Conformally dressed black hole in 2+1 dimensions

TL;DR

The paper presents an exact black hole solution in $2+1$ dimensions with a negative cosmological constant conformally coupled to a massless scalar field. The solution is static, circularly symmetric and asymptotically AdS, with a curvature singularity at the origin and a regular scalar field that carries no independent hair. The horizon temperature is fixed by regularity at the horizon to $T=\frac{9 r_+}{16 \pi l^2}$, and Euclidean thermodynamics yields $M=\frac{3 r_+^2}{32 l^2}$ and $S=\frac{\pi r_+}{3}$, giving the first law $dM= T dS$ and a entropy $S=(2/3)(1/4)A$. The entropy deviates from the standard area law, reflecting the nontrivial scalar coupling, and underscores the role of boundary terms in obtaining consistent thermodynamics for conformal matter in curved spacetime.

Abstract

A three dimensional black hole solution of Einstein equations with negative cosmological constant coupled to a conformal scalar field is given. The solution is static, circularly symmetric, asymptotically anti-de Sitter and nonperturbative in the conformal field. The curvature tensor is singular at the origin while the scalar field is regular everywhere. The condition that the Euclidean geometry be regular at the horizon fixes the temperature to be $T=\frac{9\, r_+}{16πl^2}$. Using the Hamiltonian formulation including boundary terms of the Euclidean action, the entropy is found to be $\frac{2}{3}$ of the standard value ($\frac{1}{4} A$), and in agreement with the first law of thermodynamics.