LemmaStatement: AI-adaptedProof: AI-generatedprecheck pass

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.

Pointwise local causality: closed cones, openness of the local chronological future, no local closed causal curves, corner smoothing and closedness of the local causal relation

Statement

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) for the absolutely continuous causal-curve calculus and the supplied causal results used below.

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ . For every $x \in M$ there is an open neighborhood $B$ of $x$ , a strictly convex normal ball satisfying the hypotheses of Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic with base point $x$ , such that, writing $J_{B} (x) : = {x} \cup {y \in B : a future causal curve in B joins x to y}$ and $I_{B} (x) : = {y \in B : there is a future timelike curve η : [0, 1] \to B with η (0) = x, η (1) = y}$ , and writing $C^{+}, T^{+}, N^{+}$ for the future causal or zero, future timelike, and future null nonzero vectors of $\exp_{x}^{- 1} (B)$ :

(a) $J_{B} (x) = \exp_{x} (C^{+})$ , $I_{B} (x) = \exp_{x} (T^{+})$ , and $J_{B} (x) ∖ (I_{B} (x) \cup {x}) = \exp_{x} (N^{+})$ ;

(b) $J_{B} (x)$ and $J_{B} (x) ∖ I_{B} (x)$ are closed in $B$ and $I_{B} (x)$ is open in $B$ ;

(c) there is no nonconstant future causal curve $γ : [0, 1] \to B$ with $γ (0) = γ (1) = x$ ;

(d) corner smoothing. if $γ : [0, 1] \to B$ is future causal with $γ (0) = x$ and there is $t_{0}$ at which $γ^{'} (t_{0})$ is future timelike, then $γ (1) \in I_{B} (x)$ ;

(e) closed relation at fixed base. if $q_{n} \in J_{B} (x)$ and $q_{n} \to q \in B$ , then $q \in J_{B} (x)$ . The same holds for $J_{B} (x) ∖ I_{B} (x)$ , the future null boundary including the vertex $x$ . A sequence in $I_{B} (x)$ may have a null or zero limit, and a sequence in $\exp_{x} (N^{+})$ may converge to $x$ ; in either case its limit remains in $J_{B} (x)$ .

Facts & Assumptions

Given: A point $x \in M$ and a cone-theorem ball around $x$ as in Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points and Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic.

[L1]

The cone theorem holds in $B$ with base $x$ : clauses (a)-(d) of Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic (future causal curves from $x$ have future causal or zero position vectors; the cone characterizations, with the reflexive vertex included; null rigidity; length maximality).

[L3]

The future causal cone in a tangent space is closed, the future timelike cone is open, and a limit of future timelike vectors is future causal or zero (The timelike set has exactly two convex components, described in orthonormal coordinates, Pairing of an observer with future causal vectors).

Proof

technique · direct

1.1L1givenconstruct

(Existence of $B$ and the cone characterizations.) Choose a point-normal exponential-coordinate ball around $x$ with $\nabla u_{x}^{0}$ past timelike on it; the gradient is past timelike at $x$ , and this is an open condition, so the ball of Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points can be shrunk inside that region. The cone theorem now applies to every strictly convex normal set; it gives the cone characterizations in this chosen $B$ with base point $x$ . The temporal-coordinate property is used separately below and is not a hypothesis of the cone theorem.

1.2L1L3givenalgebra

(No closed causal curves.) In the exponential chart $u$ of [L1], $\nabla u^{0}$ is past timelike throughout $B$ . Thus $d u^{0} (z) = g (\nabla u^{0}, z) > 0$ for every nonzero future causal tangent $z$ , by [L3]. Along a future causal curve the locally absolutely continuous composite $u^{0} \circ γ$ therefore has positive derivative almost everywhere. Integrating on $[0, 1]$ gives $u^{0} (γ (1)) > u^{0} (γ (0))$ , so the endpoints cannot coincide. This proves (c) and also shows that a curve from $x$ has nonzero position vector at every positive parameter.

2.1step 1.1L1L3givenalgebra

(Closedness and openness.) Under $\exp_{x} ∣_{W_{x}}$ , the set $J_{B} (x)$ corresponds to the relatively closed cone ${v \in W_{x} : g_{x} (v, v) \leq 0, g_{x} (v, N_{x}) \leq 0}$ , while $J_{B} (x) ∖ I_{B} (x)$ corresponds to ${v \in W_{x} : g_{x} (v, v) = 0, g_{x} (v, N_{x}) \leq 0}$ including $v = 0$ . The future timelike cone is relatively open in $W_{x}$ . The exponential map is a homeomorphism, so these facts prove (b) and the closed-set assertions of (e). Any limit of points in $I_{B} (x)$ or $\exp_{x} (N^{+})$ still belongs to the closed set $J_{B} (x)$ ; the limit need not stay in either of those smaller sets.

2.2step 1.1step 1.2L1givenalgebra

(Corner smoothing.) Let $γ$ have a future timelike derivative at $t_{0} \in [0, 1]$ (with the appropriate one-sided derivative at an endpoint). Its final position vector is future causal by [L1], and nonzero by step 1.2. If that vector were null, [L1]'s null rigidity, with the nonvanishing supplied by step 1.2, would give $γ (t) = \exp_{x} (φ (t) v_{0})$ on $[0, 1]$ for a fixed future null $v_{0}$ . The coefficient $φ$ is a linear coordinate of the smooth inverse exponential applied to $γ$ , and hence is differentiable at $t_{0}$ . Differentiating gives $γ^{'} (t_{0}) = φ^{'} (t_{0}) d (\exp_{x})_{φ (t_{0}) v_{0}} (v_{0})$ , a null or zero vector by the radial-geodesic character in [L1], contradicting its timelike character. Thus the final position is future timelike, proving (d).

3.1step 1.1step 2.1step 1.2step 2.2∎

Claims (a)-(e) are the preceding steps, with (e) from the closedness step.

Remarks

Locality and use. All statements are about one small ball $B$ and one base point $x$ ; in the downstream arguments they are applied at the finitely many base points that occur in a fixed compact region, and the openness of $I_{B} (x)$ together with the closedness of $J_{B} (x)$ is exactly the local input used in The existence of a Cauchy set implies strong causality and compact causal diamonds and In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact.
The closedness of $J_{B} (x)$ is not asserted for the relation with varying base point. That stronger statement is proved where it is used, from the limit curve theorem (Two-endpoint limit curve theorem: bounded lengths give joining limits, unbounded lengths give two inextendible tails with the closure relation).

Pointwise local causality: closed cones, openness of the local chronological future, no local closed causal curves, corner smoothing and closedness of the local causal relation

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources