Black hole mass/charge relation and weak no-hair theorem conjecture

A bstract We consider Einstein gravity minimally coupled to two Maxwell fields and one (real) dilaton scalar. We study the electrically-charged spherically-symmetric and static solutions that are asymptotic to Minkowski spacetime. General solutions are described by four independent parameters: mass M , two electric charges ( Q 1 , Q 2 ) and the scalar charge Σ. For black holes, the scalar charge is not independent, but a function of ( M , Q 1 , Q 2 ). We provide a set of formulae relating the mass to the charges, which allows us to determine Σ( M , Q 1 , Q 2 ) without having to solve the black hole equations. Our results confirm the weaker version of the no-hair theorem conjecture involving one real scalar: it can be turned on in a black hole, but it does not have a continuous independent hairy parameter.