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Equivalence of the absolutely-continuous and continuous causal conventions, and one-sided inextendible extensions

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 (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class). Call a map $x : I \to M$ a geodesically connectible continuous causal curve when it is continuous (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) and for every strictly convex normal ball $C$ as produced by Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points (Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function clause (b)) and all $t_{1} < t_{2}$ in $I$ with $x ([t_{1}, t_{2}]) \subseteq C$ , the endpoints $x (t_{1})$ and $x (t_{2})$ are joined by a future causal geodesic whose image is contained in $C$ . Write $\leq$ for the causal relation of Causal curves, chronological and causal reachability, and inextendibility (defined by locally absolutely continuous future causal curves) and put $n : = \frac{N}{\sqrt{- g (N, N)}}, h : = g + 2 n^{♭} \otimes n^{♭} .$

(a) Absolutely continuous implies connectible. Every future causal curve $γ : I \to M$ in the sense of Causal curves, chronological and causal reachability, and inextendibility is a geodesically connectible continuous causal curve.

(b) Connectible curves give causal relations. Let $x : I \to M$ be a geodesically connectible continuous causal curve and let $t_{1} < t_{2}$ in $I$ . If $x ∣_{[t_{1}, t_{2}]}$ is nonconstant, then for every open neighborhood $W$ of $x ([t_{1}, t_{2}])$ there is a piecewise smooth future causal curve $η : [t_{1}, t_{2}] \to W$ with $η (t_{1}) = x (t_{1})$ and $η (t_{2}) = x (t_{2})$ ; in particular $x (t_{1}) \leq x (t_{2})$ . If the restriction is constant, its endpoints coincide and reflexivity gives $x (t_{1}) \leq x (t_{2})$ without asserting a curve in the nonconstant future causal class.

(c) Equality of relations. For $p, q \in M$ the following are equivalent: $p \leq q$ ; $p = q$ or there is a geodesically connectible continuous causal curve with endpoints $p, q$ .

(d) One-sided extension. Let $γ : [0, \infty) \to M$ be a future-inextendible future causal curve in the sense of Causal curves, chronological and causal reachability, and inextendibility clause (d). Then there is an inextendible future causal curve $γ^{*} : I^{*} \to M$ with $[0, \infty) \subseteq I^{*} \subseteq R$ and $γ^{*} ∣_{[0, \infty)} = γ$ ; that is, $γ$ is contained in a curve that is both future- and past-inextendible. The dual statement holds for past-inextendible curves.

No choice principle beyond the stated assumptions is used. The extension in (d) is not asserted to be unique.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ and the objects of the four clauses.

[L1]

Strictly convex normal balls exist around every point; in such a ball $C$ with base $p$ the future causal reachability is the exponential cone: $J_{C}^{+} (p) = \exp_{p} (C^{+})$ and $I_{C}^{+} (p) = \exp_{p} (T^{+})$ , where $C^{+}$ is the set of future causal or zero vectors in the open star-shaped inverse domain $W_{p} = {u \in E_{p} : \exp_{p} (t u) \in C \forall t \in [0, 1]}$ ; the radial geodesic of $v \in C^{+}$ stays in $C$ (Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points clauses (a) and (b), 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 clauses (a) and (b), Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function clause (b)).

[L2]

Every future causal curve with image in a strictly convex normal ball $C$ starting at $p \in C$ has its endpoint in $J_{C}^{+} (p)$ (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 clause (a)).

[L3]

Finite concatenation of locally absolutely continuous causal curves needs no choice: split the finitely many test intervals at each junction, use the AC modulus on each of the finitely many pieces with error tolerance $ε / k$ , and take the minimum of these finitely many moduli. The triangle inequality bounds the concatenated coordinate increments by the sum of the piecewise errors. Velocities agree with the original velocities off the finitely many junctions, so nonzero future causal velocity holds almost everywhere. Smooth geodesic pieces are locally AC directly by their bounded coordinate derivatives on compact pieces (Causal curves, chronological and causal reachability, and inextendibility).

[L6]

The smooth observer normalization $n = N / \sqrt{- g (N, N)}$ supplies a specified future unit timelike vector $n_{q}$ at each given point $q$ . The geodesic with this initial velocity has a unique open maximal parameter domain containing $0$ , is smooth, and remains future timelike because parallel transport preserves its causal character and direction (Observers, rest spaces and the spatial projection at an event, The geodesic initial-value problem and the exponential map of the metric-selected connection clause (a), Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed clause 3).

[L7]

Every maximal geodesic with nonzero initial velocity has no point limit at either end of its open maximal parameter interval, whether the end is finite or infinite (Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal clause (b)). This is the exact no-limit statement; compact closure of an image alone does not imply geodesic extendibility.

