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Smooth vacuum constraint data obtained by reflection of a canonical end section

Statement

Let $\overline{C} = C \cup S$ be a Hausdorff second-countable smooth three-manifold with compact boundary $S$ , carrying smooth vacuum initial data $(h, K)$ up to $S$ . Here $h$ is a positive-definite Riemannian metric, $K$ is a symmetric covariant two-tensor, and vacuum initial data means that $H (h, K) = 0$ and $M (h, K) = 0$ , where $D$ is the Levi–Civita connection of $h$ , $R_{h}$ is its scalar curvature in the constraint supplier's convention, $τ = h^{i j} K_{i j}$ , $∣ K ∣_{h}^{2} = h^{i a} h^{j b} K_{i j} K_{a b}$ , $H (h, K) = R_{h} + τ^{2} - ∣ K ∣_{h}^{2}$ , and $M (h, K)_{j} = h^{i a} (D_{i} K)_{a j} - \partial_{j} τ$ . These are equations on abstract data; no ambient field equation is assumed to establish them.

Assume $(\overline{C}, h)$ is complete with its boundary included, in the intrinsic-distance sense, and has an asymptotically Euclidean end: a specified end chart $x : E \to {x \in R^{3} : ∣ x ∣ > R_{0}}$ in which $h_{i j} (x) - δ_{i j} \to 0$ uniformly in direction as $∣ x ∣ \to \infty$ . No numerical decay rate or derivative order is needed here; any supplied stronger differentiated bounds on $h$ and $K$ remain part of the data. Use the usual authoring convention of the Axiom of Choice, hence also countable choice for the global smooth-double and curvature interfaces.

Supply a smooth collar $c : [0, ϵ) \times S \to \overline{C}$ , with $c (0, y) = y$ , and the following local canonical identifications. For every $p \in S$ , supply a coordinate neighborhood $B_{p} \subset S$ , a width $0 < ϵ_{p} \leq ϵ$ , and a $W$ -invariant adapted boost patch satisfying the exact local boost and commuting-field hypotheses of the reflection-parity supplier. In that patch supply a spacelike canonical section $Q_{p} = {U = V = r}$ . Identify $c ([0, ϵ_{p}) \times B_{p})$ , with its induced data, with the positive half of $Q_{p}$ using the same collar coordinates: $(r, y)$ corresponds to $(U, V, y) = (r, r, y)$ . Thus the canonical reflection is $R_{p} (r, y) = (- r, y)$ in the signed collar coordinates. Use the convention $K (v, w) = g (\nabla_{v} n, w) = - k (v, w)$ . These identifications are supplied locally along $S$ . Form the labelled topological double $\hat{C} = C_{+} \cup_{S} C_{-}$ of two copies of $\overline{C}$ and equip it with the smooth structure induced by this collar: the signed seam chart $(r, y)$ represents $[c (r, y), +]$ for $r \geq 0$ and $[c (- r, y), -]$ for $r \leq 0$ .

On $C_{-}$ define the metric as the reflected copy of $h$ and the second fundamental form as minus the reflected copy of $K$ , using tensor pullback by the copy identification. Then these fields extend smoothly across $S$ , give a positive-definite metric and symmetric two-tensor on $\hat{C}$ , and satisfy the vacuum Hamiltonian and momentum constraints on the entire double. The premise that the full outer section has been joined to the canonical collar is explicit and is not proved here.

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: Smooth reflected vacuum constraint data on the collar-induced double.

