Exact black hole solution with a minimally coupled scalar field

TL;DR

This work constructs an exact four-dimensional black hole solution in gravity with a negative cosmological constant and a minimally coupled self-interacting scalar field, featuring a horizon of negative constant curvature and locally AdS asymptotics. Using Euclidean action techniques, it derives the thermodynamics, including a positive specific heat and finite on-shell action despite scalar back-reaction, and identifies a second-order phase transition at $T_c=\frac{1}{2\pi l}$ between a scalar-dressed black hole and a vacuum black hole, with the two branches intersecting at the massless AdS configuration. The transition is controlled by an order parameter $\lambda=\left| \tanh \sqrt{\frac{4\pi G}{3}}\,\phi(r_+) \right|$, and a conformal-frame analysis in the Appendix reveals a richer causal structure for the solution. The results have potential implications for AdS/CFT contexts and the role of scalar hair in holographic phase transitions.

Abstract

An exact four-dimensional black hole solution of gravity with a minimally coupled self-interacting scalar field is reported. The event horizon is a surface of negative constant curvature enclosing the curvature singularity at the origin, and the scalar field is regular everywhere outside the origin. This solution is an asymptotically locally AdS spacetime. The strong energy condition is satisfied on and outside the event horizon. The thermodynamical analysis shows the existence of a critical temperature, below which a black hole in vacuum undergoes a spontaneous dressing up with a nontrivial scalar field in a process reminiscent of ferromagnetism.