Black holes and asymptotics of 2+1 gravity coupled to a scalar field

TL;DR

The paper addresses how a slowly decaying scalar field coupled to $2+1$ dimensional gravity in AdS$\_3$ affects the asymptotic symmetry and conserved charges. It develops a framework with a self-interacting scalar whose fall-off $\phi \sim $ $\chi/r^{1/2}$ leaves Brown–Henneaux type boundary conditions invariant and computes the total charges via Regge–Teitelboim, showing the Virasoro algebra with central charge $c=\frac{3l}{2G}$ is retained. It then constructs an exact static black-hole solution with scalar hair for a one-parameter family of potentials $V_\nu(\phi)$, detailing the scalar profile, horizon structure $r_{+}=B\Theta_\nu$, and thermodynamics with $S=\frac{A}{4}$ and $C>0$, including a degenerate ground state at $\nu=-1$. The discussion highlights that scalar hair cannot be turned off at fixed mass, tests Cardy counting in light of the unchanged $c$, and shows the asymptotic structure remains robust under generalized potentials, contributing to the understanding of hair and microstate counting in low-dimensional gravity.

Abstract

We consider 2+1 gravity minimally coupled to a self-interacting scalar field. The case in which the fall-off of the fields at infinity is slower than that of a localized distribution of matter is analyzed. It is found that the asymptotic symmetry group remains the same as in pure gravity (i.e., the conformal group). The generators of the asymptotic symmetries, however, acquire a contribution from the scalar field, but the algebra of the canonical generators possesses the standard central extension. In this context, new massive black hole solutions with a regular scalar field are found for a one-parameter family of potentials. These black holes are continuously connected to the standard zero mass black hole.

Figures (1)

• Figure 1: Plot of $\Theta _{\nu }$ as a function of $\nu$. The intercepts are at $\Theta_{-1}=0$ and $\Theta_{0}=\frac{4}{3}$.