Charged-scalar compatibility, Maxwell constraints and Einstein gauge propagation

Differentiate $C_i$ in $s$ and group the compatibility equation from [L2]: ${\mathrm{\partial }}_s{C}_i=({\mathrm{\partial }}_s{\pi }_i-{\mathrm{\partial }}_i{\pi }_0)+i\lambda ({\mathrm{\partial }}_s{A}_i)\phi +i\lambda A_i{\mathrm{\partial }}_s\phi =(-i\lambda A_i{\pi }_0-i\lambda F_{0i}\phi )+i\lambda F_{0i}\phi +i\lambda A_i{\pi }_0=0$. Thus $C_i=0$ on the initial slice implies $C_i=0$ throughout the local time region, and the definition of ${\pi }_i$ in [L1] gives ${\pi }_i={\mathcal{D}}_i\phi$.

Let $R_{abi}:=q_{abi}-{\bar{\mathrm{\nabla }}}_i{g}_{ab}$. The first-order metric equations in [L6] include ${\bar{\mathrm{\nabla }}}_0{g}_{ab}=p_{ab}$ and ${\bar{\mathrm{\nabla }}}_0{q}_{abi}-{\bar{\mathrm{\nabla }}}_i{p}_{ab}={\mathcal{C}}_{abi}$, where ${\mathcal{C}}_{abi}=[{\bar{\mathrm{\nabla }}}_0,{\bar{\mathrm{\nabla }}}_i]g_{ab}$. Therefore ${\bar{\mathrm{\nabla }}}_0{R}_{abi}={\bar{\mathrm{\nabla }}}_i{p}_{ab}+{\mathcal{C}}_{abi}-{\bar{\mathrm{\nabla }}}_0{\bar{\mathrm{\nabla }}}_i{g}_{ab}={\bar{\mathrm{\nabla }}}_i{\bar{\mathrm{\nabla }}}_0{g}_{ab}+[{\bar{\mathrm{\nabla }}}_0,{\bar{\mathrm{\nabla }}}_i]g_{ab}-{\bar{\mathrm{\nabla }}}_0{\bar{\mathrm{\nabla }}}_i{g}_{ab}=0$. The initial metric jet gives $R_{abi}=0$, so the first-order spatial metric variables remain the actual derivatives of $g$.

First propagate the magnetic residual $U:=dF$, independently of the scalar equation and current. The normal evolution gives ${\iota }_{n}U=0$, so a three-form decomposition in an adapted frame gives $U=q{\iota }_n{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Its pullback to the initial slice is $(D_i{b}^i){\mathrm{v}\mathrm{o}\mathrm{l}}_h=0$, hence $q=0$ initially. Since $dU=d^{2}F=0$, the coordinate volume formula gives ${\mathrm{\nabla }}_a(q{n}^a)=n(q)+q{\mathrm{\nabla }}_a{n}^a=0$. On each normal integral curve the integrating factor $\exp (\int {\mathrm{\nabla }}_a{n}^a)$ makes $q$ constant times its initial value, so $q=0$ near the initial slice. Thus $dF=0$ before any current-conservation or Einstein inference. The normal is transverse to the time levels and its local flow exists with unique curves; at the stated finite Sobolev regularity it is at least $C^1$.

The full closedness now proved gives ${\mathrm{\partial }}_s{F}_{ij}-{\mathrm{\partial }}_i{F}_{0j}+{\mathrm{\partial }}_j{F}_{0i}=0$ in coordinates. Since ${\mathrm{\partial }}_s{A}_i=F_{0i}$, differentiating $B_{ij}=F_{ij}-({\mathrm{\partial }}_i{A}_j-{\mathrm{\partial }}_j{A}_i)$ gives ${\mathrm{\partial }}_s{B}_{ij}=0$. Its initial value is zero, so $B_{ij}=0$; also $A_0=0$ gives $F_{0i}={\mathrm{\partial }}_s{A}_i-{\mathrm{\partial }}_i{A}_0$. Consequently every component satisfies $F=dA$. Step 1.1 gives ${\pi }_i={\mathcal{D}}_i\phi$, and ${\mathrm{\partial }}_s\phi ={\pi }_0$ with $A_0=0$ gives ${\pi }_0={\mathcal{D}}_0\phi$, so $\pi =\mathcal{D}\phi$ in all slots. The reduced scalar divergence equation is therefore the actual scalar equation $E_{\phi }=0$ in [L3].

With $F=dA$, $\pi =\mathcal{D}\phi$, and $E_{\phi }=0$, the current and scalar-stress identities in [L3] now give ${\mathrm{\nabla }}_a{j}^a=0$ and ${\mathrm{\nabla }}_a(T^{\phi }{)}^a{}_b=F_{ba}{j}^a$. The form/source sign is fixed independently: put $v^b:={\mathrm{\nabla }}_a{F}^{ab}$. Metric compatibility preserves the oriented volume and Hodge star, while torsion-freeness identifies exterior differentiation with antisymmetrized covariant differentiation. In a positively oriented orthonormal frame, [L7] gives $(d{\star }_{g}F{)}_{123}=-v^0$ and $(d{\star }_{g}F{)}_{0ij}={ϵ}_{ijk}{v}^k$; the corresponding components of ${\iota }_v{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ are $v^0$ and $-{ϵ}_{ijk}{v}^k$. Hence $d{\star }_{g}F=-{\iota }_v{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ for any such $F$. In particular, $d{\star }_{g}F={\mu }_0{\iota }_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is equivalent to $v=-{\mu }_{0}j$; neither equation is assumed before the residual proof in the next step.

