On the Existence of Solutions with a Horizon in Pure Scalar-Gauss-Bonnet Theories

Abstract

We consider the Einstein-scalar-Gauss-Bonnet theory and assume that, at regimes of large curvature, the Ricci scalar may be ignored compared to the quadratic Gauss-Bonnet term. We then look for static, spherically-symmetric, regular black-hole solutions with a non-trivial scalar field. Despite the use of a general form of the spacetime line-element, no black-hole solutions are found. In contrast, solutions that resemble irregular particle-like solutions or completely regular gravitational solutions with a finite energy-momentum tensor do emerge. In addition, in the presence of a cosmological constant, solutions with a horizon also emerge, however, the latter corresponds to a cosmological rather than to a black-hole horizon. It is found that, whereas the Ricci term works towards the formation of the positively-curved topology of a black-hole horizon, the Gauss-Bonnet term exerts a repulsive force that hinders the formation of the black hole. Therefore, a pure scalar-Gauss-Bonnet theory cannot sustain any black-hole solutions. However, it could give rise to interesting cosmological or particle-like solutions where the Ricci scalar plays a less fundamental role.