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 n (3) -dimensional static solutions in general relativity with an (n-2) -dimensional Einstein base manifold. Assuming linear relations p ₑ ₑ and p₂ ₓ near a Killing horizon between the energy density, radial pressure p ₑ, and tangential pressure p₂ of the matter field, we prove that any non-vacuum solution satisfying ₑ0 does not admit a horizon as it becomes a curvature singularity. While many exact solutions with horizons are known for ₑ=-1, we construct asymptotic solutions near a Killing horizon for ₑ[-1/3, 0) and show that a matter field is absent on the horizon for ₑ (-1/3, 0). In contrast, there exists a matter on the horizon for ₑ=-1/3, which is of the Hawking-Ellis type II and may be interpreted as a null dust fluid. Differentiability of the metric on the horizon depends on the value of ₑ and such solutions can be attached to the Schwarzschild-Tangherlini-type vacuum solution at the Killing horizon in a C^1, 1 regular manner without a lightlike thin-shell. Additionally, some of those results are generalized in Lovelock gravity with a maximally symmetric base manifold.