ExampleConstruction: AI-generatedVerification: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Maximal common development and gluing of the flat vacuum data with its translate

Example

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 invoked general causal and maximal-development suppliers.

Let the flat vacuum data of The maximal development of the flat vacuum data is Minkowski spacetime be given ( $Σ = R^{3}$ , $h = δ$ , $K = 0$ , $Λ = 0$ , vacuum). Let $M : = (R^{1, 3}, η)$ with the embedding $j (x) : = (0, x)$ and let $M^{'} : = (R^{1, 3}, η)$ with the translated embedding $j^{'} (x) : = (1, x)$ ; both are developments of the same data (in $M^{'}$ the slice ${t = 1}$ has induced metric $δ$ and vanishing second fundamental form and is a Cauchy surface). Let $ψ : M \to M^{'}$ be the time translation $ψ (t, x) : = (t + 1, x)$ . Then:

(a) $(M, ψ)$ is a common subdevelopment of $M$ and $M^{'}$ , and it is the maximal common development;

(b) the glued quotient of Gluing two developments along their maximal common development: the quotient is a development containing both is isometric, in a data-fixing way, to Minkowski spacetime itself, the two embeddings being the identity and the inverse translation.

Facts & Assumptions

Given: Countable Choice and Dependent Choice; the flat vacuum data $Σ = R^{3}$ , $h = δ$ , $K = 0$ , $Λ = 0$ , vacuum; the developments $M = (R^{1, 3}, η)$ , $j (x) = (0, x)$ and $M^{'} = (R^{1, 3}, η)$ , $j^{'} (x) = (1, x)$ ; and the time translation $ψ (t, x) : = (t + 1, x)$ .

[L1]

Both $M$ and $M^{'}$ are developments of the data in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a) (The maximal development of the flat vacuum data is Minkowski spacetime); $ψ$ preserves $η$ , the time orientation and the matter fields, and $ψ \circ j = j^{'}$ .

[L2]

Common subdevelopments and the maximal common development: $(M, ψ)$ is a common subdevelopment of $M$ and $M^{'}$ (Common subdevelopments, maximal common developments and corresponding boundary points clause (a)), the maximal one exists and contains every common subdevelopment with the same map (Existence and uniqueness of the maximal common development of two developments clauses (a)-(b)), and the pair quotient of Gluing two developments along their maximal common development: the quotient is a development containing both is a development of the data in which both members embed by data-fixing development embeddings.

Verification

1.1givenL1algebra

( $ψ$ is a data-fixing isometry.) The translation $ψ$ is a diffeomorphism of $R^{1, 3}$ preserving $η$ , the time orientation and (vacuously) the matter fields, and $ψ \circ j = j^{'}$ by definition of $j, j^{'}$ . Hence $ψ : M \to M^{'}$ is a development embedding of the whole of $M$ , and $(M, ψ)$ is a common subdevelopment (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)).

2.1step 1.1L2algebra

(Maximality.) By Existence and uniqueness of the maximal common development of two developments the maximal common development $(U^{*}, ψ^{*})$ contains every common subdevelopment, so $U^{*} = M$ and $ψ^{*} = ψ$ by step 1.1 applied to $(U^{*}, ψ^{*})$ in the reverse direction. In particular $U$ has no boundary in $M$ , hence no corresponding boundary points, in accordance with A maximal common development has no corresponding boundary points: the spacelike restart produces a strictly larger common development. This is (a).

3.1step 2.1L2algebra∎

(The quotient.) By Gluing two developments along their maximal common development: the quotient is a development containing both the quotient $Q : = (M ⊔ M^{'}) / \sim$ with the identifications along $ψ$ is a development of the data and the maps $p_{M}, p_{M^{'}}$ are data-fixing development embeddings; here every point of $M$ is identified with its image in $M^{'}$ and conversely, so the canonical map $M \to Q$ , $p \mapsto [p]$ , is a bijection and an isometry: its inverse is the map sending the class of $m \in M$ to $m$ and the class of $m^{'} \in M^{'}$ to $ψ^{- 1} (m^{'})$ , which is well defined because $m^{'} \sim ψ^{- 1} (m^{'})$ and because $ψ$ is a bijection. Both compositions with $η$ agree, so $Q$ is isometric to $M$ by a data-fixing isometry; the two development embeddings are the identity and the inverse translation. This is (b).

Remarks

The example shows that the gluing construction is non-vacuous already when two developments are diffeomorphic but their initial embeddings differ: the identifications are dictated by the maximal common development map, not by an arbitrary isometry of the ambient manifolds. It also shows that a development embedding of one maximal development into another need not have proper image (here it is onto), so the "extension" terminology carries no properness.

Depends on

Dependency tree · 2 levels

Sources

J. Sbierski, On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification, arXiv:1309.7591v3 (2015): Theorem 2.7, whose simplest non-identical instance is the example discussed here, printed pp. 20-22