Exploring causal structures in 2+1 dimensional Lorentzian Regge–Calculus

The causal structure of Lorentzian triangles and tetrahedra in (2+1) dimensions is investigated based on their intersections with the light cones emanating from one vertex of the simplex, yielding 13 different causal positions for tetrahedra. Based on this, an algorithm to determine the light cone structure at vertices of a (2+1)-dimensional Reggetype triangulation is developed. Both a graphical interpretation and a computational implementation are provided. The algorithm is used to prove that a regular causal structure at vertices does not imply a regular causal structure along the edges of a general triangulation and vice versa. The way toward such an analysis of (3 + 1)-dimensional triangulations is discussed.