\ldots,4$ as marked on the diagram, we find a small region in the bottom left of the plot where $n=5$. This region is too small to indicate on the current figure, but can be seen in figure \ref{['win:fig:su2bh3']}. Taken from win:Baxter3.
• Figure 4: Close-up of the phase space of ${\mathfrak {su}}(2)$ black holes with $r_{h}=1$ and smaller values of $\Lambda$. In the bottom left of the plot there is a small region of solutions for which $n=7$, but the region is too small to be visible. Taken from win:Baxter3.
• Figure 5: Black hole mass $M$ and magnetic charge $Q$ for ${\mathfrak {su}}(2)$ EYM black holes with $r_{h}=1$ and varying $\Lambda$ (cf. figure 8 in win:Bjoraker2).