Globally Hyperbolic Spacetime

A globally hyperbolic spacetime is a time-oriented Lorentzian manifold that is strongly causal and has compact causal diamonds for every pair of points . Here and contain the points reached by future-directed and past-directed causal curves, respectively. Strong causality means that every point has arbitrarily small neighborhoods that a causal curve cannot leave and then reenter. In this definition, strong causality can be replaced by the absence of closed causal curves (Bernal and Sánchez 2007). Global hyperbolicity is the causal condition used in the Einstein initial-value problem.