Dressing a black hole with non-minimally coupled scalar field hair

We investigate the possibility of dressing a four-dimensional black hole with classical scalar field hair which is non-minimally coupled to the spacetime curvature. Our model includes a cosmological constant but no self-interaction potential for the scalar field. We are able to rule out black-hole hair except when the cosmological constant is negative and the constant governing the coupling to the Ricci scalar curvature is positive. In this case, non-trivial hairy black-hole solutions exist, at least some of which are linearly stable. However, when the coupling constant becomes too large, the black-hole hair becomes unstable.