LemmaStatement: AI-generatedProof: AI-generated

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The surviving source-normal ray has the outgoing affine normalization

Statement

Work in ZFC. Let $(M, g)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ with its Levi–Civita connection. Use the compact spacelike two-surface $S$ , the future/outward frame $(n, ν)$ , the oriented open exterior $E$ , and the smooth spacelike section extension $(Σ, h)$ of the outward-graph initial-jets data. Fix a smooth function $ϕ > 0$ on $S$ . Choose $ϵ > 0$ sufficiently small that $u = ϵ ϕ$ is an admissible normal graph and the inward-branch exclusion applies, and let $K$ be its fixed compact tube. Set $F_{ϵ} (y) = \exp_{y}^{h} (ϵ ϕ (y) ν_{y})$ and $S_{ϵ} = F_{ϵ} (S)$ .

Let $n_{ϵ}$ be the future unit normal of $Σ$ restricted to $S_{ϵ}$ , and let $ν_{ϵ}$ be the canonical outward unit graph normal tangent to $Σ$ , with positive pairing with the increasing normal-tube coordinate direction. Put $L_{ϵ} = n_{ϵ} + ν_{ϵ}$ and $V_{ϵ} = n_{ϵ} - ν_{ϵ}$ . Every nonzero future null normal $w$ to $S_{ϵ}$ at $p$ is a positive multiple of exactly one of $L_{ϵ} (p)$ and $V_{ϵ} (p)$ .

Suppose $γ : [0, b] \to M$ , $b > 0$ , is a future affinely parametrized null geodesic with $γ (0) = p \in S_{ϵ}$ , $\dot{γ} (0) = w$ normal to $S_{ϵ}$ , $γ ([0, b]) \subset E$ , and $γ (b) \notin K$ . Then $w = c L_{ϵ} (p)$ for a unique $c > 0$ . The positive affine rescaling $\hat{γ} (λ) = γ (λ / c)$ , $0 \leq λ \leq c b$ , has the same oriented curve image and initial tangent exactly $L_{ϵ} (p)$ . It is the unique positive affine rescaling with source parameter zero and that initial tangent. In the normal congruence initialized by the smooth field $L_{ϵ}$ , the optical expansion at parameter zero is the surface expansion $θ_{L_{ϵ}}$ in this exact outgoing gauge.

Facts & Assumptions

[F1]

For the sufficiently small graph family and fixed compact tube $K$ of the inward-branch exclusion, a future null geodesic starting on $S_{ϵ}$ , remaining entirely in the open exterior $E$ , and ending outside $K$ cannot have initial tangent a positive multiple of $V_{ϵ} = n_{ϵ} - ν_{ϵ}$ . An exterior-contained null segment to a distant target cannot start inward

[F2]

The spacelike-hypersurface data supply the smooth future unit normal of $Σ$ . On an admissible normal graph, the canonical outward graph-gauge lemma supplies a smooth unit normal $ν_{ϵ}$ tangent to $Σ$ and normal to $S_{ϵ}$ , with $g (n_{ϵ}, n_{ϵ}) = - 1$ , $g (ν_{ϵ}, ν_{ϵ}) = 1$ , and $g (n_{ϵ}, ν_{ϵ}) = 0$ . Its outward convention is positive pairing with the increasing tube coordinate direction, and its outgoing null field is $n_{ϵ} + ν_{ϵ}$ . Spacelike sections, maximality and lapse–shift data Canonical outward normals and null gauge on a graph

[F3]

For a smooth affine null normal congruence initialized by a null normal field $L$ , the initial optical expansion equals $θ_{L} = {tr}_{g ∣_{T S}} χ_{L}$ , where $χ_{L} (X, Y) = g (\nabla_{X} L, Y)$ . Null expansions, screen quotients and focal data

[F4]

On a smooth time-oriented Lorentzian four-manifold with its metric-selected connection, the geodesic initial-value theorem supplies uniqueness, an open smooth geodesic evaluation domain, and the scaling law $γ_{p, s v} (t) = γ_{p, v} (s t)$ whenever either side is defined, with matching rescaled domains. Only clauses (a)–(c) are used here. The supplier has domain mathematics. The geodesic initial-value problem and the exponential map of the metric-selected connection

