A Closeness Function on Coarse Grained Lorentzian Geometries

2025/10/22 by Sumati Surya, Surya, Sumati
Mathematics · Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Noncommutative and Quantum Gravity Theories #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2510.19403

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We construct a family of closeness functions on the space of finite volume Lorentzian geometries using the abundance of discrete intervals in the underlying random causal sets. Although strictly weaker than a Lorentzian Gromov-Hausdorff distance function, it has the advantage of being numerically calculable for large causal sets. It thus provides a concrete and quantitative measure of continuumlike behaviour in causal set theory and can be used to define a weak convergence condition for Lorentzian geometries.

Citations

• Spacetime reconstruction by order and number
• Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry
• Lorentzian Gromov-Hausdorff convergence and pre-compactness
• On the Hauptvermutung of Causal Set Theory
• Introducing Various Notions of Distances between Space-Times
• Lorentzian Gromov-Hausdorff theory and finiteness results
• The Causal Set Approach to Quantum Gravity
• The Hartle-Hawking wave function in 2d causal set quantum gravity
• On the structure of causal spaces

Related

• Lorentzian Manifolds and Causal Sets as Partially Ordered Measure Spaces
• Lorentzian causality theory
• The global properties of the finiteness and continuity of the Lorentzian distance
• Simple indicators for Lorentzian causets
• Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry