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A maximal common development has no corresponding boundary points: the spacelike restart produces a strictly larger common development

Statement

Assume the Axioms of 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, let $(U, φ)$ be a common subdevelopment (Common subdevelopments, maximal common developments and corresponding boundary points clause (a)), write $Σ : = j (Σ)$ , and let $C \subseteq \partial U$ be the set of boundary points with a corresponding boundary point. Assume that $(U, φ)$ is maximal among the common subdevelopments of $M$ and $M^{'}$ in the following sense: for every common subdevelopment $(V, χ)$ of $M$ and $M^{'}$ one has $V \subseteq U$ and $χ = φ$ on $V$ (this is the maximality property of the maximal common development, Existence and uniqueness of the maximal common development of two developments clause (b)). Then $C = \emptyset$ .

Facts & Assumptions

Given: Countable Choice, Dependent Choice, the two developments, the maximal common subdevelopment $(U, φ)$ , and the corresponding set $C$ .

[L1]

Future and past restart geometry in the original orientation. If $C \cap J^{+} (Σ) \neq \emptyset$ , An exposed boundary point: the last boundary point of a boundary null geodesic has no other boundary point in its past supplies an exposed future point, and The spacelike restart surface at an exposed boundary point: maximizers on the boundary, the level hypersurface and its radial tube supplies a normal neighborhood, precompact diamond $W$ , a whole spacelike surface $S \subseteq U \cup C$ meeting $\partial U$ , and radial tubes with clauses (c)1-4. If $C \cap J^{-} (Σ) \neq \emptyset$ , Past corresponding-boundary exposure and a spacelike restart in the original time orientation supplies the corresponding past exposure and complement localization in clause (a), the whole surface in (b), and the same thin-tube, original-Cauchy and overlap conclusions in (c)1-4. Its radial time increases toward the original future, whereas the future-side construction uses a past radial flow. Both constructions keep the original $N, N'$ , metric, initial normal convention and matter fields fixed.

[L2]

Data transfer: $Φ$ is defined and smooth on a neighbourhood of $U \cup C$ , and for the spacelike hypersurface $S \subseteq U \cup C$ the restriction $Φ ∣_{S}$ is an embedding onto its image identifying the induced metric, the second fundamental form and the matter fields with their covariant derivatives of every finite order; in particular $Φ ∣_{S}$ identifies the complete Cauchy data $({\overset{ˉ}{g}}_{S}, k_{S}, ψ_{S})$ of $S$ realized in $M$ with the corresponding data of $Φ (S)$ in $M^{'}$ (The set of boundary points with a corresponding point is open, and the extension preserves the metric, matter fields and connection on the common domain and its corresponding boundary clause (e)).

[L3]

Local complete-data geometric uniqueness is required for the fixed admissible coupled model at the actual restart data, as specified in Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality. Use Local geometric development for a fixed covariant Einstein-matter model with complete constrained data when its full initialization, hyperbolicity, compatibility recovery and stress-conservation hypotheses hold on this spacelike surface. Scalar and strict-fluid instances are 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. Geodesic nonnegative tilted dust instead uses Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry with its independent common-flow/frame reduction, including its separately prescribed velocity on zero density. Complete data, including all required matter jets, agree by [L2]; the comparison isometry matches every matter field. An arbitrary conserved stress is not a fixed-model uniqueness premise.

[L4]

The restart supplier makes $S$ a smooth embedded spacelike hypersurface without boundary. The intrinsic neighborhood theorem Local Cauchy neighborhoods in a conically bounded product chart clause (d), applied separately to the embeddings $S ↪ M$ and $Φ (S) ↪ M^{'}$ , supplies open neighborhoods on which the whole respective surface is Cauchy and which are globally hyperbolic. With the restricted solution fields these are developments of the complete data on $S$ . Neither surface needs to be globally acausal in the original ambient manifold; the neighborhoods are intrinsic.

[L5]

Embedding theorem: a data-fixing isometric immersion of one development into another is injective and hence a development embedding; its image is open and it is a diffeomorphism onto it (A data-fixing isometric immersion of developments is injective, hence a development embedding; uniqueness of data-fixing embeddings, Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps clause (a)).

[L6]

Every causal curve inextendible in $U \cup N$ meets the original $Σ$ once for a tube of The spacelike restart surface at an exposed boundary point: maximizers on the boundary, the level hypersurface and its radial tube clause (c), or of Past corresponding-boundary exposure and a spacelike restart in the original time orientation clause (c) in the past-side case. In either case clause (c)4 supplies a smooth quantitatively thin tube inside any specified open neighborhood of the whole $S$ and below any specified positive continuous ceiling, with the explicit derivative bound required for conclusions (c)1-3. Thus $U \cup N$ is a development by the local field equations and that Cauchy conclusion.

No choice principle beyond the two declared axioms is used in this item.

Proof

technique · direct; by contradiction

1.1givenL1algebra

(Both boundary sides in the fixed model.) Suppose $C \neq \emptyset$ . Since $Σ \subset U$ and the original Cauchy side decomposition exhausts $M$ , every point of $C$ lies strictly to the original future or past of $Σ$ . If $C \cap J^{+} (Σ) \neq \emptyset$ , take the future exposure and restart configuration of [L1]. Otherwise $C \cap J^{-} (Σ) \neq \emptyset$ , and take the complete past-side exposure, complement localization, spacelike surface and quantitatively thin tubes supplied by [L1]. In both cases $S \subseteq U \cup C$ , $S \cap \partial U \neq \emptyset$ , and the tube conclusions preserve the original $Σ$ Cauchy property and original-oriented overlap comparison. No development, normal or matter datum is time-reversed.

