Conditional model deduction: the adopted field law and matter model yield a maximal globally hyperbolic development of constraint-satisfying initial data

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the mathematical suppliers.

Adopt the postulates Adopted Einstein field equation in the direct-equation presentation (the Einstein field law $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$ for a Lorentzian four-metric $g$ with cosmological constant $\mathrm{\Lambda }$ and coupling $\kappa$), Adopted spacetime model and classical domain of the gravitational framework (the classical smooth Lorentzian domain, time orientation and the identification of $g$ as the gravitational field), and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions (the selected matter sectors, their equations of motion and stress-energy tensors, and the requirement that the coupled system be treated within the stated regularity).

Adopt the specified fixed matter equations and stress expression as model premises as well; for any additional model beyond the concrete sectors of the cited postulate, these are additional stipulated premises. Mathematical admissibility does not itself assert a physical matter law.

Let $\mathrm{\Sigma }$ be a connected smooth three-manifold, let $h$ be a smooth Riemannian metric and $K$ a smooth symmetric $2$-tensor on $\mathrm{\Sigma }$, and let complete smooth initial data for one fixed admissible coupled model of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality be given, satisfying the constraints of Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form. Assume its actual local existence and uniqueness/restart hypotheses, including complete matter initialization and recovery. The concrete branches of High-regularity local developments for the selected matter sectors: vacuum with Lambda, scalar, strict barotropic fluid and geodesic dust include vacuum, scalar, strict fluid and nonnegative tilted geodesic dust with velocity through zero density; any further model must supply the same fixed-model obligations. Then, as a conditional deduction from the adopted model:

(a) there exists a development $(M_{\mathrm{m}\mathrm{a}\mathrm{x}},g_{\mathrm{m}\mathrm{a}\mathrm{x}},N_{\mathrm{m}\mathrm{a}\mathrm{x}},j_{\mathrm{m}\mathrm{a}\mathrm{x}},{\psi }_{\mathrm{m}\mathrm{a}\mathrm{x}})$ of these data in which $g_{\mathrm{m}\mathrm{a}\mathrm{x}}$ solves the adopted field law with the adopted stress-energy tensor and the matter fields solve the adopted matter equations, and which is maximal among developments of the same data in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (d);

(b) every development of the same data admits a data-fixing development embedding into $M_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and any two maximal developments of the same data are related by exactly one data-fixing isometry; thus within the model the maximal development is determined by the initial data up to isometry fixing the initial surface;

(c) $M_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is globally hyperbolic with Cauchy surface $j_{\mathrm{m}\mathrm{a}\mathrm{x}}(\mathrm{\Sigma })$, so the deduction yields a deterministic initial-value formulation inside the model for those complete constrained data that also satisfy the stated admissibility and local/restart hypotheses.

The conclusion is a statement about the adopted model: it asserts that the postulated equations, together with the mathematical hypotheses on the data and the regularity requirements of the sector, imply the existence and uniqueness of the maximal development as a conditional consequence. It makes no assertion that the model or its data describe any observed system, and carries no empirical content beyond the model premises.

This item records the model-level content of the mathematical theorem Existence and uniqueness of the smooth maximal globally hyperbolic development of constrained initial data in the selected matter sectors: postulating the field law and matter models, constraint-satisfying smooth data determine a maximal globally hyperbolic development uniquely up to isometry fixing the data. The physical interpretation of $(M_{\mathrm{m}\mathrm{a}\mathrm{x}},g_{\mathrm{m}\mathrm{a}\mathrm{x}})$, the status of the idealisation of smoothness and of the specific matter sectors, the absence of any claim of global existence, completeness or genericity, and the boundary between model deduction and observation are recorded in Scope and empirical qualifications of the maximal globally hyperbolic development: model content, idealisations, and no observational claims. The mathematical supplier is a theorem of draft status in this library; if its hypotheses were to fail for data of interest, this conditional deduction would fail with them.