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Connectedness and causal convexity of a chronological exterior

Statement

Work in ZFC. Let $U$ be a nonempty connected open region in a time-oriented smooth Lorentzian four-manifold $(D, g, N)$ . Let $T$ be a smooth vector field whose flow restricts to a smooth global flow $Φ : R \times U \to U$ of $T ∣_{U}$ . Thus the line through $u \in U$ is $t \mapsto Φ (t, u)$ , with $Φ (0, u) = u$ and $\partial_{t} Φ (t, u) = T_{Φ (t, u)}$ . No causal character of $T$ on $U$ is required. Take all chronological relations in $D$ , and set $A = I_{D}^{-} (U)$ , $B = I_{D}^{+} (U)$ and $E = A \cap B$ . Then $U \subset E$ , $E$ is path connected, and every causal segment in $D$ with endpoints in $E$ stays in $E$ .

Statement references

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Causal exterior, chronological boundaries and Cauchy data

Causal curves, chronological and causal reachability, and inextendibility

Local and global flows generated by a vector field

The Axiom of Choice

Facts & Assumptions

[F1]

ZFC includes the Axiom of Choice, which implies Dependent Choice and Countable Choice. These inherited principles meet the choice hypotheses of the chronological openness and push-up suppliers; no simultaneous selection of paths indexed by all exterior points is needed. The Axiom of Choice AC implies DC implies countable choice

[F2]

The smooth Lorentzian interface supplies smooth metric coefficients and a smooth future timelike orientation field $N$ . The sets $I_{D}^{\pm} (U)$ have the causal-exterior interface's meanings for any open $U$ , without requiring it to be a timelike stationary end. The global-flow interface supplies the stated flow notation and integral-curve laws. The curve interface defines future timelike and causal curves and their reachability relations; smooth future timelike curves satisfy that convention. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model Causal exterior, chronological boundaries and Cauchy data Local and global flows generated by a vector field Causal curves, chronological and causal reachability, and inextendibility

[F3]

A topological manifold has a basis of open path-connected neighborhoods. A connected locally path-connected topological space is path connected. Topological manifolds are locally compact and locally path connected A connected, locally path-connected space is path-connected, because its path components are open

[F4]

The push-up rule gives $a ≪ b \leq c \Rightarrow a ≪ c$ and $a \leq b ≪ c \Rightarrow a ≪ c$ in the smooth Lorentzian model. Here $\leq$ includes equality and $≪$ implies $\leq$ , by the curve convention, so these rules also give chronological transitivity. Push-up rule Causal curves, chronological and causal reachability, and inextendibility

[F5]

Chronological futures and pasts of individual points are open. Their unions $I_{D}^{+} (U)$ and $I_{D}^{-} (U)$ are therefore open in $D$ . Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets Causal curves, chronological and causal reachability, and inextendibility

Proof

Given: The open-region and global-flow data in the Statement, in ZFC.

Goal: Prove inclusion of $U$ , path connectedness of $E$ , and causal convexity in $D$ .

1.1givenF1F2F5

Fix $u \in U$ and choose a smooth chart $x$ around $u$ contained in $U$ . Write $v$ for the coordinate components of $N_{u}$ , and extend them as a constant-component vector field $V$ on the chart. At $u$ , both $g (V, V)$ and $g (V, N)$ are negative. Smoothness and continuity make both inequalities remain negative on a smaller neighborhood of $u$ . For sufficiently small $ε > 0$ , the straight coordinate segment $γ (s) = x^{- 1} (x (u) + s v)$ , $- ε \leq s \leq ε$ , lies in that neighborhood and has velocity $V$ . It is a smooth future timelike curve in $U$ , so $γ (- ε) ≪_{D} u ≪_{D} γ (ε)$ . Both endpoints belong to $U$ , giving $u \in B \cap A$ . Thus $U \subset E$ , without any sign assumption on $T$ . Also $A$ and $B$ , and hence $E$ , are open by chronological openness. The choice assumptions needed for that supplier follow from ZFC.

1.2givenF3

Since $U$ is open in $D$ , every open neighborhood $V$ of a point $u$ in $U$ is also open in $D$ . The manifold neighborhood basis supplies an open path-connected neighborhood $W$ with $u \in W \subset V$ . Thus $U$ is locally path connected in its subspace topology. It is connected by hypothesis, so the connected locally path-connected theorem makes $U$ path connected.

1.3givenF1F2F4

Orient a causal segment with endpoints $p, q \in E$ from $p$ to $q$ , and let $z$ be any point on it. The segment restrictions give $p \leq_{D} z \leq_{D} q$ , allowing equality at endpoints. By the definitions of $B$ and $A$ , choose $u_{-}, u_{+} \in U$ with $u_{-} ≪_{D} p$ and $q ≪_{D} u_{+}$ . The two push-up implications give $u_{-} ≪_{D} z$ and $z ≪_{D} u_{+}$ . Therefore $z \in B \cap A = E$ , proving causal convexity. The choice assumptions of push-up are supplied by ZFC.

2.1F2step 1.1step 1.2step 1.3∎

Given $p_{0}, p_{1} \in E$ , choose $q_{0}, q_{1} \in U$ and future timelike curves from $p_{i}$ to $q_{i}$ , as permitted by $p_{i} \in A$ . Their endpoints belong to $E$ by step 1.1, so their images lie in $E$ by step 1.3. Step 1.2 supplies a continuous path inside $U \subset E$ from $q_{0}$ to $q_{1}$ . Traverse the first timelike curve, this path, and the reverse of the second timelike curve. Reparametrizing the three continuous curves onto three consecutive closed subintervals gives a continuous path in $E$ from $p_{0}$ to $p_{1}$ , since the endpoint values agree at the two joins. This proves path connectedness and completes all three claims.

Depends on

Used by · 3

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