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Existence and uniqueness of the maximal common development of two developments

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). Let $(M, g, N, j, ψ)$ and $(M^{'}, g^{'}, N^{'}, j^{'}, ψ^{'})$ be developments of the same data in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a), with $Σ : = j (Σ)$ . Then there exist $U^{*} \subseteq M$ and $ψ^{*} : U^{*} \to M^{'}$ such that:

(a) $(U^{*}, ψ^{*})$ is a common subdevelopment of $M$ and $M^{'}$ in the sense of Common subdevelopments, maximal common developments and corresponding boundary points clause (a);

(b) for every common subdevelopment $(U, φ)$ of $M$ and $M^{'}$ one has $U \subseteq U^{*}$ and $ψ^{*} ∣_{U} = φ$ ; in particular $(U^{*}, ψ^{*})$ is a maximal common development in the sense of clause (b) of that definition, and it contains $j (Σ)$ ;

(c) uniqueness: if $(U^{†}, ψ^{†})$ also satisfies (a) and (b), then $U^{†} = U^{*}$ and $ψ^{†} = ψ^{*}$ .

Facts & Assumptions

Given: Countable Choice, Dependent Choice, two developments $(M, g, N, j, ψ)$ , $(M^{'}, g^{'}, N^{'}, j^{'}, ψ^{'})$ of the same data, with $Σ = j (Σ)$ Cauchy in both.

[L1]

Common subdevelopments form a set: a common subdevelopment is a pair $(U, φ)$ with $U \subseteq M$ and $φ : U \to M^{'}$ ; since $U$ is an element of the power set of $M$ and the graph of $φ$ is an element of the power set of $M \times M^{'}$ , the collection of all common subdevelopments is a set by separation (The power set $P (x) = {z : z \subseteq x}$ , The Axiom Schema of Separation: for each formula $φ$ , $\forall \overset{ˉ}{p} \forall x \exists y \forall z (z \in y \leftrightarrow (z \in x \land φ (z, \overset{ˉ}{p})))$ ).

[L2]

Restriction and gluing of development data: each $U_{α}$ is open, is a development of the same data with the restricted fields, and $ψ_{α}$ is a development embedding, hence in particular an isometric immersion fixing the data and the time orientation (Common subdevelopments, maximal common developments and corresponding boundary points clause (a), Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clauses (b)-(c)).

[L3]

Agreement on overlaps is Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps clause (c). Nonemptiness follows from the local complete-data uniqueness theorem for the selected fixed model: the admissible fixed coupled-model theorem Local geometric development for a fixed covariant Einstein-matter model with complete constrained data (including its vacuum instance), Local Einstein-scalar evolution: smooth constrained data admit a local development, unique up to a data-fixing isometry, Local evolution for a strict barotropic perfect fluid: smooth constrained data admit a development, unique up to a data-fixing isometry or a specified independent reduction, including the tilted nonnegative dust theorem Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry. With their Countable Choice and Dependent Choice assumptions, two developments agree near the entire initial surface by a future data-fixing isometry matching all matter fields. Apply the intrinsic Cauchy-neighborhood construction Local Cauchy neighborhoods in a conically bounded product chart to a smaller common isometry collar, and transport that neighborhood by the isometry. Its source and image have the entire initial surface Cauchy, so this restriction is a common subdevelopment.

[L4]

Portion inextendibility and Cauchy property: if a causal curve is inextendible in $U^{*}$ , then for every component $J$ of its preimage under one of the sets $U_{α}$ the restriction is inextendible in $U_{α}$ (A maximal connected portion of an inextendible causal curve inside an open subset is inextendible there); and every causal curve inextendible in $U_{α}$ meets $Σ$ in exactly one parameter value (Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a)5).

[L5]

Two crossings in the ambient development: if a causal curve in $M$ met $Σ$ in two distinct parameter values, then by the segment-extension clause the compact segment between them lies on an inextendible causal curve of $M$ meeting $Σ$ twice, contradicting that $Σ$ is a Cauchy set of $M$ (Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal clause (c), Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (c), Causal curves, chronological and causal reachability, and inextendibility).

[L6]

Local differential geometry on $U^{*}$ : a map whose restriction to an open cover of its domain is a smooth isometric immersion is a smooth isometric immersion, and the field equations of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a)4 are local, so they hold on $U^{*}$ ; injectivity of a data-fixing isometric immersion of developments follows from A data-fixing isometric immersion of developments is injective, hence a development embedding; uniqueness of data-fixing embeddings clause (b).

