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Smooth reflection of the stationary vacuum Killing initial data

Statement

In the preceding data-double construction, suppose $T$ is Killing in an actual vacuum neighborhood of the positive section and commutes with the exact boost field on the canonical collar. Assume that throughout the entire positive section the supplied $h$ and $K$ are the data induced by this same ambient metric $g$ , with future unit normal $n$ , $h (U, V) = g (U, V)$ and $K (U, V) = g (\nabla_{U} n, V) = - k (U, V)$ for tangent $U, V$ . Its induced lapse and shift $(N, X)$ extend to $C_{-}$ as minus the reflected lapse and the reflected shift. These are smooth on $\hat{C}$ and satisfy the vacuum Killing-initial-data equations everywhere: $L_{X} h + 2 N K = 0$ and $L_{X} K + \nabla_{h}^{2} N - N ({Ric}_{h} + ({tr}_{h} K) K - 2 K \circ K) = 0$ . This is a statement about initial data, not yet a global Killing flow or an identification of the original exterior.

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: Smooth reflected lapse and shift satisfying both vacuum KID equations.

[F1]

The complete reflected metric and second fundamental form are smooth vacuum constraint data. Smooth vacuum constraint data obtained by reflection of a canonical end section

[F2]

The canonical lapse and shift jets have odd lapse and reflected-vector shift parities. Reflection parities of canonical section data and stationary lapse–shift

[F3]

The positive-side KID equations are the imported necessary identities recorded below; the constraint equations alone do not establish the second identity.

[F4]

Here $K (U, V) = g (\nabla_{U} n, V) = - k (U, V)$ and $T = N n + X$ , $N = - g (T, n)$ . The normal-frame supplier uses this positive-sign $K$ convention, while the hypersurface definition uses $k$ ; Killing means $L_{T} g = 0$ . Spacelike sections, maximality and lapse–shift data Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form Killing fields of a Lorentzian metric

External input

Import the necessary identities from Beig–Chruściel, §2, (2.6), (2.13) in the Ricci-flat specialization, with their $Y = X$ and $g_{i j} = h_{i j}$ . Their convention $\partial_{t} h_{i j} = 2 K_{i j}$ matches F4. Their projections $ρ, τ_{i j}$ in (2.10) vanish in vacuum. These identities require no division by $N$ . Only these premises are imported; the reflection argument is local below.

Proof

Given: The data and hypotheses of the Statement.

1.1F1F2algebra

Extend $N$ with a minus sign and $X$ by reflected vector pullback. The all-jet collar parities give smoothness across $S$ by the componentwise pasting criterion proved in the data-double supplier (Proof 3.1); it applies to scalar and vector components as well as to the data tensors.

1.2F3F4given

Apply F3 on the positive open half with the induced data from F4: the actual ambient metric is smooth and Ricci-flat there and $T$ is Killing. Thus both displayed equations hold on that half, including at zeros of $N$ .

2.1F1step 1.1step 1.2algebra∎

On the negative open half use the charts copied from the positive half. In these charts $(h, K, N, X)$ becomes $(h, - K, - N, X)$ , with the same component functions and coordinate derivatives before the displayed sign changes. Thus the Levi–Civita connection and ${Ric}_{h}$ have the same components, while $D^{2} N$ and $L_{X} K$ change sign. The terms $({tr}_{h} K) K$ and $K \circ K$ do not change. Consequently the first KID expression is unchanged and the entire second expression changes by a common minus sign; both vanish by step 1.2. Smoothness of all data and of $(N, X)$ makes the KID expressions continuous across $S$ , so vanishing on the dense union of the open halves proves vanishing on the seam. This establishes precisely the initial-data conclusion.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

Robert Beig and Piotr T. Chruściel, Killing Initial Data · Section 2, printed pp. 3–4 (PDF pages 3–4), equations (2.3)–(2.13), especially (2.6), (2.10), (2.13).

The induced pair obeys $\mathcal L_Xh+2NK=0$ and $\mathcal L_XK+D^2N=N(\operatorname{Ric}_h+(\operatorname{tr}_hK)K-2K\circ K)$ in vacuum; $(K\circ K)_{ij}=h^{ab}K_{ia}K_{bj}$.

Hypotheses: A smooth spacelike hypersurface in a smooth Ricci-flat Lorentzian four-manifold; a smooth ambient Killing field $T=Nn+X$, $N=-g(T,n)$, $K(U,V)=g(\nabla_U n,V)$.

Reading limits: Inspected §2, (2.3)–(2.13), including the Gaussian-coordinate sign convention and Einstein-tensor projections. Importing the necessary identities, without reconstructing their derivation or importing converse/global extension claims.

Sources

Robert Beig and Piotr T. Chruściel, Killing Initial Data · Section 2, printed pp. 3–4 (PDF pages 3–4), equations (2.3)–(2.13), especially (2.6), (2.10), (2.13).