时空因果结构与斯科特拓扑学

IF 0.9 Q2 MATHEMATICS
Langelihle Mazibuko, Dharmanand Baboolal, Rituparno Goswami
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引用次数: 0

摘要

本文通过随机结构将时空流形建立为偏序集。我们证明了这些部分有序集合是自然连续的,因为可以通过时间顺序来建立下面的关系。我们进一步考虑那些可以通过物理定义连接点和连接点来赋予晶格结构的时空类。通过考虑空时流形上零测地线的物理性质,我们证明了这些格是必然分布的。这些格是连续的,因为下面的关系和时间顺序是等价的。这使我们能够在时空流形上定义斯科特拓扑,并在与任何其他连续晶格相同的基础上描述它。我们进一步证明Scott拓扑是Alexandroff拓扑的一个适当子集,它必须是强因果时空的流形拓扑(因此是比Alexandroff更粗糙的拓扑)。在这个过程中,我们发现了一些关于这些流形的清醒性的有趣结果。证明了它们在Scott拓扑下不清醒,但在Alexandroff拓扑下恢复清醒。通过赋予这些流形双连续偏置结构,我们定义了对偶Scott拓扑,并证明了Scott拓扑与对偶的连接是Alexandroff拓扑。我们还讨论了在这个主题中所做的以前的工作,以及目前的工作如何在某种程度上概括这些结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Causal structure of spacetime and Scott topology

In this paper we establish the spacetime manifold as a partially ordered set via the casual structure. We show that these partially ordered sets are naturally continuous as a suitable way below relation can be established via the chronological order. We further consider those classes of spacetimes on which a lattice structure can be endowed by physically defining the joins and meets. By considering the physical properties of null geodesics on the spacetime manifold we show that these lattices are necessarily distributive. These lattices are then continuous as a result of the equivalence between the way below relation and chronology. This enables us to define the Scott topology on the spacetime manifold and describe it on an equal footing as any other continuous lattice. We further show that the Scott topology is a proper subset of Alexandroff topology, which must be the manifold topology for the strongly causal spacetimes, (and hence a coarser topology than Alexandroff). In the process we find some interesting results on the sobriety of these manifolds. We prove that they are necessarily not sober under the Scott topology but regain their sobriety under Alexandroff topology. We also define a dual Scott topology on these manifolds by endowing them with bicontinuous poset structure and show that the join of the Scott topology with the dual is the Alexandroff topology. We also discuss the previous works done in this topic and how the present work generalises those results to some extent.

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来源期刊
Afrika Matematika
Afrika Matematika MATHEMATICS-
CiteScore
2.00
自引率
9.10%
发文量
96
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