Black hole uniqueness theorems and their violation due to scalar fields

Abstract

In General Relativity, the no-hair theorem states that the exterior geometry of a black hole is completely determined by its mass, M, charge, Q, and angular momentum, L. In this work, we first revisit the black hole uniqueness theorems within the Einstein-Maxwell theory, which establish the uniqueness of the Reissner-Nordstr¨om metric. We then consider the presence of a probe scalar field in the gravitational theory, located in the vicinity of a black hole. Our goal is to evaluate the validity of the no-hair theorem in this scenario. We establish that all interacting non-singular field solutions must be trivial, provided their squared mass is positive, m2> 0. Subsequently, we find a non-trivial solution in Anti-de Sitter spacetime (where fields with m2< 0 are allowed), which depends on a parameter determined by the field. Thus, we conclude that black hole solutions with scalar hair, which depend on parameters other than M, Q, and L, may exist in this spacetime