Causal Set Topology

TL;DR

This paper reviews the current understanding of topology in Causal Set Theory, exploring how discrete posets can approximate continuum spacetimes and presenting new results linking poset and spacetime topologies.

Contribution

It introduces new findings on the relationship between poset and spacetime topologies within CST, enhancing the theoretical framework for quantum gravity models.

Findings

New results relating poset and spacetime topologies

Analysis of topological properties necessary for continuum approximation

Bridging CST concepts with broader mathematical community

Abstract

The Causal Set Theory (CST) approach to quantum gravity is motivated by the observation that, associated with any causal spacetime (M,g) is a poset (M,<), with the order relation < corresponding to the spacetime causal relation. Spacetime in CST is assumed to have a fundamental atomicity or discreteness, and is replaced by a locally finite poset, the causal set. In order to obtain a well defined continuum approximation, the causal set must possess the requisite intrinsic topological and geometric properties that characterise a continuum spacetime in the large. The continuum approximation thus sets the stage for the study of topology in CST. We review the status of causal set topology and present some new results relating poset and spacetime topologies. The hope is that in the process, some of the ideas and questions arising from CST will be made accessible to the larger community of computer scientists and mathematicians working on posets.