Causal set topology

The Causal Set Theory (CST) approach to quantum gravity is motivated by the observation that, associated with any causal spacetime (M,g) is a poset (M,≺), with the order relation ≺ corresponding to the spacetime causal relation. Spacetime in CST is assumed to have a fundamental atomicity or discreteness, and is replaced by a locally finite poset, the causal set. In order to obtain a well defined continuum approximation, the causal set must possess the requisite intrinsic topological and geometric properties that characterise a continuum spacetime in the large. The continuum approximation thus sets the stage for the study of topology in CST. We review the status of causal set topology and present some new results relating poset and spacetime topologies. The hope is that in the process, some of the ideas and questions arising from CST will be made accessible to the larger community of computer scientists and mathematicians working on posets. 1 The Spacetime Poset and Causal Sets In Einstein’s theory of gravity, space and time are merged into a single entity represented by a pseudo-Riemannian or Lorentzian geometry g of signature −+++ on a differentiable four dimensional manifold M . Unlike a Riemannian space in which the distance between two distinct points is always positive definite, a Lorentzian geometry is characterised by the fact that this distance can be positive definite (spacelike), negative definite (timelike) or zero (null).