[Paper Review] A menagerie of hairy black holes

Elizabeth Winstanley|arXiv (Cornell University)|Oct 6, 2015

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates hairy black hole solutions in four-dimensional Einstein-Yang-Mills (EYM) theory with a negative cosmological constant, focusing on static, spherically symmetric and topological black holes. It examines the validity of Bizon's modified no-hair conjecture—asserting that stable black holes are uniquely characterized by global charges—and finds that stable purely magnetic EYM black holes in asymptotically anti-de Sitter space are indeed characterized by such charges, while the status for dyonic and topological solutions remains open.

ABSTRACT

According to the no-hair conjecture, equilibrium black holes are simple objects, completely determined by global charges which can be measured at infinity. This is the case in Einstein-Maxwell theory due to beautiful uniqueness theorems. However, the no-hair conjecture is not true in general, and there is now a plethora of matter models possessing hairy black hole solutions. In this note we focus on one such matter model: Einstein-Yang-Mills (EYM) theory, and restrict our attention to four-dimensional, static, non-rotating black holes for simplicity. We outline some of the menagerie of EYM solutions in both asymptotically flat and asymptotically anti-de Sitter space. We attempt to make sense of this black hole zoo in terms of Bizon's modified no-hair conjecture.

Motivation & Objective

• To investigate the validity of Bizon's modified no-hair conjecture in the context of Einstein-Yang-Mills (EYM) theory with a negative cosmological constant.
• To determine whether stable black hole solutions in EYM theory are uniquely determined by global conserved charges measurable at infinity.
• To explore the existence and stability of purely magnetic and dyonic black holes in spherically symmetric and topological configurations.
• To assess whether the rich menagerie of EYM black hole solutions can be systematically classified by global charges despite their complex gauge field hair.
• To identify open questions regarding characterization by global charges for dyonic and topological black holes in EYM theory.

Proposed method

• Analyzes static, spherically symmetric and topological black hole solutions of the Einstein-Yang-Mills (EYM) equations in four-dimensional asymptotically anti-de Sitter (adS) spacetime.
• Uses the action $ S = \frac{1}{2}\int d^4x \sqrt{-g} \left[ R - 2\Lambda - \text{Tr}(F_{\alpha\beta}F^{\alpha\beta}) \right] $ to derive the field equations for EYM theory with $ \mathfrak{su}(N) $ gauge group.
• Applies a static, spherically symmetric ansatz for the gauge potential $ A_\alpha $, parameterized by $ N-1 $ electric and magnetic gauge field functions $ h_j(r) $, $ \omega_j(r) $, and a constant matrix $ D $.
• Considers purely magnetic configurations with $ h_j(r) \equiv 0 $, focusing on solutions with $ \omega_j(r) $ having no nodes for stability.
• Employs numerical and analytical techniques to study existence and linear stability of black hole solutions under perturbations.
• Evaluates global charges such as mass and Yang-Mills charge at spatial infinity to test uniqueness of solutions.

Experimental results

Research questions

• RQ1 Are stable, static, spherically symmetric black holes in $ \mathfrak{su}(N) $ Einstein-Yang-Mills theory in anti-de Sitter space uniquely determined by global conserved charges?
• RQ2 Do purely magnetic, nodeless $ \omega_j(r) $ configurations in $ \mathfrak{su}(N) $ EYM adS black holes remain stable under linear perturbations?
• RQ3 Can dyonic black holes in $ \mathfrak{su}(N) $ EYM theory in adS be uniquely characterized by global charges, and what is their stability status?
• RQ4 What is the role of the cosmological constant magnitude $ |\Lambda| $ in enabling stable, nodeless hairy black hole solutions?
• RQ5 To what extent do topological black holes (with $ k = 0 $ or $ k = -1 $) in $ \mathfrak{su}(N) $ EYM adS satisfy the modified no-hair conjecture?

Key findings

• Stable, purely magnetic, spherically symmetric $ \mathfrak{su}(N) $ EYM black holes in anti-de Sitter space exist for any $ N $ when $ |\Lambda| $ is sufficiently large.
• For such stable purely magnetic black holes, there is analytic and numerical evidence that they are uniquely characterized by global charges (mass and Yang-Mills charge) at infinity.
• Stable topological black holes with purely magnetic $ \mathfrak{su}(N) $ gauge fields and $ k = 0 $ or $ k = -1 $ event horizons exist when $ |\Lambda| $ is large enough, as proven for $ \omega_j(r) $ with no zeros.
• The existence of stable spherically symmetric dyonic black holes with $ \mathfrak{su}(2) $ gauge group has been recently established, but their characterization by global charges remains unproven.
• For topological and dyonic EYM black holes, the question of whether they are uniquely determined by global charges is still open and uninvestigated in the literature.
• Despite their complex gauge field hair with unbounded degrees of freedom, stable EYM black holes in adS space appear to satisfy Bizon's modified no-hair conjecture, suggesting global charges may fully classify them.