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

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $Σ \subseteq M$ be a spacelike embedded hypersurface in the sense of Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a), so that $Σ$ is an embedded three-dimensional submanifold (Embedded submanifolds and slice charts) and the restriction of $g_{p}$ to $T_{p} Σ$ is positive definite for every $p \in Σ$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space). In the notation of Spacelike initial data, the constraint equations, developments and maximal developments clause (a), whose claims this lemma establishes:

(a) The future unit normal. For every $p \in Σ$ the orthogonal complement $T_{p} Σ^{⊥}$ is one-dimensional, nondegenerate for $g_{p}$ , and negative definite: every nonzero $u \in T_{p} Σ^{⊥}$ is timelike, $g_{p} (u, u) < 0$ . Hence there is exactly one $n_{p} \in T_{p} Σ^{⊥}$ with $g_{p} (n_{p}, n_{p}) = - 1$ and $g_{p} (n_{p}, N_{p}) < 0$ .

(b) Smoothness along the hypersurface. The assignment $p \mapsto n_{p}$ is smooth along $Σ$ : for every $p \in Σ$ there are an open neighborhood $O$ of $p$ in $M$ and a smooth vector field $\tilde{n}$ on $O$ (A smooth vector field is a smooth section of the tangent bundle) with $\tilde{n} ∣_{Σ \cap O} = n ∣_{Σ \cap O}$ , and for any two such local extensions the covariant derivative at a point of $Σ$ in a direction tangent to $Σ$ agrees. Thus for $V \in T_{p} Σ$ the vector $\nabla_{V} n \in T_{p} M$ is well defined as $\nabla_{\tilde{X}} \tilde{n} ∣_{p}$ for any smooth extensions $\tilde{n}$ of $n$ and $\tilde{X}$ of a tangent field with ${\tilde{X}}_{p} = V$ (Covariant derivative along a curve, Metric compatibility, torsion freedom and the metric-selected connection class).

(c) The induced metric and second fundamental form. The induced metric $h_{p} (V, W) : = g_{p} (V, W)$ is a smooth Riemannian metric on $Σ$ , and $K_{p} (V, W) : = g_{p} (\nabla_{V} n, W) (V, W \in T_{p} Σ)$ (the second fundamental form) defines a smooth symmetric covariant $2$ -tensor field $K$ on $Σ$ (Smooth covariant 2-tensor fields on a smooth manifold) with $K_{p} (V, W) = - g_{p} (\nabla_{V} W, n_{p}),$ where $\nabla_{V} W$ is the ambient covariant derivative at $p$ of any smooth extension of a tangent field with value $W$ at $p$ , in the direction of any extension with value $V$ at $p$ .

(d) The shape operator. The endomorphism $S_{p} : T_{p} Σ \to T_{p} Σ$ defined by $h_{p} (S_{p} V, W) = K_{p} (V, W)$ for all $W \in T_{p} Σ$ is well defined, linear and self-adjoint for the positive definite form $h_{p}$ , and the Weingarten relation $\nabla_{V} n = S_{p} V$ holds for every $V \in T_{p} Σ$ .

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ and a spacelike embedded hypersurface $Σ \subseteq M$ .

[L1]

For each $p \in Σ$ there is a slice chart $(U, y)$ around $p$ with $Σ \cap U = {y^{4} = 0}$ , and $(y^{1}, y^{2}, y^{3}) ∣_{Σ \cap U}$ are coordinates of $Σ$ ; the restriction of $g_{p}$ to $T_{p} Σ$ is positive definite (Embedded submanifolds and slice charts, Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a)).

