A General Theorem on Temporal Foliations of Causal Sets

Abstract

Causal sets (or causets) are a particular class of partially ordered sets, which are proposed as basic models of discrete space-time, specially in the field of quantum gravity. In this context, we show the existence of temporal foliations for any causal set, or more generally, for a causal space. Moreover, we show that (order-preserving) automorphisms of a large class of infinite causal sets fall into two classes 1) Automorphisms of spacelike hypersurfaces in some given foliation (i.e. spacelike automorphisms), or 2) Translations in time. More generally, we show that for any automorphism $\Phi$ of a generic causal set ${\mathcal {C}}$, there exists a partition of ${\mathcal {C}}$ into finitely many subcausets, on each of which (1) or (2) above hold. These subcausets can be assumed connected if, in addition, there are enough distinct orbits under $\Phi$.