Static black hole solutions with a self-interacting conformally coupled scalar field

We study static, spherically symmetric black hole solutions of the Einstein equations with a positive cosmological constant and a conformally coupled self-interacting scalar field. Exact solutions for this model found by Mart\'nez, Troncoso, and Zanelli were subsequently shown to be unstable under linear gravitational perturbations, with modes that diverge arbitrarily fast. We find that the moduli space of static, spherically symmetric solutions that have a regular horizon---and satisfy the weak and dominant energy conditions outside the horizon---is a singular subset of a two-dimensional space parametrized by the horizon radius and the value of the scalar field at the horizon. The singularity of this space of solutions provides an explanation for the instability of the Mart\'nez, Troncoso, and Zanelli spacetimes and leads to the conclusion that, if we include stability as a criterion, there are no physically acceptable black hole solutions for this system that contain a cosmological horizon in the exterior of its event horizon.