Temporal-gauge first-order reduction of the charged scalar equation

Expanding the divergence equation and retaining only derivatives of the independent $\pi$ variables gives $g^{00}{\mathrm{\partial }}_s{\pi }_0+g^{0i}{\mathrm{\partial }}_s{\pi }_i+g^{i0}{\mathrm{\partial }}_i{\pi }_0+g^{ij}{\mathrm{\partial }}_i{\pi }_j$. Substitute ${\mathrm{\partial }}_s{\pi }_i={\mathrm{\partial }}_i{\pi }_0-i\lambda A_i{\pi }_0-i\lambda F_{0i}\phi$ from the compatibility equations. Since $g^{i0}=g^{0i}$, the derivative terms become $g^{00}{\mathrm{\partial }}_s{\pi }_0+2g^{0i}{\mathrm{\partial }}_i{\pi }_0+g^{ij}{\mathrm{\partial }}_i{\pi }_j$; the substituted source terms contain no derivatives of $\pi$. Multiplying this row by $-1$ gives the top row of the stated principal system.

For each output row $r$, summing the $i$th compatibility equation against $g^{ri}$ gives time term $g^{ri}{\mathrm{\partial }}_s{\pi }_i$ and spatial term $-g^{ri}{\mathrm{\partial }}_i{\pi }_0$. Thus the coefficient of ${\mathrm{\partial }}_k{\pi }_0$ in row $r$ is $-g^{rk}$. The complete $\pi$-block has exactly the displayed $A^0,A^k$. The time block is positive because $-g^{00}>0$ and $(g^{ij})$ is positive definite. Each spatial block is symmetric because $g^{ij}=g^{ji}$, including the off-diagonal entries $-g^{kj}$ and $-g^{jk}$. The same calculation applies separately to the real and imaginary parts.

The equations ${\mathrm{\partial }}_s\phi ={\pi }_0$ and ${\mathrm{\partial }}_s{A}_i=F_{0i}$ append identity time blocks and zero spatial blocks. Write $v\in {\mathbb{R}}^{13}$ for, in order, the real parts of the four ${\pi }_a$, their imaginary parts, $\mathrm{Re}\phi ,\mathrm{Im}\phi$, and the three real $A_i$. Denote these scalar-potential matrices by $S^{\mu }$: $S^0=\mathrm{diag}(A^0,A^0,I_5)$ and $S^k=\mathrm{diag}(A^k,A^k,0_5)$ in this component order. They are symmetric, with $S^0$ positive definite.

To verify the full coefficient domain in [L1], introduce independent real coordinates $z=(G_{ab},Q_{cab},E_i)\in {\mathbb{R}}^{53}$, with $G_{ab}=G_{ba}$ and $Q_{cab}=Q_{cba}$, for $0\le a,b,c\le 3$ and $1\le i\le 3$. Here $Q$ represents the first metric derivatives and $E_i$ represents $F_{0i}$. Let $O\subseteq {\mathbb{R}}^{66}$ consist of $u=(v,z)$ such that $G$ has Lorentzian signature $(-,+,+,+)$, $G^{00}<0$, $(G^{ij})$ is positive definite, and $\mathrm{\mid }\phi {\mathrm{\mid }}^2$ lies in the open domain of $V$. This is open: inversion is smooth by the adjugate-over-determinant formula, and invertibility, fixed signature, and strict definiteness persist under sufficiently small changes of a symmetric matrix. Set ${\mathrm{\Gamma }}^c{}_{ab}=\frac{1}{2}{G}^{cd}(Q_{abd}+Q_{bad}-Q_{dab})$, $A_0=0$, $c_i=-i\lambda A_i{\pi }_0-i\lambda E_i\phi$, and $R=G^{ab}{\mathrm{\Gamma }}^c{}_{ab}{\pi }_c+i\lambda G^{ab}{A}_a{\pi }_b+V^{\mathrm{\prime }}(\mathrm{\mid }\phi {\mathrm{\mid }}^2)\phi$. The transformed scalar source vector $b(v,z)$ has complex components $R-G^{0i}{c}_i$ in the ${\pi }_0$ row and $-G^{ri}{c}_i$ in the ${\pi }_r$ rows, components $-{\pi }_0$ in the $\phi$ rows, and $-E_i$ in the $A_i$ rows, all split into real coordinates. Indeed the original divergence equation is $G^{ab}{\mathrm{\partial }}_a{\pi }_b=R$, so these signs follow by the substitution in step 1.1. All these maps are smooth on $O$, using only finite sums, products, inversion and composition with the smooth function $V^{\mathrm{\prime }}$.

On $\mathbb{R}\times {\mathbb{T}}_{2\pi }^3\times O$ define ${\widehat{A}}^0=\mathrm{diag}(S^0,I_{53})$, ${\widehat{A}}^k=\mathrm{diag}(S^k,0_{53})$, and $\widehat{B}=(b,0_{53})$, with every formula independent of the base point $(s,x)$. The lifted maps are therefore smooth and periodic on the entire base, and the matrices are symmetric with positive-definite time matrix. Restricting the time base to any interval $J$ of [L1] gives a coefficient family exactly of its stated type. Evaluating its first thirteen rows along $(G,Q,E)=(g,\mathrm{\partial }g,F_{0i})$ on $U$ gives precisely the claimed scalar-potential reduction. The extra identity and zero blocks are a coefficient completion only: no auxiliary evolution equation, global extension of the given fields, or existence of a coupled solution is asserted. Thus the local block has the required coefficient realization, while the actual metric and Maxwell evolution blocks must be supplied separately in a coupled system.