Gauge-Symmetry Breakdown at the Horizon of Extreme Black Holes

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)×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 Λ 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. Finally, some complete spacetime solutions are constructed whose horizons have the structure found by our method. There are also abelian solutions of a new type. In some cases there are different spacetimes having the same type of horizon.