 Solution

A globally hyperbolic spacetime admits a Cauchy surface: every inextendible causal trajectory meets this spacelike initial-data surface exactly once. Physically, suitable field data on it determine evolution everywhere in its domain without additional data arriving from an unprescribed external boundary. This includes absence of causal loops and pathological almost-loops. An equivalent standard characterization is strong causality together with compact causal diamonds $J^{+}(p)\cap J^{-}(q)$. The Cauchy condition concerns all causal curves, not merely the timelike geodesics that happen to remain in a bounded spatial region.