I am been studying differential geometry and spacetime and I keep coming across the term globally hyperbolic. I am having a hard time coming up with an intuitive understanding of this idea. What is an intuitive meaning of globally hyperbolic.
My background is mostly mathematics at a higher undergraduate level.
Answer
Globally hyperbolic refers to the fact that hyperbolic equations always have locally a well defined Cauchy problem, that is, a unique development given initial conditions. Which means that, given a matter field at a time $t_1$, there exists a unique solution of that field at a time $t_2$. It is boosted up to globally hyperbolic if that property holds globally.
Globally hyperbolic isn't directly related to this notion (you can still have hyperbolic equations for which this property does not hold up globally in a globally hyperbolic spacetime), but the two notions are close. A globally hyperbolic is, formally, a spacetime that is both causal and such that the intersection of the past and future of two different points is compact, which roughly translates to having no naked singularities (a point removed from spacetime might cause that set to fail to be compact).
Alternatively, global hyperbolicity can be defined by the existence of a Cauchy surface - an achronal (cannot be linked by any causal curve) spacelike hypersurface such that every causal curve crosses it exactly once. No more (no curves going to the same time more than once) and no less (no curve appearing out of nowhere).
Breaks in causality and naked singularities may cause a lack of existence or uniqueness of solutions for matter fields - corresponding either to matter going round in a circle, so to speak, or coming out of nowhere or disappearing. Globally hyperbolic spacetimes benefit from a theorem due to Hawking stating that, for such a spacetime with a reasonable enough matter field (basically if it already behaves nicely in flat space), any hyperbolic equation is globally hyperbolic.
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