A spacelike hypersurface of a globally hyperbolic model that is not a Cauchy hypersurface

(The set is spacelike.) By [L3] the set $\mathrm{\Sigma }$ is an embedded three-dimensional submanifold of $A$ with tangent space spanned by ${\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3$, on which $\eta =\mathrm{diag}(1,1,1)$ is positive definite; hence $\mathrm{\Sigma }$ is a spacelike embedded hypersurface.

(An inextendible causal curve missing it.) Define $u:\mathbb{R}\to A$, $u(s):=x^{-1}(s,z_0)$. This is a smooth curve with constant tangent ${\mathrm{\partial }}_0=N$, which satisfies $\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1<0$ and is future-directed by clause (c) of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets, so $u$ is a future timelike (hence causal) curve; its $t$-coordinate is the identity of $\mathbb{R}$, so it has no point limit at either infinite parameter end: such a limit would force its continuous real time coordinate to have a finite limit. Thus $u$ is inextendible by Causal curves, chronological and causal reachability, and inextendibility clause (d). But $t(u(s))=s$ and $z(u(s))=z_0$ with $\mathrm{\mid }{z}_0{\mathrm{\mid }}_{\mathrm{e}}=2>1$, so $u(s)\mathrm{\notin }\mathrm{\Sigma }$ for every $s$; the parameter set $\{s:u(s)\in \mathrm{\Sigma }\}$ is empty and in particular not a singleton.

(Conclusion.) By step 1.2 there is an inextendible causal curve that meets $\mathrm{\Sigma }$ in no parameter value; by [L4] the set $\mathrm{\Sigma }$ is not a Cauchy set, and by step 1.1 it is a spacelike embedded hypersurface, so it is not a Cauchy hypersurface. Since the ambient model is globally hyperbolic by [L1], this refutes the displayed statement: in a globally hyperbolic model a spacelike hypersurface need not be Cauchy.