LemmaStatement: AI-generatedProof: AI-generated

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Future timelike cylinder translations make their chronological pasts increase

Statement

Assume Countable Choice and Dependent Choice for the causal-curve calculus and the causal results used below.

Let $(M, \hat{g}, N)$ be a time-oriented smooth Lorentzian four-manifold without boundary, of signature $(-, +, +, +)$ . All chronological pasts and curves are taken in $M$ , with the locally absolutely continuous curve convention.

Let $T$ be a smooth vector field on $M$ with a supplied local flow $ϕ^{T} : Ω \to M$ , where $Ω \subseteq R \times M$ . Supply a subset $C_{0} \subseteq M$ such that $R \times C_{0} \subseteq Ω$ . Define $C_{t} = {ϕ^{T} (t, c) : c \in C_{0}}$ for every $t \in R$ , and let the complete cylinder be $C = ⋃_{t \in R} C_{t}$ . Suppose $T$ is future timelike throughout $C$ , meaning $\hat{g} (T, T) < 0$ and $\hat{g} (T, N) < 0$ there.

Then $A_{t} = I_{\hat{g}}^{-} (C_{t})$ is open in $M$ and $A_{s} \subseteq A_{t}$ for $s < t$ . If a point $p$ has a future $\hat{g}$ -timelike curve ending at a point of $C_{a}$ , it belongs to $A_{t}$ for every $t \geq a$ .

Statement references

Causal exterior, chronological boundaries and Cauchy data

Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

Push-up rule

Local and global flows generated by a vector field

The Axiom of Countable Choice (AC_ω)

The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

Metric compatibility, torsion freedom and the metric-selected connection class

Causal curves, chronological and causal reachability, and inextendibility

Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

Facts & Assumptions

[F1]

Countable Choice and Dependent Choice are assumed exactly as required by the two causal suppliers; their meanings are the stated choice principles. These assumptions are inherited from those suppliers, rather than a claim that the present finite orbit argument requires a new infinite selection. The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

[F2]

Time orientation and chronological reachability have the causal-data interface meanings, applied with metric $\hat{g}$ on $M$ . The metric-selected connection required by the push-up supplier is the connection constructed for this smooth metric in clause (d) of the metric-selected-connection interface. Causal exterior, chronological boundaries and Cauchy data Metric compatibility, torsion freedom and the metric-selected connection class

[F3]

For a time-oriented smooth Lorentzian four-manifold under [F1], every point chronological past is open. If $p ≪ q$ and $q \leq r$ , the push-up rule gives $p ≪ r$ ; a future timelike curve is in particular future causal. Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets Push-up rule

[F4]

A supplied local flow is smooth, has interval fibres, follows its smooth vector field and satisfies the flow law wherever both sides are defined. Here $R \times C_{0} \subseteq Ω$ explicitly supplies every orbit segment used in the proof. Local and global flows generated by a vector field

[F5]

For every subset $S \subseteq M$ , its chronological past is $I_{\hat{g}}^{-} (S) = {p \in M : \exists r \in S, p ≪ r} = ⋃_{r \in S} I_{\hat{g}}^{-} (r)$ , using the reachability definition in clause (c); no openness of $S$ is required. Causal curves, chronological and causal reachability, and inextendibility

[F6]

In the manifold topology of $M$ , any union of open sets is open by topology axiom (T2), and the empty set is open by (T1), including the empty-union case. Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

Proof

Given: The data and hypotheses of the Statement.

Goal: Prove openness, monotonicity and the arrival-time assertion.

1.1givenF2F4

Fix $s < t$ and $q \in C_{s}$ . By the definition of $C_{s}$ , there is $c \in C_{0}$ with $q = ϕ^{T} (s, c)$ . The curve $γ (u) = ϕ^{T} (u, c)$ for $s \leq u \leq t$ is defined because $R \times C_{0} \subseteq Ω$ . Its image lies in $C$ , its velocity is $T_{γ (u)}$ , and the two timelikeness inequalities hold throughout. It is therefore a future timelike curve from $q$ to $r = ϕ^{T} (t, c) \in C_{t}$ .

2.1givenF1F2F3F5step 1.1

For $p \in A_{s}$ there is $q \in C_{s}$ with $p ≪ q$ . Step 1.1 gives $q ≪ r$ for some $r \in C_{t}$ , hence $q \leq r$ . All hypotheses of the causal suppliers hold by the stated smooth four-dimensional setting, [F1] and [F2]. Applying [F3] gives $p ≪ r$ , so $p \in A_{t}$ . Thus $A_{s} \subseteq A_{t}$ for every $s < t$ .

3.1F1F2F3F5F6step 2.1

By the definition of chronological past, $A_{t} = ⋃_{r \in C_{t}} I_{\hat{g}}^{-} (r)$ . Each set in this union is open by [F3] under the same hypotheses checked in step 2.1, and an arbitrary union of open sets is open. This includes $C_{t} = \emptyset$ , when $A_{t} = \emptyset$ .

4.1givenF2F5step 2.1step 3.1∎

A future timelike curve from $p$ to a point of $C_{a}$ gives $p \in A_{a}$ by definition. For $t > a$ , step 2.1 gives $A_{a} \subseteq A_{t}$ , while for $t = a$ membership was already established. Hence $p \in A_{t}$ for every $t \geq a$ , completing all conclusions.

Depends on

Used by · 2

Dependency tree · 2 levels

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