[F1]

In each supplied canonical patch let $h_{p}, K_{p}$ be the smooth induced data on $Q_{p}$ , and let $R_{p} (r, y) = (- r, y)$ . The local clause of the reflection-parity supplier gives $j_{{r = 0}}^{\infty} R_{p}^{*} h_{p} = j_{{r = 0}}^{\infty} h_{p}$ and $j_{{r = 0}}^{\infty} R_{p}^{*} K_{p} = - j_{{r = 0}}^{\infty} K_{p}$ , including every mixed normal and tangential derivative. By the specified coordinate identification their positive restrictions are the original collar data $h, K$ . These local identities provide the jet matching in each seam chart; the supplier's separate whole-surface reflection clause is not invoked. The induced $K$ has the opposite sign from the datum $k$ in the hypersurface convention. Reflection parities of canonical section data and stationary lapse–shift Spacelike sections, maximality and lapse–shift data

[F2]

The vacuum constraint expressions are $H = R_{h} + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2}$ and $M = {div}_{h} K - d {tr}_{h} K$ , with the contractions specified in the Statement. The normal-frame theorem identifies these expressions with vacuum constraints for induced data; their vanishing on the supplied abstract data is a hypothesis, not a consequence of an unstated ambient solution. Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form

[F3]

The labelled double identifies only corresponding boundary points. A smooth collar supplies signed seam charts, which together with the two interior atlases give a smooth boundaryless manifold structure, under countable choice. The double of a smooth manifold with boundary The double has a well-defined smooth structure Smooth collars of a manifold boundary

[F4]

A Riemannian metric is a smooth positive-definite symmetric two-tensor, also at boundary points where allowed. Covariant tensor pullback inserts the differential in every slot. Smoothness of a tensor field is equivalent to smoothness of its chart components. Riemannian metric and riemannian manifold The pullback of a covariant tensor field Smoothness of a tensor field is equivalent to smooth coordinate components

[F5]

A smooth Riemannian metric, including at a boundary, has a unique smooth Levi–Civita connection, whose coordinate coefficients are $Γ^{k}_{i j} = \frac{1}{2} h^{k ℓ} (\partial_{i} h_{j ℓ} + \partial_{j} h_{i ℓ} - \partial_{ℓ} h_{i j})$ . With $R (\partial_{i}, \partial_{j}) \partial_{k} = R^{ℓ}_{k i j} \partial_{ℓ}$ , the curvature components are $R^{ℓ}_{k i j} = \partial_{i} Γ^{ℓ}_{j k} - \partial_{j} Γ^{ℓ}_{i k} + Γ^{m}_{j k} Γ^{ℓ}_{i m} - Γ^{m}_{i k} Γ^{ℓ}_{j m}$ . This uses the same curvature-operator sign convention as F2. Its scalar curvature is the smooth inverse-metric trace of Ricci curvature. Fundamental theorem of riemannian geometry Christoffel formula for the levi civita connection Scalar curvature Coordinate formula for the curvature tensor

[F6]

A scalar function continuous on a closed real interval and differentiable in its interior satisfies the mean value theorem. Existence and continuity of all first coordinate partials imply total differentiability. The $C^{m}$ convention requires continuous iterated partials for every coordinate word of length at most $m$ . The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a) If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative Cᵏ maps and multi-index derivative notation in Euclidean space

[A1]

The authoring convention assumes the Axiom of Choice. Applying a choice function to the family of sets occurring in a sequence of nonempty sets supplies a choice sequence, hence the countable choice required in F3 and the scalar-curvature interface. The Axiom of Choice

Proof

1.1F3A1given

Use the smooth structure of the specified collar, not an arbitrary smoothing of the topological double. The seam-chart transitions have the form $(r, y) \mapsto (r, \tilde{y} (y))$ ; overlaps with the original interior charts lie in one open half, where the collar and its inverse are smooth. F3 therefore gives a smooth boundaryless $\hat{C}$ , with each copied half smooth up to its boundary. The supplied collar is already fixed; no canonical choice or collar-independence assertion is needed.

