Spacelike sections, maximality and lapse–shift data

Let $(M,g)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$ with its Levi–Civita connection $\mathrm{\nabla }$, and let $\mathrm{\Sigma }\subset M$ be a smooth embedded spacelike hypersurface: its tangent spaces are three-dimensional and $g$ restricts positively on each of them. Let $n$ be the future-directed unit normal, so $g(n,n)=-1$ and $g(n,U)=0$ for tangent $U$. The induced Riemannian metric is $h(U,V)=g(U,V)$, and we use the second fundamental form convention $k(U,V)=-g({\mathrm{\nabla }}_{U}n,V)$.

The mean curvature in this convention is $H=-{\mathrm{tr}}_{h}k$, where the trace is contraction with the inverse of $h$. Equivalently, $H={\mathrm{div}}_{g}n$ along $\mathrm{\Sigma }$: here the divergence is the trace of $V\mapsto {\mathrm{\nabla }}_V\tilde{n}$ for any smooth local unit timelike extension $\tilde{n}$ of $n$, satisfying $g(\tilde{n},\tilde{n})=-1$ throughout its ambient neighborhood. Such extensions exist locally by normalization. The hypersurface is maximal when $H=0$ everywhere.

For any smooth ambient vector field $T$, its lapse along $\mathrm{\Sigma }$ is $N=-g(T,n)$ and its shift is $X=T-Nn$. Thus $X$ is tangent to $\mathrm{\Sigma }$ and $T=Nn+X$. These names impose no sign on $N$.

For the boundary convention, suppose separately that $\bar{\mathrm{\Sigma }}\subset M$ is a connected smooth embedded three-dimensional submanifold with compact boundary, supplied with its manifold-with-boundary smooth structure and smooth inclusion, whose interior is $\mathrm{\Sigma }$. Suppose that $g$ restricts to a positive-definite smooth metric $\bar{h}$ on all tangent spaces of $\bar{\mathrm{\Sigma }}$, including its boundary. Its intrinsic distance is the infimum of $\bar{h}$-lengths of piecewise $C^1$ curves in $\bar{\mathrm{\Sigma }}$ joining the two points. Complete up to the compact boundary means that this metric space, with the boundary included, is complete: every Cauchy sequence converges in $\bar{\mathrm{\Sigma }}$. It does not assert completeness of the interior with its own intrinsic distance. For a disconnected extension, the convention applies separately to each connected component.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface

Metric compatibility, torsion freedom and the metric-selected connection class

Riemannian metric and riemannian manifold

Riemannian distance on a connected manifold

Riemannian distance is a metric

The Axiom of Choice

Embedded smooth submanifolds with boundary

Cauchy sequence in a metric space

Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ

The normal, its tangential covariant derivatives, $h$ and $k$ are well defined and smooth by the stated hypersurface supplier; its second fundamental form has the opposite sign from $k$. The inverse of positive-definite $h$ defines the trace, so maximality is a definite condition. Choose a smooth local extension $v$ of $n$ supplied by the normal lemma. Near each point of $\mathrm{\Sigma }$, continuity permits shrinking its ambient neighborhood so that $g(v,v)<0$. Then $\tilde{n}=v\mathrm{/}\sqrt{-g(v,v)}$ is smooth, has unit timelike length throughout that neighborhood and agrees with $n$ along $\mathrm{\Sigma }$.

For any such unit timelike extension, metric compatibility gives $g({\mathrm{\nabla }}_{\tilde{n}}\tilde{n},\tilde{n})=\frac{1}{2}\tilde{n}(g(\tilde{n},\tilde{n}))=0$ throughout the neighborhood. On $\mathrm{\Sigma }$, in a $g$-orthonormal frame $n,e_1,e_2,e_3$ with tangent $e_i$, the divergence trace is $-g({\mathrm{\nabla }}_n\tilde{n},n)+{\sum }_{i=1}^{3}g({\mathrm{\nabla }}_{e_i}\tilde{n},e_i)$. Its normal term vanishes and its tangent term equals $-{\mathrm{tr}}_{h}k$. The normal lemma makes the tangential derivatives independent of extension, so every local unit timelike extension gives the same divergence on $\mathrm{\Sigma }$.

Finally, $g(X,n)=g(T,n)-Ng(n,n)=0$, so the normal supplier's orthogonal decomposition makes $X$ tangent; the formula gives the unique such decomposition. The embedded-submanifold-with-boundary supplier specifies the smooth structure and smooth inclusion used in the boundary convention. The positive-definiteness hypothesis then gives a Riemannian metric smooth up to the boundary, and the distance suppliers, whose metric argument includes boundary half-charts, make the stated Cauchy-completeness condition well defined. No existence of a maximal hypersurface or complete extension is asserted.