High-regularity local developments for the selected matter sectors: vacuum with Lambda, scalar, strict barotropic fluid and geodesic dust

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

Let $\mathrm{\Lambda }\in \mathbb{R}$, $\kappa >0$ and $c>0$ be constants, let $\mathrm{\Sigma }$ be a smooth three-manifold with smooth Riemannian metric $h$ and symmetric $2$-tensor $K$, and suppose smooth matter initial data and a matter sector are given in one of the following four branches, with the constraint equations of Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form satisfied:

1. vacuum with cosmological constant: no matter fields; the constraints read $S_h+({\mathrm{tr}}_{h}K{)}^2-\mathrm{\mid }K{\mathrm{\mid }}_h^2=2\mathrm{\Lambda }$ and ${\mathrm{div}}_{h}K-d{\mathrm{tr}}_{h}K=0$;
2. scalar field: a model $(\phi ,V,K_{\varphi })$ as in The scalar-field matter model of the classical gravitational framework with smooth potential and initial data $({\phi }_0,{\pi }_0)$;
3. strict barotropic perfect fluid: a model with smooth $p=p(ϵ)$ and $ϵ+p>0$, $0<p^{\mathrm{\prime }}(ϵ)<1$, with initial data $({ϵ}_0,v_0)$ with $h(v_0,v_0)<1$, equivalently the abstract bundle velocity $U_0=(1-h(v_0,v_0){)}^{-1\mathrm{/}2}(N+v_0)$, as specified in Local evolution for a strict barotropic perfect fluid: smooth constrained data admit a development, unique up to a data-fixing isometry;
4. geodesic dust: an energy-coefficient density ${\epsilon }_0\ge 0$ with prescribed smooth future unit velocity $U_0$, as a section of $\mathbb{R}N\oplus T\mathrm{\Sigma }$ with metric $-ab+h(X,Y)$. On positive-density regions this is the model $(\epsilon ,u)$ of The dust matter model: rest density, four-velocity, stress-energy and dust system; through zero-density regions its smooth unit velocity is separately prescribed and required to remain geodesic. Its source is $T=\epsilon u^{\mathrm{♭}}\otimes u^{\mathrm{♭}}$.

Then in each branch there is a smooth local development with the properties established in the corresponding item, Local Einstein-scalar evolution: smooth constrained data admit a local development, unique up to a data-fixing isometry (branch 2), Local evolution for a strict barotropic perfect fluid: smooth constrained data admit a development, unique up to a data-fixing isometry (branch 3), Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry (branch 4) and, for branch 1, the vacuum instance of Local geometric development for a fixed covariant Einstein-matter model with complete constrained data: a smooth Lorentzian metric $g$ with $\mathrm{\Sigma }$ spacelike of induced data $(h,K)$, solving $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$ with the branch's stress tensor and matter equations, with $\mathrm{\Sigma }$ in the interior of its domain of dependence, and with uniqueness near $\mathrm{\Sigma }$ up to a data-fixing time-orientation-preserving isometry.

• Role. This is the mathematical form of the Gr14 headline claim: the full high-regularity symmetric hyperbolic reduction yields local Einstein-matter developments in the selected branches. The physical reading and the scope exclusions are recorded in Local Cauchy development of the adopted Einstein field law for smooth constrained data in the selected matter sectors and Scope and status of the Gr14 high-regularity symmetric hyperbolic material.