Global hyperbolicity

Global hyperbolicity is a condition on a spacetime manifold that guarantees the well-posedness of the initial value problem for field equations. A globally hyperbolic spacetime possesses a Cauchy surface — a spacelike hypersurface that every inextendible timelike curve intersects exactly once — and has the topology of the product of that surface with the real line. This structure ensures that the entire history of the spacetime is determined by data on the Cauchy surface, with no causal pathologies such as closed timelike curves.

The condition was introduced to rescue determinism from the exotic topologies permitted by General Relativity. In a globally hyperbolic spacetime, causality is not merely local but globally coherent: no signal can loop backward in time, and the arrow of time is well-defined everywhere. The Chronology Protection Conjecture can be understood as the claim that physical dynamics select globally hyperbolic solutions over non-hyperbolic ones, not by fiat but through instability mechanisms that destroy horizons where predictability would fail.

Global hyperbolicity is the geometric precondition for field quantization, statistical mechanics, and the very concept of a system with bounded history and determinate future.