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 $n(\ge 3)$-dimensional static solutions in general relativity with an $(n-2)$-dimensional Einstein base manifold. Assuming linear relations $p_{\mathrm{r}}\simeq {\chi }_{\mathrm{r}}\rho$ and $p_2\simeq {\chi }_{\mathrm{t}}\rho$ near a Killing horizon between the energy density $\rho$, radial pressure $p_{\mathrm{r}}$, and tangential pressure $p_2$ of the matter field, we prove that any non-vacuum solution satisfying ${\chi }_{\mathrm{r}}<-1\mathrm{/}3$ (${\chi }_{\mathrm{r}}\mathrm{\ne }-1$) or ${\chi }_{\mathrm{r}}>0$ does not admit a horizon as it becomes a curvature singularity. For ${\chi }_{\mathrm{r}}=-1$ and ${\chi }_{\mathrm{r}}\in [-1\mathrm{/}3,0)$, non-vacuum solutions admit Killing horizons, on which there exists a matter field only for ${\chi }_{\mathrm{r}}=-1$ and $-1\mathrm{/}3$, which are of the Hawking-Ellis type~I and type~II, respectively. Differentiability of the metric on the horizon depends on the value of ${\chi }_{\mathrm{r}}$, and non-analytic extensions beyond the horizon are allowed for ${\chi }_{\mathrm{r}}\in [-1\mathrm{/}3,0)$. 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.