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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?
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?