Static black holes with axial symmetry in asymptotically AdS$_4$ spacetime

The known static electro-vacuum black holes in a globally AdS$_4$ background have an event horizon which is geometrically a round sphere. In this work we argue that the situation is different in models with matter fields possessing an explicit dependence on the azimuthal angle $φ$, which, however, does not manifest at the level of the energy-momentum tensor. As a result, the full solutions are axially symmetric only, possessing a single (timelike) Killing vector field. Explicit examples of such static black holes are constructed in Einstein--(complex) scalar field and Einstein--Yang-Mills theories. The basic properties of these solutions are discussed, looking for generic features. For example, we notice that the horizon has an oblate spheroidal shape for solutions with a scalar field and a prolate one for black holes with Yang-Mills fields. The deviation from sphericity of the horizon geometry manifests itself in the holographic stress-tensor. Finally, based on the results obtained in the probe limit, we conjecture the existence in the Einstein-Maxwell theory of static black holes with axial symmetry only.

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