Proof

technique · direct

1.1L1L3construct

(The union and the patched map.) Let ${(U_{α}, ψ_{α})}_{α \in A}$ be the set of all common subdevelopments, which exists by [L1] and is nonempty by [L3]. Put $U^{*} : = ⋃_{α \in A} U_{α}$ , an open subset of $M$ containing $j (Σ)$ ; define $ψ^{*} : U^{*} \to M^{'}$ by $ψ^{*} (p) : = ψ_{α} (p)$ for $p \in U_{α}$ . This is well defined by [L3]: if $p \in U_{α} \cap U_{β}$ , then $ψ_{α} (p) = ψ_{β} (p)$ .

2.1step 1.1L2L4L5L6algebra

( $U^{*}$ is a development with Cauchy surface $Σ$ .) Let $γ : I \to U^{*}$ be a future causal curve inextendible in $U^{*}$ , choose $t_{0} \in I$ with $γ (t_{0}) \in U_{α_{0}}$ , and let $J$ be the connected component of $γ^{- 1} [U_{α_{0}}]$ containing $t_{0}$ . By [L4] the restriction $γ ∣_{J}$ is inextendible in $U_{α_{0}}$ and therefore meets $Σ$ in exactly one parameter value of $J \subseteq I$ ; hence $γ$ meets $Σ$ at least once. If it met $Σ$ at two distinct parameter values, then by [L5] applied in $M$ this would contradict that $Σ$ is a Cauchy set of $M$ ; hence $γ$ meets $Σ$ in exactly one parameter value, and the same holds for past causal curves by the time reverse. Thus $(U^{*}, g ∣_{U^{*}}, N ∣_{U^{*}}, j, ψ ∣_{U^{*}})$ is a development of the same data: it is a time-oriented smooth Lorentzian four-manifold with $j (Σ)$ a spacelike embedded hypersurface (as $U^{*}$ is open in $M$ ), the field equations hold by [L6], the induced data are those of $j$ and $ψ$ , and $Σ$ is Cauchy by the above.

3.1step 2.1L2L6algebra

( $ψ^{*}$ is a data-fixing development embedding.) By [L2] each $ψ_{α}$ is an isometric immersion fixing the data and the time orientation; these are local properties, so $ψ^{*}$ is a smooth isometric immersion with $ψ^{*} \circ j = j^{'}$ and $g_{ψ^{*} (p)}^{'} (d ψ_{p}^{*} N_{p}, N_{ψ^{*} (p)}^{'}) < 0$ for every $p \in U^{*}$ , matching every matter field by [L2]. Hence $ψ^{*}$ is a data-fixing isometric immersion of the development $U^{*}$ into $M^{'}$ ; by [L6] it is injective and therefore a data-fixing development embedding. So $(U^{*}, ψ^{*})$ is a common subdevelopment, which is clause (a).

4.1step 1.1step 3.1algebra

(Universality.) Every $(U_{α}, ψ_{α})$ satisfies $U_{α} \subseteq U^{*}$ by definition of the union, and $ψ^{*} ∣_{U_{α}} = ψ_{α}$ by definition of $ψ^{*}$ ; in particular the pair $(U^{*}, ψ^{*})$ satisfies the universal property of clause (b), and $j (Σ) \subseteq U^{*}$ by step 1.1.

5.1step 4.1algebra∎

(Uniqueness.) Let $(U^{†}, ψ^{†})$ also satisfy (a) and (b). Since $(U^{†}, ψ^{†})$ is itself one of the common subdevelopments, (b) applied to $(U^{†}, ψ^{†})$ gives $U^{†} \subseteq U^{*}$ ; conversely (b) for $(U^{*}, ψ^{*})$ applied to $(U^{†}, ψ^{†})$ gives $U^{*} \subseteq U^{†}$ . Hence $U^{†} = U^{*}$ , and then for every $p \in U^{*}$ the two maps agree because $p$ lies in some common subdevelopment $(U_{α}, ψ_{α})$ and both $ψ^{†}$ and $ψ^{*}$ restrict to $ψ_{α}$ there. This is clause (c).

Remarks

The construction is Sbierski's; the only structural ingredients are the agreement of the maps on overlaps (Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps clause (c)), the fact that a maximal connected portion of an inextendible curve inside one member remains inextendible there (A maximal connected portion of an inextendible causal curve inside an open subset is inextendible there), and the acyclicity of "two crossings of a Cauchy set" (Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal clause (c)). No Zorn's lemma and no selection of representatives of isometry classes is used.

Existence and uniqueness of the maximal common development of two developments

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources