Globally hyperbolic spacetime

Definition

A time-oriented Lorentzian manifold $(M,g)$ is globally hyperbolic if it is strongly causal and every causal diamond

J^+(p)\cap J^-(q)

is compact.

The Cauchy-hypersurface characterization and the smooth splitting theorem are equivalent global descriptions and consequences, not additional clauses in this definition. Global hyperbolicity is also the geometric hypothesis in the global Cauchy theorem for normally hyperbolic operators.

Examples and non-examples

Minkowski spacetime is globally hyperbolic; each constant-time hyperplane is a Cauchy hypersurface. The static cylinder $\mathbb R\times\Sigma$ with metric $-dt^2+h$ , for complete Riemannian $(\Sigma,h)$ , is globally hyperbolic. A spacetime with a closed timelike curve is not strongly causal and therefore is not globally hyperbolic.

References

Antonio N. Bernal and Miguel Sánchez, “Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes,” Communications in Mathematical Physics 257 (2005), 43–50. Journal record.
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §§1.3 and 3.2.