Local geometric development for a fixed covariant Einstein-matter model with complete constrained data

(Complete harmonic initial value.) On a small initial coordinate ball set $g_{00}=-1$, $g_{0i}=0$, $g_{ij}=h_{ij}$, $p_{ij}=2K_{ij}$ and $q_{abi}={\mathrm{\partial }}_i{g}_{ab}^0$. For the flat comparison metric the Christoffel contraction at the surface is $H^0=\frac{1}{2}{p}_{00}+{\mathrm{tr}}_{h}K,H^{\mathrm{\ell }}=-h^{\mathrm{\ell }i}{p}_{0i}+h^{jk}\mathrm{\Gamma }(h{)}^{\mathrm{\ell }}{}_{jk}\mathrm{.}$ Thus prescribe $p_{00}=-2{\mathrm{tr}}_{h}K$ and $p_{0i}=h_{i\mathrm{\ell }}{h}^{jk}\mathrm{\Gamma }(h{)}^{\mathrm{\ell }}{}_{jk}$, giving $H=0$ initially. The spatial derivative definitions and the initialization $w=\mathcal{I}(h,K,d)$ supply every remaining initial value and every auxiliary compatibility constraint. In this coordinate gauge ${\mathrm{\partial }}_0=n$ initially, so $K_{ij}=\frac{1}{2}{p}_{ij}$ and the reconstructed matter data are exactly $d$. The generally nonzero $p_{0i}$ is essential for the flat comparison metric; the special background $-d{t}^2+h$ instead gives $p_{0i}=0$.

(Local coupled reduced existence.) Shrink about each initial point so the initial state values lie in a compact ball strictly inside a convex admissible state ball $O_{\ast }⋐O$. Extend the smooth data to a torus by a spatial cutoff interpolating to a fixed value in $O_{\ast }$; convexity keeps all interpolated values admissible. On a smaller initial cylinder extend the coefficient functions by a smooth spatial cutoff, keeping the original functions on the smaller patch and interpolating $A^0$ to the identity, $A^i$ to zero and $B$ to zero outside the larger patch. Symmetry persists and the convex positive-matrix cone preserves positivity. Choose compact sets $C\subset \mathrm{int}{C}^{\mathrm{\prime }}⋐O_{\ast }$ containing the extended data and a finite $H^s$ norm bound $R$ for one integer $s\ge 5$. On the compact time, spatial and state ranges the coefficient derivatives are bounded and $A^0\ge aI$ for some $a>0$. These are the actual hypotheses of [L1]; the periodic existence and smooth persistence theorems give a smooth solution on a nonzero interval. Restrict to the original patch, where it solves the original coupled system. No theorem for a prescribed spacetime tensor $T$ has been substituted for this coupled existence argument.

(Removal of extensions.) For two smooth solutions of systems agreeing near a smaller initial cone, subtract their equations and put $v=z_1-z_2$. On a smaller compact cylinder their states lie in a common convex admissible compact neighborhood. The coefficient mean-value formula gives $A^{\mu }(x,z_1){\mathrm{\partial }}_{\mu }v+C(x)v=0,$ where $C$ includes the bounded terms obtained from coefficient differences multiplying ${\mathrm{\partial }}_{\mu }{z}_2$ and from the source difference. Let $b={\sup }_{\mathrm{\mid }\xi \mathrm{\mid }=1}\mathrm{\parallel }{\sum }_i{\xi }_i{A}^i(x,z_1)\mathrm{\parallel }$ there and choose $\lambda >b\mathrm{/}a$. On the shrinking ball of radius $r-\lambda t$, [L5] gives $\frac{d}{dt}{\int }_{B_{r-\lambda t}}{v}^{\mathsf{T}}{A}^{0}v\le c{\int }_{B_{r-\lambda t}}{v}^{\mathsf{T}}{A}^{0}v-{\int }_{\mathrm{\partial }{B}_{r-\lambda t}}{v}^{\mathsf{T}}(\lambda A^0+{\nu }_i{A}^i)v\le c{\int }_{B_{r-\lambda t}}{v}^{\mathsf{T}}{A}^{0}v\mathrm{.}$ The boundary matrix is positive since $\lambda a>b$. Equal initial values give zero energy by Gronwall. Reverse time for the other half. This proves independence of the extensions on an actual open initial neighborhood, with the combined spatial-symbol bound rather than an incorrect bound on each separate $A^i$.

