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 pᵣχᵣ ρ and p₂χₜ ρ near a Killing horizon between the energy density ρ, radial pressure pᵣ, and tangential pressure p₂ of the matter field, we prove that any non-vacuum solution satisfying χᵣ<-1/3 (χᵣ≠ -1) or χᵣ>0 does not admit a horizon as it becomes a curvature singularity. While many exact solutions with horizons are known for χᵣ=-1, we construct asymptotic solutions near a Killing horizon for χᵣ∈[-1/3,0) and show that a matter field is absent on the horizon for χᵣ∈(-1/3,0). In contrast, there exists a matter on the horizon for χᵣ=-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 χᵣ and such solutions can be attached to the Schwarzschild-Tangherlini-type vacuum solution at the Killing horizon in a C1,1 regular manner without a lightlike thin-shell. Additionally, some of those results are generalized in Lovelock gravity with a maximally symmetric base manifold.