X iv : g r - qc / 9 50 40 03 v 1 4 A pr 1 99 5 CU - TP - 688 gr - qc / 9504003 Are All Static Black Hole Solutions Spherically Symmetric ?

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-Nordström 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. This work was supported in part by the US Department of Energy. One of the many remarkable aspects of black holes is the high degree of symmetry of the classical black hole solutions. Both of the static solutions that were discovered in the early days of general relativity — the Schwarzschild and the Reissner-Nordström — are spherically symmetric. At one time, one might have thought that this simply reflected the fact that spherically symmetric solutions are easier to find. However, it was shown two decades ago [1] that these are in fact the only static electrovac black hole solutions. Does this result generalize to gravity coupled to other types of matter; i.e., are static black holes always spherically symmetric? The answer [2], as we will describe below, is no. The restrictions on the possible electrovac black holes can be viewed as just one instance of the “no-hair” results that limit the possible structure of black holes in a number of matter theories. This suggests that in seeking solutions that depart from spherical symmetry one should look to theories that do not admit no-hair theorems; these tend to be [3] theories that possess static soliton solutions in the absence of gravity. An important class of such theories is the spontaneously broken gauge theories that possess nonsingular magnetic monopole solutions [4] in the absence of gravity; the simplest example is the SU(2) gauge theory with the symmetry broken to the U(1) of electromagnetism by a triplet Higgs field. The elementary particles of this theory include, in addition to the massless photon, a spin-one particle with mass m and electric charge e and an electrically neutral spinless particle. This theory possesses spherically symmetric black hole solutions [5] with nontrivial matter fields outside the horizon; these can be constructed by numerical integration of the field equations. They carry magnetic charge 1/e, and may be viewed as Schwarzschild-like black holes embedded in the center of ’t Hooft-Polyakov monopoles. In fact, one does not need a detailed examination of the field equations to show that such solutions exist. An analysis [6] of the small fluctuations about the Reissner-Nordström solution reveals an instability leading to the development of massive vector meson fields just outside the horizon whenever the horizon radius is less than a critical value of order m; the existence of this instability indicates that there must be a nearby static solution with hair. The physical basis for this instability is easily understood. Charged particles with nonzero spin in general have magnetic moments. In a sufficiently strong magnetic field, the energetic cost of producing a cloud of such particles can be more than offset by the energy gained by aligning their magnetic moments so as to partially shield the magnetic field. Hence, the essential features needed for these new black holes are captured by a theory that includes, in addition to the electromagnetic and gravitational fields, a massive charged vector field Wμ with a magnetic moment fixed by an arbitrary parameter g. This theory has the flat spacetime Lagrangian L = − 1 4 FμνF μν − 1 2 |DμWν −DνWμ| 2 −mW ∗ μW μ