标题： 类时测地线在渐近平坦时空中的存在性：一种广义的拓扑准则 显示英文标题

标题： Existence of Time-like Geodesics in Asymptotically Flat Spacetimes: A Generalized Topological Criterion

摘要： 本文研究了渐近平坦时空中类时测地线的存在性和性质，并提出了一个新颖的广义拓扑准则，用于判断类时测地线的存在性。 其有效性通过诸如Jordan-Brouwer分离定理、Raychaudhuri方程以及微分几何的关键要素进行证明。 更具体地说，证明主要依赖于时空流形的一个闭合、单连通子集和一个连续映射，导致从该子集的边界到单位圆的第一同调群的非平凡归纳。 数学分析明确证实了这些测地线的存在，并在所述时空子集中横向相交。 研究结果强调了这些测地线对渐近平坦时空结构的重要影响，包括稳定性以及虫洞假设存在的意义。 广义拓扑准则也对洛伦兹度量存在性的障碍问题以及爱因斯坦约束方程具有影响。 未来的研究应将这一拓扑准则扩展到其他类型的时空，包括具有非平凡拓扑或非零宇宙常数的时空。 此外，该准则在研究复杂动力系统（如引力波或旋转黑洞）中的应用可能会提供重要的见解。 显示英文摘要

摘要： This paper examines the issue of the existence and nature of time-like geodesics in asymptotically flat spacetimes and proposes a novel generalized topological criterion for the existence of time-like geodesics. Its validity is proved using theorems such as the Jordan-Brouwer Separation Theorem, the Raychaudhuri Equation, and key elements of Differential Geometry. More specifically, the proof primarily hinges on a closed, simply-connected subset of the spacetime manifold and a continuous map, causing a non-trivial induction on the first homology groups, from the boundary of this subset to a unit circle. The mathematical analysis conclusively affirms the presence of these geodesics, intersecting transversally within the said subset of spacetime. Findings underscore these geodesics' significant implications for the structure of asymptotically flat spacetimes, including stability, and hypothetical existence of wormholes. The generalized topological criterion also has implications on the problem of obstructions for the existence of Lorentzian metrics, and Einstein's Constraint Equations. Future research should extend this topological criterion to other classes of spacetimes, including those with non-trivial topologies or non-zero cosmological constants. Also, the criterion's application to study complex dynamical systems, such as gravitational waves or rotating black holes, could offer significant insights.