Black Hole Mass/Charge Relation and Weak No-hair Theorem Conjecture

TL;DR

This paper investigates Einstein gravity with two Maxwell fields and a scalar, deriving relations among black hole parameters that support a weak no-hair theorem, showing the scalar charge is not independent.

Contribution

It provides explicit formulae linking mass and charges in black holes with scalar hair, confirming the weak no-hair theorem for a specific scalar field.

Findings

Scalar charge depends on mass and electric charges

Explicit relations allow determining scalar charge without solving equations

Supports the weak no-hair theorem in this model

Abstract

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 $\Sigma$. 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 $\Sigma(M,Q_1,Q_2)$ 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.