LemmaStatement: AI-generatedProof: AI-generated

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every negative-time cut has a closed selected domain and smooth endpoint extensions

Statement

Work in ZFC. Use the hypotheses and uniformly localized cuts of the translated-cut localization lemma, with its target $Q = C$ , inside $R_{v} \times Ω$ and cylinder $C = R_{v} \times Γ$ . In addition assume that $E$ is globally hyperbolic, $0 < δ \leq δ_{0}$ , and $\min_{C} v > 2 δ$ . Here $C = C_{t}$ is a nonempty compact smooth spacelike two-surface forming the fixed stationary-time section of $C$ , meeting each cylinder $T$ -orbit exactly once. Assume $T$ is future timelike on $C$ . Use stationary section time $v$ , so $T v = 1$ , and put $P = I_{E}^{-} (C)$ , $Σ_{z} = {v = z}$ and $D_{z} = ({z} \times Ω) \cap \partial_{E} P$ .

Choose a smooth nonzero future outward null-normal vector field $L_{c}$ on $C$ . Here outward is relative to the specified inside $R_{v} \times Ω$ : for every local smooth defining function $r$ with $C = {r = 0}$ , $d r \neq 0$ on $C$ and inside ${r < 0}$ , require $d r_{c} (L_{c}) > 0$ . Thus the sign is measured toward the outside ${r > 0}$ . Let $γ_{c} : I_{c} \to E$ be the unique maximal affinely parametrized geodesic with $γ_{c} (0) = c$ and ${\dot{γ}}_{c} (0) = - L_{c}$ ; its portion with $s \in I_{c} \cap [0, \infty)$ is the selected past ray. Assume the prescribed-branch generator input: every inside point $p \in \partial_{E} P$ with $v (p) < \min_{C} v$ has a future null geodesic segment to some $c \in C$ , lying on $\partial_{E} P$ and having final tangent on the positive ray of $L_{c}$ .

For $∣ z ∣ \leq 2 δ$ , define $A_{z} = {c \in C : γ_{c} (s) \in D_{z} for some s \in I_{c} \cap [0, \infty)}$ and let $F_{z} (c)$ be that hit. Then the hit is unique, $A_{z}$ is compact and Borel, and $F_{z} : A_{z} \to D_{z}$ is onto and has local smooth extensions into $Σ_{z}$ . If $z < 0$ , then $A_{z} \subset A_{0}$ and $F_{0} (c)$ is the same selected ray's intermediate endpoint at $v = 0$ for every $c \in A_{z}$ .

Facts & Assumptions

[A1]

Work in ZFC. AC implies Dependent Choice and Countable Choice, meeting the choice assumptions of the localization, endpoint, geodesic and causal suppliers. The Axiom of Choice AC implies DC implies countable choice

[F1]

With the localization lemma's hypotheses and target $Q = C$ , the cuts $D_{z}$ are compact and strictly inside the specified cylinder for $∣ z ∣ \leq 2 δ$ . Its $Ω$ and $Γ$ specify the inside and its spatial boundary in $Σ^{\circ}$ . Uniform localization of translated past-boundary cuts

[F2]

For a compact smooth target surface having strictly positive minimum temporal value relative to the zero-level slice, smooth future null initial vectors and a compact temporal zero-level cut in a globally hyperbolic spacetime, the selected hitting domain is closed. It is compact and Borel, and the unique-hit map has local smooth extensions into the zero-level slice. The extension conclusion requires no regularity of the cut as a surface. Compact causal diamonds close the selected-generator endpoint domain The closed selected domain has smooth endpoint extensions

[F3]

A prescribed initial vector determines a unique maximal geodesic. Affine reversal and rescaling preserve geodesics; a nonzero initially null tangent remains nonzero and null, and its time orientation is constant. Short geodesics with prescribed timelike initial tangent exist smoothly. The geodesic initial-value problem and the exponential map of the metric-selected connection Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed

[F4]

Chronological pasts are open, and push-up gives $x \leq_{E} y ≪_{E} z \Rightarrow x ≪_{E} z$ and $x ≪_{E} y \leq_{E} z \Rightarrow x ≪_{E} z$ . Restrictions of smooth null segments are causal curves; smooth future timelike segments are timelike curves in the stipulated curve class. Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets Push-up rule Chart independence, concatenation, reparametrization and transitivity for causal curves

[F5]

Causal relations, temporal functions and relative boundaries are computed in $E$ . Smooth spacelike two-surfaces and nonzero future null normals have the supplied geometric meanings. The outward convention is the defining-function sign in the Statement, with the inside specified by [F1]. In a boundary chart any two such defining functions have positively proportional differentials on the cylinder, so this sign is independent of the function. Only the surface and normal definitions are used; no focal-exclusion conclusion is invoked. Causal exterior, chronological boundaries and Cauchy data Null expansions, screen quotients and focal data

[F6]

Stationary section time satisfies $T v = 1$ and the spatial coordinates are constant along stationary orbits. On the product, the flow is defined for every stationary parameter. Stationary section time and translated maximal sections

Proof

Given: The hypotheses and prescribed-branch generator input in the Statement.

