Curved spacetime in the causal set approach to quantum gravity.

Causal Set theory is an approach to quantum gravity. In the viewpoint of causal set theory, the spacetime continuum is discrete rather than continuous at the most fundamental level. The discrete points in a causal set can approximate a spacetime continuum if they can be embedded in a manifold such that the causal structure between every pair of points is preserved. We sprinkled points inside a causal diamond uniformly with the use of a computer-generated random number r ∈ [0 1]. These sprinklings were done for the number of points ranging from 100 to 1000 at an interval of 100 points. The sprinkling of points randomly and uniformly inside the causal diamond, which was dictated by the spacetime metric, was used to obtain the relations matrix R. Finally, with the use of this relations matrix R, chain-length distributions were obtained. Chains of length greater than two were used to acquire information about the dimension and curvature of the manifold the causal set was embedded into. The dimension was estimated with the aid of chains of length three for two and higher-dimensional flat spacetime manifold. Similarly, the scalar curvature of several spacetime manifolds (Minkowski, de Sitter, and Anti-de Sitter) was approximated using chains of length three. These results averaged over fifty causal sets with the number of points ranging from 100 to 1000 with 100-point increments, demonstrated excellent accuracy with minimal fluctuations from the continuum limit even for a 100-point causal set.