Maxwell's equations in potential form: the Lorenz divergence, the decoupled wave equations and residual generators

Regularity bookkeeping. The pair $(\phi ,A)$ is $C^2$, so by [F1] the field pair $(E,B)$ and the Lorenz divergence $G$ are of class $C^1$ and the expressions $W\phi$, $WA$, ${\mathrm{\partial }}_{t}G$ and $\mathrm{\nabla }G$ are defined and continuous on $\mathrm{\Omega }\times I$; at included endpoints of a non-open $I$ all derivatives are those of the restriction convention [F2]. Hence both identities displayed in (a) are equalities of continuous functions on $\mathrm{\Omega }\times I$.

First identity. By [F1] and the linearity of the divergence [F4], $\mathrm{div}E=\mathrm{div}(-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A)=-\mathrm{div}\mathrm{\nabla }\phi -\mathrm{div}{\mathrm{\partial }}_{t}A$. The definition of the Laplacian [F6] gives $\mathrm{div}\mathrm{\nabla }\phi =\mathrm{\Delta }\phi$, and the componentwise commutation of the time and space partial derivatives of the $C^2$ field $A$ [F5] gives $\mathrm{div}{\mathrm{\partial }}_{t}A={\mathrm{\partial }}_t\mathrm{div}A$. On the other hand, expanding $G$ [F1] and using the linearity of ${\mathrm{\partial }}_t$ and of $\mathrm{\Delta }$ [F4] with [F5], $W\phi -{\mathrm{\partial }}_{t}G=(c^{-2}{\mathrm{\partial }}_t^2\phi -\mathrm{\Delta }\phi )-({\mathrm{\partial }}_t\mathrm{div}A+c^{-2}{\mathrm{\partial }}_t^2\phi )=-\mathrm{\Delta }\phi -{\mathrm{\partial }}_t\mathrm{div}A$. The two right-hand sides agree, so $\mathrm{div}E=W\phi -{\mathrm{\partial }}_{t}G$ on $\mathrm{\Omega }\times I$.

Second identity. By [F1] and the linearity of the curl and of ${\mathrm{\partial }}_t$ [F4], $\mathrm{curl}B-c^{-2}{\mathrm{\partial }}_{t}E=\mathrm{curl}\mathrm{curl}A+c^{-2}{\mathrm{\partial }}_t\mathrm{\nabla }\phi +c^{-2}{\mathrm{\partial }}_t^{2}A$. The curl-of-curl identity [F6] rewrites the first term as $\mathrm{\nabla }\mathrm{div}A-\mathrm{\Delta }A$; the componentwise commutation [F5] for the $C^2$ function $\phi$ rewrites $c^{-2}{\mathrm{\partial }}_t\mathrm{\nabla }\phi$ as $c^{-2}\mathrm{\nabla }{\mathrm{\partial }}_t\phi =\mathrm{\nabla }(c^{-2}{\mathrm{\partial }}_t\phi )$; and the last term is one of the two summands of $WA$ [F1]. Collecting terms, $\mathrm{curl}B-c^{-2}{\mathrm{\partial }}_{t}E=(\mathrm{\nabla }\mathrm{div}A+\mathrm{\nabla }(c^{-2}{\mathrm{\partial }}_t\phi ))+(c^{-2}{\mathrm{\partial }}_t^{2}A-\mathrm{\Delta }A)=\mathrm{\nabla }G+WA$ by the linearity of $\mathrm{\nabla }$ [F4] and the definition of $G$, that is $\mathrm{curl}B-c^{-2}{\mathrm{\partial }}_{t}E=WA+\mathrm{\nabla }G$ on $\mathrm{\Omega }\times I$.

Automatic source-free equations. For the potential pair, $\mathrm{div}B=\mathrm{div}\mathrm{curl}A=0$ by [F7], and $\mathrm{curl}E=\mathrm{curl}(-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A)=-\mathrm{curl}\mathrm{\nabla }\phi -\mathrm{curl}{\mathrm{\partial }}_{t}A$ by linearity [F4]; the first term vanishes by [F7] applied to the $C^2$ function $\phi$, and the second equals ${\mathrm{\partial }}_t\mathrm{curl}A={\mathrm{\partial }}_{t}B$ by the componentwise commutation [F5], so $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ on $\mathrm{\Omega }\times I$.

