Asymptotically flat black holes with scalar hair: a review

TL;DR

This review discusses the existence, properties, and theorems related to asymptotically flat black holes with scalar hair, highlighting recent developments and the conditions under which scalar hair can or cannot exist.

Contribution

It provides a comprehensive overview of classical and recent results on scalar-haired black holes, clarifying the assumptions behind no-hair theorems and cataloging known solutions.

Findings

Summary of no-scalar-hair theorems and their assumptions

Catalog of known scalar-haired black hole solutions

Analysis of conditions allowing scalar hair presence

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.