Asymptotically flat black holes with scalar hair: A review

We consider the status of black hole (BH) solutions with nontrivial scalar fields but no gauge fields, in four-dimensional asymptotically flat spacetimes, reviewing both classical results and recent developments. We start by providing a simple illustration on the physical difference between BHs in electro-vacuum and scalar-vacuum. Next, we review no-scalar-hair theorems. In particular, we detail an influential theorem by Bekenstein and stress three key assumptions: (1) The type of scalar field equation; (2) the spacetime symmetry inheritance by the scalar field and (3) an energy condition. Then, we list regular (on and outside the horizon), asymptotically flat BH solutions with scalar hair, organizing them by the assumption which is violated in each case and distinguishing primary from secondary hair. We provide a table summary of the state-of-the-art.