2.1F1F4F6step 1.1algebra

Fix a seam point and use the collar chart identified with its canonical patch in the Statement. Let index $0$ denote $r$ , set $ε_{0} = - 1$ and $ε_{1} = ε_{2} = 1$ , and write $h^{+}, K^{+}$ for the original collar components. On $r < 0$ the proposed components are ${\hat{h}}_{i j} (r, y) = ε_{i} ε_{j} h_{i j}^{+} (- r, y)$ and ${\hat{K}}_{i j} (r, y) = - ε_{i} ε_{j} K_{i j}^{+} (- r, y)$ , while on $r > 0$ they are the original ones. The signs in both tensor slots come from pullback. In these same coordinates $h^{+}, K^{+}$ are the positive restrictions of the smooth canonical data $h_{p}, K_{p}$ . The local parity identities in F1 therefore make every one-sided derivative of the displayed formulas agree at $r = 0$ , including all mixed normal and tangential derivatives. This argument applies independently at every seam point and requires no common ambient negative section or global ambient reflection.

3.1F6step 2.1algebra

Here is the needed smooth-pasting argument in this chart. For one component $f$ and each finite coordinate-derivative word $w$ , paste the corresponding one-sided derivatives to a function $g_{w}$ , using their common boundary value. Matching boundary jets and one-sided smoothness make every $g_{w}$ continuous. At $(0, y)$ , its tangential partials are the derivatives of the common smooth boundary trace, hence equal $g_{w A} (0, y)$ . For its normal partial apply the mean value theorem on the segment from $0$ to $t$ in either half: $(g_{w} (t, y) - g_{w} (0, y)) / t = g_{w 0} (ξ_{t}, y)$ for a point $ξ_{t}$ strictly between $0$ and $t$ . Continuity of $g_{w 0}$ makes both limits equal $g_{w 0} (0, y)$ . Thus every $g_{w}$ has all first partials, and they are the continuous functions $g_{w i}$ . F6 makes $g_{w}$ continuously differentiable; iteration proves $f = g_{\emptyset}$ is $C^{m}$ for every $m$ . This proves smooth pasting locally, rather than importing an undeclared pasting lemma.

4.1F4step 1.1step 2.1step 3.1algebra

Apply step 3.1 to every component of $\hat{h}$ and $\hat{K}$ . On overlaps the tensor transformation laws hold away from the seam by construction, and hold on the seam by continuity, since the union of the two open halves is dense. Thus the smooth component fields define global smooth tensors by F4. At the seam $\hat{h}$ equals the original positive-definite form in the collar tangent basis; on either open half it is a pullback of $h$ . Hence it is positive definite everywhere. Symmetry of $\hat{K}$ is preserved by pullback, sign reversal and continuity.

5.1F2F4F5givenstep 4.1algebra

On the positive open half the constraints vanish by the abstract-data hypothesis. On the negative open half use the copied original charts. The metric components and their derivatives are the same as on the original half, while every component of $K$ and its derivatives has the opposite sign. The Christoffel and curvature formulas in F5 give identical connection coefficients and their first derivatives, hence identical curvature components in these copied charts. Taking the Ricci and inverse-metric traces gives the same scalar curvature; therefore $τ$ changes sign, $τ^{2}$ and $∣ K ∣_{h}^{2}$ do not, and $M$ changes sign. Equivalently, if $I : C_{-} \to C_{+}$ is the copy identification on the interiors, then $H (\hat{h}, \hat{K}) = I^{*} H (h, K) = 0$ and $M (\hat{h}, \hat{K}) = - I^{*} M (h, K) = 0$ . This calculation establishes the required covariance and sign change on both open halves.

6.1F2F5step 4.1step 5.1algebra∎

By smoothness and positive definiteness from step 4.1, the inverse metric, its Levi–Civita connection and scalar curvature, and the constraint fields of F2 are smooth, in particular continuous, across $S$ . Step 5.1 makes both constraint fields zero on the dense union of the open halves, so they are zero on the seam as well. This proves smooth vacuum constraint data on the whole double. Completeness and the end assumptions on the original half were retained as supplied data but were not needed in this local gluing argument; no spacetime extension or identification of its exterior is concluded.

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