LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Timelike transfer supplies access to every sufficiently distant stationary cylinder

Statement

Work in ZFC. Let $(M, g, N, T)$ be smooth stationary Lorentzian data. Supply an open end $U \subset M$ and its chronological exterior $E = I_{M, g}^{-} (U) \cap I_{M, g}^{+} (U)$ . In addition, supply a smooth three-manifold $B$ , an open subset $B_{\infty} \subset B$ , and diffeomorphisms $F : R \times B \to E$ and $e : {x \in R^{3} : ∣ x ∣ > a} \to B_{\infty}$ , where $a > 0$ , with $F_{*} \partial_{t} = T$ and $U = F (R \times B_{\infty})$ . Require $T$ to be future $g$ -timelike on $U$ . These are explicit product-end hypotheses, rather than consequences of stationarity. Put $π (F (t, b)) = b$ and $ϕ_{s}^{T} (F (t, b)) = F (t + s, b)$ for every $s \in R$ .

Let $h$ be a smooth time-oriented Lorentzian metric on $E$ . Assume that every original future $g$ -timelike curve $γ : [0, 1] \to E$ has a future $h$ -timelike transfer $\tilde{γ} : [0, 1] \to E$ with $\tilde{γ} (0) = γ (0)$ and $π \circ \tilde{γ} = π \circ γ$ . Both curve classes use locally absolutely continuous coordinates and nonzero future timelike velocity almost everywhere. The transfer is required on the whole curve, without any continuity requirement on its dependence on $γ$ .

For $R > a$ define $S_{R} = {x : ∣ x ∣ = R}$ , $C_{R, t} = F ({t} \times e (S_{R}))$ and the stationary cylinder $Γ_{R} = ⋃_{t \in R} C_{R, t}$ . Every $p \in E$ can send a future $h$ -timelike curve, entirely in $E$ , to $Γ_{R}$ . If $T$ is future $h$ -timelike everywhere on this cylinder, every arrival point $y \in Γ_{R}$ reaches $ϕ_{s}^{T} (y)$ for every $s > 0$ . Under this same cylinder hypothesis, every compact source $K \subset E$ , in particular every compact source surface, has a finite $t_{R}$ such that for every $t \geq t_{R}$ and every $p \in K$ there is $y \in C_{R, t}$ with $p ≪_{E, h} y$ . Thus these conclusions hold at every sufficiently large radius whenever $T$ is future $h$ -timelike on a sufficiently distant product tail. No bound on $t_{R}$ uniform as $R \to \infty$ is asserted.

Facts & Assumptions

[F1]

The exterior is the chronological future and past intersection of the specified original stationary end. Causal exterior, chronological boundaries and Cauchy data

[F2]

Chronological reachability is open in its endpoints. Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

[F3]

Stationary Lorentzian data give the smooth metric, its time orientation and the Killing equation. The specified product diffeomorphisms are additional hypotheses. In these coordinates $T = \partial_{t}$ , so the Killing equation gives $\partial_{t} g_{μ ν} = 0$ on $U$ ; the displayed translations define a global flow on $E$ and preserve each $Γ_{R}$ . Stationary Lorentzian data and tensor conventions Diffeomorphisms and local diffeomorphisms of manifolds Local and global flows generated by a vector field

[F4]

Finite concatenations of future timelike curves remain future timelike; the same curve calculus includes piecewise smooth curves and increasing affine reparametrization. Chart independence, concatenation, reparametrization and transitivity for causal curves

[F5]

A compact space has a finite subcover for every open cover. The interval $[0, 1]$ is compact, and continuous real functions on it attain finite extrema. Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right A subset of ℝ is compact if and only if it is closed and bounded Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value

[F6]

In a Lorentz tangent space the timelike set has two open components. A continuous path of timelike vectors starting in the future component stays there. The timelike set has exactly two convex components, described in orthonormal coordinates

[A1]

Work in ZFC, including the countable and dependent choice required by the causal suppliers. Choice also permits selecting an arrival point for each source point in step 4.1. The Axiom of Choice

Proof

Given: The original stationary geometry, the supplied product end and exterior product, and the transfer in the Statement; the future $h$ -timelike cylinder hypothesis is used only for later translates and common-time access.

