Discrete geodesics and cellular automata

TL;DR

This paper introduces a new concept of discrete geodesics in discretized curved spacetime, providing an algorithm for their computation and validating it through numerical experiments, with discussions on cellular automata compatibility.

Contribution

It formulates a generic dynamical notion of discrete geodesics and develops an effective algorithm for their computation in discretized spacetime.

Findings

Algorithm accurately computes discrete geodesics.

Numerical validation via perihelion shift of Mercury-like planet.

Consistent with standard geodesics in the continuum limit.

Abstract

This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation---as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length. KEYWORDS: Discrete connection, parallel transport, general relativity, Regge calculus.