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

Abstract

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. However, the proofs of these theorems traditionally depend upon a collection of assumptions about the continuum spacetime (and, in the physical case, the stress-energy distribution defined over it), including its global structure, its energy conditions, the existence of trapped null surfaces and the various volume/intersection properties of geodesic congruences, that are inherently difficult to translate to the case of discrete spacetime formalisms, such as causal set theory or the Wolfram model. Some, such as the discrete analog of Raychaudhuri's equation for the volumes of geodesic congruences, are subtle to formulate due to intrinsic differences in the behavior of discrete vs. continuous geodesics; for others, such as the definition of a trapped null surface, no appropriate translation is known for the general discrete case due to a lack of a priori coordinate information. It is therefore 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.

Computational General Relativity in the Wolfram Language using Gravitas II: ADM Formalism and Numerical Relativity

“…Note that, as with the previous article [1] in this series, much of the GRAVITAS functionality that is demonstrated and discussed here is fully-documented and available via the Wolfram Function Repository, including ADMDecomposition, SolveVacuumADMEquations and DiscreteHypersurfaceDecomposition. However, there are many more functions that have not yet been documented in this way, or which have received further development beyond their documented versions; as always, an up-to-date version of the GRAVITAS framework may be obtained from its GitHub repository (currently approximately 30,000 lines of Wolfram Language code, including experimental and research functionality).…”

“…The first article in this series [1] contained a more-or-less complete and self-contained introduction to the underlying philosophy of the GRAVITAS computational framework, a summary of its core design principles, and an incomplete but high-level survey of the field of computational and numerical relativity as a whole, along with a detailed explanation of how GRAVITAS compared to the many other existing open source tools that are presently available. In the interests of brevity, we do not seek to repeat those introductory remarks here, and we instead encourage the reader to consult the previous article before proceeding with this one.…”

“…GRAVITAS's use of next-generation adaptive refinement algorithms based upon totally-unstructured hypergraph rewriting/Wolfram model evolution [22,23] [24] facilitates its handling of arbitrary (potentially overlapping) curvilinear coordinate systems and highly non-trivial spacetime topologies with ease, since one is not constrained to use a rigid Cartesian or spheroidal grid structure for the computational domain, and the hypergraph's topology can instead adapt organically to both the topology and the dynamics of the problem being solved. This hypergraph rewriting formalism [25][26][27] (including its efficient hypergraph canonicalization system [28]), and the resulting numerical algorithms that it enables, have previously been used to investigate a variety of physical problems in discrete spacetimes, including binary black hole mergers [29], entanglement entropies in curved spacetime quantum field theories [30], singularity theorems in idealized gravitational collapse scenarios [31], amongst others.…”

General Relativistic Hydrodynamics in Discrete Spacetime: Perfect Fluid Accretion onto Static and Spinning Black Holes

“…Since discreteness of the fundamental structure of spacetime is a generic feature of many proposed models of quantum gravity, including casual set theory [20][21][22][23], causal dynamical triangulations [24,25], loop quantum gravity [26][27][28], and the Wolfram model [29][30][31][32][33], it is hoped that such an investigation will eventually enable the observational investigation of certain classes of quantum gravity theories by means of astrophysical probes of near-black hole accretion regions. To this end, we make use of the GRAVITAS computational general relativity framework [34,35], which allows for the configuration, execution, visualization and analysis of complex numerical relativity simulations in both discrete and continuous spacetime settings, by combining a powerful tensor calculus and differential geometry framework on the analytical side, with a sophisticated hypergraph-based adaptive refinement system [36,37] on the numerical side. Most general relativistic simulations of black hole accretion consider a perfect relativistic fluid evolving on top of a fixed, time-independent spacetime metric (typically representing either a Schwarzschild geometry or a Kerr geometry), and thereby neglect any gravitational effects of the fluid density on the black hole itself.…”

“…Note that all of the GRAVITAS functionality necessary to reproduce the results presented within this article can be found in the GRAVITAS GitHub repository, with extensive documentation available within both the Wolfram Function Repository (e.g. ADMDecomposition and StressEnergyTensor) and within the two previous articles [34,35]. This article follows all of the same notational and terminological conventions as these two previous articles; in particular, we assume geometric units with c = G = h = 1, we employ a metric signature of (−, +, +, +) in all relevant cases, and the Einstein summation convention is assumed throughout (such that all repeated tensor indices are implicitly summed over).…”