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∈M|Every past inextendible causal curve through p intersects S}
Note that we always have S⊂D+(S)⊂J+(S).
I have to disagree with the last statement. We know that S is achronal, i.e. I+(S)∩S=∅. The relation S⊂D+(S)⊂J+(S) implies S⊂J+(S), i.e. J+(S)∩S≠∅. 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⊂D+(S)⊂¯J+(S).
This post imported from StackExchange Physics at 2015-02-15 12:13 (UTC), posted by SE-user 0celo7