The quasi-topological space of quasi-topological gravities

Abstract

We investigate general properties of quasi-topological theories of gravity involving infinite towers of polynomial densities and of their regular vacuum black hole solutions. We show that every positive-mass regular vacuum solution necessarily possesses a de Sitter core, while Minkowski and anti-de Sitter cores are excluded. Regarding the horizon structure, we prove that asymptotically flat and anti-de Sitter regular black holes always admit extremal configurations. We find that there is no upper bound on the number of horizons and that horizons of arbitrarily high degeneracy can arise. These results extend to more general theories of gravity with second-order equations on spherically symmetric backgrounds. Finally, we characterize the space of quasi-topological theories admitting regular black holes, showing that it is contractible (in particular, path-connected and simply connected) and that regularity is stable under suitably small deformations of the higher-order couplings. In the sense of Spanier, this space is naturally a contractible \emph{quasi-topological} space. Additionally, we prove that any theory defined by a set of couplings up to any arbitrarily high, but finite, curvature order can be completed into a theory yielding regular black hole solutions, while preserving the same set of lower-order couplings. These results show that regular solutions arise in a broad class of quasi-topological gravities and do not require fine-tuning of the couplings.