Einstein harmonic gauge propagation with Maxwell on shell

By [L1], the reduced Maxwell evolution and its initial divergence constraints give $dF=d{\star }_{g}F=0$. The coordinate/Hodge identity in [L2] therefore supplies the divergence hypothesis for the stress lemma with $j=0$, giving ${\mathrm{\nabla }}^a{T}_{ab}=0$ before using the unreduced Einstein equations. The Bianchi identity and the divergence identity for a trace-adjusted symmetric covector derivative are local identities in the third jet of $g$ and second jet of $H$. At each point these jets are exactly realized by smooth Taylor polynomials in a chart, whose metric remains Lorentzian nearby. Hence the unrestricted coordinate identities underlying [F1] transfer at $g\in C^3$, $H\in C^2$; substituting the actual reduced equation and conserved stress gives its homogeneous wave equation. This transfers a differential identity only, not a smooth solution.

Since $H=0$ as a field on ${\mathrm{\Sigma }}_0$, its covariant derivative in every tangential direction vanishes there. The normal-projection identities in [L3] involve only the second jet of $g$ and the algebraic stress, so the same finite-jet argument in step 1.1 applies. The stated Einstein constraints therefore give $E(n,n)=E(n,{\mathsf{e}}_i)=0$ for the unreduced residual. The residual formula of [F1], evaluated in the orthonormal frame $(n,{\mathsf{e}}_i)$, gives $E(n,n)=\frac{1}{2}({\mathrm{\nabla }}_n{H}^{\mathrm{♭}})(n)$ and $E(n,{\mathsf{e}}_i)=\frac{1}{2}({\mathrm{\nabla }}_n{H}^{\mathrm{♭}})({\mathsf{e}}_i)$: the tangential derivatives vanish, and ${\mathrm{\nabla }}_c{H}^c=-({\mathrm{\nabla }}_n{H}^{\mathrm{♭}})(n)$. Hence ${\mathrm{\nabla }}_n{H}^{\mathrm{♭}}=0$ and, by nondegeneracy of $g$, ${\mathrm{\nabla }}_{n}H=0$ initially.

Fix a point of ${\mathrm{\Sigma }}_0$ and choose local coordinates $(s,x^1,x^2,x^3)$ with spacelike $s$-levels, $s=0$ on ${\mathrm{\Sigma }}_0$, and ${\mathrm{\partial }}_s$ normal to the slice at that point. Such coordinates follow by a linear change of the spatial coordinates by multiples of $s$ in any slice chart. At that point $g^{00}<0$ and $(g^{ij})>0$; these strict inequalities persist on a smaller cylinder. Set $P_b:={\mathrm{\partial }}_s{H}_b$ and $Q_{bi}:={\mathrm{\partial }}_i{H}_b$. The wave equation in [F1] has scalar principal part $g^{cd}{\mathrm{\partial }}_c{\mathrm{\partial }}_d{H}_b$; connection and Ricci terms are lower order. In the ordering $W=(H_b,P_b,Q_{bi})$, its first-order reduction consists of ${\mathrm{\partial }}_s{H}_b-P_b=0$, the wave equation solved for ${\mathrm{\partial }}_s{P}_b$, and the compatibility equations ${\mathrm{\partial }}_s{Q}_{bi}-{\mathrm{\partial }}_i{P}_b=0$. Multiplying the wave equation by $-1$ and the compatibility equations by $g^{ji}$ gives time matrix $A^0=\mathrm{diag}(I,-g^{00}I,(g^{ij})I)$ and, for each spatial index $k$, the only nonzero spatial blocks $A_{PP}^k=-2g^{0k}I$, $A_{P{Q}_j}^k=-g^{kj}I$, and $A_{Q_{i}P}^k=-g^{ik}I$. Thus $A^0$ is positive definite on a sufficiently small cylinder with spacelike $s$-levels, and each $A^k$ is symmetric; all remaining coefficients and their first derivatives are bounded there by $g\in C^3$ and $H\in C^2$. Choose a compact coordinate ball about the chosen point and shrink the time interval so that the shrinking balls $B_s$ of radius $r-\lambda s$ remain in this cylinder. If $A^0⪰aI$ and $M:={\sup }_{\mathrm{\mid }\nu \mathrm{\mid }=1}\mathrm{\parallel }{\nu }_i{A}^i\mathrm{\parallel }$, choose $\lambda >M\mathrm{/}a$. For $s\ge 0$ set $\mathcal{E}(s)={\int }_{B_s}{W}^{\mathsf{T}}{A}^{0}Wdx$. The symmetric system's product rule and spatial integration by parts give ${\mathcal{E}}^{\mathrm{\prime }}(s)\le C\mathcal{E}(s)-{\int }_{\mathrm{\partial }{B}_s}{W}^{\mathsf{T}}(\lambda A^0+{\nu }_i{A}^i)WdS$. The boundary matrix is positive semidefinite because $\lambda a>M$, so ${\mathcal{E}}^{\mathrm{\prime }}\le C\mathcal{E}$. The zero initial values established in steps 1.1 and 2.1 and Gronwall force $\mathcal{E}=0$ in the shrinking cone. Reversing time proves the same on the past cone. Thus $H=0$ in a neighborhood of the chosen point, and the point was arbitrary.

When $H=0$, the harmonic-reduced Einstein equation differs from the unreduced equation by terms linear in $\mathrm{\nabla }H$, which now vanish. Step 1.1 already gives both Maxwell equations, so the full coupled Einstein–Maxwell equations hold near the initial surface. The stress identity is applied at its explicit classical regularity, with no smoothness improvement inferred.