Hairy AdS black holes with a toroidal horizon in 4D Einstein-nonlinear $\sigma$-model system

Abstract

An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy-momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a discrete parameter which has neither topological nor Noether charge associated with it and therefore represents a hair. A $U(1)$ gauge field interacting with Einstein gravity can also be included. The thermodynamics is analyzed. Interestingly, the hairy black hole is always thermodynamically favored with respect to the corresponding black hole with vanishing Pionic field.

15 pages, 1 figure. Accepted for publication in Physics Letters B

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• Inconsistencies of nonmetric Einstein-Dirac-Maxwell theories and a cure for geometric flows of f(Q) black ellipsoid, toroid and wormhole solutions
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• General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations
• Geometrically Regular Black Holes with Hedgehog Scalar Hair