Globally hyperbolic

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Globally hyperbolic (also global hyperbolicity) is a term describing the causal structure of a spacetime manifold in Einstein's theory of general relativity, or potentially in other metric gravitational theories.

Open sets are used in discussing the global hyperbolicity of spacetimes. An open set $U$ is said to be globally hyperbolic if the following two conditions hold^[1]

For every pair of points $p,q \in U$ , $I^-(p)\cap I^+(q)$ is compact. Here $I^\pm(S)$ is the future (past) of a set $S$ in spacetime.
"Causality" holds on $U$ (no closed timelike curves exist). Classically, a more restrictive and technical assumption is required, named strong causality (no "almost closed" timelike curves exist); but a recent result [1] shows that causality suffices.

Global hyperbolicity implies that there is a family of Cauchy surfaces for $U$ . Essentially, it means that everything that happens on $U$ is determined by the equations of motion, together with initial data specified on a surface.