Goal: Establish unique selected endpoints, compact/Borel domains, smooth local extensions, coverage and negative-time nesting.

1.1givenA1F1F2F3F5

By [F3], $γ_{c}$ has nonzero past null tangent on its selected portion. Thus $(v \circ γ_{c})^{'} < 0$ , so it meets a level $v = z$ at most once. Because $v (c) > 2 δ \geq z$ , every hit on $D_{z}$ has positive parameter. This makes $A_{z}$ and $F_{z}$ precisely the selected domain and endpoint map of [F2] for the shifted temporal function $v - z$ . That function has differential $d v$ , zero-level $Σ_{z}$ and $\min_{C} (v - z) > 0$ . The smooth compact target and initial field are as assumed, while [F1] supplies compact $D_{z}$ . Apply both endpoint suppliers to conclude that $A_{z}$ is compact and Borel and $F_{z}$ has local smooth extensions into $Σ_{z}$ .

2.1givenF1F3F5step 1.1

Fix $p \in D_{z}$ . It is inside the cylinder and $v (p) = z < \min_{C} v$ , so the assumed generator input gives a future null segment from $p$ to some $c \in C$ ending on the positive ray of $L_{c}$ . Choose an affine parameter on that segment, reverse it at $c$ , and rescale by a positive constant to give initial tangent $- L_{c}$ . By maximal-geodesic uniqueness in [F3], it is a restriction of $γ_{c}$ . Hence $c \in A_{z}$ and $F_{z} (c) = p$ by step 1.1. Together with the definition $F_{z} (A_{z}) \subset D_{z}$ , this proves surjectivity. The segment is used for one fixed $p$ at a time.

2.2givenA1F3F4F5step 1.1

Fix any $c \in A_{z}$ , put $p = F_{z} (c)$ and orient the selected segment from $p$ to $c$ towards the future. For each point $r$ on it, [F4] gives $p \leq_{E} r \leq_{E} c$ , including endpoints by reflexivity. As $P$ is open and $p \in \partial_{E} P$ , $p \notin P$ . If $r \in P$ , there is $c^{'} \in C$ with $r ≪_{E} c^{'}$ , and push-up gives $p ≪_{E} c^{'}$ , a contradiction. Thus every $r$ is outside $P$ . To show $r \in {\overline{P}}^{E}$ , take a short geodesic $η$ with $η (0) = r$ and initial tangent $- N_{r}$ , where $N$ is the future timelike orientation field. By smoothness it is past timelike for small positive parameter. For such $ε > 0$ , its reversed restriction is a future timelike curve from $r_{ε} = η (ε)$ to $r$ . Push-up with $r \leq_{E} c$ gives $r_{ε} ≪_{E} c$ , including when $r = c$ , and $r_{ε} \to r$ . Hence $r \in {\overline{P}}^{E} ∖ P = \partial_{E} P$ . This proves boundary containment for every selected segment, including ones other than the segment used for coverage.

3.1givenF1F4F5F6step 1.1step 2.2∎

Let $z < 0$ and $c \in A_{z}$ . Along the finite segment from $F_{z} (c)$ to $c$ , strict temporal monotonicity and $z < 0 < \min_{C} v$ give a unique point $p_{0}$ with $v (p_{0}) = 0$ . If $p_{0}$ were outside the inside region, including on its boundary, continuity of the spatial projection would give a first exit $y \in C$ between $F_{z} (c)$ and $p_{0}$ ; the inside and boundary here are the $Ω$ and $Γ$ of [F1]. Temporal monotonicity gives $v (y) \leq 0$ . The unique point $c_{y} \in C_{t}$ on its cylinder orbit has $v (c_{y}) > 2 δ > 0$ . Since $T v = 1$ , the stationary flow reaches $c_{y}$ from $y$ after parameter $v (c_{y}) - v (y) > 0$ . This orbit segment stays on the cylinder and is future timelike by hypothesis and [F4], so $y \in I_{E}^{-} (C) = P$ , contradicting step 2.2. Therefore $p_{0}$ is inside; step 2.2 also puts it on $\partial_{E} P$ , so $p_{0} \in D_{0}$ . It lies on the same selected ray from $c$ , and the unique-hit definition gives $c \in A_{0}$ and $F_{0} (c) = p_{0}$ .

Depends on

Used by · 2

Dependency tree · 2 levels

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