Relational Causal Simplicial Complexes

• Relational causal simplicial complexes are combinatorial-geometric models that encode causal and topological structures in piecewise-flat Lorentzian manifolds.
• The framework classifies 13 distinct causal types by analyzing the intersection of vertex lightcones using barycentric coordinates in (2+1)-dimensional simplicial gravity.
• This rigorous approach informs discrete quantum gravity methods, guiding models like CDT, Spin Foam, and Group Field Theory by distinguishing regular from singular configurations.

A relational causal simplicial complex is a combinatorial-geometric model for encoding causal and topological structure in piecewise-flat Lorentzian manifolds, specifically developed for (2+1)-dimensional simplicial gravity. This framework analyzes the intersection patterns of lightcones emanating from a vertex with its surrounding simplicial neighborhood, resulting in a rigorous classification of local causal types, topological configurations, and their dynamical implications. The formalism enables systematic identification of regular and irregular causal arrangements and establishes their correspondence with singularities and discrete topology change, with direct applications to discrete quantum gravity approaches such as Causal Dynamical Triangulations, Spin Foam models, and Group Field Theory (Asante et al., 14 Jul 2025).

1. The Thirteen Causal Types of Lorentzian Tetrahedra

Each Lorentzian tetrahedron $\Delta(v_0v_1v_2v_3)$ in $\mathbb{R}^{1,2}$ is determined by its six squared edge-lengths $\mathcal{S}=(s_1,s_2,s_3,s_{12},s_{13},s_{23})$, with $s_i=\langle v_i,v_i\rangle$ and $s_{ij}=\langle v_i-v_j,v_i-v_j\rangle$. The causal type at reference vertex $v_0$ is deduced from the intersection of the two lightcones $\mathcal{C}^\pm$ with the triangle $\Delta(v_1v_2v_3)$.

This intersection, analyzed in barycentric coordinates and classified by Proposition 3.1 (Asante et al., 14 Jul 2025), produces 13 distinct causal types (excluding configurations with null faces). These types are organized into four major groups (A–D) according to the signs of $(s_1, s_2, s_3)$:

Group	Vertex Types	Description
A	A1 ("no intersection"),	All $v_i$ timelike: All on same cone, or two on future one on past (A2)
	A2 ("two intersections")
B	B1 ("one intersection"),	Two timelike, one spacelike: $v_i$ sharing timelike regions (B1), or split cones (B2)
	B2 ("two intersections")
C	C1–C3	One timelike, two spacelike: distinguished by edge intersection with lightcones
D	D1–D6	All $v_i$ spacelike: classified by the number of edges intersecting the cones

Each type is associated with complex dihedral angles $\Theta_{e,\tau}$ (equation 2.9) encoding geometric and causal information.

2. Vertex Causal-Topological Classification

The star $\mathrm{St}(v)$ of a bulk vertex $v$ has a boundary $\partial \mathrm{St}(v) \cong S^2$ triangulated by the opposite triangles of incident tetrahedra. Each tetrahedron contributes a null-conic to the boundary, separating "timelike" regions ($\langle x,x\rangle < 0$) from "spacelike" regions ($\langle x,x\rangle > 0$). The number of connected components—$N_{\text{tl}}$ (timelike) and $N_{\text{sl}}$ (spacelike)—serves as a local causal-topological invariant.

• Definition 3.3 (Vertex causal structure):
  • Causally regular: $(N_{\text{sl}}, N_{\text{tl}}) = (1, 2)$ (two timelike caps separated by one spacelike belt)
  • Causally irregular: Otherwise

The regular configuration replicates the standard division of the tangent space $T_vM$ into disjoint time-lobes and a spacelike sheet, representing manifold-like behavior.

