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I've been studying singularities in GR, and (obviously), came across PST.

Let us state it as the following:

Let $(M, g)$ be a connected globally hyperbolic spacetime with a noncompact Cauchy hypersurface $S$, satisfying the null energy condition. If $S$ contains a trapped surface $\Sigma$, then $(M,g)$ is singular.

I'll suppose the reader is familiar with the definitions used above. For the purpose of my question, I do believe intuition on those is probably sufficient. If my assumptions turn out to be inappropriate, I'll be happy to edit the question.

Anyway, I'm having a hard time coming around as to why $S$ needs to not be compact. Simple as that.

I'm trying to get actual physical intuition here, so, being simply a necessary condition to prove the theorem; well, it's not amusing.

What does it represent, for the theorem as a whole?

Any help will be greatly appreciated.

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"Penrose’s theorem concerns a globally hyperbolic space- time M with a noncompact Cauchy hypersurface S. For example, in any spacetime that is asymptotic at spatial infinity to Minkowski space, a Cauchy hypersurface (if there is one) is not compact. Thus, Penrose’s theorem applies to any space- time that is asymptotically flat and also globally hyperbolic."

Sentence taken from 'Light rays, singularities, and all that', E. Witten 2020. https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.92.045004

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