Without specifying a matter field nor imposing energy conditions, we study Killing horizons in $n(\ge 3)$-dimensional static solutions in general relativity with an $(n-2)$-dimensional Einstein base manifold. Assuming linear relations $p_{ \rm r}\simeq\chi_{ \rm r} \rho$ and $p_2\simeq\chi_{ \rm t} \rho$ near a Killing horizon between the energy density $\rho$, radial pressure $p_{ \rm r}$, and tangential pressure $p_2$ of the matter field, we prove that any non-vacuum solution satisfying $\chi_{ \rm r}<-1/3$ ($\chi_{ \rm r}\ne -1$) or $\chi_{ \rm r}>0$ does not admit a horizon as it becomes a curvature singularity. For $\chi_{ \rm r}=-1$ and $\chi_{ \rm r}\in[-1/3,0)$, non-vacuum solutions admit Killing horizons, on which there exists a matter field only for $\chi_{ \rm r}=-1$ 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 $\chi_{ \rm r}$, and non-analytic extensions beyond the horizon are allowed for $\chi_{ \rm r}\in[-1/3,0)$. 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.
Categories: gr-qc hep-th