Proper time and Minkowski structure on causal graphs

TL;DR

This paper introduces a new way to define proper time on causal graphs that transitions from volume-based to length-based measures across scales, enabling a continuum limit with Minkowski structure.

Contribution

It proposes a scale-dependent proper time definition on causal graphs, providing an alternative to coarse graining and establishing Minkowski structure in the continuum limit.

Findings

Proper time transitions from volume to length at a scale related to a dynamical clock.

The approach yields Minkowski structure on regular causal lattices.

Illustrative example with Sine-Gordon breather solutions demonstrates the method.

Abstract

For causal graphs we propose a definition of proper time which for small scales is based on the concept of volume, while for large scales the usual definition of length is applied. The scale where the change from "volume" to "length" occurs is related to the size of a dynamical clock and defines a natural cut-off for this type of clock. By changing the cut-off volume we may probe the geometry of the causal graph on different scales and therey define a continuum limit. This provides an alternative to the standard coarse graining procedures. For regular causal lattice (like e.g. the 2-dim. light-cone lattice) this concept can be proven to lead to a Minkowski structure. An illustrative example of this approach is provided by the breather solutions of the Sine-Gordon model on a 2-dimensional light-cone lattice.