Retarded Maxwell fields for spatially compact smooth histories

Fix $(x,t)$. The source integrands vanish off ${\bar{B}}_a$. On this compact ball the retarded time $t-\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}c$ lies in the bounded interval $[t-(\mathrm{\mid }x\mathrm{\mid }+a)\mathrm{/}c,t]$, so smoothness bounds both source functions there by a finite constant $M$. The triangle inequality and [F5] give ${\int }_{B_a}\mathrm{\mid }h(y,t-\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}c)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x-y\mathrm{\mid }dy\le M{\int }_{B(x,\mathrm{\mid }x\mathrm{\mid }+a)}\mathrm{\mid }x-y{\mathrm{\mid }}^{-1}dy<\mathrm{\infty }$ for $h=\rho$ and each component of $J$; the possible point $y=x$ is harmless because $\mathrm{\mid }z{\mathrm{\mid }}^{-1}$ is locally integrable in three dimensions. Thus both retarded integrals are absolutely convergent.

Fix an event $(x_0,t_0)$ and choose a bounded open ball $O$ around it. Its closure is compact by [F3], as is the source ball ${\bar{B}}_a$. On $\bar{O}\times {\bar{B}}_a$, both observation times and all values $t-\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}c$ lie in one bounded interval because $\mathrm{\mid }x-y\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }+a$; take a closed finite interval containing these times. By [F3] choose a smooth compactly supported time cutoff $\chi$ equal to one on a neighborhood of this interval. Then $h_{\rho }(y,s):=\chi (s)\rho (y,s)$ and $h_J(y,s):=\chi (s)J(y,s)$ belong to $C_c^{\mathrm{\infty }}({\mathbb{R}}^3\times \mathbb{R})$ by [F1], since their supports are contained in ${\bar{B}}_a\times \mathrm{supp}\chi$. On $O$, every retarded source value and every observation-time value sees $\chi =1$, so the potentials in the Statement equal ${\mu }_0{c}^{2}K\ast h_{\rho }$ and ${\mu }_{0}K\ast h_J$, and the localized sources equal $\rho$ and $J$. By [F2] these potentials are smooth on $O$ and satisfy $W\phi ={\mu }_0{c}^2\rho =\rho \mathrm{/}{\epsilon }_0$ and $WA={\mu }_{0}J$ there. Since $(x_0,t_0)$ was arbitrary, the potentials are smooth globally and the two wave equations hold everywhere.

On a smaller neighborhood of the same event, change variables $z=x-y$ in the localized convolution: $K\ast h(x,t)=\int h(x-z,t-\mathrm{\mid }z\mathrm{\mid }\mathrm{/}c)\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })dz$. For a spatial difference quotient, the mean value theorem bounds the quotient of $h(x-z,t-\mathrm{\mid }z\mathrm{\mid }\mathrm{/}c)$ by the supremum of $\mathrm{\mid }{\mathrm{\nabla }}_{y}h\mathrm{\mid }$ on a common compact neighborhood, times $1\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$; for a time difference quotient the same bound uses the supremum of $\mathrm{\mid }{\mathrm{\partial }}_{t}h\mathrm{\mid }$. These bounds are integrable on a fixed ball by [F5]. The pointwise limits are the corresponding derivatives of $h$, so [F4] permits dominated convergence and yields ${\mathrm{\partial }}_{x_i}(K\ast h)=K\ast ({\mathrm{\partial }}_{y_i}h)$ and ${\mathrm{\partial }}_t(K\ast h)=K\ast ({\mathrm{\partial }}_{t}h)$ on that neighborhood. For $h_J=\chi J$, ${\mathrm{div}}_y{h}_J=\chi {\mathrm{div}}_{y}J$; for $h_{\rho }=\chi \rho$, ${\mathrm{\partial }}_t{h}_{\rho }={\chi }^{\mathrm{\prime }}\rho +\chi {\mathrm{\partial }}_t\rho$. At every retarded argument sampled on the neighborhood, $\chi =1$ and ${\chi }^{\mathrm{\prime }}=0$ by the cutoff choice in step 2.1. Therefore ${\displaystyle \mathrm{div}A+c^{-2}{\mathrm{\partial }}_t\phi ={\mu }_{0}K\ast ({\mathrm{div}}_{y}J+{\mathrm{\partial }}_t\rho )=0}$ there by the assumed continuity equation. The event was arbitrary, so the Lorenz condition holds everywhere.

The pair is jointly smooth by step 2.1, satisfies the two wave equations there, and is in Lorenz gauge by step 3.1. Applying [F6] on ${\mathbb{R}}^3\times \mathbb{R}$ gives the two sourced Maxwell equations; the other two equations are the automatic identities of the potential representation included in [F6]. The selection postulate [F7] identifies this retarded solution with the model's physical branch and excludes an independent homogeneous addition.