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Harmonic initial data from geometric data: prescribing the metric jet of a spacelike slice in a time-independent background

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $Σ \subseteq M$ be a spacelike embedded hypersurface with induced metric $h$ , future unit normal field $n$ and second fundamental form $K (V, W) = g (\nabla_{V} n, W)$ (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). Fix smooth product coordinates $(s, x^{1}, x^{2}, x^{3})$ on a neighborhood of $Σ$ such that $s = 0$ exactly on $Σ$ and $\partial_{s} ∣_{Σ} = n$ . Let $π_{Σ}$ denote the spatial projection of this product neighborhood onto $Σ$ , and set $h_{0} : = π_{Σ}^{*} h$ . Thus $h_{0}$ is a smooth covariant tensor with zero time and mixed components, $h_{0} ∣_{Σ} = h$ and $\partial_{s} h_{0} = 0$ . Put $\overset{ˉ}{g} : = - d s^{2} + h_{0}$ as background metric. Define a smooth metric germ $g_{0}$ on $Σ$ by the initial jet $g_{i j} = h_{i j}, g_{00} = - 1, g_{0 i} = 0, \partial_{s} g_{i j} = 2 K_{i j}, \partial_{s} g_{0 i} = 0, \partial_{s} g_{00} = - 2 {tr}_{h} K .$ Then:

(a) $g_{0}$ is the germ of a smooth Lorentzian metric for which $Σ$ is spacelike with induced metric $h$ , future unit normal $n = \partial_{s}$ and second fundamental form $K$ ;

(b) the harmonic-gauge vector of $(g_{0}, \overset{ˉ}{g})$ vanishes on $Σ$ : $H^{a} ∣_{Σ} = 0$ .

Facts & Assumptions

Given: the manifold, the spacelike hypersurface and its normal-coordinate setup, the background metric $\overset{ˉ}{g} = - d s^{2} + h_{0}$ with $h_{0} = π_{Σ}^{*} h$ , and the prescribed jet.

[L1]

On the slice, $g_{0 i} = 0$ makes its tangential derivatives zero. The Christoffel formula gives $g (\nabla_{\partial_{i}} \partial_{s}, \partial_{j}) = \frac{1}{2} (\partial_{i} g_{s j} + \partial_{s} g_{i j} - \partial_{j} g_{i s}) = \frac{1}{2} \partial_{s} g_{i j} = K_{i j}$ . Together with $g_{00} = - 1$ and the chosen time orientation, this identifies $\partial_{s}$ as the future unit normal and gives the stated second fundamental form (The future unit normal, induced metric and second fundamental form of a spacelike hypersurface, Christoffel symbols of an affine connection).

[L2]

Christoffel symbols on $Σ$ in the coordinate frame $(\partial_{s}, \partial_{i})$ : $g^{00} = - 1$ , $g^{0 i} = 0$ and $g^{i j} = h^{i j}$ there, so $Γ^{0}_{00} = - \frac{1}{2} \partial_{s} g_{00} = {tr}_{h} K$ , $Γ^{0}_{0 i} = \frac{1}{2} \partial_{i} g_{00} = 0$ , $Γ^{0}_{i j} = \frac{1}{2} \partial_{s} g_{i j} = K_{i j}$ , $Γ^{i}_{00} = 0$ and $Γ^{i}_{j k} = Γ^{h}^{i}_{j k}$ on $Σ$ (Christoffel symbols of an affine connection, Metric compatibility, torsion freedom and the metric-selected connection class).

[L3]

Background Christoffel symbols: $\overset{ˉ}{g} = - d s^{2} + h_{0}$ with $\partial_{s} h_{0} = 0$ gives ${\overset{ˉ}{Γ}}^{0}_{00} = 0$ , ${\overset{ˉ}{Γ}}^{0}_{0 i} = 0$ , ${\overset{ˉ}{Γ}}^{0}_{i j} = 0$ , ${\overset{ˉ}{Γ}}^{i}_{00} = 0$ and ${\overset{ˉ}{Γ}}^{i}_{j k} = Γ^{h}^{i}_{j k}$ on $Σ$ (because $h_{0} ∣_{Σ} = h$ and the tangential derivatives of $g_{j k}$ and $h_{0, j k}$ agree on $Σ$ ) (Christoffel symbols of an affine connection, $C^{r}$ and smooth maps between smooth manifolds).

[L4]

The gauge vector is $H^{a} = g^{b c} (Γ^{a}_{b c} - {\overset{ˉ}{Γ}}^{a}_{b c})$ (The harmonic gauge vector, the reduced Einstein equation and the Einstein residual).

Proof

technique · direct

1.1L1algebra

(Existence of the metric germ and its slice geometry (a).) Prescribe a smooth metric on a product neighborhood whose value and $s$ -derivative on $Σ$ are the displayed jet and whose higher jets are chosen arbitrarily smooth; such a metric germ exists because a smooth function and its first normal derivative can be prescribed on a hypersurface and extended smoothly. On $Σ$ , $g_{00} = - 1$ , $g_{0 i} = 0$ , $g_{i j} = h_{i j}$ , so $g$ restricted to $T Σ$ is $h$ and $\partial_{s}$ is a unit timelike vector orthogonal to $Σ$ ; by [L1] it is the future unit normal (with the orientation of the statement) and the second fundamental form is $\frac{1}{2} \partial_{s} g_{i j} = K_{i j}$ . This is (a).

1.2L2L3L4algebra

(Computation of $H^{0}$ .) Insert the Christoffel components [L2], [L3] into [L4]: $H^{0} = g^{00} (Γ^{0}_{00} - {\overset{ˉ}{Γ}}^{0}_{00}) + g^{i j} (Γ^{0}_{i j} - {\overset{ˉ}{Γ}}^{0}_{i j}) + 2 g^{0 i} (Γ^{0}_{0 i} - {\overset{ˉ}{Γ}}^{0}_{0 i}) = - ({tr}_{h} K - 0) + h^{i j} (K_{i j} - 0) + 0 = - {tr}_{h} K + {tr}_{h} K = 0$ on $Σ$ .

2.1step 1.2L2L3L4algebra∎

(Computation of $H^{i}$ .) Similarly $H^{i} = g^{00} (Γ^{i}_{00} - {\overset{ˉ}{Γ}}^{i}_{00}) + g^{j k} (Γ^{i}_{j k} - {\overset{ˉ}{Γ}}^{i}_{j k}) + 2 g^{0 j} (Γ^{i}_{0 j} - {\overset{ˉ}{Γ}}^{i}_{0 j}) = - (0 - 0) + h^{j k} (Γ^{h}^{i}_{j k} - Γ^{h}^{i}_{j k}) + 2 \cdot 0 = 0$ on $Σ$ , using that the tangential Christoffel symbols of $g$ on $Σ$ agree with those of $h$ and hence with those of the background because $h_{0} ∣_{Σ} = h$ and tangential derivatives agree. This is (b).

Remarks

Role. This construction produces the Cauchy data $(g^{0}, g^{1})$ to which Local existence for the reduced Einstein equation from Cauchy data on a spacelike slice is applied in Local geometric development for a fixed covariant Einstein-matter model with complete constrained data; the background $\overset{ˉ}{g}$ is chosen by the constructor and disappears from the final geometric conclusions via Local geometric uniqueness for a fixed covariant Einstein-matter model with complete data.

Harmonic initial data from geometric data: prescribing the metric jet of a spacelike slice in a time-independent background

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources