Table of Contents
Journal of Gravity
Volume 2016 (2016), Article ID 6151726, 6 pages
http://dx.doi.org/10.1155/2016/6151726
Research Article

Spacetime Causal Structure and Dimension from Horismotic Relation

Horia Hulubei National Institute for Physics and Nuclear Engineering, No. 30, Strada Reactorului, P.O. BOX MG-6, 077125 Bucharest, Romania

Received 11 March 2016; Accepted 8 May 2016

Academic Editor: Anzhong Wang

Copyright © 2016 O. C. Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 with both 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, and 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.