Axially Symmetric Monopoles and Black Holes in Einstein-Yang-Mills-Higgs Theory

TL;DR

This work analyzes static axially symmetric monopole and black hole solutions in SU(2) Einstein-Yang-Mills-Higgs theory with magnetic charge $n>1$, revealing gravity-induced binding of multimonopoles in the BPS limit and the existence of non-spherical hairy black holes. Using an axially symmetric, purely magnetic ansatz in isotropic coordinates and the isolated horizon formalism, the authors derive mass relations, horizon charges, and thermodynamic quantities, demonstrating that hairy black holes are bound states of regular solutions and Schwarzschild (or RN) horizons. A key finding is that static non-abelian black holes violate the proposed quasilocal uniqueness conjecture by admitting multiple solutions with the same horizon area and charges, and by mapping the horizon data to binding energies and horizon masses within the IH framework. The results illustrate gravitational attraction can overcome flat-space monopole repulsion, characterize the domain of existence across coupling parameters, and motivate further study of stability and non-spherical black holes in EYMH theory.

Abstract

We investigate static axially symmetric monopole and black hole solutions with magnetic charge n > 1 in Einstein-Yang-Mills-Higgs theory. For vanishing and small Higgs selfcoupling, multimonopole solutions are gravitationally bound. Their mass per unit charge is lower than the mass of the n=1 monopole. For large Higgs selfcoupling only a repulsive phase exists. The static axially symmetric hairy black hole solutions possess a deformed horizon with constant surface gravity. We consider their properties in the isolated horizon framework, interpreting them as bound states of monopoles and black holes. Representing counterexamples to the ``no-hair'' conjecture, these black holes are neither uniquely characterized by their horizon area and horizon charge.

Figures (43)

• Figure 3: The domain of existence of the hairy black hole solutions in the BPS limit is shown in the $x_{\Delta}$-$\alpha$-plane. The solid line shows the maximal values $x_{\Delta, \rm max}$ obtained for the $n=1$ hairy black hole solutions, while the crosses represent the maximal values $x_{\Delta, \rm max}$ obtained for $n=2$ hairy black hole solutions. The asterisk marks the value $\hat{\alpha}(2) = \sqrt{3}/2$, conjectured to separate the two regions with distinct critical behaviour. Also shown are the extremal RN solutions with unit charge and charge two.
• Figure 4: The dependence of the mass $\mu/\alpha^2$ of the $n=1$ hairy black hole solutions on the area parameter $x_{\Delta}$ is shown in the BPS limit for $\alpha=0.5$ and $\alpha=1$. For comparison, the mass of the corresponding RN solutions is also shown.
• Figure 5: The dependence of the surface gravity $\kappa$ of the $n=1$ hairy black hole solutions on the area parameter $x_{\Delta}$ is shown in the BPS limit for $\alpha=0.5$ and $\alpha=1$. For comparison, the surface gravity of the corresponding RN solutions is also shown.
• Figure 6: The dependence of the horizon mass $\mu_{\Delta}/\alpha^2$ of the $n=1$ hairy black hole solutions on the area parameter $x_{\Delta}$ is shown in the BPS limit for $\alpha=0.5$ and $\alpha=1$.
• Figure 7: The dependence of the binding energy $\mu_{\rm bind}/\alpha^2$ of the $n=1$ hairy black hole solutions on the area parameter $x_{\Delta}$ is shown in the BPS limit for $\alpha=0.5$ and $\alpha=1$. For comparison, the binding energy of the corresponding RN solutions is also shown.