Proof

technique · direct

1.1L1L3givenconstruct

(Clause (b).) Let $x ∣_{[t_{1}, t_{2}]}$ be nonconstant and let $W$ be an open neighborhood of its compact image. Include a parameter $t_{*}$ with $x (t_{*})$ different from $x (t_{1})$ in a finite partition. Cover the image by finitely many strictly convex normal balls $C_{1}, \dots, C_{m} \subseteq W$ and refine the partition to a Lebesgue-number subdivision so each subarc is contained in one ball. Each successive pair of endpoint values is joined there by its unique causal geodesic. If a pair coincides, that geodesic is constant and is omitted; the chosen parameter $t_{*}$ ensures at least one nonconstant segment remains. Reparametrize the remaining segments affinely, in their original order, onto consecutive subintervals of positive length partitioning $[t_{1}, t_{2}]$ . Their concatenation is piecewise smooth with nonzero future causal velocity away from finitely many joints, so it is a future causal curve by [L3]. It lies in $W$ and has the required endpoints, proving (b).

1.2L1L2given

(Clause (a).) Let $γ : I \to M$ be a future causal curve and let $C$ be a strictly convex normal ball with $γ ([t_{1}, t_{2}]) \subseteq C$ ; put $p : = γ (t_{1})$ . The restricted curve $s \mapsto γ (t_{1} + s)$ , $s \in [0, t_{2} - t_{1}]$ , is a future causal curve with image in $C$ starting at $p$ , so by [L2] its endpoint $γ (t_{2})$ lies in $J_{C}^{+} (p)$ ; by [L1] there is $v \in C^{+} \subseteq W_{p}$ with $\exp_{p} (v) = γ (t_{2})$ , and the radial geodesic $s \mapsto \exp_{p} (s v)$ , $s \in [0, 1]$ , is a future causal geodesic whose image is contained in $C$ because $W_{p}$ is star-shaped and $\exp_{p} (W_{p}) = C$ . This proves (a).

1.3givenL6construct

(One past maximal geodesic.) Let $γ : [0, \infty) \to M$ be the given future-inextendible curve, and put $q = γ (0)$ . The normalized time-orientation field specifies the nonzero future timelike vector $n_{q}$ , so no choice of a family of vectors is involved. Let $ρ : J \to M$ be the unique maximal geodesic with $ρ (0) = q$ and $ρ^{'} (0) = n_{q}$ . Write $α = \inf J \in [- \infty, 0)$ and put $I^{*} : = (α, \infty)$ . Define $γ^{*} (t) = ρ (t)$ for $α < t \leq 0$ and $γ^{*} (t) = γ (t)$ for $t \geq 0$ . The pieces agree at $0$ .

2.1step 1.1step 1.2given

(Clause (c).) If $p \leq q$ then either $p = q$ or there is a locally absolutely continuous future causal curve from $p$ to $q$ , which is geodesically connectible by step 1.2; conversely, if a geodesically connectible continuous causal curve joins $p \neq q$ , apply step 1.1 with $W = M$ to obtain a future causal locally absolutely continuous curve from $p$ to $q$ , so $p \leq q$ by Causal curves, chronological and causal reachability, and inextendibility clause (c). This proves (c).

2.2step 1.3L3L6L7given

(Causal gluing and both ends.) Every compact subinterval of $I^{*}$ meets at most these two pieces. The geodesic piece is smooth and future timelike by [L6], and the other piece is the given future causal curve, so [L3] proves the glued map is locally AC with nonzero future causal velocity almost everywhere. By [L7], $ρ (t)$ has no point limit as $t \to α^{+}$ , including $α = - \infty$ ; hence the identical past tail of $γ^{*}$ is past-inextendible in the exact no-limit convention. Its future tail is $γ$ , so it is future-inextendible by hypothesis. Thus $γ^{*}$ is inextendible, $[0, \infty) \subseteq I^{*}$ , and the original parametrization is unchanged on that half-line. Reversing time gives the dual claim. This proves (d).

3.1step 1.1step 1.2step 2.1step 2.2∎

(Conclusion.) Clauses (a), (b), (c) and (d) are proved in steps 1.2, 1.1, 2.1 and 2.2, respectively.

Remarks

What the conventions have in common. Clause (b) shows that the causal relation defined in Causal curves, chronological and causal reachability, and inextendibility is unchanged if one uses the classical "locally connected by future causal geodesics" convention; clause (a) shows that every curve of the library convention fits that convention. This is the content of Minguzzi's Theorem 2.12 and Remark 2.14, proved here from the local cone theorem.
Not every connectible curve is locally absolutely continuous. A continuous causal curve can be so singularly reparametrized that it has zero derivative almost everywhere, so clauses (b) and (c) replace it by a piecewise geodesic curve with the same endpoints; the parametrization itself is not a library curve. Clause (d) preserves the given half-line parametrization exactly and adds one maximal geodesic past tail.
Scope. The lemma is a local-to-global convention statement inside the smooth Lorentzian model; it uses no completeness and no global causal condition.

Equivalence of the absolutely-continuous and continuous causal conventions, and one-sided inextendible extensions

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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