3. Regge Action and Deficit Angle Singularities

Curvature in (2+1)-dimensional simplicial gravity is localized on edges ("hinges"), quantified via the Lorentzian Regge action:

$$S_{\textrm{Regge}} = \imath \sum_{e \in \mathcal{T} \; \mathrm{timelike}} \sqrt{|s_e|} \, \epsilon_e^E + \imath \sum_{e \in \mathcal{T} \; \mathrm{spacelike}} \sqrt{|s_e|} \, \epsilon_e^L$$

Deficit angles for edges ($\epsilon_e^E$, $\epsilon_e^L$, equation 2.11) depend on the sum of dihedral angles around each hinge. $\theta^E \in [0, \pi]$ are Euclidean dihedral angles for timelike edges, while $\theta^L = \imath\beta - m\frac{\pi}{2}$ (for integer $m$) encode Lorentzian boosts and lightray crossings for spacelike edges.

Whenever a face or edge becomes null, the associated $\theta$ encounters a branch cut, resulting in divergence or discontinuity in $\epsilon_e$. At precisely the transition where a vertex changes from regular to irregular, $S_{\textrm{Regge}}$ becomes non-analytic, reflecting a discrete topological singularity.

4. Classification of Regular and Causally Irregular Configurations

Employing $(N_{\text{sl}}, N_{\text{tl}})$ yields four principal types of causal irregularity:

• Yarmulke-like (birth/death): $(N_{\text{tl}} < 2)$, e.g. $(1,0)$, $(1,1)$, $(0,1)$. These generalize the 1+1-dimensional "caps," corresponding to universe creation or annihilation.
• Trouser-like (splitting): $(N_{\text{sl}} = 1, N_{\text{tl}} > 2)$. Multiple timelike lobes meet at $v$, signifying local time branching; analogue to 1+1 "trousers."
• Spatial-splitting: $(N_{\text{sl}} > 1, N_{\text{tl}} \geq 1)$. Multiple spacelike regions punctuate a timelike zone, producing higher-genus features on $S^2$.
• Indistinguishable lightcones: Certain gluings (A2, B2, C3, D6) result in merging future and past cones, causing time-ambiguity and closed timelike curve (CTC)-like artefacts.

Irregular vertices are interpreted as point-like singularities; conically irregular edges as line-like singularities.

5. Causal Structure Transitions and Independence of Hinge/Vertex Causality

Investigating families of edge-lengths (notably in the 1–4 Pachner move), changes in $(N_{\text{sl}}, N_{\text{tl}})$ correspond sharply to divergences or discontinuities in deficit angles $\epsilon_e$, marking Morse-type topology change events. Notably, hinge-causality (triangle classification at edges) and vertex-causality (local $N_{\text{sl}}, N_{\text{tl}}$) are logically independent; regular hinges may coexist with irregular vertices, or vice versa.

This independence substantiates the description of irregular vertices as discrete point-singularities, and irregular edges as line-singularities, contributing distinct topological and causal aberrations.

6. Implications for Discrete Quantum Gravity Approaches

Relational causal simplicial complexes have immediate consequences for discrete models of quantum gravity:

• Causal Dynamical Triangulations (CDT): Enforces $(N_{\text{sl}}, N_{\text{tl}}) = (1,2)$ with global foliation, yielding semiclassical de Sitter-like universes. Permitting only vertex-regular but hinge-irregular moves increases phase space without degrading large-scale behavior.
• Spin Foam Models: The 4-simplex amplitude's semiclassical behavior hinges on Lorentzian type (A–D). The (2+1)-dimensional causal classification serves as a testbed for analyzing the contribution of different causal types to amplitudes.
• Group Field Theory: In condensate-state construction, restrictions on the distribution of vertex types $(N_{\text{sl}}, N_{\text{tl}})$ may favor spacetimes with regular, manifold-like causal structure.

In these frameworks, the invariants $(N_{\text{sl}}, N_{\text{tl}})$ and the catalogue of 13 tetrahedron types provide coordinate-free criteria distinguishing "healthy" Lorentzian configurations from pathological ones, guiding continuum limits toward physically meaningful Lorentzian geometry (Asante et al., 14 Jul 2025).