论文标题
Wolfram模型的一些相对论和重力特性
Some Relativistic and Gravitational Properties of the Wolfram Model
论文作者
论文摘要
Wolfram模型是对Stephen Wolfram在新科学（NKS）中首先引入的模型的轻微概括，它是一种离散的时代形式主义，其中空间由set系统上的抽象替换操作确定的超图表示，在集合系统上的动力学确定了空间，并且在该空间的相结合结构中代表了causal图表。本文的目的是在NKS中首次讨论，包括大量遵守特殊和一般相对性的离散形式，以呈现此类模型的许多关键特性的严格数学推导。首先，我们证明因果不变性（即，所有因果图都是同构的要求，无论选择HyperGraph Updating Order Order Order Order Order订单如何）等于一般协方差的一般协方差，并且对与离散量规转换相对应的更新顺序的更改。然后，这一事实允许人们推断出Lorentz协方差的离散类似物，以及离散Lorentz转换的结果物理后果。我们还为超图引入了Riemann和RICCI曲率的离散概念，并证明在因果图中离散时空锥体的校正因子的校正因子与固定尺寸的弯曲时空相对应的校正因子与固定尺寸的弯曲时空相对应与该规则的时间表的预测相同，并将其定义为dive the dimevers，并随后将其保存（随后的ricci tensor）（随后的dimersive）（随后又是该尺寸）。在限制情况下的因果图），以证明离散时空RICCI张量上最一般的约束集对应于爱因斯坦磁场方程的离散形式。
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.
