I use Wald's notation: $I^+$ is the chronological future and $J^+$ is the causal future.

My confusion arises from the following passage in Wald (1984):

Now, let $S$ be a closed, achronal set (possibly with edge). We define the *future domain of dependence* of $S$, denoted $D^+(S)$, by $$D^+(S)=\{p\in M|\, \text{Every past inextendible causal curve through $p$ intersects $S$}\}$$ Note that we always have $S\subset D^+(S)\subset J^+(S)$.

I have to disagree with the last statement. We know that $S$ is achronal, i.e. $I^+(S)\cap S=\emptyset$. The relation $S\subset D^+(S)\subset J^+(S)$ implies $S\subset J^+(S)$, i.e. $J^+(S)\cap S\ne\emptyset.$ But I cannot see how a set can be both achronal and contained in its causal future. Hence the title of my question.

I *think* Wald meant to write $S\subset D^+(S)\subset \overline{J^+(S)}$.

This post imported from StackExchange Physics at 2015-02-15 12:13 (UTC), posted by SE-user 0celo7