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Outgoing expansion depends smoothly on the second graph data

Statement

Assume countable choice ${A C}_{ω}$ . Use the normal-graph data: a smooth spacelike hypersurface $Σ$ in a time-oriented smooth Lorentzian four-manifold, its induced metric $h = g ∣_{T Σ}$ , a smooth embedded two-surface $S \subset int Σ$ without boundary with a supplied smooth outward $h$ -unit normal $ν$ , and a supplied normal-graph domain $Ω$ on which $T (p, r) = \exp_{p}^{h} (r ν_{p})$ is a diffeomorphism onto its image. A smooth $u$ is admissible when its graph lies in $Ω$ , and $F_{u} (p) = T (p, u (p))$ . The domain may be local; no uniform tube for noncompact $S$ is assumed.

For an admissible graph in the fixed spacelike hypersurface and its canonical gauge, let $q_{u} = F_{u}^{*} h$ and $Θ (u) = F_{u}^{*} (θ_{L_{u}})$ . The pullback means that each scalar is evaluated at the point $F_{u} (p)$ and regarded as a function of $p \in S$ . Locally there is a smooth scalar function $A$ such that $Θ (u) (x) = A (x, u (x), \partial_{a} u (x), \partial_{a} \partial_{b} u (x)) .$ Thus the expansion uses only the second graph data (the value, first partials and symmetric second partials at the same point). Its coordinate expressions agree because it is the trace of a geometric tensor. No ambient extension of the normal fields is part of this construction.

More explicitly, choose coordinates $x^{a}$ on $S$ and $y^{i}$ on $Σ$ , with $a, b = 1, 2$ and $i, j, k, ℓ = 1, 2, 3$ . Put $E_{a}^{i} = \partial_{a} F_{u}^{i}$ , write $q_{u}^{a b}$ for the inverse of $q_{u, a b} = h_{i j} (F_{u}) E_{a}^{i} E_{b}^{j}$ , let $K (X, Y) = g (\nabla_{X} n, Y)$ , and let $Γ^{i}_{j k}$ be the coefficients of the Levi–Civita connection of $h$ . Repeated indices are summed. The exact coordinate formula is $Θ (u) = q_{u}^{a b} (K_{i j} (F_{u}) E_{a}^{i} E_{b}^{j} - h_{i ℓ} (F_{u}) ν_{u}^{ℓ} (\partial_{a} \partial_{b} F_{u}^{i} + Γ^{i}_{j k} (F_{u}) E_{a}^{j} E_{b}^{k})) .$ Writing $T_{r} = \partial_{r} T$ evaluated at $(x, u (x))$ , this can equivalently be expressed as $Θ (u) (x) = - q_{u}^{a b} h (ν_{u}, T_{r}) \partial_{a} \partial_{b} u + R (x, u, \partial u),$ where $R$ is a smooth real function of the first graph data on the coordinate domain. Thus the displayed second-partial term is the only dependence on second derivatives of $u$ . At the zero graph, $T_{r} (p, 0) = ν_{p}$ and $ν_{0} = ν$ , so $h (ν_{0}, T_{r} (p, 0)) = 1$ .

Facts & Assumptions

[F1]

Canonical outward normals and null gauge on a graph supplies the smooth canonical null gauge and first-data dependence of $ν_{u}$ .

[F2]

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface supplies $K (X, Y) = g (\nabla_{X} n, Y)$ on $Σ$ , with $K$ smooth, and Fundamental theorem of riemannian geometry supplies the Levi–Civita connection $D$ of $h$ .

[F3]

Null normal pairs and screen geometry defines $θ_{L}$ as the trace of $g (\nabla_{X} L, Y)$ on the spacelike surface, with tangential extension independence.

[F4]

The chain rule for differentials of smooth maps supplies differentiation of composites.

[F5]

Admissible domains for outward normal graphs supplies the normal-graph domain $Ω$ , the smooth diffeomorphism $T$ , admissibility, and the embedded spacelike parametrization $F_{u} (p) = T (p, u (p))$ , under the stated countable-choice hypothesis.

[F6]

Adopt countable choice ${A C}_{ω}$ in the sense of The Axiom of Countable Choice (AC_ω). This is the hypothesis inherited by the graph construction in [F5] and the smooth Lorentzian and screen conventions in [F3]; no stronger choice principle is assumed.

[F7]

Metric compatibility, torsion freedom and the metric-selected connection class, clauses (a)–(d), defines and constructs the ambient metric-selected connection $\nabla$ : it is $g$ -compatible and torsion-free, with $X (g (Y, Z)) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z)$ and $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ . Its countable-choice hypothesis is [F6].

Proof

[F1] Canonical outward normals and null gauge on a graph supplies the smooth canonical null gauge and first-data dependence of $ν_{u}$ .

[F2] The future unit normal, induced metric and second fundamental form of a spacelike hypersurface supplies $K (X, Y) = g (\nabla_{X} n, Y)$ on $Σ$ , with $K$ smooth, and Fundamental theorem of riemannian geometry supplies the Levi–Civita connection $D$ of $h$ .

