Narrow Results

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.

We obtain spinning boson star solutions and hairy black holes with synchronized hair in the Einstein–Klein–Gordon model, wherein the scalar field is massive, complex and with a nonminimal coupling to the Ricci scalar. The existence of these hairy black holes in this model provides yet another manifestation of the universality of the synchronization mechanism to endow spinning black holes with hair. We study the variation of the physical properties of the boson stars and hairy black holes with the coupling parameter between the scalar field and the curvature, showing that they are, qualitatively, identical to those in the minimally coupled case. By discussing the conformal transformation to the Einstein frame, we argue that the solutions herein provide new rotating boson star and hairy black hole solutions in the minimally coupled theory, with a particular potential, and that no spherically symmetric hairy black hole solutions exist in the nonminimally coupled theory, under a condition of conformal regularity.

We consider a general Einstein–scalar–Gauss-Bonnet theory with a coupling function f (ϕ) between the scalar field and the quadratic gravitational Gauss-Bonnet term. We show that the existing no-hair theorems are easily evaded, and therefore black holes may emerge in the context of this theory. Indeed, we demonstrate that, under mild only assumptions for f (ϕ), asymptotic solutions describing either a regular black-hole horizon or an asymptotically-flat solution always emerge. We then show, through numerical integration, that the field equations allow for the smooth connection of these asymptotic solutions, and thus for the construction of a complete, regular black-hole solution with non-trivial scalar hair. We present and discuss the physical characteristics of a large number of such solutions for a plethora of coupling functions f (ϕ). Finally, we investigate whether pure scalar-Gauss-Bonnet black holes may arise in the context of our theory when the Ricci scalar may be altogether ignored.