Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Seth K. Asante, Björn Borgolte

TL;DR

The paper develops a coordinate-free method to extract local causal structure at a bulk vertex in (2+1)-dimensional Lorentzian triangulations by analyzing lightcone intersections with the vertex neighbourhood. It identifies 13 distinct causal tetrahedron types and introduces boundary-sphere invariants $N_{ m sl}$ and $N_{ m tl}$ to classify vertex causality into regular and irregular regimes, linking these to discrete topology change and singularities. The work connects vertex causality with Regge dynamics, showing how deficit angles and the Regge action respond to causal transitions and highlighting possible independence between vertex and hinge causality. These results provide a concrete framework for incorporating causality into discrete quantum gravity approaches and for exploring topology change and singular structures in triangulated spacetimes.

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.

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

TL;DR

The paper develops a coordinate-free method to extract local causal structure at a bulk vertex in (2+1)-dimensional Lorentzian triangulations by analyzing lightcone intersections with the vertex neighbourhood. It identifies 13 distinct causal tetrahedron types and introduces boundary-sphere invariants and to classify vertex causality into regular and irregular regimes, linking these to discrete topology change and singularities. The work connects vertex causality with Regge dynamics, showing how deficit angles and the Regge action respond to causal transitions and highlighting possible independence between vertex and hinge causality. These results provide a concrete framework for incorporating causality into discrete quantum gravity approaches and for exploring topology change and singular structures in triangulated spacetimes.

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.

Paper Structure

This paper contains 16 sections, 2 theorems, 28 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Lorentzian simplicial geometry
  3. Lorentzian Regge Calculus
  4. Lightcone analysis in Lorentzian triangulation
  5. Causal structure of a simplex
  6. Causal structure in a Lorentzian triangulation
  7. Local vertex--causality conditions in a triangulation
  8. Classification of triangles in Minkowski spacetime
  9. Vertex causality in 2+1 dimensions
  10. Causal classifications of Lorentzian tetrahedra
  11. Causal structure at a vertex in a triangulation
  12. Causal irregular configurations
  13. Causal structure in simplicial gravity
  14. Examples: Deficit angles and vertex causal structure
  15. Discussion
  16. ...and 1 more sections

Key Result

Proposition 1

A set of $\tfrac{1}{2} d(d+1)$ real numbers $\{s_{ij}\}_{1\leq i,j\leq d}$ is realizable as the set of squared edge lengths of a simplex in Minkowski spacetime $\mathbb R^{1,d-1}$ if and only if the corresponding Gram matrix $G(s)$ has signature $(-,+,\dots,+)$, i.e. it has exactly one negative and

Figures (16)

  • Figure 1: Lightcones at a vertex in (2+1)--dimensional Minkowski spacetime.
  • Figure 2: Types of triangles (in black) in Minkowski spacetime $\mathbb R^{1,1}$. Each panel shows the lightrays (in red) through the vertex $v_0$ placed at the origin. The timelike regions are shaded in gray.
  • Figure 3: Causal tetrahedra types A1 and A2 via lightcone intersections with the opposite triangle $\Delta(v_1v_2v_3)$ relative to vertex $v_0$. The timelike regions are represented in gray, while the white regions are spacelike with respect to the Minkowski norm centered at $v_0$. The red curves indicate the locus of lightlike points where the triangle intersects the lightcones at $v_0$.
  • Figure 4: Tetrahedra of type B1 and B2 via lightcone intersections with the opposite triangle $\Delta(v_1v_2v_3)$ relative to vertex $v_0$.
  • Figure 5: Tetrahedra types C1, C2 and C3 via lightcone intersections with the opposite triangle $\Delta(v_1v_2v_3)$ relative to vertex $v_0$.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Proposition 1: Realizability, c.f. ShoenbergDexter:1978
  • Definition 2: Causal structure of a simplex relative to a vertex
  • Definition 3: Causal structure of a vertex neighbourhood
  • Proposition 4: Causal classification of Lorentzian tetrahedra
  • Example 1
  • Example 2
  • Example 3
  • proof