(Recovery without circularity.) By hypothesis 2, the auxiliary constraints propagate and the reconstructed reduced solution solves the original matter equations, giving the exact initial stress and data. Hypothesis 3 therefore gives ${\mathrm{\nabla }}^a{T}_{ab}=0$ on this reduced solution before Einstein recovery. Its initial geometric jet gives $H=0$ by step 1.1 and the Gauss-Codazzi constraint equations give $E(n,n)=E(n,e_i)=0$. On a reduced solution one has $E_{ab}=\frac{1}{2}({\mathrm{\nabla }}_a{H}_b+{\mathrm{\nabla }}_b{H}_a-g_{ab}{\mathrm{\nabla }}_c{H}^c)$; since tangential derivatives of $H$ vanish initially, this reads $E_{00}=\frac{1}{2}{\mathrm{\nabla }}_0{H}_0$ and $E_{0i}=\frac{1}{2}{\mathrm{\nabla }}_0{H}_i$ in the initial orthogonal frame. Thus ${\mathrm{\nabla }}_{n}H=0$ initially. Conservation gives the homogeneous subsidiary equation and [L2] forces $H=0$, hence $E=0$, on a smaller neighborhood. The local solution is now a full physical development.

(Common normal collar representatives.) For each local full development define $E_i(s,p)={\exp }_{j_i(p)}(s{n}_i(p))$. Its differential at $(0,p)$ is $(c,V)\mapsto c{n}_i+d{j}_i(V)$, an isomorphism. The inverse function theorem [L3] gives, after local shrinking, an injective normal collar over a smaller initial ball. Pull its metric and matter fields back to its normal domain in $\mathbb{R}\times \mathrm{\Sigma }$. If two initial balls intersect, [L3] gives a future data-fixing isometry germ on the intersection. Its differential is the identity on the slice tangent directions and sends $n_i$ to $n_j$ by uniqueness of the future unit normal. Preservation and uniqueness of affinely parametrized geodesics give $\mathrm{\Phi }(E_i(s,p))=E_j(s,p)$ for every sufficiently small segment staying in its domain. Consequently the two pulled-back full fields agree on an open neighborhood of the zero section over the intersection. This uses uniqueness of isometry germs and normal parametrization; fixing the slice alone does not make a general map the identity off it. No geometric uniqueness is used to create the reduced solution or to propagate its constraints.

(Locally finite literal patching.) Choose [L4]'s countable locally finite shrinking $\overline{W_i}=K_i\subset V_i\subset \overline{V_i}\subset O_i$, with $O_i$ normal-collar existence patches from step 4.1 and $K_i$ compact. Each $K_i$ meets only finitely many $K_j$, since $(V_j)$ is locally finite and $K_j\subset V_j$. Compactness supplies $t_i>0$ for which the $i$th normal representative exists on $(-t_i,t_i)\times K_i$. On each nonempty compact $K_i\cap K_j$, step 4.1 supplies an open equality neighborhood; a finite subcover gives a symmetric width $t_{ij}=t_{ji}>0$ on which the representatives agree over that compact intersection. Countable Choice selects widths for the countably many indices and pairs. Set ${\delta }_i=\frac{1}{2}\min (\{t_i\}\cup \{t_{ij}:K_i\cap K_j\mathrm{\ne }\mathrm{\varnothing }\})>0.$ The minimum is finite. The normal representatives agree on every overlap of $(-{\delta }_i,{\delta }_i)\times W_i$ and so define single smooth tensor fields on their literal open union $\mathrm{\Omega }\subset \mathbb{R}\times \mathrm{\Sigma }$. Ordinary equality supplies the triple-overlap cocycle, with no quotient construction or Hausdorffness issue. For a subordinate locally finite partition $({\rho }_i)$ set $\alpha (p)=\frac{1}{2}{\sum }_i{\delta }_i{\rho }_i(p)>0$. At each point the finite weighted average is at most the largest active width, whose $W_i$ contains that point; hence $\{\mathrm{\mid }s\mathrm{\mid }<\alpha (p)\}\subset \mathrm{\Omega }$. Restriction gives a smooth positive variable-width collar and the exact full data. No positive uniform lower bound is required.

(Causal neighborhood and uniqueness.) Apply [L5]'s generic embedded-spacelike clause to the surface $\{0\}\times \mathrm{\Sigma }$ in this glued development. It furnishes a further temporal variable-width collar $U$ and an open set $W={\mathrm{int}}_U(D_U^{+}(\mathrm{\Sigma })\cup D_U^{-}(\mathrm{\Sigma }))$ containing the entire initial surface, where $\mathrm{\Sigma }$ is Cauchy and $W$ is globally hyperbolic. The domains are intrinsic to $U$; no assertion that the one-sided domain itself is open is needed. Finally the fixed model, same reconstruction and complete data meet all hypotheses of [L3]'s uniqueness theorem. Two full developments are therefore isometric near the whole initial surface, matching all matter fields and future orientation. These are the asserted existence, causal and uniqueness conclusions.