Hypersurfaces, Extrinsic Curvature, and Embeddings

A hypersurface divides ambient geometry into tangential and normal directions. That split supplies an induced metric, an intrinsic derivative, extrinsic curvature, curvature compatibility identities, and—when hypersurfaces form a foliation—the lapse and shift. The chapter develops these structures as geometry. Gravitational constraint equations, junction dynamics, numerical evolution systems, and action principles require additional physical input and are not derived here.

Spacelike and timelike hypersurfaces admit unit normals and nondegenerate induced metrics. Null hypersurfaces do not: their normal is tangent, their induced metric is degenerate, and their transverse data carry a normalization freedom. Keeping these cases separate is the chapter’s most important structural discipline.

Helpful background. Calculus and Linear Algebra for Relativity supplies level sets, projectors, and block matrices; Affine Connections and Covariant Derivatives supplies the covariant derivative that will be projected; The Riemann Curvature Tensor supplies the ambient curvature tensor and index convention; Pseudo-Riemannian Metrics and Signature supplies causal character; Exterior Algebra and Differential Forms supplies forms and orientation algebra, while the generalized Stokes theorem supplies the bulk–boundary identity.

Geometry of an embedded slice

For a non-null unit normal $n^\mu$, write

$$n^\mu n_\mu=\epsilon, \qquad \epsilon=\pm1, \qquad h_{\mu\nu}=g_{\mu\nu}-\epsilon n_\mu n_\nu.$$

The induced metric $h$ is intrinsic to the hypersurface. Its Levi-Civita derivative $D$ is obtained by tangentially projecting the ambient derivative. Extrinsic curvature measures normal turning. This chapter adopts

$$K_{\mu\nu} =-h_\mu{}^\alpha h_\nu{}^\beta\nabla_\alpha n_\beta.$$

For a spacelike foliation with future unit normal, this convention gives

$$K_{ij} =-\frac1{2N} \left(\partial_t h_{ij}-\mathcal L_\beta h_{ij}\right).$$

Declaring the sign before use is essential because an equally common convention reverses $K$. The induced metric is invariant under $n\mapsto-n$, while $K$ changes sign.

Choose a route

• Begin with Hypersurface Normals and Tangential Projectors to classify regular level sets, fix coorientation, and separate the null case.
• Read Intrinsic Geometry of Hypersurfaces to construct the pullback metric and induced Levi-Civita derivative.
• Continue to Extrinsic Curvature and the Weingarten Map for the declared sign, shape operator, and normal metric evolution.
• Use Gauss–Codazzi–Ricci Relations to connect ambient, intrinsic, and extrinsic curvature.
• Follow Foliations, Lapse, and Shift when a region is sliced by spacelike hypersurfaces.
• Finish with Boundaries, Corners, and Orientation Conventions before applying Stokes’ theorem to regions with mixed causal boundary pieces.

The first four pages form the embedding route. The foliation page is the shortest route toward a geometric spacetime split. The final page is the boundary route and can be read earlier by anyone doing integral identities.

Recurring checks

A flat plane has $K=0$ and zero intrinsic curvature. A cylinder in Euclidean space has zero intrinsic curvature but nonzero extrinsic curvature. A round sphere with outward normal has

$$K_{ab}=-\frac1R h_{ab}$$

in this convention and positive intrinsic curvature $1/R^2$. Together, these examples separate intrinsic curvature from bending and check the Gauss sign.

Constant-time Minkowski slices have $K_{ij}=0$. Hyperboloidal Milne slices in the same flat spacetime instead have

$$K_{ij}=-\frac1\tau h_{ij},$$

while their induced geometry has negative constant curvature. The ambient Riemann tensor remains zero; the Gauss relation balances intrinsic and extrinsic terms. Rindler slices add a third pattern: $K=0$ but the future normal congruence accelerates because the lapse varies spatially.

For boundaries, an oriented rectangle supplies a reliable sign test. The top and bottom caps have opposite outward timelike normals; the right and left walls have opposite outward spacelike normals. Outward-normal-first orientation makes every corner cancel in the boundary of the boundary.

These constructions and flat checks follow the hypersurface and foliation treatment in Gourgoulhon 2007, §§2.2.3–2.5, pp. 18–38; §3.3, pp. 41–46; §4.2, pp. 54–59.

Distinctions to preserve

Intrinsic curvature can be calculated from $h$ without knowing the embedding. Extrinsic curvature requires the embedding and a chosen coorientation. Ambient curvature is a third object. None can be identified with another without an equation such as Gauss and its hypotheses.

Lapse and shift describe a selected foliation and coordinate threading. The kinematical metric reconstruction does not supply Einstein constraints or evolution equations. Similarly, Gauss–Codazzi identities become gravitational constraints only after a field equation is imposed.

Null boundaries are never obtained by setting $\epsilon=0$ in a unit-normal formula. Their normal cannot be normalized, and a screen or auxiliary null direction is required. At joints involving a null face, normalization and orientation conventions must be fixed before assigning geometric corner data Lehner et al. 2016, §§II.B–II.H, pp. 7–25.

Check your understanding

In Milne coordinates,

$$ds^2=-d\tau^2+\tau^2\gamma_{ij}dx^idx^j,$$

where $\gamma$ is the unit hyperbolic metric. Find the lapse, shift, and extrinsic curvature of a constant-$\tau$ slice. Explain why nonzero $K_{ij}$ does not imply nonzero ambient curvature.

Solution

The foliation is adapted to $\tau$, so $N=1$ and $\beta^i=0$. Its induced metric is $h_{ij}=\tau^2\gamma_{ij}$. With the convention used throughout the chapter,

$$K_{ij} =-\frac12\partial_\tau h_{ij} =-\tau\gamma_{ij} =-\frac1\tau h_{ij}.$$

Milne coordinates cover a region of Minkowski spacetime, whose ambient Riemann tensor vanishes. The nonzero extrinsic curvature measures how the hyperboloidal slices bend in that flat spacetime. Their negative intrinsic curvature and the quadratic $K$ terms cancel exactly in the Gauss relation.

What this chapter enables

After the embedding route, you should be able to classify a hypersurface, compute its projector and induced derivative, declare and calculate extrinsic curvature, and verify Gauss–Codazzi signs on elementary embeddings. The foliation route adds reconstruction of the metric from lapse and shift, while the boundary route adds consistent measures and corner orientations.

The next conceptual step depends on purpose. Geometric variational problems continue to Variational and Symplectic Geometry. Einstein constraints, junction conditions, evolution formulations, and numerical algorithms belong to the dedicated gravity and computational volumes.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94, 084046 (2016). arXiv:1609.00207.