Are non-point spacetime events partially ordered?

Question

When describing events in spacetime, we usually use points. We then phrase the relation between points as a trichotomy: either they are timelike, spacelike, or lightlike separated, based on the Minkowski metric. We can also put points into a partial order based on the metric, representing the causal past and future of points. However, this doesn't seem to hold for any pointfree generalizations of events, even balls or neighborhoods, because it seems possible for an event to be in the future and past of another event simply by surrounding it in spacetime.

Is there any nice causal structure on non-point events? Does it help if they're disjoint, bounded, connected, etc.?

Answer 1

There is a well-developed theory of the causal structure of spacetime in general relativity. This is discussed in detail in the books by Wald (General Relativity) and Hawking and Ellis (The Large Scale Structure of Spacetime), for example.

The basic construction we need to make is the definition of the causal future or causal past. Let $S \subseteq M$ be a set, where $M$ denotes spacetime. We define the causal future of $S$, $J^+(S)$, by $$J^+(S) = \{p \in M|\text{there exists a future-directed causal curve $\gamma$ from a point $q \in S$ to $p$}\}.$$ A causal curve is a curve that is everywhere either timelike or null. One can define the chronological future of $S$, $I^+(S)$, by replacing "causal" with "timelike" in the above definition.

You then have a partial order defined by $S \preccurlyeq S'$ if, and only if, $S' \subseteq J^+(S)$.