Spacelike distance from discrete causal order

TL;DR

This paper proposes methods to define spacelike distances in causal set theory, enabling the extraction of continuum spatial properties from discrete quantum gravity models, and supports the Hauptvermutung conjecture.

Contribution

It introduces novel approaches to measure spacelike distances in causal sets, bridging the gap between discrete structures and continuum spacetime.

Findings

Numerical evidence for a spatial nearest neighbor relation

Method reproduces spatial distance in Minkowski space

Supports defining curves in curved spacetime from causal sets

Abstract

Any discrete approach to quantum gravity must provide some prescription as to how to deduce continuum properties from the discrete substructure. In the causal set approach it is straightforward to deduce timelike distances, but surprisingly difficult to extract spacelike distances, because of the unique combination of discreteness with local Lorentz invariance in that approach. We propose a number of methods to overcome this difficulty, one of which reproduces the spatial distance between two points in a finite region of Minkowski space. We provide numerical evidence that this definition can be used to define a `spatial nearest neighbor' relation on a causal set, and conjecture that this can be exploited to define the length of `continuous curves' in causal sets which are approximated by curved spacetime. This provides evidence in support of the ``Hauptvermutung'' of causal sets.