Causal structure and topology change in (2+1)-dimensional simplicial gravity

TL;DR

This paper introduces a systematic method to analyze local causal structures in (2+1)-dimensional Lorentzian simplicial gravity, classifies causal types of tetrahedra, and explores implications for topology change and quantum gravity.

Contribution

It provides a novel classification of causal types in (2+1)D simplicial gravity and links local causal structures to topology change and curvature behavior.

Findings

Identified 13 distinct causal types of Lorentzian tetrahedra.

Discovered that vertex causality transitions coincide with curvature discontinuities.

Found that vertex and hinge causality are generally independent.

Abstract

We develop a systematic method for analyzing the causal structure at vertices in (2+1)-dimensional Lorentzian simplicial gravity. By examining the intersection patterns of lightcones emanating from a vertex with its simplicial neighbourhood, we identify 13 distinct causal types of Lorentzian tetrahedra -- excluding configurations with null faces. This classification forms the basis for a topological characterization of the local causal structure in terms of the number of connected regions on the triangulated 2-sphere that are spacelike and timelike separated from a bulk vertex. These local causal data allow us to identify regular causal configurations and new types of irregular causal configurations, generalizing well-known (1+1)-dimensional topologies, such as `Trousers' and `Yarmulkes', to higher dimensions. We further investigate the dynamical implications of vertex causality by analyzing the behaviour of deficit angles and the Regge action in explicit configurations. Transitions in `vertex-causality' coincide with discontinuities in the curvature, suggesting discrete topology change. Causally irregular hinges correspond to discrete conical singularities, while vertex causal irregularities manifest as point-like singularities. Interestingly, we find that vertex and hinge causality are generally independent. These results have direct implications for discrete quantum gravity approaches where the emergence of semiclassical spacetime depends on how causal structures are encoded within the triangulation.