Discretely Causal

Physicists have long considered spacetime to consist of a smooth real manifold equipped with a fancy sort of metric. One condition on a spacetime that physicists such as Roger Penrose favour as being "physically reasonable" is global hyperbolicity. What this condition essentially means is that some sort of "global" time coordinate exists. In particular, it makes sense to speak about causality.

Rafael Sorkin initiated the study of causal sets. A causal set is a partially ordered set that is locally finite. That is each set of the form {x : a <= x <= b} is finite:
Spacetime and Causal Sets, Rafael Sorkin.

What is intriguing is that we now have a discrete object, namely a causal set, which in some way encodes information about spacetime. Researchers have extended this in a few different ways. In one direction, one may wonder what a causal set looks like to an observer based at one point of the set:

The internal description of a causal set: What the universe looks like from the inside. Fontini Markapoulou.

The basic thrust of the above paper is that whenever you are at some point in a causal set, you know what events have happened in the past. Then, as you shift around the causal set, the set of events that are in your past changes. Markapoulou characterises this with some simple category theory that is explained very nicely in her paper.

For those familiar with branching temporal logics, such as CTL*, there isn't much new here. Most developments of branching temporal logics assume that the past is linear and that the future is branching. That is, when you stand at a moment in time, there is only one possible history, but many possible futures. An exception is one of the logics in the following paper, which augments CTL(*) with a branching past operator:

Once and for all, Kupferman, O.; Pnueli, A (you or your institution needs an IEEE Explore account to view this paper).

Now, a natural concern is whether the notion of a causal set is intrinsically artificial. That is, given a causal set, is it possible to reconstruct a continuous spacetime? The following paper answers this in the affirmative:

A domain of spacetime intervals in general relativity. Keye Martin and Prakash Panangaden

Martin and Panangaden do something very cool in this paper. Given a spacetime M, and a point x in M, let F(x) be all the points that are in the future of x and reachable by a timelike curve starting at x. Define a partial order on m by x <= y iff y is in F(x). Then, given a countable dense subset of points of some spacetime, together with the "causality relation" (which I have defined via F), we can reconstruct the entire spacetime, topology and all!Super cool! The techniques are based on domain theory, which was introduced by Scott in order to give a semantics to the lambda calculus. What the above paper shows is that techniques developed in theoretical computer science are applicable to something seemingly unrelated: general relativity. One point to note is that they drop the local finiteness condition on the "causal sets". This is because they work with continuous posets (well, bicontinuous posets actually), where one has a sort of interpolation condition on the order, which negates the possibility of local finiteness.

So, what is the global hyperbolicity condition anyway? If we have some spacetime M, then for each p and q in M, let I(p,q) be all the events in M that are in the "causal future" p as well as the "causal past of q". The "Alexandroff topology" (or rather, one topology with that name..) is obtained by taking as a basis all of the sets I(p,q) for p,q in M. We say that M is strongly causal if its Alexandroff topology is Hausdorff, which amounts to saying that its Alexandroff topology is the same as the manifold topology. The spacetime M is globally hyperbolic if it is strongly causal and for all a,b in M, the intersection of up(a) and down(b) is compact in the manifold topology on M. Here, up(a) are all those elements "above" a in the partial order and similarly for down(b).

Along the way, they show that every globally hyperbolic spacetime can be realised as a bicontinuous poset, where the order is the one I defined above. Moreover, the interval topology on the poset is the same as the original manifold topology.

Martin and Panangaden give a very convincing argument for swapping the "local finiteness" condition for a "bicontinuity" condition in the description of a causal set, though they remain diplomatic and don't go so far as to say this. Anyway, it's a pretty cool application of some ideas originally developed for "logic in computer science", so you should check it out!