# Difference between chronological future and domain of dependence

Given the following standard definitions of the two concepts, I fail to see how the chronological future differs from the future domain of dependence?

Chronological future: The chronological future of $$S$$, denoted $$I^+(S)$$, is the set of points that can be reached from $$S$$ by following a future-directed timelike curve.

Future domain of dependence: Let $$(\mathcal{M},g)$$ be a spacetime and $$S \subset \mathcal{M}$$ such that no two points on $$S$$ can be connected by a timelike curve (we also say that $$S$$ is achronal). The future domain of dependence of $$S$$, denoted $$D^+(S)$$ is the set of all points $$p \in \mathcal{M}$$ with the property that every past-directed inextendible (i.e. with no endpoints) timelike curve starting at $$p$$ intersects $$S$$.

• It is the word every that makes them different. In the first definition it says that a point can be reached by a timelike curve, but it doesn't mean that every timelike curve that goes through it must also go through S.
– MBN
Jun 25, 2020 at 12:03

The chronological future is the light cone. It's a cone that expands outwards as time goes on, because it consists of all the events that $$S$$ can influence. In other words, the events in $$I^+(S)$$ have at least one point of $$S$$ in their past: On the other hand, the domain of dependence, also called the Cauchy development, consists of those events that are completely determined by what happens in $$S$$, because all their past intersects $$S$$. This means that it's a cone that closes: as you go far enough into the future you can't predict anymore, because information from events outside $$S$$ has had enough time to arrive: 