Simpler embeddings of causal sets into Minkowski spacetime

TL;DR

This paper introduces a simplified method for embedding causal sets into Minkowski spacetime, improving accuracy over previous techniques by utilizing spacetime volumes to derive coordinates and distances.

Contribution

A new, simpler embedding method for causal sets into Minkowski spacetime that yields better results than prior approaches.

Findings

Effective embeddings in 2, 3, and 4 dimensions

Uses spacetime volumes for coordinate assignment

Achieves improved embedding quality

Abstract

We present a new method for embedding a causal set into Minkowski spacetime. The method is similar to a previously presented method, but is simpler and provides better embedding results. The method uses spacetime volumes to define causal set analogs of time coordinates for all elements, and spatial distances for pairs of causally related elements. The spatial distances for causally related pairs are then used to derive spatial distances for spacelike separated pairs by applying the triangle inequality. The result is a matrix of spatial distances for all pairs of elements in the causal set. This distance matrix can be decomposed to give coordinates in Minkowski spacetime. Results are presented showing good quality embeddings into Minkowski spacetime for dimensions d=2,3,4.