Dynamical-Space Regular-Time Lattice and Induced Gravity

Abstract

It is proposed that gravity may arise in the low energy limit of a model of matter fields defined on a special kind of a dynamical random lattice. Time is discretized into regular intervals, whereas the discretization of space is random and dynamical. A triangulation is associated to each distribution of the spacetime points using the flat metric of the embedding space. We introduce a diffeomorphism invariant, bilinear scalar action, but no ``pure gravity'' action. Evidence for the existence of a non-trivial continuum limit is provided by showing that the zero momentum scalar excitation has a finite energy in the limit of vanishing lattice spacing. Assuming the existence of localized low energy states which are described by a natural set of observables, we show that an effective curved metric will be induced dynamically. The components of the metric tensor are identified with quasi-local averages of certain microscopic properties of the quantum spacetime. The Planck scale is identified with the highest mass scale of the matter sector.