No Scalar Hair Theorem for a Charged Spherical Black Hole

TL;DR

This paper proves that charged spherical black holes cannot have scalar hair in general relativity and most scalar-tensor theories, except when matter fields are coupled to scalars, like in dilaton gravity.

Contribution

It establishes a no scalar hair theorem for charged black holes across various theories, clarifying conditions under which scalar hair can or cannot exist.

Findings

No scalar hair for charged black holes in general relativity.

The theorem extends to all scalar-tensor theories with positive effective Newtonian constant.

Exception identified in cases where matter fields couple to scalar fields, such as in dilaton gravity.

Abstract

This paper consolidates noscalar hair theorem for a charged spherically symmetric black hole in four dimension in general relativity as well as in all scalar tensor theories, both minimally and nonminimally coupled, when the effective Newtonian constant of gravity is positive. However, there is an exception when the matter field itself is coupled to the scalar field, such as in dilaton gravity.