Gauge shift and the Lorenz condition. Let $\chi$ be a gauge function of class $C^2$ jointly. By [F1] the transform $T_{\chi }(\phi ,A)=(\phi -{\mathrm{\partial }}_t\chi ,A+\mathrm{\nabla }\chi )$ is a potential pair of class $C^1$, so its Lorenz divergence is defined; expanding it with the linearity of $\mathrm{div}$ [F4] and the definitions [F1], $G(T_{\chi }(\phi ,A))=\mathrm{div}A+\mathrm{div}\mathrm{\nabla }\chi +c^{-2}{\mathrm{\partial }}_t\phi -c^{-2}{\mathrm{\partial }}_t^2\chi =G+\mathrm{\Delta }\chi -c^{-2}{\mathrm{\partial }}_t^2\chi =G-W\chi$, the middle identity using $\mathrm{div}\mathrm{\nabla }\chi =\mathrm{\Delta }\chi$ [F6] and the last the definition of $W$ [F1]. Thus $G(T_{\chi }(\phi ,A))=0$ if and only if $W\chi =G$ on $\mathrm{\Omega }\times I$; and the field pair is unchanged by [F9], so the transformed pair represents the same $(E,B)$ of [F3].

Equivalence with the sourced system. By [F8] the sourced Maxwell equations for $(E,B,\rho ,J)$ are the four identities $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$ and $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$ on $\mathrm{\Omega }\times I$; the pair $(E,B)$ is of class $C^1$ by step 1.1 with all derivatives required by those equations existing and continuous, and $\rho ,J$ are continuous by [F3]. The second and third identities hold identically by step 2.3. The first is equivalent, by step 2.1, to $W\phi -{\mathrm{\partial }}_{t}G=\rho \mathrm{/}{\epsilon }_0$, and the fourth is equivalent, by step 2.2, to $WA+\mathrm{\nabla }G={\mu }_{0}J$. Hence the sourced Maxwell equations hold for $(E,B,\rho ,J)$ if and only if the two residual equations of (a) hold.

Residual generators. If in addition to $G=0$ the gauge function satisfies $W\chi =0$ on $\mathrm{\Omega }\times I$, then step 2.4 gives $G(T_{\chi }(\phi ,A))=G-W\chi =0$, so the transformed pair is again in Lorenz gauge, and by [F9] it has the same field pair. For nonempty $\mathrm{\Omega }$, take $\chi (x,t)=x_1$: all its second partial derivatives vanish, hence $W\chi =0$, while ${\mathrm{\partial }}_t\chi =0$ and $\mathrm{\nabla }\chi =(1,0,0)$, so $T_{\chi }(\phi ,A)=(\phi ,A+(1,0,0))$ is distinct from $(\phi ,A)$ on the nonempty product. Thus the Lorenz condition does not determine the pair on a nonempty region. On empty $\mathrm{\Omega }$ all identities are vacuous and there is only one potential pair.

Decoupling under the Lorenz condition. Assume additionally $G=0$ on $\mathrm{\Omega }\times I$. Then the $C^1$ function $G$ has ${\mathrm{\partial }}_{t}G=0$ and $\mathrm{\nabla }G=0$ on $\mathrm{\Omega }\times I$ by [F4]: each spatial restriction is identically zero, and the time restriction is identically zero on the nondegenerate interval $I$, so its interior and relative endpoint difference quotients vanish. Hence the residual equations of step 3.1 read $W\phi =\rho \mathrm{/}{\epsilon }_0$ and $WA={\mu }_{0}J$ there; conversely, if these two wave equations hold then the residual equations hold and, by step 3.1, the sourced Maxwell system holds for the pair. This equivalence is asserted only under the explicitly imposed hypothesis $G=0$: for a general representative the equivalent conditions are the residual equations of step 3.1, which contain the derivatives ${\mathrm{\partial }}_{t}G$ and $\mathrm{\nabla }G$ of the representative's own Lorenz divergence, and no step infers $G=0$ from the Maxwell equations.

Conclusion and scope. Step 2.1 and step 2.2 give the exact pointwise identities of (a), and step 3.1 identifies the two residual equations with the sourced Maxwell system; step 2.3 records the two automatic source-free equations. Step 2.4 gives the gauge shift and the equivalence of the Lorenz condition for the transform with $W\chi =G$; step 4.1 gives the decoupling inside $G=0$, with the explicit warning that Maxwell alone does not impose that condition; and step 3.2 gives the residual generators. Throughout, no existence of a solution of $W\chi =G$ or of a generator with prescribed data is asserted, no boundary, decay or initial condition is imposed, and $C^2$ is the exact standing regularity: for a merely $C^1$ pair the expressions $W\phi$, $WA$, $\mathrm{\Delta }$ and ${\mathrm{\partial }}_{t}G$ of the Statement need not exist. No choice principle is used, and no hypothesis is made on the shape of $\mathrm{\Omega }$ beyond openness.