Embedding Causal Sets into Minkowski Spacetime

TL;DR

This paper introduces a novel method for embedding causal sets into Minkowski spacetime using spacetime volumes and matrix factorization, enabling accurate embeddings without needing spacelike distances as input.

Contribution

The authors develop a new embedding technique for causal sets into Minkowski spacetime that does not rely on spacelike distances and works in any dimension.

Findings

Good quality embeddings achieved in 2, 3, and 4 dimensions

Method does not require spacelike distances as input

Provides a new way to define spatial orientation and distances

Abstract

We present a new method for embedding a causal set into an interval of Minkowski spacetime. The method uses spacetime volumes for causally related elements to define causal set analogs of Minkowski inner products. These are used to construct matrices of inner products which are then factored using the singular value decomposition to give coordinates in Minkowski spacetime. Results are presented showing good quality embeddings into Minkowski spacetime for dimensions d=2,3,4. The method applies in any dimension and does not require spacelike distances to be used as an input. It offers a new way to define spatial orientation and spacelike distances in a causal set.