Black holes in general relativity with scalar field There is famous  Reissner–Nordström solution in general relativity coupled with Maxwell theory.
One can naturally be intrested in similar solutions in general relativity coupled to scalar field. I have found some examples of such solutions:

*

*Conformally coupled scalar field:




*Other examples are here
What is known about such solutions? Are there any interesting examples of such solutions? Are there any general statements about such solutions?
 A: The interesting part is that the scalar field might bring scalar hair in the black hole that can be detected at infinity by an observer.
There are two types of scalar hair:
Primary scalar hair: There exists a constant (the scalar charge) that appears in the scalar field that is not related to any of the usual parameters of a black hole (mass, charge, angular momentum).
Secondary scalar hair: There exists a parameter in the scalar field that is related to a black hole parameter. For example the BBMB black hole in the photo is a black hole with secondary scalar hair since the parameter of the scalar field $m$ is related to the black hole mass.
A Black hole with primary scalar hair is reported in https://arxiv.org/abs/1309.2161. The parameter $\nu$ is the scalar charge that controls the behavior of the scalar field. In equation (30) the scalar charge $\nu$ appears in an $\mathcal{O}(r)$ term. The parameter $\nu$ is independent of the black hole mass.
The BBMB Black hole, the one reported at https://arxiv.org/abs/hep-th/0205319, or the MTZ black hole https://arxiv.org/abs/hep-th/0406111 are all black holes with secondary scalar hair. The $\mu$ parameter of the scalar field in the MTZ black hole is related with the black hole mass which also happens with the parameter  $m$ in the BBMB black hole and de Sitter black hole with a conformally coupled scalar field in four dimensions.
If you try to solve the BBMB action, the scalar field configuration yields:
\begin{equation} \phi(r) = \cfrac{A}{r+B} \end{equation}
where $A,B$ are the scalar charges. In order for the function to be a solution of the field equations, $A$ and $B$ take specific values related to the black hole mass and therefore these solutions are solutions with secondary scalar hair.
For a review i suggest this paper https://arxiv.org/abs/1504.08209.
