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I am currently trying to grasp the nuance between a compactly generated future Cauchy horizon (as per Hawking's chronological protection conjecture) and a compactly constructed causality violating region (as per Ori's spacetime).

From what I understand, a compactly generated one is just when the past geodesics of the region lead back to a compact region in some achronal spacelike hyperslice. So if we have some big partial Cauchy surface $S$, something of the form $J^-(H^+(S)) \cap S $ being compact. Basically the creation of the chronology violating region stems from finite region of space.

On the other hand, compactly constructed causality violating region seem to be defined by $\overline{D^+(U)}\cap \mathcal{V}(M) \neq \varnothing$, with $U$ some compact subset of $S$ and $\mathcal{V}(M)$ the causality violating region. So there are CTCs in the closure of the Cauchy horizon (so equivalently I suppose, $H^+(U) \cap \mathcal{V}(M) \neq \varnothing$). The definition comes from "Geometric analysis of Ori-type spacetimes" as the concept doesn't seem to be written down in Ori's papers.

While I can appreciate that their definitions are different, I am having a hard time grasping how a compactly constructed CTC isn't also compactly generated. Ori does go on to say that

"We emphasize again that although probably not ”compactly generated”, our model does demonstrate the formation of closed causal loops from the initial data on a compact vacuum region $S_0$."

This paper also mentions :

"How can this spacetime evade the theorems of Tipler and Hawking? The answer is that N is not in fact the fountain where the causality violating region originates, but rather a place where the Cauchy horizon terminates. The origin of the causality violation lies outside the region where the weak energy condition is obeyed."

So I am not sure exactly what is going on there and where the subtelty lies between the two.

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