Put $W:=d{\star }_{g}F-{\mu }_0{\iota }_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Its normal contraction vanishes by the sourced evolution, so $W=r{\iota }_n{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Initially, the Hodge formula in [L7] gives $({\star }_{g}F){\mathrm{\mid }}_{\mathrm{\Sigma }}={\iota }_e{\mathrm{v}\mathrm{o}\mathrm{l}}_h$, while $({\iota }_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g){\mathrm{\mid }}_{\mathrm{\Sigma }}=j^0{\mathrm{v}\mathrm{o}\mathrm{l}}_h$. Thus $W{\mathrm{\mid }}_{\mathrm{\Sigma }}=(D_i{e}^i-{\mu }_0{j}^0){\mathrm{v}\mathrm{o}\mathrm{l}}_h=0$. By [L8] and step 3.1, $dW=-{\mu }_0({\mathrm{\nabla }}_a{j}^a){\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$. The same normal-volume computation as step 1.3 gives $n(r)+r{\mathrm{\nabla }}_a{n}^a=0$ with zero initial value, and the integrating factor proves $r=0$. Thus the full sourced Maxwell equation holds, with the divergence sign established in step 3.1. For the classical finite-Sobolev solutions, these identities also hold distributionally; the continuous residuals and their initial traces give the same zero solution in the $C^1$ normal flow coordinates.

Combining the on-shell scalar identity from step 3.1 with [L4] gives ${\mathrm{\nabla }}_a{T}^a{}_b=F_{ba}{j}^a-F_{ba}{j}^a=0$, equivalently ${\mathrm{\nabla }}^a{T}_{ab}=0$ because $T$ is symmetric. This conservation holds before the Einstein equation is recovered.

Let $E_{ab}:=G_{ab}+\mathrm{\Lambda }{g}_{ab}-\kappa T_{ab}$. By [L9]–[L10], $E_{ab}=\frac{1}{2}({\mathrm{\nabla }}_a{H}_b+{\mathrm{\nabla }}_b{H}_a-g_{ab}{\mathrm{\nabla }}_c{H}^c)$. Since $H{\mathrm{\mid }}_{\mathrm{\Sigma }}=0$, its tangential covariant derivatives vanish there. In an orthonormal frame $(n,e_i)$, ${\mathrm{\nabla }}_c{H}^c=-{\mathrm{\nabla }}_n{H}_n$ on $\mathrm{\Sigma }$, so $E(n,n)=\frac{1}{2}{\mathrm{\nabla }}_n{H}_n$ and $E(n,e_i)=\frac{1}{2}{\mathrm{\nabla }}_n{H}_i$. The normal Einstein projections in [L11] and the assumed initial constraints set both residual components to zero. Therefore $H$ and its normal derivative vanish on $\mathrm{\Sigma }$.

The identities in [L9]–[L11] remain classical at the stated regularity: their coordinate expressions depend on at most the third jet of $g$, the second jet of $H$ and the first jet of $T$. At any fixed point these jets are realized exactly by their finite Taylor polynomials in a smooth chart; the polynomial metric stays Lorentzian near that point by nondegeneracy. The unrestricted coordinate Bianchi identity and the divergence identity for the trace-adjusted symmetric derivative of an arbitrary covector therefore transfer pointwise from smooth fields with the same jets. Substituting the reduced residual and ${\mathrm{\nabla }}^a{T}_{ab}=0$ gives the same homogeneous subsidiary wave equation. This finite-jet argument transfers a local differential identity only; it neither replaces the solution by a smooth one nor asserts smooth persistence for nonsmooth data. Its curvature and lower-order coefficients are continuous and bounded on compact subcylinders. The inherited inequalities $g^{00}<0$ and $(g^{ij})>0$ hold in the reduced region. In a smaller spacelike coordinate cylinder, write its principal part as $g^{00}{\mathrm{\partial }}_s^2{H}_b+2g^{0i}{\mathrm{\partial }}_s{\mathrm{\partial }}_i{H}_b+g^{ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_j{H}_b$ and introduce $v_b={\mathrm{\partial }}_s{H}_b$ and $w_{bi}={\mathrm{\partial }}_i{H}_b$. The compatibility equations are ${\mathrm{\partial }}_s{w}_{bi}-{\mathrm{\partial }}_i{v}_b=0$; substituting them into the wave equation gives the symmetric matrices of a scalar wave reduction, with time matrix $\mathrm{diag}(-g^{00},(g^{ij}))$ repeated over $b$. Thus the first-order state $Y=(H,v,w)$ has a positive time matrix and symmetric spatial matrices, while connection and curvature terms are bounded zero-order coefficients on a compact cylinder. Choose a ball shrinking at speed $\sigma$ greater than ${\sup }_{\mathrm{\mid }\xi \mathrm{\mid }=1}\mathrm{\parallel }{\xi }_i{A}^i\mathrm{\parallel }\mathrm{/}{\lambda }_{\min }(A^0)$. For $E(s)={\int }_{B_{r-\sigma s}}{Y}^{\mathsf{T}}{A}^{0}Ydx$, multiplying the system by $2Y^{\mathsf{T}}$ and using symmetry gives a divergence identity whose interior terms are bounded by $CE(s)$; the side flux is nonnegative because $\sigma A^0+{\nu }_i{A}^i$ is positive definite. The finite-domain divergence theorem [L12] therefore gives $E^{\mathrm{\prime }}(s)\le CE(s)$. The initial data from step 6.1 give $Y=0$ and $E(0)=0$, so Gronwall [L12] yields $E(s)=0$; reverse time for $s<0$. Hence $H=0$ near every initial point, and the reduced equation has $E_{ab}=0$, which is the full Einstein equation. For the stated Sobolev regularity, [L13] supplies the classical derivatives in these identities.