Spacetime causal structure and dimension from horismotic relation

TL;DR

This paper presents a method to reconstruct the causal and dimensional structure of spacetime from the horismos relation, applicable to both continuous and discrete models, with implications for quantum gravity and causal set theory.

Contribution

It introduces a natural construction of spacetime causal structure from lightlike relations, enabling dimension determination and manifold reconstruction in a unified framework.

Findings

Reconstruction of causal structure from horismos relation

Dimension determination in discrete and continuous spacetimes

Potential application to causal set quantum gravity

Abstract

A reflexive relation on a set can be a starting point in defining the causal structure of a spacetime in General Relativity and other relativistic theories of gravity. If we identify this relation as the relation between lightlike separated events (the horismos relation), we can construct in a natural way the entire causal structure: causal and chronological relations, causal curves, and a topology. By imposing a simple additional condition, the structure gains a definite number of dimensions. This construction works both with continuous and discrete spacetimes. The dimensionality is obtained also in the discrete case, so this approach can be suited to prove the fundamental conjecture of causal sets. Other simple conditions lead to a differentiable manifold with a conformal structure (the metric up to a scaling factor) as in Lorentzian manifolds. This structure provides a simple and general reconstruction of the spacetime in relativistic theories of gravity, which normally requires topological structure, differential structure, geometric structure (which decomposes in the conformal structure, giving the causal relations, and the volume element). Motivations for such a reconstruction come from relativistic theories of gravity, where the conformal structure is important, from the problem of singularities, and from Quantum Gravity, where various discretization methods are pursued, particularly in the causal sets approach.