Black Holes with a Conformally Coupled Scalar Field

We consider gravity in presence of a cosmological constant in arbitrary spacetime dimensions with a conformally coupled scalar field and a self-interacting potential. The energy-momentum tensor is traceless when the constant appearing in front of the non-minimal coupling term and the power of the self-interacting potential are properly chosen. First, configurations with a constant scalar field are studied. In the general case, the spacetime is required to be an Einstein space. However, for a special value of the scalar field this condition can be relaxed and it is enough that the spacetime has a constant scalar curvature fixed by the cosmological constant. In this case the cosmological constant and the self-interacting coupling constant are related. The second part is devoted to searching black holes dressed with a conformally coupled scalar field in dimensions greater than four. Since the existence of a no-go theorem discarding static and asymptotically flat black holes in higher dimensions, we introduce a cosmological constant and a self-interacting potential in the action. Using a standard static ansatz for the metric, which includes both spherically symmetric as topological black holes, and a scalar field depending only on the radial coordinate, it is shown that there are no higher-dimensional counterparts of the known black holes in three and four dimensions.