Gauge-Symmetry Breakdown at the Horizon of Extreme Black Holes

TL;DR

This paper investigates the boundary conditions and field configurations at the horizons of extreme black holes in Einstein-Yang-Mills-Higgs systems, revealing restrictions based on gauge symmetry and the nature of the centralizer, with explicit solutions for certain cases.

Contribution

It provides a detailed analysis of gauge-symmetry restrictions at black hole horizons, including explicit solutions for SU(3) cases and the impact of the cosmological constant.

Findings

Fields at the horizon are restricted to the centralizer of the electric field.

Two types of centralizers in SU(3): abelian and non-abelian, with different horizon geometries.

Explicit solutions for spherically symmetric horizons in the non-abelian case.

Abstract

Static solutions of the Einstein-Yang-Mills-Higgs system containing extreme black holes are studied. The field equations imply strong restrictions on boundary values of all fields at the horizon. If the Yang-Mills radial electric field $E$ is non-zero there, then all fields at the horizon take values in the centralizer of $E$. For the particular case of SU(3), there are two different kinds of centralizers: two-dimensional abelian (Cartan subalgebra) and four-dimensional (su(2)$\times$u(1)) ones. The two-dimensional centralizer admits only constant fields: even the geometry of the horizon is that of constant curvature. If the cosmological constant $\Lambda$ is negative, a two-surface of any genus is possible; for positive curvature, only spherically symmetrical horizons are allowed. For the four-dimensional centralizer, all spherically symmetrical horizons are explicitly given.