Asymptotically flat black holes with scalar hair: a review

TL;DR

This review maps the landscape of four-dimensional, asymptotically flat black holes with scalar hair, detailing no-hair theorems and the broad set of loopholes that yield hairy solutions. It systematizes results by which no-hair assumption is violated (minimally coupled vs non-minimal scalars, energy conditions, or symmetry inheritance) and surveys canonical, conformal, Horndeski/Galileon, and multi-scalar frameworks, including Skyrme and boson-star analogs. Key takeaways include that scalar hair often appears as secondary and is tied to specific couplings or symmetries, with Kerr black holes hosting primary hair only in particular complex-scalar setups; many hairy solutions are still subject to stability questions and depend on the underlying model. The article also highlights the potential phenomenological relevance and outlines the current state-of-the-art against which future observations and simulations can be benchmarked.

Abstract

We consider the status of black hole solutions with non-trivial scalar fields but no gauge fields, in four dimensional asymptotically flat space-times, reviewing both classical results and recent developments. We start by providing a simple illustration on the physical difference between black holes 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; 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.