基于Wolfram模型演化的广义相对论柯西问题超图离散化
Hypergraph Discretization of the Cauchy Problem in General Relativity via Wolfram Model Evolution
摘要：
虽然传统形式的爱因斯坦场方程本质上是四维的，但数值广义相对论领域的重点是将这些方程重新表述为3 + 1维柯西问题，其中柯西初始数据在三维空间超曲面上指定，然后随时间向前演化。Wolfram模型将爱因斯坦场方程的固有离散形式作为先验柯西问题，其中柯西初始数据被指定在单个空间超图上，然后通过超图替换规则进行演化，由此产生的因果网络对应于时空的共形结构。本文介绍了一种新的数值广义相对论程序，它基于具有约束破坏阻尼的共形和协变Z4（CCZ 4）公式，如果需要，可以选择简化为标准的BSSN公式，并将柯西数据定义在超图上;该代码结合了由Berger和Colella提出的自适应网格细化技术的非结构化概括，其中基于局部共形曲率项来细化或粗化超图的拓扑。我们验证这个代码对各种标准的时空，包括史瓦西黑洞，克尔黑洞，最大限度地扩大史瓦西黑洞，和二进制黑洞合并（旋转和非旋转），并明确说明离散超图拓扑结构和连续黎曼几何之间的关系，被近似。最后，我们比较这个代码产生的结果，通过纯Wolfram模型演化（没有潜在的PDE系统），使用可证明满足连续极限的爱因斯坦场方程的超图替换规则得到的结果。
英文摘要：
Although the traditional form of the Einstein field equations is intrinsically four-dimensional, the field of numerical general relativity focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy problem, in which Cauchy initial data is specified on a three-dimensional spatial hypersurface, and then evolved forwards in time. The Wolfram model offers an inherently discrete formulation of the Einstein field equations as an a priori Cauchy problem, in which Cauchy initial data is specified on a single spatial hypergraph, and then evolved by means of hypergraph substitution rules, with the resulting causal network corresponding to the conformal structure of spacetime. This article introduces a new numerical general relativity code based upon the conformal and covariant Z4 (CCZ4) formulation with constraint-violation damping, with the option to reduce to the standard BSSN formalism if desired, with Cauchy data defined over hypergraphs; the code incorporates an unstructured generalization of the adaptive mesh refinement technique proposed by Berger and Colella, in which the topology of the hypergraph is refined or coarsened based upon local conformal curvature terms. We validate this code numerically against a variety of standard spacetimes, including Schwarzschild black holes, Kerr black holes, maximally extended Schwarzschild black holes, and binary black hole mergers (both rotating and non-rotating), and explicitly illustrate the relationship between the discrete hypergraph topology and the continuous Riemannian geometry that is being approximated. Finally, we compare the results produced by this code to the results obtained by means of pure Wolfram model evolution (without the underlying PDE system), using a hypergraph substitution rule that provably satisfies the Einstein field equations in the continuum limit.
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