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

TL;DR

This paper investigates static black hole solutions with a conformally coupled scalar field, revealing that physically acceptable solutions with a cosmological horizon are unstable or non-existent when stability is considered.

Contribution

It characterizes the solution space of such black holes, explains the instability of known solutions, and shows no stable solutions exist with a cosmological horizon.

Findings

The solution space is a singular subset of a two-dimensional parameter space.

The known MTZ solutions are unstable under linear perturbations.

No physically acceptable stable black hole solutions with a cosmological horizon exist in this model.

Abstract

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{\'\i}nez, Troncoso, and Zanelli, (MTZ), were subsequently shown to be unstable under linear 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 parameterized 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 MTZ 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.