LemmaStatement: AI-generatedProof: AI-generated

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Linear and path-independent normal-graph variation of expansion

Statement

Assume countable choice ${A C}_{ω}$ . Let $Σ$ be a smooth spacelike hypersurface in a time-oriented smooth Lorentzian four-manifold $(M, g)$ , let $h = g ∣_{T Σ}$ , and let $S \subset int Σ$ be a smooth embedded two-surface without boundary with a supplied smooth outward $h$ -unit normal $ν$ . Use a supplied open normal-graph domain $Ω$ containing the relevant zero section, on which $T (p, r) = \exp_{p}^{h} (r ν_{p})$ is a diffeomorphism onto its image. The domain may be local, with the graph constructions then restricted to its open base patch in $S$ ; a global graph uses such a domain over all of $S$ . For compact $S$ a uniform tube exists, while no uniform tube is presumed for noncompact $S$ . A smooth $u$ is admissible when its graph lies in $Ω$ , and $F_{u} (p) = T (p, u (p))$ .

Use the canonical gauge of the second-graph-data construction: $n$ is the future unit normal to $Σ$ , $ν_{u}$ is the outward graph unit normal, $L_{u} = n ∣_{F_{u} (S)} + ν_{u}$ , and $Θ (u) = F_{u}^{*} (θ_{L_{u}})$ is the pulled-back outgoing expansion. For this gauge on admissible graphs, define $L ϕ = {\frac{d}{d ϵ} ∣}_{ϵ = 0} Θ (ϵ ϕ) .$ For compact $S$ this is defined for every $ϕ \in C^{\infty} (S)$ ; on a noncompact surface the definition applies on a supplied tube to every $ϕ$ admitting a common parameter interval of admissible graphs, in particular every compactly supported smooth $ϕ$ . Independently, the local formulas define the same differential operator on all smooth functions as a germ at each point. The operator is linear with smooth coefficients, of order at most two. Here linear means $L (a ϕ + b ψ) = a L ϕ + b L ψ$ for real constants $a, b$ wherever the relevant graph paths are admissible.

If $u (t, p)$ is a jointly smooth admissible graph path with $u (0, p) = 0$ and $\partial_{t} u (0, p) = ϕ (p)$ , then $\partial_{t} Θ (u (t)) ∣_{0} = L ϕ$ . Thus the first derivative does not depend on the higher parameter behavior of the chosen graph path.

Facts & Assumptions

[F1]

Under the geometric and graph-domain hypotheses of the Statement and the countable-choice hypothesis in [F5], Admissible domains for outward normal graphs supplies a common parameter interval for each compactly supported direction in a supplied tube, and for every smooth direction when $S$ is compact.

[F2]

For the stated normal-graph data and canonical gauge, Outgoing expansion depends smoothly on the second graph data gives the smooth local expression $A (x, u, \partial u, \partial^{2} u)$ for pulled-back expansion.

[F3]

The chain rule for differentials of smooth maps gives the chain rule for this finite-dimensional smooth function.

[F4]

Clairaut--Schwarz theorem for continuous second partial derivatives gives equality of mixed coordinate partial derivatives for a $C^{2}$ scalar function on an open Euclidean set. Joint smoothness makes this applicable to $u (t, x)$ and each spatial first derivative $\partial_{b} u (t, x)$ .

[F5]

Adopt countable choice ${A C}_{ω}$ in the sense of The Axiom of Countable Choice (AC_ω). This meets the countable-choice requirements of the graph-domain and second-graph-data suppliers in [F1] and [F2]; their geometric and graph-domain hypotheses are those explicitly stated above. No stronger choice principle is assumed.

[F6]

Under the geometric and graph-domain hypotheses of the Statement and the countable-choice assumption in [F5], Canonical outward normals and null gauge on a graph supplies the smooth future unit normal $n$ , the outward graph unit normal $ν_{u}$ with $ν_{0} = ν$ , and the smooth future null normal $L_{u} = n ∣_{F_{u} (S)} + ν_{u}$ used in the Statement.

[F7]

Null normal pairs and screen geometry defines the outgoing expansion on each embedded spacelike graph by $θ_{L_{u}} = {tr}_{q_{u}} χ_{L_{u}}$ , where $q_{u}$ is the induced positive metric and $χ_{L_{u}} (X, Y) = g (\nabla_{X} L_{u}, Y)$ for graph-tangent $X, Y$ , with tangential extension independence. Its countable-choice convention is met by [F5]. The canonical gauge in [F6] supplies the future null partner $n ∣_{F_{u} (S)} - ν_{u}$ , which pairs with $L_{u}$ to $- 2$ , as required by this definition; this choice of partner leaves $L_{u}$ unchanged. No congruence or compactness assumption is needed for this surface expansion.

Proof

Given: The geometric and normal-graph data of the Statement, the countable-choice hypothesis in [F5], and a smooth graph direction, with an admissible parameter interval whenever a global graph path is asserted.

Goal: Establish the linear differential expression and its path independence.

1.1F1F2F3F5givenalgebra

The geometric and graph-domain hypotheses in the Statement, together with the countable-choice assumption in [F5], meet the supplier requirements in [F1] and [F2]. At a fixed chart point, [F2] and [F3] give $L ϕ = A_{z} (x, 0, 0, 0) ϕ + \sum_{a} A_{p_{a}} (x, 0, 0, 0) \partial_{a} ϕ + \sum_{a \leq b} A_{B_{a b}} (x, 0, 0, 0) \partial_{a} \partial_{b} ϕ .$ Here $z$ is the scalar value, $p_{a}$ the first partials and $B_{a b}$ the independent symmetric second-partial coordinates; these symbols denote coordinates of the smooth function $A$ , not extra geometric fields. The coefficient functions are smooth. This formula is linear in $ϕ$ and its derivatives. Chart formulas agree since they differentiate the same pulled-back geometric scalar; locally all sufficiently small graph parameters are admissible. It therefore defines a global differential operator on germs, without requiring a uniform tube on a noncompact surface.

2.1F1F2F3F4F5step 1.1algebra∎

For a jointly smooth path $u (t, x)$ , apply [F4] to $u$ and to each $\partial_{b} u$ , all of which are $C^{2}$ on the open parameter-coordinate domain. Commuting the parameter derivative first with $\partial_{a}$ on $\partial_{b} u$ and then with $\partial_{b}$ on $u$ gives $\partial_{t} \partial_{a} u ∣_{0} = \partial_{a} ϕ$ and $\partial_{t} \partial_{a} \partial_{b} u ∣_{0} = \partial_{a} \partial_{b} ϕ$ . Differentiating [F2] therefore gives precisely the expression in step 1.1. In the compact or compactly supported case [F1] supplies an actual interval for $u = t ϕ$ . For two such directions, their union of supports is compact; a common smaller interval handles each linear combination. If global admissibility is otherwise supplied, only that supplied interval is used. This proves the path statement and all asserted linearity without a claim of differentiability in an unspecified infinite-dimensional function-space norm.

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