Discrete Spacetime, Emergent Geometry and Computable Quantum Gravity

Closely related to the question of whether spacetime should best be modeled as a discrete or a continuous mathematical structure, an important open question remains regarding the extent to which quantum gravity will end up being a computable theory. I will begin by presenting a fully discrete formalism for classical gravity, in which all of the necessary mathematical structures are a priori computable (with hypergraphs replacing Cauchy surfaces, directed acyclic graphs replacing the conformal structure of Lorentzian manifolds, hypergraph rewriting rules replacing the ADM evolution equations, hypergraph consistency conditions replacing the ADM constraint equations, etc.). I will then proceed to show how classical GR may nevertheless be recovered in the macroscale (continuum) limit of this formalism, with a particular focus on black hole spacetimes and simple gravitational collapse models as illustrative examples. Some potentially observable discrepancies from continuum GR in the mesoscale regime of the formalism will also be discussed. I will conclude with a mention of how a discrete gravitational path integral may be conjecturally formulated by allowing the rewriting dynamics to become non-deterministic and multi-threaded, and present some speculations regarding potential implications for the foundations of quantum gravity.