  The static black hole solutions to the Einstein-Maxwell equations are all
spherically symmetric, as are many of the recently discovered black hole
solutions in theories of gravity coupled to other forms of matter. However,
counterexamples demonstrating that static black holes need not be spherically
symmetric exist in theories, such as the standard electroweak model, with
electrically charged massive vector fields. In such theories, a magnetically
charged Reissner-Nordstrom solution with sufficiently small horizon radius is
unstable against the development of a nonzero vector field outside the horizon.
General arguments show that, for generic values of the magnetic charge, this
field cannot be spherically symmetric. Explicit construction of the solution
shows that it in fact has no rotational symmetry at all.
