Cauchy hypersurface

Definition

Let $(M,g)$ be a time-oriented Lorentzian manifold. A Cauchy hypersurface is a subset $\Sigma\subset M$ that every inextendible timelike curve meets exactly once.

This is a global causal condition, not merely a statement that $\Sigma$ is codimension one. A smooth spacelike Cauchy hypersurface is a smooth embedded spacelike hypersurface that also has the Cauchy property. Its future-directed unit normal supplies the two traces used in the Cauchy problem for a normally hyperbolic operator.

A spacetime admits a Cauchy hypersurface exactly when it is globally hyperbolic; this is the Cauchy-hypersurface characterization of global hyperbolicity.