Local Cauchy development of the adopted Einstein field law for smooth constrained data in the selected matter sectors

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).

Adopt the direct-equation presentation of the classical gravitational framework for a model satisfying the spacetime and classical-domain postulate Adopted spacetime model and classical domain of the gravitational framework and the field-and-matter postulate Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, with the field law Adopted Einstein field equation in the direct-equation presentation: on the matter domain of the model, $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab},\kappa =\frac{8\pi G_N}{c^4},$ with $T$ the stress-energy tensor of the model's selected matter sector. Let the sector be one of:

1. vacuum with cosmological constant ($T=0$);
2. the scalar-field model $(\phi ,V,K_{\varphi })$ of The scalar-field matter model of the classical gravitational framework with a chosen smooth potential whose scalar equation and stress tensor are those defined there, writing $K_{\varphi }$ for that definition's scalar normalization constant;
3. the perfect-fluid model of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy with a smooth barotropic equation of state satisfying $ϵ+p>0$ and $0<p^{\mathrm{\prime }}(ϵ)<1$ on the range of the data;
4. an energy-coefficient dust density $\epsilon \ge 0$ and prescribed smooth future unit velocity, with $T=\epsilon u^{\mathrm{♭}}\otimes u^{\mathrm{♭}}$. On positive-density regions this is The dust matter model: rest density, four-velocity, stress-energy and dust system; on zero-density regions the velocity is separately prescribed and the geodesic equation remains imposed. Equivalently the mass-normalized density is $\rho =\epsilon \mathrm{/}{c}^2$.

Use the SI and coordinate conventions of SI unit and coordinate conventions of the classical gravitational framework: in a length-coordinate chart $x^0=ct$, the metric is dimensionless, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[\mathrm{\Lambda }]={\mathrm{m}}^{-2}$, $[G_N]={\mathrm{m}}^3{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and $[\kappa ]={\mathrm{m}}^{-1}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{s}}^2$, so $\kappa T$ has curvature units. Every velocity denoted $u$ in this theorem is normalized by $g(u,u)=-1$. Its SI proper-time four-velocity is $U_{\mathrm{S}\mathrm{I}}=cu$, with $g(U_{\mathrm{S}\mathrm{I}},U_{\mathrm{S}\mathrm{I}})=-c^2$; the initial $U_0$ in the mathematical bundle formulation is normalized, and its SI counterpart is $c{U}_0$. The mass-normalized density is $\rho ={\rho }_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}:=\epsilon \mathrm{/}{c}^2$ with units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$. The dust source uses $\epsilon ={\rho }_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}{c}^2$ and $\epsilon u^{\mathrm{♭}}\otimes u^{\mathrm{♭}}={\rho }_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}{U}_{\mathrm{S}\mathrm{I}}^{\mathrm{♭}}\otimes U_{\mathrm{S}\mathrm{I}}^{\mathrm{♭}}$; fluid $ϵ$ and $p$ have units $\mathrm{J}{\mathrm{m}}^{-3}$ and $p^{\mathrm{\prime }}$ is dimensionless. For the scalar branch choose an arbitrary field unit $\mathrm{\Phi }$ and use $[\phi ]=\mathrm{\Phi }$, $[V]={\mathrm{\Phi }}^2{\mathrm{m}}^{-2}$, and $[K_{\varphi }]=\mathrm{J}{\mathrm{m}}^{-1}{\mathrm{\Phi }}^{-2}$; then ${\mathrm{\square }}_g\phi$ and $V^{\mathrm{\prime }}(\phi )$ have units $\mathrm{\Phi }{\mathrm{m}}^{-2}$ and $T_{ab}$ has units $\mathrm{J}{\mathrm{m}}^{-3}$.

Let $(\mathrm{\Sigma },h,K)$ with the sector's smooth initial data (respectively nothing, $({\phi }_0,{\pi }_0)$, $({ϵ}_0,v_0)$, $({\epsilon }_0,U_0)$) satisfy the constraint equations Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form for the sector's stress tensor. Here fluid $v_0\in \mathrm{\Gamma }(T\mathrm{\Sigma })$ obeys $h(v_0,v_0)<1$, with full initial unit velocity $U_0=(1-h(v_0,v_0){)}^{-1\mathrm{/}2}(N+v_0)$ in $\mathbb{R}N\oplus T\mathrm{\Sigma }$, exactly as in High-regularity local developments for the selected matter sectors: vacuum with Lambda, scalar, strict barotropic fluid and geodesic dust. Then the model admits a local Cauchy development: there are a smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of the framework and a spacelike embedded image of $\mathrm{\Sigma }$ inducing $(h,K)$ and the stated matter data, solving the field law $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$ and the sector's matter equations on a neighborhood of $\mathrm{\Sigma }$; $\mathrm{\Sigma }$ lies in the interior of its domain of dependence; and any two such developments of the same data agree near $\mathrm{\Sigma }$ up to an isometry fixing the data and the time orientation. The development is local in time and no uniform-in-the-data existence time or global statement is asserted.

• Model versus nature. This theorem is a model deduction: it establishes that the adopted field law and matter definitions, plus the stated mathematical hypotheses, imply the existence of local solutions. It does not establish that the model describes nature, and no evidence from observation is used.
• Scope. The limitations are collected in Scope and status of the Gr14 high-regularity symmetric hyperbolic material.