[F3] Null normal pairs and screen geometry defines $θ_{L}$ as the trace of $g (\nabla_{X} L, Y)$ on the spacelike surface, with tangential extension independence.

[F4] The chain rule for differentials of smooth maps supplies differentiation of composites.

[F5] Admissible domains for outward normal graphs supplies the normal-graph domain $Ω$ , the smooth diffeomorphism $T$ , admissibility, and the embedded spacelike parametrization $F_{u} (p) = T (p, u (p))$ , under the stated countable-choice hypothesis.

[F6] Adopt countable choice ${A C}_{ω}$ in the sense of The Axiom of Countable Choice (AC_ω). This is the hypothesis inherited by the graph construction in [F5] and the smooth Lorentzian and screen conventions in [F3]; no stronger choice principle is assumed.

Given: The normal-graph data and admissible graph from [F5], the canonical gauge from [F1], and the countable-choice assumption in [F6].

Goal: Establish the exact expansion formula, its second-partial coefficient and the smooth local second-data expression.

1.1F1F2F3F4F5F6F7givenalgebra

The assumption in [F6] supplies the inherited countable-choice hypotheses of [F5] and [F3]. In any ambient coordinate chart $y^{i}$ on $Σ$ write $E_{a}^{i} = \partial_{a} F_{u}^{i}$ and $q_{a b} = h_{i j} (F_{u}) E_{a}^{i} E_{b}^{j}$ . The $E_{a}$ depend smoothly on $(x, u, \partial u)$ by [F4]. Since the graph is an immersion into a Riemannian hypersurface, $\det q > 0$ ; hence the cofactor formula makes $q^{a b}$ a smooth function of those same data. The normal $ν_{u}$ has the same dependence by [F1]. Let $Γ^{i}_{j k}$ denote the smooth Christoffel coefficients of $D$ in this ambient chart. The connection with coefficients $D$ is the tangential part of the ambient metric connection on $Σ$ : that projection is metric compatible and torsion free by [F7], hence equals $D$ by uniqueness in [F2]. Consequently, for $X$ tangent to the graph $F_{u} (S)$ and $Y$ tangent to $Σ$ along that graph, $g (\nabla_{X} ν_{u}, Y) = h (D_{X} ν_{u}, Y)$ . Both derivatives are pullback covariant derivatives along the graph, so no off-graph values of $ν_{u}$ are required.

2.1F1F2F3F4F5step 1.1algebra

Differentiating $h (ν_{u}, E_{b}) = 0$ along the graph using metric compatibility gives $h (D_{E_{a}} ν_{u}, E_{b}) = - h (ν_{u}, D_{E_{a}} E_{b})$ . The covariant derivative along the parametrization has components $\partial_{a} \partial_{b} F_{u}^{i} + Γ^{i}_{j k} (F_{u}) E_{a}^{j} E_{b}^{k}$ . Splitting $L_{u} = n + ν_{u}$ and taking the trace in [F3] therefore gives exactly the coordinate formula in the Statement: $Θ (u) = q_{u}^{a b} (K_{i j} (F_{u}) E_{a}^{i} E_{b}^{j} - h_{i ℓ} (F_{u}) ν_{u}^{ℓ} (\partial_{a} \partial_{b} F_{u}^{i} + Γ^{i}_{j k} (F_{u}) E_{a}^{j} E_{b}^{k})) .$ Here $q_{u}^{a b}$ is the inverse matrix denoted $q^{a b}$ in step 1.1. All factors other than the displayed second derivatives of $F_{u}$ depend smoothly only on $(x, u, \partial u)$ .

3.1F1F2F3F4F5step 1.1step 2.1algebra∎

The chain rule for $F_{u}^{i} = T^{i} (x, u (x))$ gives $\partial_{a} \partial_{b} F_{u}^{i} = T_{r}^{i} \partial_{a} \partial_{b} u + C_{a b}^{i} (x, u, \partial u)$ , where $C_{a b}^{i}$ is smooth: differentiating $\partial_{b} F_{u}^{i} = T_{b}^{i} + T_{r}^{i} \partial_{b} u$ leaves only derivatives of $T$ and products of first partials of $u$ after removing the displayed second-partial term. Substitution into step 2.1 gives the stated smooth remainder $R$ and coefficient $- q_{u}^{a b} h (ν_{u}, T_{r})$ . At $u = 0$ , [F5] gives $T_{r} (p, 0) = ν_{p}$ , and [F1] gives $ν_{0} = ν$ , whose $h$ -norm is one. The coefficient pairing is therefore one there. The same formula defines the asserted smooth $A (x, u, \partial u, \partial^{2} u)$ . Derivatives of normal fields occur only along the graph, so [F3] gives extension independence; the trace is a scalar, so its coordinate expressions agree. This proves both the original second-data assertion and the exact formula.

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