Spacetime reconstruction by order and number

TL;DR

This paper proves that the law of adjacency matrices derived from causal relations and samples uniquely determines the spacetime's geometry, confirming a key causal set theory hypothesis and advancing the formalization of the spacetime reconstruction conjecture.

Contribution

It establishes that spacetime can be reconstructed from order and number data, relaxing previous isometry assumptions in Lorentzian geometry.

Findings

Random adjacency matrices coincide iff spacetimes are smoothly isometric.

Results support the causal set theory paradigm that spacetime is recoverable from order and number.

Contributes to the formal proof of the Hauptvermutung in causal set theory.

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-S\"amann 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.