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Note on scalar fields non-minimally coupled to \((2+1)\)-gravity. (English) Zbl 0983.83037

Summary: Scalar fields nonminimally coupled to \((2+1)\)-gravity, in the presence of cosmological constant term, are considered. Nonminimal 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 Martínez-Zanelli black hole for \(\zeta=1/8\). We obtain also two new solutions with nontrivial 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.

MSC:

83C80 Analogues of general relativity in lower dimensions
81T20 Quantum field theory on curved space or space-time backgrounds
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