标题： 非真空解，引力坍缩和沃尔夫拉姆模型系统中的离散奇点定理 显示英文标题

标题： Non-Vacuum Solutions, Gravitational Collapse and Discrete Singularity Theorems in Wolfram Model Systems

摘要： 著名的雷乔德里-共形方程，以及它所导致的彭罗斯和霍金的奇点定理，提供了一组高度普遍的条件，预期在这些条件下，时空（或更一般地说，伪黎曼流形）将变得测地不完整。 询问是否存在等效的普遍条件，以预期离散时空会变得测地不完整，以及这些条件与连续情况下的条件有何不同，这是一个非平凡的问题。 本文基于之前的工作，其中在真空时空的情况下，通过在离散（空间）超图上的Wolfram模型演化，定义了爱因斯坦场方程的柯西问题的共形和协变Z4（CCZ4）公式，并进一步考虑了对非真空情况的最小扩展，通过引入一个质量标量场分布，在球对称或轴对称情况下定义。 在适当的假设下，该标量场分布可以解释为坍缩的（在轴对称情况下，均匀旋转的）尘埃，并且我们能够通过严格数学分析和显式数值模拟相结合的方法，证明所得的离散时空分别渐近收敛到非旋转的施瓦茨希尔德黑洞解或最大旋转（极端）克尔黑洞解。 尽管获得这些初步结果所使用的假设非常强，但它们仍然提供了希望，或许最终可以证明一个更普遍的、可能类似于彭罗斯的奇点定理，同样适用于离散时空的情况。 显示英文摘要

摘要： The celebrated geodesic congruence equation of Raychaudhuri, together with the resulting singularity theorems of Penrose and Hawking that it enabled, yield a highly general set of conditions under which a spacetime (or, more generically, a pseudo-Riemannian manifold) is expected to become geodesically incomplete. It is a non-trivial question to ask whether (and to what extent) there exist equivalently general conditions under which one expects discrete spacetimes to become geodesically incomplete, and how these conditions might differ from those in the continuum. This article builds upon previous work, in which the conformal and covariant Z4 (CCZ4) formulation of the Cauchy problem for the Einstein field equations, with constraint-violation damping, was defined in terms of Wolfram model evolution over discrete (spatial) hypergraphs for the case of vacuum spacetimes, and proceeds to consider a minimal extension to the non-vacuum case by introducing a massive scalar field distribution, defined in either spherical or axial symmetry. Under appropriate assumptions, this scalar field distribution admits a physical interpretation as a collapsing (and, in the axially-symmetric case, uniformly rotating) dust, and we are able to show, through a combination of rigorous mathematical analysis and explicit numerical simulation, that the resulting discrete spacetimes converge asymptotically to either non-rotating Schwarzschild black hole solutions or maximally-rotating (extremal) Kerr black hole solutions, respectively. Although the assumptions used in obtaining these preliminary results are very strong, they nevertheless offer hope that a more general, perhaps ultimately ``Penrose-like'', singularity theorem may be provable in the discrete spacetime case too.