A finite action for three dimensional gravity with a minimally coupled scalar field

TL;DR

This paper investigates three-dimensional gravity coupled with a scalar field, deriving finite actions with specific boundary conditions, exploring solutions, and analyzing their thermodynamics and holographic properties.

Contribution

It introduces a method to find counterterms for scalar fields in 3D gravity ensuring finite action and explores scalar black hole solutions and their thermodynamics.

Findings

Counterterms depend explicitly on the scalar field.

Scalar black holes can decay into BTZ black holes.

The central charge matches that of pure gravity in AdS/CFT.

Abstract

Three-dimensional gravity with a minimally coupled self-interacting scalar is considered. The fall-off of the fields at infinity is assumed to be slower than that of a localized distribution of matter, so that the asymptotic symmetry group is the conformal group. The counterterm Lagrangian needed to render the action finite is found by demanding that the action attain an extremum for the boundary conditions implied by the above fall-off of the fields at infinity. These counterterms explicitly depend on the scalar field. As a consequence, the Brown-York stress-energy tensor acquires a non trivial contribution from the matter sector. Static circularly symmetric solutions with a regular scalar field are explored for a one-parameter family of potentials. Their masses are computed via the Brown-York quasilocal stress-energy tensor, and they coincide with the values obtained from the Hamiltonian approach. The thermal behavior, including the transition between different configurations, is analyzed, and it is found that the scalar black hole can decay into the BTZ solution irrespective of the horizon radius. It is also shown that the AdS/CFT correspondence yields the same central charge as for pure gravity.