[L2]

For a smooth function $f$ on an open set $U$ and the metric-selected connection, there is a unique smooth vector field $ξ = ♯ d f$ on $U$ with $g (ξ, X) = d f (X)$ for every smooth field $X$ ; in components $ξ^{a} = \sum_{b} g^{a b} \partial_{b} f$ , so $ξ$ is smooth whenever $f$ and $g$ are, by smoothness of the inverse metric components (The inverse metric, musical isomorphisms and index raising, Smoothness and transformation of the inverse-metric components, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L3]

For an observer $n$ at $p$ (so $g_{p} (n, n) = - 1$ ) the rest space $n^{⊥} = {v : g_{p} (v, n) = 0}$ has dimension three and the restriction of $g_{p}$ to it is positive definite (The observer rest-space decomposition and the spatial projection, Observers, rest spaces and the spatial projection at an event). Consequently, if $u$ is a nonzero timelike vector at $p$ and $v ⊥ u$ is also timelike, then applying this to the observer $\pm u / \sqrt{- g_{p} (u, u)}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space) exhibits a nonzero timelike vector in a positive definite space, a contradiction; hence $g_{p} (u, v) \neq 0$ for two timelike vectors $u, v$ , and $u^{⊥}$ is positive definite.

[L4]

$\nabla_{X} Y ∣_{p}$ depends only on $X_{p}$ and on the values of $Y$ along any smooth curve through $p$ with velocity $X_{p}$ ; in particular two smooth fields that agree along such a curve have the same covariant derivative in the direction $X_{p}$ (Covariant derivative along a curve, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L5]

$\nabla$ is torsion-free and metric-compatible: $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ and $X (g (Y, Z)) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z)$ for smooth fields (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L6]

If smooth fields $X, Y$ on $M$ are tangent to $Σ$ along $Σ$ , then the bracket $[X, Y]$ is tangent to $Σ$ along $Σ$ : in a slice chart of [L1], write $X = X^{i} \partial_{i} + X^{4} \partial_{4}$ , $Y = Y^{i} \partial_{i} + Y^{4} \partial_{4}$ with $X^{4} ∣_{Σ \cap U} = Y^{4} ∣_{Σ \cap U} = 0$ ; then the fourth component of $[X, Y]$ is $X (Y^{4}) - Y (X^{4})$ , and at $q \in Σ \cap U$ this is $d Y^{4} (X_{q}) - d X^{4} (Y_{q}) = 0$ because $X_{q}, Y_{q} \in T_{q} Σ$ and $d Y^{4}, d X^{4}$ annihilate $T_{q} Σ$ (Embedded submanifolds and slice charts).

Proof

technique · direct

1.1givenL1L2

(A smooth nonzero normal field on a slice chart.) Fix $p \in Σ$ and a slice chart $(U, y)$ as in [L1], and put $f : = y^{4}$ , so that $Σ \cap U = {f = 0}$ and $d f_{q} \neq 0$ for $q \in Σ \cap U$ ; let $ξ : = ♯ d f$ on $U$ be the metric dual field of [L2]. Then $ξ$ is smooth on $U$ and nonzero there, and for $q \in Σ \cap U$ and $X \in T_{q} Σ$ one has $g_{q} (ξ_{q}, X) = d f_{q} (X) = 0$ because $X$ is tangent to the level set ${f = 0}$ ; hence $ξ_{q} \in T_{q} Σ^{⊥} ∖ {0}$ .

