Reconstructing Minkowski geometry from causal separations

TL;DR

This paper demonstrates that in Minkowski space, the causal relations and distances between causally related points fully determine the entire geometric structure, advancing the understanding of causal set theory.

Contribution

It shows that distances between causally related points suffice to reconstruct the full Minkowski geometry, extending previous causal relation results to include spatial relations.

Findings

Distances between causally related points determine all spatial distances.

Causal relations and related distances fully specify Minkowski space.

Supports the causal set approach to spacetime reconstruction.

Abstract

Aleksandrov, and then Zeeman, showed that the causal relations among the set of points in a Minkowski space of dimension greater than 2 determine the Minkowski space structure of the set up to a global conformal factor. We show that in any dimension the distances between causally related pairs of points determine the distances between spatially related pairs of points, and thus completely determine the Minkowski space structure of the set. This is a step in the direction of proving that causal sets arising from a Poisson process in a Lorentzian manifold determine that manifold up to the degree of approximation inherent in the intensity of the Poisson process -- the Hauptvermutung of causal set theory.