Note on scalar fields non-minimally coupled to (2+1)-gravity

Scalar fields nonminimally coupled to (2+1)-gravity, in the presence of cosmological constant term, are considered. Nonminimal couplings are described by the term ζ R Ψ2 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 ζ=0, and the Martı́nez–Zanelli black hole for ζ=1/8. We obtain also two new solutions with nontrivial scalar field, for ζ=1/6 and ζ=1/8, respectively, nevertheless, the corresponding space–times can be reduced, via coordinate transformations, to the standard anti-de Sitter space.