Ultralocal nature of geometrogenesis
Résumé
Ultralocal state of gravity, characterized by decoupling of the space points, is one of the main consequences of the Belinskii–Khalatnikov–Lifshitz (BKL) conjecture. In this article we investigate relation between the ultralocality and crumpled phase of gravity (observed e.g. in causal dynamical triangulations) characterized by high connectivity of the associated graph structure. By applying the anisotropic scaling (parametrized by ) of the Hořava–Lifshitz theory to CDT-like ring graph we prove that in the ultralocal scaling limit () the graph representing connectivity structure of space becomes complete. In consequence, transition from the ultralocal phase () to the standard relativistic scaling () is implemented by the process of geometrogenesis, which transforms nongeometric field configuration into the one possessing a geometric character. By coupling Ising spin matter to the considered graph we show that the process of geometrogenesis can be associated with critical behavior. Based on both analytical and numerical analysis phase diagram of the system is reconstructed showing that (for a ring graph) symmetry broken phase occurs at . Finally, cosmological consequences of the considered process of geometrogenesis are briefly discussed.