[F5]

For the compact surface $S$ in the smooth spacelike section extension, the normal-graph domain lemma supplies a Riemannian normal tube and embedded spacelike graphs $F_{u}$ whenever the graph of $u$ lies in that tube domain. For fixed smooth $ϕ$ on $S$ , $u = ϵ ϕ$ is admissible for all sufficiently small $∣ ϵ ∣$ . Admissible domains for outward normal graphs

[F6]

The outward-graph initial-jets data supply the oriented exterior and the section extension used here, the family $F_{ϵ} (y) = \exp_{y}^{h} (ϵ ϕ (y) ν_{y})$ for fixed smooth $ϕ > 0$ , and the future/outward graph fields $n_{ϵ}, ν_{ϵ}$ . For sufficiently small positive $ϵ$ these graphs lie in $E$ , with the unhalved inward field $V_{ϵ} = n_{ϵ} - ν_{ϵ}$ and outgoing field $L_{ϵ} = n_{ϵ} + ν_{ϵ}$ . Small outward normal graphs have controlled inward null initial jets

[A1]

Work in ZFC, including the choice assumptions inherited from the graph, geodesic, compactness and causal suppliers. No additional choice principle is used in the normal-plane calculation or the uniquely determined affine rescaling. The Axiom of Choice

Proof

Given: The geometric data and the segment hypotheses of the Statement.

Goal: Classify the initial normal ray and establish its unique outgoing affine scale and initial expansion.

1.1F2F5F6givenalgebra

By [F5]–[F6] the chosen graph is an embedded spacelike two-surface, and [F2] supplies its smooth future/outward orthonormal frame. The independent vectors $n_{ϵ}, ν_{ϵ}$ are orthogonal to its two-dimensional tangent space, so they span its Lorentzian normal plane. Write $w = α n_{ϵ} + β ν_{ϵ}$ . Nullness gives $β^{2} = α^{2}$ , while future direction gives $α = - g (w, n_{ϵ}) > 0$ . Thus $w = α L_{ϵ}$ or $w = α V_{ϵ}$ , and these rays are distinct.

2.1F1step 1.1givenalgebra

The segment in the Statement starts at this graph with initial vector $w$ , remains in the open exterior, and ends outside the fixed tube. These are precisely the conditions in [F1]; its inward possibility is therefore excluded. Hence $w = c L_{ϵ} (p)$ with $c = α > 0$ , uniquely by the frame decomposition.

3.1F4step 2.1algebra

Apply [F4], clause (c), with initial vector $w$ and scaling factor $1 / c$ . It gives the affine geodesic $\hat{γ} (λ) = γ (λ / c)$ on $[0, c b]$ with initial tangent $w / c = L_{ϵ} (p)$ . The increasing bijection $λ \mapsto λ / c$ preserves the oriented curve image and both endpoints. An affine parameter change preserving source parameter zero has the form $t = a λ$ ; its initial tangent is $a w$ , so the required tangent forces $a = 1 / c$ . This proves uniqueness of the positive rescaling.

4.1F2F3F4step 3.1algebra∎

The field $L_{ϵ}$ along the graph is smooth by [F2]. The open smooth evaluation domain in [F4], clauses (a)–(b), supplies, near each source point and parameter zero, the affine normal geodesic congruence initialized by this field. Its initial optical expansion is $θ_{L_{ϵ}}$ by [F3]. In particular the expansion is in the unhalved outgoing gauge, with no target-cylinder rescaling. The usual positive normal-scale rule also follows from the same definition: for a positive smooth function $f$ on the source, $g (\nabla_{X} (f L_{ϵ}), Y) = f g (\nabla_{X} L_{ϵ}, Y)$ for source-tangent $X, Y$ , because $g (L_{ϵ}, Y) = 0$ ; hence $θ_{f L_{ϵ}} = f θ_{L_{ϵ}}$ .

Depends on

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