Radiation data at fixed retarded time

Let the laboratory frame and SI conventions be those of the framework, and let $\rho ,J$ be a jointly smooth conserved volume source supported in ${\bar{B}}_a(0)$ in space at every time, for some $a>0$. Let $E,B$ be the retarded fields generated by that source. Write $S^2=\{n\in {\mathbb{R}}^3:\mathrm{\mid }n\mathrm{\mid }=1\}$, $r=\mathrm{\mid }x\mathrm{\mid }$, and $t=u+r\mathrm{/}c$. If maps $E_{\mathrm{r}\mathrm{a}\mathrm{d}},B_{\mathrm{r}\mathrm{a}\mathrm{d}}:S^2\times \mathbb{R}\to {\mathbb{R}}^3$ satisfy

$rE(rn,u+r\mathrm{/}c)⟶E_{\mathrm{r}\mathrm{a}\mathrm{d}}(n,u),rB(rn,u+r\mathrm{/}c)⟶B_{\mathrm{r}\mathrm{a}\mathrm{d}}(n,u)$

uniformly in $n\in S^2$ and $u$ in every compact interval as $r\to \mathrm{\infty }$, then $E_{\mathrm{r}\mathrm{a}\mathrm{d}}$ and $B_{\mathrm{r}\mathrm{a}\mathrm{d}}$ are the electric and magnetic radiation data at fixed retarded time. The scaled fields carry units $\mathrm{V}$ and $\mathrm{T}\mathrm{m}$, respectively. This definition by itself does not assert that the limits exist.

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