发表机构
Università degli Studi di Pisa（比萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文介绍连续偏序集的时空几何方法，补充相关证明并获新结果，提出拓扑Kronheimer-Penrose因果空间概念，证明相关完备化等价性，建议时空本质为余连续偏序集。
AI 中文摘要
本文向读者介绍时空几何的连续偏序集方法，该主题由Martin和Panangaden开创。我们提供所有来自标准域论文献的相关证明，以方便来自洛伦兹几何和广义相对论领域的读者过渡到该主题。随后，我们呈现时空解释的新结果：在将 way-below 关系等同于时序关系 I 的前提下，我们确定了 ≤ 的唯一选择以及因果条件，以获得特定的偏序集性质。连续偏序集要求 ≤ = D_p，对应“过去区分且未来反射”；双连续偏序集是具有 ≤ = D 的因果连续时空；全局双曲偏序集恰好是具有 ≤ = J 的全局双曲时空，证明了超出Martin和Panangaden充分性结果的必要性。该理论完全以偏序集表述，具有低正则性。我们引入拓扑Kronheimer-Penrose因果空间的概念，其足够通用以涵盖文献中的洛伦兹长度空间，并给出使其成为（双）连续偏序集的弱条件。一旦时空成为连续偏序集，域论构造可直接应用，例如完备化方案可生成时空边界。我们回顾其中几种；最适合保持连续性的Lawson圆理想完备化被证明等于未来GKP完备化。最近Gigli等人研究的定向完备化也被证明等价于未来GKP完备化；然而，我们能够移除其额外的SC条件和向前近似。最后，基于过去反射性在黑洞蒸发中近期确立的关键作用的物理考量，我们提出时空在基本层面上是余连续偏序集。
英文摘要
We introduce the reader to the continuous posets approach to spacetime geometry, a topic pioneered by Martin and Panangaden (MP), supplying the relevant domain-theoretic proofs to ease the transition for readers from Lorentzian geometry and general relativity. We then present new results for the spacetime interpretation. Identifying the way-below relation with the chronological relation $I$, we determine the unique choice of $\le$ and of causality condition yielding each poset property: continuous posets require $\le\,=D_p$ and ``past-distinction, (SC) and future-reflectivity'', bicontinuous posets are causally continuous spacetimes with $\le\,=D$, and globally hyperbolic posets are precisely globally hyperbolic spacetimes with $\le\,=J$, a necessity result beyond MP's sufficiency. Causal simplicity is recovered through a chain realizability condition, stable causality through the Hausdorffness of a topology $τ$, so the causal ladder is fully determined at the abstract level, solving a problem posed by MP. Being phrased entirely in terms of a poset, the theory is of low regularity. We introduce topological Kronheimer--Penrose causal spaces, general enough to encompass the Lorentzian length spaces in the literature, and give weak conditions making them (bi)continuous posets. Domain-theoretic constructions then apply directly; e.g., Lawson's round-ideal completion, the most natural for preserving continuity, is proved to equal the future GKP completion, as is the directed completion of Gigli et al., for which we remove the (SC) assumption under strong causality. Reversing the logical order of the theory, we characterize the relations that induce a continuous poset having them as way-below relation. Finally, physical considerations on the key role of past-reflectivity in black hole evaporation lead us to suggest that spacetime is, at the fundamental level, a co-continuous poset.
Comments98 pages. v3: considerably expanded to develop the full causal ladder for continuous posets and other results
