Existence and absence of Killing horizons in static solutions with symmetries

Without specifying a matter field nor imposing energy conditions, we study Killing horizons in -dimensional static solutions in general relativity with an -dimensional Einstein base manifold. Assuming linear relations and near a Killing horizon between the energy density ρ, radial pressure , and tangential pressure p2 of the matter field, we prove that any non-vacuum solution satisfying ( ) or does not admit a horizon as it becomes a curvature singularity. For and , non-vacuum solutions admit Killing horizons, on which there exists a matter field only for and , which are of the Hawking–Ellis type I and type II, respectively. Differentiability of the metric on the horizon depends on the value of , and non-analytic extensions beyond the horizon are allowed for . In particular, solutions can be attached to the Schwarzschild–Tangherlini-type vacuum solution at the Killing horizon in at least a regular manner without a lightlike thin shell. We generalize some of those results in Lovelock gravity with a maximally symmetric base manifold.