2.1step 1.1givenL1L3

(The normal line is one-dimensional and negative definite.) Let $q \in Σ$ . If $v \in T_{q} Σ \cap T_{q} Σ^{⊥}$ , then $g_{q} (v, v) = 0$ because $v$ is orthogonal to itself, while positive definiteness of $g_{q} ∣_{T_{q} Σ}$ forces $v = 0$ ; hence $T_{q} Σ \cap T_{q} Σ^{⊥} = {0}$ . Therefore the quotient map $T_{q} M \to T_{q} M / T_{q} Σ$ restricts injectively on $T_{q} Σ^{⊥}$ , so $\dim T_{q} Σ^{⊥} \leq \dim (T_{q} M / T_{q} Σ) = 1$ , and $ξ_{q} \neq 0$ from step 1.1 shows that the dimension is exactly one and that $\dim (T_{q} Σ + T_{q} Σ^{⊥}) = 4$ , that is $T_{q} M = T_{q} Σ \oplus T_{q} Σ^{⊥}$ . If $u \in T_{q} Σ^{⊥}$ is nonzero and $g_{q} (u, u) \geq 0$ , then on the direct sum $T_{q} Σ \oplus R u = T_{q} M$ the form $g_{q}$ is a block sum of a positive definite form and the form $g_{q} (u, u) \geq 0$ on the line $R u$ , hence positive semidefinite with the only possible zero values on $R u$ ; this contradicts the existence of the timelike time-orientation vector $N_{q}$ with $g_{q} (N_{q}, N_{q}) < 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Hence $g_{q} (u, u) < 0$ for every nonzero $u \in T_{q} Σ^{⊥}$ : the normal line is negative definite, and in particular nondegenerate.

3.1step 2.1givenL2L3

(Existence and uniqueness of the future unit normal.) Let $q \in Σ$ . The vector $ξ_{q}$ is nonzero and, by step 2.1, timelike; so is $N_{q}$ . By [L3] two timelike vectors at $q$ are never orthogonal, so $g_{q} (ξ_{q}, N_{q}) \neq 0$ , and $n_{q} : = - \frac{sign (g_{q} (ξ_{q}, N_{q}))}{\sqrt{- g_{q} (ξ_{q}, ξ_{q})}} ξ_{q}$ is a well-defined element of $T_{q} Σ^{⊥}$ with $g_{q} (n_{q}, n_{q}) = - 1$ and $g_{q} (n_{q}, N_{q}) = - ∣ g_{q} (ξ_{q}, N_{q}) ∣ / \sqrt{- g_{q} (ξ_{q}, ξ_{q})} < 0$ . If $m \in T_{q} Σ^{⊥}$ satisfies $g_{q} (m, m) = - 1$ and $g_{q} (m, N_{q}) < 0$ , then $m = t ξ_{q}$ for some $t \in R$ because the line is spanned by $ξ_{q}$ ; $g_{q} (m, m) = t^{2} g_{q} (ξ_{q}, ξ_{q}) = - 1$ gives $t = \pm 1 / \sqrt{- g_{q} (ξ_{q}, ξ_{q})}$ , and the sign of $g_{q} (m, N_{q}) = t g_{q} (ξ_{q}, N_{q})$ is the product of the signs of $t$ and $g_{q} (ξ_{q}, N_{q})$ ; the condition $g_{q} (m, N_{q}) < 0$ therefore selects exactly $t = - sign (g_{q} (ξ_{q}, N_{q})) / \sqrt{- g_{q} (ξ_{q}, ξ_{q})}$ , that is $m = n_{q}$ . This proves (a).

4.1step 1.1step 2.1step 3.1givenL2L4

(Smoothness of the normal along $Σ$ .) On the slice chart $(U, y)$ of step 1.1, the functions $q \mapsto g_{q} (ξ_{q}, ξ_{q})$ and $q \mapsto g_{q} (ξ_{q}, N_{q})$ are smooth on $U$ ; on $Σ \cap U$ the first is negative and the second nonzero by steps 2.1 and 3.1. Shrinking $U$ to a smaller connected neighborhood of a given point of $Σ$ , we may suppose $g (ξ, ξ) < 0$ on all of $U$ and, on the connected set $Σ \cap U$ , $sign (g (ξ, N))$ constant; then the formula of step 3.1 defines a smooth ambient field $\tilde{n} : = - sign (g (ξ, N)) ξ / \sqrt{- g (ξ, ξ)}$ on $U$ whose restriction to $Σ \cap U$ is $n$ . Such local extensions cover $Σ$ and agree on overlaps because all restrict to $n$ on $Σ$ . Independence of the extensions: let ${\tilde{n}}_{1}, {\tilde{n}}_{2}$ be local extensions of $n$ near $p$ and let $\tilde{X}$ be a smooth extension of a tangent field with ${\tilde{X}}_{p} = V$ ; choose a smooth curve $γ$ in $Σ$ with $γ (0) = p$ and $γ^{'} (0) = V$ . Then ${\tilde{n}}_{1} \circ γ = {\tilde{n}}_{2} \circ γ$ as curves in $T M$ , so by [L4] $\nabla_{\tilde{X}} {\tilde{n}}_{1} ∣_{p} = \nabla_{\tilde{X}} {\tilde{n}}_{2} ∣_{p}$ . For a fixed extension $\tilde{n}$ , the map $X \mapsto \nabla_{X} \tilde{n} ∣_{p}$ is additive and $C^{\infty} (M)$ -linear by the affine-connection laws (Affine connection on a smooth manifold), so its value depends only on $X_{p}$ ; hence $\nabla_{V} n : = \nabla_{\tilde{X}} \tilde{n} ∣_{p}$ is independent of both extensions and is additive in $V$ . This proves (b).

