Static hairy black hole in 4D general relativity

In four-dimensional vacuum general relativity the only known static, exact and analytical black hole solution is given by the Schwarzschild spacetime. In this paper this renowned metric is generalized by adding another integrating constant, a hair that switches the metric from the Petrov type <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>D</a:mi> </a:math> to the type <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>I</c:mi> </c:math> . This new parameter represents the intensity of an external gravitational field, which can be considered the hyperbolic generalization of the Witten’s bubble of nothing. No curvature or conical singularities are present outside the event horizon. The no hair arguments are circumvented because the metric is not asymptotically flat, and neither is the black hole spherical. The gravitational hair continuously deforms the Schwarzschild geometry: the horizon becomes oblate, while its area is reduced. Conserved charges and thermodynamic properties of the black hole are studied.