Global Hyperbolicity

Global hyperbolicity is the property of a spacetime that guarantees the initial value problem is well-posed: given complete data on a spacelike surface, the entire future and past of the spacetime are uniquely determined. A globally hyperbolic spacetime admits a Cauchy surface — a slice that every inextendible timelike curve intersects exactly once — and this surface serves as the stage on which the dynamical laws of general relativity play out predictably.

The loss of global hyperbolicity at a Cauchy horizon is the central concern of the strong cosmic censorship hypothesis. Without it, determinism fails: new information can enter from regions not constrained by initial data, and the future becomes contingent rather than necessary. The blue-shift instability near inner horizons is nature's mechanism for restoring global hyperbolicity by destroying the Cauchy horizon before it can be reached — converting a locus of unpredictability into a spacelike singularity that remains causally inert.

Global hyperbolicity is not merely a technical condition. It is the precondition for the arrow of time, for entropy increase, and for the very possibility of learning from the past. A universe without it would be one in which memory is unreliable and consequence is local — a universe in which systems, in any meaningful sense, could not exist.