Abstract
The known static electrovacuum black holes in a globally ${\mathrm{AdS}}_{4}$ 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 $\ensuremath{\varphi}$, 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 Einstein-Maxwell theory of static black holes with axial symmetry only.
Highlights
The asymptotically flat black holes (BHs) in d 1⁄4 4 spacetime dimensions are rather special objects
Based on the results obtained in the probe limit, we conjecture the existence in EinsteinMaxwell theory of static black holes with axial symmetry only
The topology of a spatial section of the event horizon of a stationary BH is necessarily that of a two-sphere, S2 [2,3]
Summary
The asymptotically flat black holes (BHs) in d 1⁄4 4 spacetime dimensions are rather special objects. The configurations are no longer spherically symmetric, and the full solutions (geometry þ matter fields) possess a single Killing vector ∂=∂t (with t the time coordinate). The main purpose of this work is to investigate static BHs with axial symmetry in asymptotically AdS4 spacetime As expected, such solutions retain many of the properties of their Λ 1⁄4 0 counterparts. There we speculate about the possible existence of static BHs with axial symmetry only in the Einstein-Maxwell theory This conjecture is based on the results found in the Appendix for Maxwell fields in a fixed Schwarzschild-AdS (SAdS) BH background. There, apart from the Maxwell case, we show the existence of static solutions for (nonlinear–)KleinGordon and Yang-Mills equations in a SAdS background In both cases, the basic properties of the matter fields in the presence of backreaction are already present
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