2.1step 1.1L2L3L4algebra

(Local uniqueness on the whole restart surface.) The complete data of $S$ realized in $M$ and in $M^{'}$ via $Φ ∣_{S}$ agree by [L2]. Apply [L4] to obtain intrinsic Cauchy neighborhoods of each surface, carrying the restricted metric and full matter solution. The applicable fixed-model theorem [L3] gives an open neighborhood $O$ of the whole $S$ in the source and a future isometry $Θ : O \to M^{'}$ matching all matter fields, with $Θ ∣_{S} = Φ ∣_{S}$ . Replace $O$ by $O \cap N$ . This construction uses the actual branch uniqueness theorem and its admissibility at the realized restart data.

3.1step 2.1L1L6construct

(A smooth admissible thinner tube.) Apply the applicable restart clause (c)4 with the prescribed open neighborhood $O \subseteq N \subseteq W$ of $S$ and width ceiling $δ / 2$ . It gives a smooth positive width $δ_{0} \leq δ / 2$ with $∣ d δ_{0} ∣_{q_{0}} \leq 1 / (8 \sqrt{2})$ and $N_{0} = F ({(s, t) : ∣ t ∣ < δ_{0} (s)}) \subseteq O$ . The comparison metric bounds and the signed-face endpoint argument therefore retain restart conclusions (c)1-3. No scaling of an arbitrary continuous width is used.

4.1step 2.1step 3.1L1L2L5algebra

(Agreement on the actual overlap.) The future local isometry $Θ$ is defined on $N_{0}$ , equals $Φ$ on $S$ , and matches the matter fields there and on $N_{0}$ by step 2.1. The applicable restart clause (c)3 now gives $Θ = φ$ on $N_{0} \cap U$ : the punctured radial segment lies in $U$ , and its limiting value and velocity at its possibly boundary base agree through $Φ$ . Connection transport and uniqueness of the two image geodesics prove that comparison, without requiring the base to lie in $U$ .

5.1step 4.1algebra

(The patched map.) Define $Ψ : U \cup N_{0} \to M^{'}$ by $Ψ : = φ$ on $U$ and $Ψ : = Θ$ on $N_{0}$ . This is well defined by step 4.1 and smooth and is a data-fixing isometric immersion preserving the time orientation and the matter fields, because these are local properties and hold for $φ$ on $U$ and for $Θ$ on $N_{0}$ (Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clauses (b)-(c), Common subdevelopments, maximal common developments and corresponding boundary points clause (a)).

6.1step 5.1L6algebra

( $U \cup N_{0}$ is a development.) By [L6] every causal curve in $U \cup N_{0}$ that is inextendible in $U \cup N_{0}$ meets $Σ$ in exactly one parameter value; $U \cup N_{0} \subseteq M$ is open with the restricted metric, time orientation and matter fields, the field equations hold (they are local and hold on $M$ ), and $j : Σ \to U \cup N_{0}$ is an embedding with the induced data $(h, K, ψ_{0})$ . Hence $U \cup N_{0}$ is a development of the same data (Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a)); in particular it is globally hyperbolic with Cauchy surface $Σ$ by The existence of a Cauchy set implies strong causality and compact causal diamonds.

7.1step 6.1L5algebra

( $Ψ$ is a development embedding.) By step 5.1, $Ψ$ is a data-fixing isometric immersion of the development $U \cup N_{0}$ into $M^{'}$ ; hence it is injective and a diffeomorphism onto the open set $Ψ (U \cup N_{0})$ by [L5].

8.1step 7.1algebracontradiction

(Strictly larger.) Pick $x \in S \cap \partial U$ (nonempty by the applicable surface clause (b) in [L1]). Then $x \in S \subseteq N_{0}$ and $x \notin U$ , so $N_{0} \subseteq̸ U$ and $U ⊊ U \cup N_{0}$ . Thus $(U \cup N_{0}, Ψ)$ is a common subdevelopment of $M$ and $M^{'}$ strictly larger than $(U, φ)$ , contradicting the maximality of $(U, φ)$ assumed in the statement.

9.1step 1.1step 8.1discharge-contradiction∎

(Conclusion.) Therefore the assumption $C \neq \emptyset$ is false: $C = \emptyset$ .

Remarks

This is Theorems 3.6 and 3.13 of Sbierski's paper. The two corrections recorded in The set of boundary points with a corresponding point is open, and the extension preserves the metric, matter fields and connection on the common domain and its corresponding boundary (no isometry claim for the normal-coordinate formula outside $\overline{U}$ , and the connection compatibility replacing it) enter through the data transfer [L2] and through the geodesy of the extension used in An exposed boundary point: the last boundary point of a boundary null geodesic has no other boundary point in its past. The patching step 4.1 is the place where the source writes "the same continuity argument as in the proof of Lemma 3.1, applied to $N \cap U$ ": it is made precise here through the limiting first jet at the base point $s \in S$ of the radial segment, using clause (c)3 of The spacelike restart surface at an exposed boundary point: maximizers on the boundary, the level hypersurface and its radial tube instead of a density argument. Step 1.1 treats the two sides separately. The full past-side geometric construction is Past corresponding-boundary exposure and a spacelike restart in the original time orientation, with future radial flow and its signed future-endpoint argument. The local uniqueness invocation in step 2.1 uses the actual complete data on $S$ with the original future unit normal in both branches. Neither $N$ , the initial $K$ , nor any scalar, fluid, dust or additional model field is reversed; no invariance of a fixed model under time reversal is assumed.

A maximal common development has no corresponding boundary points: the spacelike restart produces a strictly larger common development

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources