Spacetime reconstruction by order and number

2025/07/02 by Mathias Braun, Braun, Mathias · 1 citation
Mathematics · Physics and Astronomy · #51K10 #53C23 #60B20 #60G55 #83C99 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Primary 51G05 #Probability (math.PR) #Secondary 60A10

paper · pdf · doi:10.48550/arxiv.2507.01907

openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the random adjacency matrices induced by the chronological relations and i.i.d. samples of two spacetimes coincide in law if and only if the spacetimes in question are smoothly isometric. A similar result holds for weighted spacetimes. In the smooth framework of our article, this relaxes the hypotheses of the recent Gromov reconstruction theorem in Lorentzian signature by Braun-Sämann from a.s. isometry of the respective time separation functions to a.s. order isometry. In a probabilistic way, our result makes a key paradigm of causal set theory rigorous: spacetime can be recovered by only knowing "order" and "number" of its points. It confirms a weak version of Bombelli's conjecture; therefore, it contributes to recent efforts of formalizing the Hauptvermutung (viz. fundamental conjecture) of causal set theory.

Citations

• Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry
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• The Large N Limit of Superconformal Field Theories and Supergravity
• On the structure of causal spaces

Cited by

• A Closeness Function on Coarse Grained Lorentzian Geometries

Related

• Lorentzian causality theory
• Remarks on antichains in the causality order of space-time
• The mathematics of causal sets
• Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry
• Causality and time order -- relativistic and probabilistic aspects