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

**Authors:** Jonathan Gorard  
**arXiv:** [2303.07282](https://arxiv.org/abs/2303.07282)  
**Date:** 2023-03-10  
**Summary page:** https://www.summarizepaper.com/en/paper/2303.07282v1/

## Key points

- Geodesic congruence equation of Raychaudhuri and singularity theorems of Penrose and Hawking provide conditions for spacetime to become geodesically incomplete in general relativity
- Question remains whether equivalent conditions exist for discrete spacetimes and how they differ from those in the continuum
- Article builds upon previous work on CCZ4 formulation of Cauchy problem for Einstein's field equations over discrete spatial hypergraphs for vacuum spacetimes
- Authors extend this work to consider non-vacuum cases by introducing a massive scalar field distribution defined in either spherical or axial symmetry, which can be interpreted as collapsing dust under appropriate assumptions
- Explicit numerical simulations show that discrete spacetimes converge asymptotically to non-rotating Schwarzschild black hole solutions or maximally-rotating Kerr black hole solutions, respectively
- Preliminary results offer hope that a more general "Penrose-like" singularity theorem may be provable in the discrete spacetime case too
- Study provides insight into how discrete models can complement traditional continuous approaches to explore gravitational collapse and singularities
- Research contributes to understanding gravity at macroscopic and microscopic scales with implications for future studies on quantum gravity and cosmology.

## Summary

In the field of general relativity, the celebrated geodesic congruence equation of Raychaudhuri and the resulting singularity theorems of Penrose and Hawking have provided a highly general set of conditions under which a spacetime is expected to become geodesically incomplete. However, it remains a non-trivial question whether equivalently general conditions exist for discrete spacetimes and how they might differ from those in the continuum. This article builds upon previous work that defined the conformal and covariant Z4 (CCZ4) formulation of the Cauchy problem for Einstein's field equations over discrete spatial hypergraphs for vacuum spacetimes. The authors extend this work to consider non-vacuum cases by introducing a massive scalar field distribution defined in either spherical or axial symmetry. Under appropriate assumptions, this scalar field distribution can be interpreted as collapsing dust, which allows for explicit numerical simulations to show that discrete spacetimes converge asymptotically to either non-rotating Schwarzschild black hole solutions or maximally-rotating Kerr black hole solutions, respectively. While these preliminary results rely on strong assumptions, they offer hope that a more general "Penrose-like" singularity theorem may be provable in the discrete spacetime case too. The study provides insight into how discrete models can be used to explore gravitational collapse and singularities in ways that complement traditional continuous approaches. Overall, this research contributes to our understanding of how gravity behaves at both macroscopic and microscopic scales and has implications for future studies on quantum gravity and cosmology.