Relativistic emitted power of a prescribed point charge at a general emission event

Weighted flux limit (a). The definition of $P_{\mathrm{e}\mathrm{m}}(s_0)$ in the Statement is the limit of the ${\kappa }_0$-weighted far-zone fluxes, and the corresponding limit and its value $\frac{C^2}{{\mu }_0{c}^5}{\int }_{S^2}\mathrm{\mid }n\times ((n-{\beta }_0)\times a){\mathrm{\mid }}^2{\kappa }_0^{-5}d\sigma$ are exactly clause (f) of [F1], whose proof establishes the uniform pointwise convergence of ${\kappa }_0{r}^{2}n\cdot S$ to the displayed integrand and the domination needed to pass the limit through the sphere integral [F6]; the same clause uses ${\kappa }_0=1-n\cdot {\beta }_0$ and $r=c(t_r-s_0)$ on the emitted spheres [F2]. The interpretation of the weight as the emission-time/observation-time factor is ${\mathrm{\partial }}_{t}s=1\mathrm{/}\kappa$ of [F4], evaluated at the emission event, and the boundary-corrected reading of $S\cdot n-(z^{\mathrm{\prime }}\cdot n)u$ is the leading-order form of that factor. This is (a).

Closed form and four-acceleration form (b). Clause (f) of [F1] evaluates the integral as ${\gamma }_0^6(\mathrm{\mid }a{\mathrm{\mid }}^2-\mathrm{\mid }{\beta }_0\times a{\mathrm{\mid }}^2)q^2\mathrm{/}(6\pi {\epsilon }_0{c}^3)$, which is the first displayed equality; at ${\beta }_0=0$ the weight is $1$ and the formula reduces to the rest-event Larmor statement of [F1]. For the four-acceleration form, apply [F3] to the chart worldline $t\mapsto (z(t),t)$: its coordinate velocity is $w=z^{\mathrm{\prime }}(s_0)=c{\beta }_0$, its Lorentz factor is ${\gamma }_w={\gamma }_0$, and its coordinate acceleration is $a=z^{\mathrm{\prime }\mathrm{\prime }}(s_0)$, so $\eta (A,A)={\gamma }_0^4\mathrm{\mid }a{\mathrm{\mid }}^2+{\gamma }_0^6(c{\beta }_0\cdot a{)}^2\mathrm{/}{c}^2={\gamma }_0^4\mathrm{\mid }a{\mathrm{\mid }}^2+{\gamma }_0^6({\beta }_0\cdot a{)}^2$. By [F5], ${\gamma }_0^4\mathrm{\mid }a{\mathrm{\mid }}^2+{\gamma }_0^6({\beta }_0\cdot a{)}^2={\gamma }_0^6[(1-\mathrm{\mid }{\beta }_0{\mathrm{\mid }}^2)\mathrm{\mid }a{\mathrm{\mid }}^2+({\beta }_0\cdot a{)}^2]={\gamma }_0^6(\mathrm{\mid }a{\mathrm{\mid }}^2-\mathrm{\mid }{\beta }_0\times a{\mathrm{\mid }}^2)$, since $\mathrm{\mid }{\beta }_0\times a{\mathrm{\mid }}^2=\mathrm{\mid }{\beta }_0{\mathrm{\mid }}^2\mathrm{\mid }a{\mathrm{\mid }}^2-({\beta }_0\cdot a{)}^2$. Hence $P_{\mathrm{e}\mathrm{m}}=q^2\eta (A,A)\mathrm{/}(6\pi {\epsilon }_0{c}^3)$. This is (b).

Low-speed comparison (c). The eventwise inequalities $P_{\mathrm{L}}\le P_{\mathrm{e}\mathrm{m}}\le {\gamma }_0^6{P}_{\mathrm{L}}$ and the uniform bounds $P_{\mathrm{L}}\le P_{\mathrm{e}\mathrm{m}}\le (1-{\delta }^2{)}^{-3}{P}_{\mathrm{L}}$, $\mathrm{\mid }{P}_{\mathrm{e}\mathrm{m}}-P_{\mathrm{L}}\mathrm{\mid }\le 3{\delta }^2(1-{\delta }^2{)}^{-3}{P}_{\mathrm{L}}$ for $\mathrm{\mid }{\beta }_0\mathrm{\mid }\le \delta <1$ are clause (f) of [F1], transported verbatim to this setting; the case $\mathrm{\mid }{\beta }_0\mathrm{\mid }=0$ gives the exact rest formula and is the $\delta =0$ endpoint of the comparison. The statement adds no smallness assertion beyond the explicit hypothesis $\mathrm{\mid }{\beta }_0\mathrm{\mid }\le \delta$. This is (c).

Scope (d). The definition is per unit emission time at the single event $s_0$: the weight is the retarded Jacobian of [F4], not a dynamical rate, and no worldline evolution, self-force, radiation reaction or mass is introduced anywhere in steps 1.1-3.1. The near (velocity) part of the fields is $O(r^{-2})$ with the uniform remainder of [F2], so its product contributions to $S$ are $O(r^{-3})$ and $O(r^{-4})$, and after multiplication by $r^2$ and integration over the compact sphere they contribute $O(r^{-1})$ and $O(r^{-2})$ respectively, vanishing in the limit; this is why the weighted definition isolates the acceleration-field radiation. The ${\kappa }_0$-weighting is declared as part of the definition of emitted power, not inferred as a physical law, and no empirical or measurement content is added. This is (d).