Let $(M,g)$ be a globally hyperbolic Lorentzian spacetime with a non-compact Cauchy hypersurface $S \subset M$. Let $ \Omega \Subset S$ be an open subset. Is it true that the future Cauchy development of $\Omega$, denoted by $D^+(\Omega)$, is a compact subset in $M$?



Globally hyperbolic spacetimes and future Cauchy developement
No.


Let $M$ be the maximally extended Schwarzschild solution. Let $S$ be (for example) the $t = 0$ hypersurface that goes through the bifurcate sphere. If you take $\Omega$ sufficiently large, then $D^+(\Omega)$ touches the inner singularity, and hence is not compact.




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For a more trivial example: let $(M,g)$ be the portion of Minkowski space with $t\in (-1,1)$ and let $S = \{t = 0\}$. Take $\Omega = B(0,10)$.


On the other hand, the above example shows that it is not enough to ask that $M$ is Cauchy inextendible.




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The statement is true if you add the condition that there exists a sequence of events $p\_i\in M$ with $p\_i \prec p\_{i+1}$ such that $\cup (J^-(p\_i)\cap S)$ covers $S$. But this is quite strong.


