Exact hairy black holes and their modification to the universal law of gravitation

In this paper two things are done. First, it is pointed out the existence of exact asymptotically flat, spherically symmetric black holes when a self-interacting, minimally coupled scalar field is the source of the Einstein equations in four dimensions. The scalar field potential is recently found to be compatible with the hairy generalization of the Plebanski-Demianski solution of general relativity. This paper describes the spherically symmetric solutions that smoothly connect the Schwarzschild black hole with its hairy counterpart. The geometry and scalar field are everywhere regular except at the usual Schwarzschild-like singularity inside the black hole. The scalar field energy momentum tensor satisfies the null-energy condition in the static region of spacetime. The first law holds when the parameters of the scalar field potential are fixed under thermodynamical variation. Second, it is shown that an extra, dimensionless parameter, present in the hairy solution, allows to modify the gravitational field of a spherically symmetric black hole in a remarkable way. When the dimensionless parameter is increased, the scalar field generates a flat gravitational potential that, however, asymptotically matches the Schwarzschild gravitational field. Finally, it is shown that a positive cosmological constant can render the scalar field potential convex if the parameters are within a specific rank.