Existence and absence of Killing horizons in static solutions with symmetries

Hideki Maeda; Cristian Martinez

doi:10.1088/1361-6382/ad8ea4

paper

Existence and absence of Killing horizons in static solutions with symmetries

arXiv:2402.11012 · doi:10.1088/1361-6382/ad8ea4

Abstract

Without specifying a matter field nor imposing energy conditions, we study Killing horizons in $n (\geq 3)$ -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 $p_{2}$ of the matter field, we prove that any non-vacuum solution satisfying $χ_{r} < - 1/3$ ( $χ_{r} \neq = - 1$ ) or $χ_{r} > 0$ does not admit a horizon as it becomes a curvature singularity. For $χ_{r} = - 1$ and $χ_{r} \in [- 1/3, 0)$ , 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 $χ_{r} \in [- 1/3, 0)$ . 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.

arXiv comment51 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 $χ_{r}$ and $χ_{t}$ can be attached regularly at the horizon

Existence and absence of Killing horizons in static solutions with symmetries

Abstract

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