Spacelike slices

The definition of spacelike slices used here is directly analogous to what is used in traditional relativity theory (typically under names like spacelike hypersurfaces and Cauchy surfaces). There will normally be many different possible choices of spacelike slices, but in all cases a particular such slice is set up to represent what can consistently be thought of as all of space at a given time. One definition of a spacelike slice is then a maximal set of points in which no pair are causally related (corresponding to a maximal antichain in a poset). Another definition (equivalent for any connected causal network) is that spacelike slices are what consistently divide a causal network into a past and a future. And an intermediate definition is that a spacelike slice contains points that are not themselves causally related, but which appear in either the past or the future of every other point. Given a spacelike slice in a causal network, it is always possible to construct another such slice by finding all those points whose immediate predecessors are all included either in the original slice or its predecessors.