Asymptotically flat black holes with scalar hair: a review

Understand

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.