2021/02/14 by Jonathan Gorard, Gorard, Jonathan
Physics and Astronomy · #Black Holes and Theoretical Physics #Galaxies: Formation, Evolution, Phenomena #Cosmology and Gravitation Theories
paper · pdf · doi:10.48550/arxiv.2102.09363
Although the traditional form of the Einstein field equations is\nintrinsically four-dimensional, the field of numerical general relativity\nfocuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy\nproblem, in which Cauchy initial data is specified on a three-dimensional\nspatial hypersurface, and then evolved forwards in time. The Wolfram model\noffers an inherently discrete formulation of the Einstein field equations as an\na priori Cauchy problem, in which Cauchy initial data is specified on a single\nspatial hypergraph, and then evolved by means of hypergraph substitution rules,\nwith the resulting causal network corresponding to the conformal structure of\nspacetime. This article introduces a new numerical general relativity code\nbased upon the conformal and covariant Z4 (CCZ4) formulation with\nconstraint-violation damping, with the option to reduce to the standard BSSN\nformalism if desired, with Cauchy data defined over hypergraphs; the code\nincorporates an unstructured generalization of the adaptive mesh refinement\ntechnique proposed by Berger and Colella, in which the topology of the\nhypergraph is refined or coarsened based upon local conformal curvature terms.\nWe validate this code numerically against a variety of standard spacetimes,\nincluding Schwarzschild black holes, Kerr black holes, maximally extended\nSchwarzschild black holes, and binary black hole mergers (both rotating and\nnon-rotating), and explicitly illustrate the relationship between the discrete\nhypergraph topology and the continuous Riemannian geometry that is being\napproximated. Finally, we compare the results produced by this code to the\nresults obtained by means of pure Wolfram model evolution (without the\nunderlying PDE system), using a hypergraph substitution rule that provably\nsatisfies the Einstein field equations in the continuum limit.\n