Note on Scalar Fields Non-Minimally Coupled to (2+1)-Gravity

TL;DR

This paper investigates scalar fields non-minimally coupled to (2+1)-dimensional gravity with a cosmological constant, revealing known solutions like anti-de Sitter space and the Martinez-Zanelli black hole, as well as new solutions reducible to anti-de Sitter space.

Contribution

It identifies and analyzes solutions for scalar fields with specific non-minimal couplings in (2+1)-gravity, including new solutions that are equivalent to anti-de Sitter space.

Findings

Anti-de Sitter space for ζ=0

Martinez-Zanelli black hole for ζ=1/8

Two new solutions for ζ=1/6 and ζ=1/8 that reduce to anti-de Sitter space

Abstract

Scalar fields non--minimally coupled to (2+1)-gravity, in the presence of cosmological constant term, are considered. Non-minimal couplings are described by the term $\zeta R \Psi^2$ in the Lagrangian. Within a class of static circularly symmetric space-times, it is shown that the only existing physically relevant solutions are the anti-de Sitter space-time for $\zeta=0$, and the Martinez-Zanelli black hole for $\zeta=1/8$. We obtain also two new solutions with non-trivial scalar field, for $\zeta=1/6$ and $\zeta=1/8$ respectively, nevertheless, the corresponding space-times can be reduced, via coordinate transformations, to the standard anti-de Sitter space.