Spacetime reconstruction by order and number

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 is proven 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. Moreover, among our contributions is a new extension theorem for chronology-preserving maps between spacetimes.