Goal: Prove cylinder access, access to every later translate, and a common sufficiently late time for each fixed cylinder and compact source.

1.1givenA1F1F3F4choose

Fix $p \in E$ and $R > a$ . By [F1] there are $u, q \in U$ with $u ≪_{M, g} p ≪_{M, g} q$ . Choose a future $g$ -timelike curve from $p$ to $q$ . Every interior point $z$ of it satisfies $u ≪_{M, g} p ≪_{M, g} z ≪_{M, g} q$ , by restriction and concatenation, so $z \in E$ . Its endpoints are also in $E$ , with $q \in U \subset E$ by the supplied product. Thus this curve is contained in the transfer domain. Write $q = F (t_{0}, e (x_{0}))$ .

1.2givenF3F4construct

Now suppose $T$ is future $h$ -timelike everywhere on $Γ_{R}$ . For any $y = F (t, b) \in Γ_{R}$ and $s > 0$ , the curve $v \mapsto F (t + v, b)$ , $0 \leq v \leq s$ , exists throughout that interval, stays on $Γ_{R}$ and has future $h$ -timelike tangent $T$ . Thus $y ≪_{E, h} ϕ_{s}^{T} (y)$ . The full product, not timelikeness alone, supplies every flow time used here.

2.1givenF3F5F6step 1.1constructalgebra

Join $x_{0}$ to radius $R$ by $χ (s) = ((1 - s) + s R / ∣ x_{0} ∣) x_{0}$ for $0 \leq s \leq 1$ . Its radius lies between $∣ x_{0} ∣$ and $R$ , hence stays above $a$ . Write $W (s) = χ^{' i} (s) \partial_{x^{i}}$ in the end coordinates. The coefficient functions along $(0, χ (s))$ are continuous. By [F5] and future timelikeness of $T$ there are $c > 0$ and $b \geq 0$ with $g (T, T) \leq - c$ , $∣ g (T, W) ∣ \leq b$ and $∣ g (W, W) ∣ \leq b$ along this path. By [F3] these bounds hold at every stationary time over the same path. Choose $λ > 0$ so large that $- c + b (2 / λ + 1 / λ^{2}) < 0$ . For every $0 \leq ε \leq 1 / λ$ , $g (T + ε W, T + ε W) \leq - c + 2 b ε + b ε^{2} < 0$ ; [F6] puts all these vectors in the future component containing $T$ . Therefore $η (s) = F (t_{0} + λ s, e (χ (s)))$ is future $g$ -timelike. This proves the required finite end travel directly. The same bounds prove timelike lifting of any fixed smooth finite spatial path with compact parameter interval; finite piecewise smooth paths follow by concatenation.

3.1givenF3F4step 1.1step 2.1construct

Concatenate the curve from step 1.1 with $η$ , and reparametrize onto $[0, 1]$ . By [F4] it is a future $g$ -timelike curve in $E$ . Apply the assumed transfer to this whole curve. Its initial point is $p$ , and its final spatial point is $e (χ (1)) \in e (S_{R})$ , so its endpoint lies on $Γ_{R}$ at a finite product time. This proves access for every $p$ and $R > a$ , without needing a global radius on the inner region or a crossing argument there.

4.1givenA1F2F3F4F5step 3.1step 1.2construct∎

If $K$ is empty the common-time assertion is vacuous. Otherwise choose, for each $p \in K$ , an arrival $y_{p} \in C_{R, t_{p}}$ from step 3.1. By [F2], $O_{p} = I_{E, h}^{-} (y_{p})$ is an open neighborhood of $p$ . The sets $K \cap O_{p}$ are an open cover in the subspace topology of $K$ , so [F5] gives a finite subcover indexed by $p_{1}, \dots, p_{m}$ . Put $t_{R} = 1 + \max_{i} t_{p_{i}}$ . For $t \geq t_{R}$ and $z \in K$ , choose $i$ with $z \in O_{p_{i}}$ . Step 1.2 advances $y_{p_{i}}$ along its cylinder orbit to $F (t, π (y_{p_{i}})) \in C_{R, t}$ , and [F4] concatenates that orbit with the curve from $z$ to $y_{p_{i}}$ . This proves the common-time conclusion. Every construction is for fixed $R$ ; no radius-uniform time estimate has been used.

Depends on

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.