Wolfram模型的一些相对论性质和引力性质
Some Relativistic and Gravitational Properties of the Wolfram Model
摘要：
沃尔夫勒姆模型（英语：Wolfram Model），是斯蒂芬·沃尔夫勒姆在《一种新科学》（A New Kind of Science，NKS）中首次提出的模型的一个轻微推广，是一种离散时空形式主义，其中空间由超图表示，超图的动力学由集合系统上的抽象置换运算确定，时空的共形结构由因果图表示。这篇文章的目的是提出严格的数学推导的许多关键性质的连续极限，如第一次讨论的NKS，包括事实上，他们的大类服从离散形式的狭义和广义相对论。首先，我们证明了因果不变性（即要求所有因果图是同构的，无论选择超图更新顺序）是等价于一般协方差的离散版本，与离散规范变换对应的更新顺序的变化。这一事实允许我们推导出洛伦兹协方差的离散模拟，以及离散洛伦兹变换的物理结果。我们还引入了超图的离散Riemann曲率和Ricci曲率的概念，并证明了对应于固定维数弯曲时空的因果图中离散时空锥的体积的校正因子与离散时空Ricci张量的类时投影成正比，随后利用这一事实（以及更新规则在极限情况下保持因果图的维度的假设）证明离散时空的Ricci张量的最一般的约束集合对应于爱因斯坦场方程的离散形式。
英文摘要：
The Wolfram Model, which is a slight generalization of the model first introduced by Stephen Wolfram in A New Kind of Science (NKS), is a discrete spacetime formalism in which space is represented by a hypergraph whose dynamics are determined by abstract replacement operations on set systems, and in which the conformal structure of spacetime is represented by a causal graph. The purpose of this article is to present rigorous mathematical derivations of many key properties of such models in the continuum limit, as first discussed in NKS, including the fact that large classes of them obey discrete forms of both special and general relativity. First, we prove that causal invariance (namely, the requirement that all causal graphs be isomorphic, irrespective of the choice of hypergraph updating order) is equivalent to a discrete version of general covariance, with changes to the updating order corresponding to discrete gauge transformations. This fact then allows one to deduce a discrete analog of Lorentz covariance, and the resultant physical consequences of discrete Lorentz transformations. We also introduce discrete notions of Riemann and Ricci curvature for hypergraphs, and prove that the correction factor for the volume of a discrete spacetime cone in a causal graph corresponding to curved spacetime of fixed dimensionality is proportional to a timelike projection of the discrete spacetime Ricci tensor, subsequently using this fact (along with the assumption that the updating rules preserve the dimensionality of the causal graph in limiting cases) to prove that the most general set of constraints on the discrete spacetime Ricci tensor corresponds to a discrete form of the Einstein field equations.
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