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 $n(\ge 3)$-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 ${\chi }_{\mathrm{r}}<-1/3$ (${\chi }_{\mathrm{r}}\ne -1$) or ${\chi }_{\mathrm{r}}>0$ does not admit a horizon as it becomes a curvature singularity. For ${\chi }_{\mathrm{r}}=-1$ and ${\chi }_{\mathrm{r}}\in [-1/3,0)$, non-vacuum solutions admit Killing horizons, on which there exists a matter field only for ${\chi }_{\mathrm{r}}=-1$ 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 ${\chi }_{\mathrm{r}}$, and non-analytic extensions beyond the horizon are allowed for ${\chi }_{\mathrm{r}}\in [-1/3,0)$. 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.

51 pages, 1 figure, 2 tables; v3, this arXiv version has Appendix D based on the addendum (2025 Class. Quantum Grav. 42, 129401) to the published version to clarify that even two solutions with different values of ${\chi }_{\mathrm{r}}$ and ${\chi }_{\mathrm{t}}$ can be attached regularly at the horizon

References in corpus (12)

• A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
• Lovelock black holes with maximally symmetric horizons
• Gauss-Bonnet black holes with non-constant curvature horizons
• Horizons vs. singularities in spherically symmetric space-times
• On the smoothness of static multi-black hole solutions of higher-dimensional Einstein-Maxwell theory
• Black holes can have curly hair
• Buchdahl-like transformations for perfect fluid spheres
• Criteria for energy conditions
• The singularity theorems of General Relativity and their low regularity extensions
• Truly naked spherically-symmetric and distorted black holes
• Scalar field as a null dust
• Vacuum-dual static perfect fluid obeying $p=-(n-3)\rho /(n+1)$ in $n(\ge 4)$ dimensions

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