Harmonic static multicenter electrovac model

Let $N\ge 1$ be finite, let $a_1,\dots ,a_N\in {\mathbb{R}}^3$ be distinct length-coordinate points, choose SI mass parameters $m_{\ast ,A}>0$, and fix one sign $\sigma \in \{-1,+1\}$. Set $m_A:=\frac{G_N{m}_{\ast ,A}}{c^2},D:={\mathbb{R}}^3\setminus \{a_1,\dots ,a_N\},H(x):=1+{\sum }_{A=1}^N\frac{m_A}{\mathrm{\mid }x-a_A\mathrm{\mid }}\mathrm{.}$ The positive length parameters $m_A$ make $H$ dimensionless and smooth positive harmonic on $D$. The future unit field is $N_H=H{\mathrm{\partial }}_{x^0}$, with $g_H(N_H,N_H)=-1$. Here static means the metric coefficients are independent of $x^0$, have no $d{x}^{0}d{x}^i$ terms, and are invariant under $x^0\mapsto -x^0$; the displayed metric has all three properties. On $\mathcal{M}={\mathbb{R}}_{x^0}\times D$, with $x^0=ct$ and the orientation of Positive harmonic conformastatic electrovac data, take $g_H=-H^{-2}(d{x}^0{)}^2+H^2{\delta }_{ij}d{x}^{i}d{x}^j,{\mathcal{A}}_{\sigma }=\sigma (H^{-1}-1)d{x}^0,{\mathcal{F}}_{\sigma }=d{\mathcal{A}}_{\sigma },F:={\alpha }^{-1}{\mathcal{F}}_{\sigma },$ where $\alpha =\sqrt{4\pi G_N\mathrm{/}({\mu }_0{c}^4)}$ as in SI and geometric solution parameter identifications. Under the adopted equation-based model of Equation-based Einstein–Maxwell model with $\mathrm{\Lambda }=0$, $j=0$, and $T^m=0$, these data satisfy the source-free SI equations $dF=0,{\mathrm{\nabla }}_a^g{F}^{ab}=0,G_{ab}={\kappa }_E{T}_{ab}^{\mathrm{E}\mathrm{M}}\mathrm{.}$ For each $A$ and $r>0$, define the coordinate sphere and its closed ball by $S_A(r):=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x-a_A\mathrm{\mid }=r\}\mathrm{.}$

${\bar{B}}_A(r):=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x-a_A\mathrm{\mid }\le r\}\mathrm{.}$ Orient $S_A(r)$ by its outward unit normal relative to the standard orientation of ${\mathbb{R}}^3$. For every such sphere $S\subset D$, fix any $s\in {\mathbb{R}}_{x^0}$ and use $i_{s,S}:S\hookrightarrow \mathcal{M}$, $i_{s,S}(x)=(s,x)$, with the transported outward orientation. Every flux written over $S$ below means the integral of the spacetime form pulled back by $i_{s,S}$; by Finite positive multicenter MP exterior and fluxes these values are independent of $s$. Whenever ${\bar{B}}_A(r)$ contains no other center, the geometric fluxes are $q_e(S_A(r))=\sigma m_A,q_m(S_A(r))=0.$ The corresponding SI electric-flux parameter is $Q_{e,A}:={\epsilon }_{0}c{\int }_{S_A(r)}{\star }_{g}F=\sigma \sqrt{4\pi {\epsilon }_0{G}_N}{m}_{\ast ,A},$ and its magnetic flux ${\int }_{S_A(r)}F$ is zero. Define the outward-oriented origin-centered sphere by $S_R:=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }=R\}$, where the orientation is induced by the outward unit normal relative to the standard orientation of ${\mathbb{R}}^3$. For $R>{\max }_A\mathrm{\mid }{a}_A\mathrm{\mid }$, it has the sum of these electric fluxes. These are model parameters and regular-sphere fluxes, not measurements or evidence of regular material sources at the removed points.