齐次洛伦兹叶状结构的时空闭包定理
Spacetime closedness theorem for homogeneous Lorentzian foliations
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中文总结 AI 辅助
该文提出齐次洛伦兹叶状结构的时空闭包定理，通过分析步骤从大爆炸和时间控制得有限切片体积，再经几何步骤提升为紧致性，并给出Lean 4形式化证明。
中文摘要 AI 辅助
受空间齐次宇宙学模型几何解释的启发，我们直接在洛伦兹层次上提出了一个时空闭包定理。经典的空间形式分类定理组织了齐次且各向同性的空间几何，但仅凭它本身并不能得出关于环境时空的洛伦兹陈述。主要技术步骤是从叶状结构的真正洛伦兹控制假设中推导出有限的切片体积。这在一个强版本中得以实现，适用于具有有限时间大爆炸和齐次完备类空切片的极大正则全局双曲$(n+1)$维时空；同时也在一个较弱版本中实现，其中极大正则性被累积膨胀率的时间可积性所取代。在两种情况下，论证都区分了一个分析步骤（从大爆炸和时间控制推导出有限切片体积）和一个几何步骤（通过齐次性和完备性将有限体积提升为紧致性）。该定理在任意时空维度下陈述，并附有强和弱抽象陈述的Lean~4形式化。
英文摘要
Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic $(n+1)$--spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean~4 formalization of the strong and weak abstract statements.
发表机构
- CUNEF Universidad（CUNEF大学）
- Universidad de Chile（智利大学）
机构由 AI 辅助整理，请以论文原文为准。
