Coulomb-exterior support-radius energy bound

Let an extended static charged body have a nonempty compact rest-frame stress-energy support ${\mathrm{supp}}_{0}T$ in the flat-body moment convention of Body moments and transport convention, total charge $q\in \mathbb{R}$ coulombs, and finite total rest energy $M{c}^2$ with $M>0$ kilograms. Use a momentum-rest inertial frame $(o;e_0,e_1,e_2,e_3)$ of Inertial frames and inertial charts of the affine Minkowski model, with $p=Mc{e}_0$ and $o$ the stipulated body center. Its rest spatial chart has coordinates $\mathbf{x}=(x^1,x^2,x^3)$; define $r:=\mathrm{\mid }\mathbf{x}\mathrm{\mid }:=\sqrt{(x^1{)}^2+(x^2{)}^2+(x^3{)}^2}$, $\widehat{\mathbf{r}}:=\mathbf{x}\mathrm{/}r$ for $r>0$, and ${\bar{B}}_{\mathrm{\ell }}(0):=\{\mathbf{x}:r\le \mathrm{\ell }\}$. Define the body support radius $\mathrm{\ell }:={\max }_{\mathbf{x}\in {\mathrm{supp}}_{0}T}\mathrm{\mid }\mathbf{x}\mathrm{\mid }>0$, in metres. Thus the material support lies in ${\bar{B}}_{\mathrm{\ell }}(0)$; $\mathrm{\ell }$ is its exact radius about $o$, and the exterior region below is defined from this support, not inferred from spherical symmetry. Use the SI constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, including $c$ in metres per second and ${\epsilon }_0>0$. On the open exterior spatial region ${\mathrm{\Omega }}_{\mathrm{\ell }}:=\{\mathbf{x}\in {\mathbb{R}}^3:r>\mathrm{\ell }\}$, the static field is assumed to obey $\mathbf{E}(\mathbf{x})=\frac{q}{4\pi {\epsilon }_0{r}^2}\widehat{\mathbf{r}},\mathbf{B}=0.$ Let $u(\mathbf{x})$ be the time-independent vacuum energy density of Vacuum electromagnetic energy density and energy flux (the Poynting vector) on this exterior region and define $E_{\mathrm{e}\mathrm{x}\mathrm{t}}:={\int }_{{\mathrm{\Omega }}_{\mathrm{\ell }}}u(\mathbf{x})d^{3}x$. Assume the total rest energy is the sum of this exterior field energy and a remaining material-plus-interior contribution $E_{\mathrm{r}\mathrm{e}\mathrm{m}}\ge 0$. Then $M{c}^2\ge \frac{q^2}{8\pi {\epsilon }_0\mathrm{\ell }},\mathrm{\ell }\ge \frac{q^2}{8\pi {\epsilon }_{0}M{c}^2}\mathrm{.}$ The exact Coulomb exterior and nonnegative energy accounting are hypotheses. The theorem does not infer an exterior field from spherical symmetry alone. No identification of field energy with an ideal point particle's rest mass is made.

The energy is a field contribution on the exterior region only. Nonnegative interior/material energy and the specified total-energy accounting are load-bearing assumptions. The point-charge divergence result Divergence of the point-charge field energy and the limits of mass inference explicitly does not identify this field energy with particle rest mass.