Expanded evasion of the black hole no-hair theorem in dilatonic Einstein-Gauss-Bonnet theory
We study a hairy black hole solution in the dilatonic Einstein-Gauss-Bonnet theory of gravitation, in which the Gauss-Bonnet term is nonminimally coupled to the dilaton field. Hairy black holes with spherical symmetry seem to be easily constructed with a positive Gauss-Bonnet (GB) coefficient α within the coupling function, f(ϕ)=αeγϕ, in an asymptotically flat spacetime; i.e., no-hair theorem seems to be easily evaded in this theory. Therefore, it is natural to ask whether this construction can be expanded into the case with the negative coefficient α. In this paper, we numerically present the dilaton black hole solutions with a negative α, and we analyze the properties of GB terms through the aspects of the black hole mass. We construct the new integral constraint allowing the existence of the hairy solutions with the negative α. Through this procedure, we expand the evasion of the no-hair theorem for hairy black hole solutions.