5.1step 4.1givenL1L2L4

( $h$ and $K$ are smooth tensor fields.) On $Σ \cap U$ of a slice chart, the components $h_{i j} = g_{i j} ∣_{Σ \cap U}$ ( $i, j \in {1, 2, 3}$ ) are smooth functions of the slice coordinates and the matrix $(h_{i j})$ is positive definite by [L1]; since $Σ$ is covered by such charts and the local expressions agree on overlaps (both are restrictions of the smooth field $g$ ), $h$ is a smooth symmetric positive definite covariant $2$ -tensor field on $Σ$ , that is, a Riemannian metric. For $K$ , fix $p \in Σ$ and choose a local smooth extension $\tilde{n}$ of $n$ and smooth tangent extensions $\tilde{X}, \tilde{Y}$ of fields with ${\tilde{X}}_{p} = V$ , ${\tilde{Y}}_{p} = W$ ; by step 4.1 the number $g_{p} (\nabla_{V} n, W) : = g_{p} (\nabla_{\tilde{X}} \tilde{n}, \tilde{Y})$ does not depend on the three extensions. It is additive and $R$ -bilinear in $(V, W)$ because $\nabla$ and $g$ are, and it is $C^{\infty} (Σ)$ -linear in each slot: replacing $\tilde{X}$ by $φ \tilde{X}$ for a smooth function $φ$ on $Σ$ extended locally changes $\nabla_{\tilde{X}} \tilde{n}$ by a factor $φ$ by the connection law $\nabla_{φ \tilde{X}} \tilde{n} = φ \nabla_{\tilde{X}} \tilde{n}$ , and replacing $\tilde{Y}$ by $φ \tilde{Y}$ multiplies $g (\nabla_{\tilde{X}} \tilde{n}, \tilde{Y})$ by $φ$ by bilinearity of $g$ ; the chart components of $K$ in slice coordinates, $K_{i j} = g (\nabla_{\partial_{i}} \tilde{n}, \partial_{j})$ , are then smooth in the slice coordinates because the connection coefficients and $\tilde{n}$ are smooth. Hence $K$ is a smooth covariant $2$ -tensor field on $Σ$ .

6.1step 5.1step 4.1givenL4L5L6

(Symmetry and the Gauss relation.) Fix $p \in Σ$ and $V, W \in T_{p} Σ$ ; extend $V, W$ to smooth ambient fields $X, Y$ tangent to $Σ$ along $Σ$ , and extend $n$ to a smooth ambient field $\tilde{n}$ near $p$ (step 4.1). The function $g (\tilde{n}, Y)$ vanishes identically on $Σ$ because $Y$ is tangent and $\tilde{n}$ normal there; differentiating it along $V$ at $p$ and using [L5] gives $0 = g (\nabla_{V} \tilde{n}, Y) + g (n, \nabla_{V} Y)$ at $p$ , that is $K_{p} (V, W) = - g_{p} (\nabla_{V} W, n_{p})$ ; the same argument with $V$ and $W$ exchanged gives $K_{p} (W, V) = - g_{p} (\nabla_{W} V, n_{p})$ . By torsion-freedom in [L5], $\nabla_{V} W - \nabla_{W} V = [X, Y]$ at $p$ , and $[X, Y]$ is tangent to $Σ$ at $p$ by [L6]; since $n_{p}$ is normal, $g_{p} ([X, Y], n_{p}) = 0$ . Therefore $K_{p} (V, W) - K_{p} (W, V) = - g_{p} ([X, Y], n_{p}) = 0$ , so $K$ is symmetric and the displayed Gauss relation holds.

