arXiv 1508.005312015-08math.DGgr-qc

时空上的距离

Null distance on a spacetime

Christina Sormani, Carlos Vega

备注：: v2: Small changes/improvements, mainly in the Introduction, Section 3.7, and the opening of Section 4, references added, 39 pages, 8 figures. To appear in Classical and Quantum Gravity
发表信息：: Classical and Quantum Gravity, 33 (2016) no 8
DOI：: 10.1088/0264-9381/33/7/085001

摘要：

给定时空$M$上的一个时间函数$τ$，我们定义了一个“零距离函数”，$\hat{d}_τ$，它是从$M$的因果结构中建立的，并且与之密切相关。在具有类时$\nabla τ$的基本模型中，我们证明了：1）$\hat{d}_τ$是一个确定的距离函数，它导致了流形拓扑; 2）$M$的因果结构完全编码在$\hat{d}_τ$和$τ$中.一般来说，$\hat{d}_τ$是共形不变的伪度量，它可以是不定的。我们给出了一个“反Lipschitz”的条件$τ$，这确保了$\hat{d}_τ$是明确的，并显示这个条件得到满足时，$τ$的梯度向量$\nabla τ$几乎处处不在，与$\nabla τ$本地“有界远离光锥”。结果表明，[1]的宇宙学时间函数在“正则”时是反Lipschitz的，从而导出了一个确定的零距离函数.这提供了可以解释为时空上的正则度量空间结构的东西，它源自一个共同的初始奇点，例如一个“大爆炸”。

英文摘要：

Given a time function $τ$ on a spacetime $M$, we define a `null distance function', $\hat{d}_τ$, built from and closely related to the causal structure of $M$. In basic models with timelike $\nabla τ$, we show that 1) $\hat{d}_τ$ is a definite distance function, which induces the manifold topology, 2) the causal structure of $M$ is completely encoded in $\hat{d}_τ$ and $τ$. In general, $\hat{d}_τ$ is a conformally invariant pseudometric, which may be indefinite. We give an `anti-Lipschitz' condition on $τ$, which ensures that $\hat{d}_τ$ is definite, and show this condition to be satisfied whenever $τ$ has gradient vectors $\nabla τ$ almost everywhere, with $\nabla τ$ locally `bounded away from the light cones'. As a consequence, we show that the cosmological time function of [1] is anti-Lipschitz when `regular', and hence induces a definite null distance function. This provides what may be interpreted as a canonical metric space structure on spacetimes which emanate from a common initial singularity, e.g. a `big bang'.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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