# Wald's definition of the Cauchy Horizon

1. Nov 20, 2011

### Matterwave

Hello, Wald defines, on page 203 the future Cauchy Horizon of a set $S\subset M$ as:

$$H^+(S)=\overline{D^+(S)}-I^-[D^+(S)]$$

Where the overline means the closure of the set. D+ is the future domain of dependence (i.e. all points in the manifold which can be connected to S by a past inextendible causal curve), and I- is the chronological past.

It seems to me that the closure of the set D+ does not include some parts of the set I-(D+) since the second term is the entire chronological past including the chronological past of the points on S.

Is there a mistake in this definition? I would have thought (intuitively) that the future Cauchy Horizon of a set S would simply be the boundary of the Future domain of dependence of S minus S itself. In that case, Wald defined the interior of D+ as:

$$int(D^+(S))=I^-[D^+(S)]\cap I^+(S)$$

Perhaps he meant this as the second term? Am I missing something here?

2. Nov 20, 2011

### Matterwave

Nevermind, I figured it out......my set theory knowledge was weak apparently haha.