Local Einstein-scalar evolution: smooth constrained data admit a local development, unique up to a data-fixing isometry

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). Fix $\mathrm{\Lambda }\in \mathbb{R}$, $\kappa >0$ and one scalar model $(\phi ,V,K_{\varphi })$ of The scalar-field matter model of the classical gravitational framework, with its fixed smooth potential and normalization. Let $\mathrm{\Sigma }$ be a smooth three-manifold with smooth Riemannian metric $h$, symmetric tensor $K_2$ and complete scalar data $({\phi }_0,{\pi }_0)$. Suppose the constraints of Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form hold for the scalar stress computed from these data, with ${\pi }_0$ the prescribed future-normal derivative. Then there is a smooth local Einstein-scalar development $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab},{\mathrm{\square }}_g\phi -V^{\mathrm{\prime }}(\phi )=0,$ with induced geometric data $(h,K_2)$, $\phi {\mathrm{\mid }}_{\mathrm{\Sigma }}={\phi }_0$ and $n(\phi ){\mathrm{\mid }}_{\mathrm{\Sigma }}={\pi }_0$. The neighborhood may have position-dependent time width. The initial surface is interior to its intrinsic domain of dependence; restricting to the interior Cauchy neighborhood gives a globally hyperbolic development. Every two developments of this same fixed model and complete data are related near the entire initial surface by a data-fixing time-orientation-preserving isometry matching $\phi$. No potential positivity, mass sign, global existence or uniform noncompact slab is asserted (Local geometric development for a fixed covariant Einstein-matter model with complete constrained data).

The complete scalar equation is used to obtain conservation. Conservation of the scalar stress alone is not substituted for that equation at critical scalar values.