LemmaStatement: AI-generatedProof: AI-generated

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Compact normalized source data give a target-independent energy bound

Statement

Work in ZFC. Use stationary Lorentzian data $(M, g, T)$ of signature $(-, +, +, +)$ with their specified time orientation and Levi–Civita connection. Let $U \subset M$ be a designated open end on which $T$ is future timelike. Let $Σ$ be a smooth embedded spacelike hypersurface with induced metric $h$ and smooth future unit normal $n_{Σ}$ . Let $S \subset int Σ$ be a nonempty compact smooth embedded two-surface without boundary, carrying a supplied smooth outward $h$ -unit normal $ν_{0}$ tangent to $Σ$ . Fix a smooth $ϕ > 0$ on $S$ and one $ϵ > 0$ sufficiently small that $u = ϵ ϕ$ is admissible in the Riemannian normal tube. Set $F_{ϵ} (y) = \exp_{y}^{h} (ϵ ϕ (y) ν_{0} (y))$ and $A = F_{ϵ} (S)$ . On this fixed outward graph put $n = n_{Σ} ∣_{A}$ and let $ν$ be its canonical smooth outward unit normal tangent to $Σ$ , with positive pairing with the increasing normal-tube coordinate direction.

The number $E_{\max} = 1 + \max_{p \in A} ∣ g_{p} (T_{p}, n_{p} + ν_{p}) ∣$ is finite and depends only on the fixed source data. For every future affinely parametrized null geodesic segment $γ : [0, b] \to M$ , $b > 0$ , with $γ (0) \in A$ , $\dot{γ} (0) = n + ν$ at its source point, and $γ (b) \in U$ , its stationary energy $E = - g (T, \dot{γ})$ satisfies $0 < E \leq E_{\max}$ throughout the segment. The bound is independent of target radius and target stationary translate. In particular it applies to relevant boundary segments put in this scale by the outward affine-normalization lemma under that lemma's hypotheses.

Facts & Assumptions

[F1]

Under its exterior-containment and outside-tube endpoint hypotheses, the outward affine-normalization lemma gives the relevant future source-normal null segments the unique outgoing affine scale $\dot{γ} (0) = n + ν$ , preserving their oriented images. The surviving source-normal ray has the outgoing affine normalization

[F2]

For an affine null geodesic in the stationary metric, $E = - g (T, \dot{γ})$ is constant, with no timelike assumption on $T$ along the full segment. Where $T$ is future timelike and the tangent is future null, $E$ is strictly positive. The stationary Killing energy is constant along every affine null segment

[F3]

The spacelike-hypersurface data specify the smooth future unit normal $n_{Σ}$ and the smooth positive induced metric $h = g ∣_{T Σ}$ . Spacelike sections, maximality and lapse–shift data

[F4]

Stationary Lorentzian data supply the smooth metric $g$ , its time orientation and Levi–Civita connection, and the smooth Killing field $T$ . A designated stationary end requires $T$ to be future timelike there. Stationary Lorentzian data and tensor conventions

[F5]

For the supplied smooth unit normal $ν_{0}$ on compact $S$ , the graph-domain lemma supplies a Riemannian normal tube and a smooth embedded spacelike graph $F_{ϵ}$ for every sufficiently small $ϵ$ in the fixed smooth family $u = ϵ ϕ$ . Admissible domains for outward normal graphs

[F6]

On an admissible graph in that tube, the canonical outward graph gauge supplies the smooth unit normal $ν$ tangent to $Σ$ , normal to $A$ and positively paired with the increasing tube direction. The field $n + ν$ is a nonzero future null normal. Canonical outward normals and null gauge on a graph

[F7]

A smooth map between smooth manifolds is continuous; this applies to the smooth graph parametrization and to the smooth real-valued contraction of the supplied fields. Smooth maps are continuous

[F8]

A continuous image of a compact space is compact (claim 1). A continuous real-valued function on a nonempty compact space attains a maximum and a minimum (claim 2). A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism

[A1]

Work in ZFC, including the countable and dependent choice required by the invoked geodesic, compactness and causal interfaces. The Axiom of Choice

Proof

Given: The fixed stationary, hypersurface and graph data and the segment hypotheses in the Statement.

Goal: Establish the finite source maximum and the positive target-independent energy bound.

1.1F3F5F6F7F8given

By [F3] and [F5], the chosen $F_{ϵ} : S \to Σ$ is a smooth embedded graph parametrization, so it is continuous by [F7]. Claim 1 of [F8] applied to the compact domain $S$ makes $A = F_{ϵ} (S)$ compact. Since $S$ is nonempty, its image $A$ is nonempty. The hypotheses of [F6] hold for this admissible graph in the supplied tube, so $n$ and $ν$ are smooth along $A$ and $n + ν$ is its specified future null normal.

2.1F3F4F6F7F8step 1.1algebra

The real-valued contraction $q (p) = g_{p} (T_{p}, n_{p} + ν_{p})$ is smooth on $A$ : in any surface chart its expression is a finite sum of products of the smooth restricted metric and field coefficients supplied by [F3], [F4] and [F6]. It is continuous by [F7]. The inequality $∣ ∣ a ∣ - ∣ b ∣ ∣ \leq ∣ a - b ∣$ shows that $f (p) = ∣ q (p) ∣$ is continuous as well. Apply claim 2 of [F8] to $f : A \to R$ on the nonempty compact space from step 1.1. It attains a finite maximum $m \geq 0$ , so $E_{\max} = 1 + m > 0$ . Its defining data are fixed before a target or segment is selected.

3.1F1F2F4step 2.1givenalgebra∎

For any segment in the Statement, its initial normalization gives $E (0) = - q (γ (0))$ , hence $∣ E (0) ∣ \leq m < E_{\max}$ by step 2.1. The stationary data in [F4] and the affine null-segment hypotheses permit [F2], so $E$ is constant. At $γ (b) \in U$ , the field $T$ is future timelike and the tangent is future null, so [F2] gives $E (b) > 0$ . Thus $0 < E \leq E_{\max}$ everywhere, without requiring $T$ timelike away from the end. The same estimate applies to the boundary segments normalized by [F1] whenever its hypotheses hold. No target radius or target translate enters the source maximum.

Depends on

Used by · 2

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