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A maximal connected portion of an inextendible causal curve inside an open subset is inextendible there

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $V \subseteq U \subseteq M$ be open subsets with the induced metric and time orientation, and let $γ : I \to U$ be a future causal curve (Causal curves, chronological and causal reachability, and inextendibility) that is inextendible in the Lorentzian manifold $U$ : $γ$ does not converge to any point of $U$ as its parameter tends to either end of $I$ (clause (d) of Causal curves, chronological and causal reachability, and inextendibility, read with ambient set $U$ ). Let $J$ be a connected component of $γ^{- 1} [V] : = {t \in I : γ (t) \in V} .$ Then $γ ∣_{J}$ , regarded as a future causal curve in the Lorentzian manifold $V$ with the restricted fields, is inextendible in $V$ .

Facts & Assumptions

Given: open sets $V \subseteq U \subseteq M$ , a future causal curve $γ : I \to U$ inextendible in $U$ , and a connected component $J$ of $γ^{- 1} [V]$ .

[L1]

Regularity and components: a future causal curve is locally absolutely continuous, hence continuous on its interval (Causal curves, chronological and causal reachability, and inextendibility, Chart independence, concatenation, reparametrization and transitivity for causal curves, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length); therefore $γ^{- 1} [V]$ is relatively open in $I$ and each of its connected components is of the form $I \cap (α, β)$ for some $α < β$ in $[- \infty, + \infty]$ .

[L2]

Inextendibility: for a future causal curve $δ$ on an interval $K$ in a Lorentzian manifold $W$ , and for a real interval $K$ with $\sup K$ (respectively $\inf K$ ) finite, the curve is future-inextendible (respectively past-inextendible) exactly when it does not converge to a point of $W$ as the parameter tends to that end; at an unbounded end, it is one-sided inextendible exactly when it does not converge as the parameter tends to $+ \infty$ or $- \infty$ (Causal curves, chronological and causal reachability, and inextendibility clause (d)). Convergence to a point $y \in V$ and convergence to the same point $y \in U \supseteq V$ are the same condition, since $V$ and $U$ are open subsets of the same manifold.

No choice principle is used in this item.

Proof

technique · direct

1.1L1algebra

(Ends of a component.) Let $J = I \cap (α, β)$ be the component of [L1]. If $α \in I$ then $α \notin J$ and $γ (α) \notin V$ : otherwise $γ (α) \in V$ with $V$ open would make $γ^{- 1} [V]$ contain a relative neighbourhood of $α$ in $I$ , so that $J$ would extend strictly to the left of $α$ , contradicting that $J$ is a component. Similarly, if $β \in I$ then $β \notin J$ and $γ (β) \notin V$ .

2.1step 1.1givenL2contradiction

(The left end.) Suppose $α \in I$ (finite and in the domain). By step 1.1, $γ (t) \to γ (α) \notin V$ as $t \to α$ along $J$ ; since limits in the ambient manifold are unique, $γ ∣_{J}$ does not converge to any point of $V$ at its past end. Suppose instead that $α = \inf I$ with $α$ finite or $- \infty$ , and that $γ ∣_{J}$ converged to some $y \in V$ as $t \to \inf I$ . Then $y \in V \subseteq U$ , so $γ$ would converge to the point $y \in U$ of its ambient manifold at its past end, contradicting the hypothesis that $γ$ is inextendible in $U$ by [L2]. Hence in both cases $γ ∣_{J}$ does not converge to a point of $V$ at its past end.

3.1step 2.1L2contradiction

(The right end.) The same argument with $β$ and $\sup I$ in place of $α$ and $\inf I$ , and with "past" replaced by "future", shows that $γ ∣_{J}$ does not converge to a point of $V$ at its future end.

4.1step 2.1step 3.1L1L2∎

(Conclusion.) The restricted curve $γ ∣_{J}$ has image in $V$ , is locally absolutely continuous, and has causal velocity almost everywhere, so it is a future causal curve of the Lorentzian manifold $V$ ([L1]); by steps 2.1 and 3.1 it converges to no point of $V$ at either end, so by [L2] it is inextendible in $V$ . This is the assertion.

Remarks

The ambient inextendibility assumption is used when a component reaches an end of the original domain. At a finite interior endpoint of a component, continuity supplies an ambient value outside the open subset, so Hausdorff uniqueness rules out a limit inside that subset. These are distinct cases; no continuation through a point outside the subset is claimed to stay within it.

A maximal connected portion of an inextendible causal curve inside an open subset is inextendible there

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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