AI 中文总结
本文针对极大延拓伪黎曼流形，建立了奇点与无穷远点可分性的充分条件，构造时空包络并结合端点定理分析纯奇点，最终将结果应用于史瓦西时空。
AI 中文摘要
理想附着点是广义相对论中针对伪黎曼流形的核心概念，而时空能否以特定性质延拓是选择理想附着点时的核心考量。本文针对任意极大延拓伪黎曼流形，建立了奇点与无穷远点可分性的充分条件。我们聚焦于时空$(\textit{M},g)$的不完备测地线，构造了该时空的一个包络$(\textit{M},g,\tilde{\textit{M}})$，使得不完备测地线$\boldsymbol{\tau}:[0,1) \rightarrow \textit{M}$在$\tilde{\textit{M}}$中存在端点$q$。若不存在一对趋近于$q$且相互缠绕的测地线，则$q$为奇点。此外，无穷仿射参数的测地线不会趋近于$q$，因此$q$无法覆盖无穷远点，从而在抽象边界框架中成为“纯奇点”。我们将端点定理应用于Graf与Beld-Serrano在arXiv:2307.11034中引入的极大g边界，还给出了方向奇点与纯奇点可分性的结果，并将该分析应用于史瓦西时空。
英文摘要
Ideal attached points are a core concept in general relativity for pseudo-Riemannian manifolds, and whether the spacetime can be extended with certain properties is a central consideration in their choice. This paper establishes a sufficient condition for the separability between singularities and points at infinity for any maximally extended pseudo-Riemannian manifold. We focus on the incomplete geodesics of $(\mathcal{M},g)$, and produce an envelopment $(\mathcal{M},g,\hat{\mathcal{M}})$ of the spacetime such that an incomplete geodesic $γ:[0,1) \rightarrow \mathcal{M}$ has an endpoint $q$ in $\hat{\mathcal{M}}$. If there is no pair of geodesics approaching $q$ which is intertwined, then $q$ is a singularity. Additionally, $q$ will not be approached by any geodesic with infinite affine parameter, and therefore cannot cover a point at infinity, thereby rendering it a {\it pure singularity} in the abstract boundary framework. We apply the Endpoint Theorem to the maximal g-boundary introduced by Graf and Beld-Serrano in arXiv:2307.11034, and also provide a result on the separability between directional singularities and pure singularities. This analysis is then applied to the Schwarzschild spacetime.
Comments37 pages, 4 figures
