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An open chronological exterior is causally convex without a timelike-end assumption

Statement

For any open subset $U$ of a smooth time-oriented spacetime put $E = I^{-} (U) \cap I^{+} (U)$ . Then $U \subset E$ , and any ambient causal segment with endpoints in $E$ is contained in $E$ . If $p \in E$ is chronologically before some $z$ in a specified open far tail $U_{f} \subset U$ , a future timelike path from $p$ to $z$ can be replaced, with the same endpoints, by finitely many smooth future timelike geodesic pieces inside $E$ . This lemma does not prove the existence of such a $z \in U_{f}$ for every $p \in E$ .

Statement references

Causal exterior, chronological boundaries and Cauchy data

Causal curves, chronological and causal reachability, and inextendibility

Push-up rule

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

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

The geodesic initial-value problem and the exponential map of the metric-selected connection

The Axiom of Choice

Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points

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

Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover

Chart independence, concatenation, reparametrization and transitivity for causal curves

AC implies DC implies countable choice

Facts & Assumptions

[F1]

Chronology, the causal exterior and causal curves have the stated meanings; timelike--causal concatenation obeys push-up, and chronological futures and pasts are open. Causal exterior, chronological boundaries and Cauchy data Causal curves, chronological and causal reachability, and inextendibility Push-up rule Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

[F2]

Local smooth geodesics and their exponential maps exist. Every point in an open neighborhood $O$ has a strictly convex normal neighborhood $C \subset O$ that is normal at every one of its points, with a unique contained geodesic between each pair of points. For every $x \in C$ , the cone theorem gives $I_{C}^{+} (x) = \exp_{x} (T_{x}^{+})$ ; thus endpoints chronologically related by a timelike curve in $C$ are joined by the smooth future timelike radial geodesic in $C$ . The fixed-base ball lemma also records this cone characterization at its specified base point. The geodesic initial-value problem and the exponential map of the metric-selected connection Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points 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 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

[F3]

A closed bounded real interval is compact by Heine--Borel. Every relatively open cover of that interval has a finite subcover and a Lebesgue number $δ > 0$ : every nonempty subset of diameter less than $δ$ lies in one cover member. Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover

[F4]

Restrictions to nondegenerate compact subintervals preserve local absolute continuity and the almost-everywhere timelike or causal velocity condition directly from the curve definition. Increasing affine reparametrizations preserve these curve classes, and finite concatenations of smooth future timelike pieces are future timelike curves. Causal curves, chronological and causal reachability, and inextendibility Chart independence, concatenation, reparametrization and transitivity for causal curves

[A1]

Work in ZFC. AC implies Dependent Choice and Countable Choice, supplying the inherited choice hypotheses of the causal results in [F1], [F2] and [F4]. The Axiom of Choice AC implies DC implies countable choice

Proof

Given: The stated data and explicit hypotheses.

Goal: Establish the stated causal convexity and conditional geodesic replacement.

1.1F1F2F4A1

Every point of the open set $U$ has a short future and past timelike segment remaining in $U$ , by local geodesic existence in [F2], so it lies in $I^{-} (U) \cap I^{+} (U)$ . If a causal segment has endpoints $p, q \in E$ and passes through $y$ , its restrictions give $p \leq y \leq q$ , including equality at the endpoints. Choose $u_{-} ≪ p$ and $q ≪ u_{+}$ with $u_{\pm} \in U$ . Push-up gives $u_{-} ≪ y ≪ u_{+}$ , so $y \in E$ . Also $I^{\pm} (U)$ are unions of open pointwise chronological futures or pasts, so $E$ is open. This proves causal convexity without a timelike stationary-field assumption on $U$ .

2.1givenF1F2F3A1step 1.1

Under the explicit accessibility hypothesis choose a future timelike curve $γ : [a, b] \to M$ , $a < b$ , with endpoints $p$ and $z$ . Both endpoints lie in $E$ by step 1.1, so its entire image lies in $E$ . Since $E$ is open, [F2] supplies a cover of this image by strictly convex normal neighborhoods $C \subset E$ . The inverse images $γ^{- 1} (C)$ form a relatively open cover of $[a, b]$ , because $γ$ is continuous. By [F3] take a finite subcover and a Lebesgue number $δ > 0$ for that subcover on the compact metric interval $[a, b]$ . Choose a finite partition $a = t_{0} < \dots < t_{m} = b$ of mesh less than $δ$ . Each $[t_{j - 1}, t_{j}]$ then lies in some $γ^{- 1} (C_{j})$ , so $γ ([t_{j - 1}, t_{j}]) \subset C_{j} \subset E$ .

3.1F2F4A1step 2.1

The restriction of $γ$ to $[t_{j - 1}, t_{j}]$ , affinely reparametrized to $[0, 1]$ , is future timelike in $C_{j}$ by [F4]. Its endpoints are therefore chronologically related in $C_{j}$ , not merely in the ambient spacetime. Apply the cone theorem of [F2] with base point $γ (t_{j - 1})$ , which is legitimate because $C_{j}$ is normal at every one of its points. The inverse-exponential displacement to $γ (t_{j})$ is future timelike, and the radial geodesic is smooth, future timelike and contained in $C_{j}$ . Replace each restricted curve by this geodesic and affinely parametrize it on $[t_{j - 1}, t_{j}]$ .

4.1F4step 3.1∎

The finitely many geodesic pieces have matching endpoints. Their concatenation is a future timelike curve by [F4], is smooth on each piece, stays in $E$ and has the original endpoints $p, z$ . This establishes the asserted conditional replacement. No access to the far tail was inferred merely from access to $U$ .

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