7.1step 5.1step 6.1givenL3L5∎

(The shape operator.) Fix $p \in Σ$ . Since $h_{p}$ is a positive definite symmetric form on the three-dimensional space $T_{p} Σ$ , there is an $h_{p}$ -orthonormal basis $e_{1}, e_{2}, e_{3}$ of $T_{p} Σ$ (Every finite-dimensional real or complex inner product space has an orthonormal basis); define $S_{p} V : = \sum_{i = 1}^{3} K_{p} (V, e_{i}) e_{i}$ for $V \in T_{p} Σ$ . Then $h_{p} (S_{p} V, e_{j}) = K_{p} (V, e_{j})$ for each $j$ , and since ${e_{j}}$ is a basis the identity $h_{p} (S_{p} V, W) = K_{p} (V, W)$ holds for all $W \in T_{p} Σ$ ; this also shows that $S_{p} V$ does not depend on the chosen orthonormal basis. The assignment $V \mapsto S_{p} V$ is linear because $K_{p}$ is bilinear. Self-adjointness is immediate from symmetry of $K$ and of $h_{p}$ : $h_{p} (S_{p} V, W) = K_{p} (V, W) = K_{p} (W, V) = h_{p} (S_{p} W, V) = h_{p} (V, S_{p} W)$ . Finally, $\nabla_{V} n \in T_{p} M$ satisfies $g_{p} (\nabla_{V} n, n_{p}) = \frac{1}{2} V (g (\tilde{n}, \tilde{n})) = 0$ because $g (\tilde{n}, \tilde{n}) \equiv - 1$ on $Σ$ and $V$ is tangent, so $\nabla_{V} n \in n_{p}^{⊥}$ ; by [L3] $n_{p}^{⊥}$ is three-dimensional, and it contains $T_{p} Σ$ , also three-dimensional, so $n_{p}^{⊥} = T_{p} Σ$ and $\nabla_{V} n \in T_{p} Σ$ . For every $W \in T_{p} Σ$ one has $g_{p} (\nabla_{V} n, W) = K_{p} (V, W) = h_{p} (S_{p} V, W)$ ; since $h_{p}$ is nondegenerate, $\nabla_{V} n = S_{p} V$ . This proves (d) and completes the proof.

Remarks

Sign and orientation conventions. With signature $(-, +, +, +)$ and future normal normalized by $g (n, n) = - 1$ , this library uses $K (V, W) = g (\nabla_{V} n, W) = - g (\nabla_{V} W, n)$ . The global sign of the extrinsic curvature is a convention: Gourgoulhon defines it as the pullback of $- \nabla n$ , which is $- K$ in the present normalization (his sphere example gives $K_{i j} = - (1 / R) γ_{i j}$ where the present convention gives $+ (1 / R) γ_{i j}$ ). Replacing $n$ by $- n$ reverses the sign of $K$ . $K$ is not claimed to be positive or negative definite, and no mean curvature sign is fixed here. Consumers must use the definition displayed in this item, not a differing textbook sign.
Scope. The lemma is local and kinematic: it constructs the normal from a slice chart and uses only the metric-selected connection, torsion freedom and metric compatibility. No field equation, no causal condition on the model beyond time orientation and no compactness or completeness hypothesis enters.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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