Existence and absence of Killing horizons in static solutions with symmetries

Abstract

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 r ≃ χ r ρ and p 2 ≃ χ t ρ near a Killing horizon between the energy density ρ, radial pressure p r , and tangential pressure p2 of the matter field, we prove that any non-vacuum solution satisfying χ r < − 1 / 3 ( χ r ≠ − 1 ) or χ r > 0 does not admit a horizon as it becomes a curvature singularity. For χ r = − 1 and χ r ∈ [ − 1 / 3 , 0 ) , non-vacuum solutions admit Killing horizons, on which there exists a matter field only for χ r = − 1 and − 1 / 3 , which are of the Hawking-Ellis type I and type II, respectively. Differentiability of the metric on the horizon depends on the value of χ r , and non-analytic extensions beyond the horizon are allowed for χ r ∈ [ − 1 / 3 , 0 ) . In particular, solutions can be attached to the Schwarzschild-Tangherlini-type vacuum solution at the Killing horizon in at least a C 1 , 1 regular manner without a lightlike thin shell. We generalize some of those results in Lovelock gravity with a maximally symmetric base manifold.