Exact smooth coupled counterpoised dust and observables

Assume the indexed Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), recorded as the mathematical premise [A2]. Use the selected smooth SI Einstein–Maxwell charged-dust model of [A1], with length coordinates $x^0=ct$ and $x^i$ in metres, positive constants $c,{\mu }_0,G_N$, ${ϵ}_0:=({\mu }_0{c}^2{)}^{-1}$, and $\kappa :=8\pi G_N\mathrm{/}{c}^4$. Use the positive-root convention of Explicit rational and real power regimes to set $Q_s:=\sqrt{4\pi {ϵ}_0{G}_N}>0$ (units $\mathrm{C}\mathrm{k}{\mathrm{g}}^{-1}$). For an open $D\subseteq {\mathbb{R}}^3$, let $H\in C^{\mathrm{\infty }}(D,(0,\mathrm{\infty }))$ be dimensionless and satisfy the Euclidean condition $\mathrm{\Delta }H:={\sum }_{i=1}^3{\mathrm{\partial }}_i^{2}H\le 0$. On $\mathcal{M}:={\mathbb{R}}_{x^0}\times D$, use the coordinate metric, orientation, and time orientation $g_H=-H^{-2}(d{x}^0{)}^2+H^2{\sum }_{i=1}^3(d{x}^i{)}^2,n:=H{\mathrm{\partial }}_0\text{ future},e_i:=H^{-1}{\mathrm{\partial }}_i\mathrm{.}$ Use the coordinate orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. Define the smooth potential, field, dust velocity, rest-mass and rest-charge densities, current, and matter stress by $A:=s\frac{c}{Q_s}{H}^{-1}d{x}^0,F:=dA,U:=cH{\mathrm{\partial }}_0,\rho :=-\frac{c^2\mathrm{\Delta }H}{4\pi G_N{H}^3},\sigma :=s{Q}_s\rho ,j^a:=\sigma U^a,T_m^{ab}:=\rho U^a{U}^b,$ where $s\in \{-1,+1\}$ and the electromagnetic stress $T_{ab}^{\mathrm{E}\mathrm{M}}$ is the one adopted in [A1]. Then $g_H$ is smooth Lorentzian, $n$ is a future unit observer, $U=cn$ is future timelike with $g_H(U,U)=-c^2$, all displayed fields are smooth, and $\rho \ge 0$. This tuple satisfies every equation of the selected dust branch: $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, ${\mathrm{\nabla }}_a(\rho U^a)=0$, $\rho U^a{\mathrm{\nabla }}_a{U}^b=\sigma F^b{}_c{U}^c$, ${\mathrm{\nabla }}_a{j}^a=0$, ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$, and the adopted Einstein equation with $\mathrm{\Lambda }=0$. These are exact conditional model deductions. On $\{\rho >0\}$ the ratio $\sigma \mathrm{/}\rho =s{Q}_s$ is constant along $U$; no division by $\rho$ is made on $\{\rho =0\}$. The observer electric components are $E_i:=cF(e_i,n)=-s{c}^2{H}_i\mathrm{/}(Q_s{H}^2)$, where $H_i:={\mathrm{\partial }}_{i}H$, and the coordinate acceleration is $a^0=0$, $a^i=-c^2{H}_i\mathrm{/}{H}^3$.

Here $\rho$ is rest-mass density in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $\sigma$ is rest-charge density in $\mathrm{C}{\mathrm{m}}^{-3}$, and the stress components have energy-density units $\mathrm{J}{\mathrm{m}}^{-3}$. The orthonormal components of $F$ have tesla units and $E_i$ has units $\mathrm{V}{\mathrm{m}}^{-1}$, under the SI convention of [A1].

The total stress $T_{ab}:=T_{m,ab}+T_{ab}^{\mathrm{E}\mathrm{M}}$ satisfies the weak, null, dominant, and strong energy conditions pointwise: respectively $T(v,v)\ge 0$ for every future timelike $v$; $T(\mathrm{\ell },\mathrm{\ell })\ge 0$ for every nonzero future null $\mathrm{\ell }$; $-T^a{}_b{v}^b$ is future causal or zero for every future timelike $v$; and the weak condition holds for $T_{ab}-\frac{1}{2}(g^{cd}{T}_{cd})g_{ab}$.

For the spherical specialization on all ${\mathbb{R}}^3$, let $a,b>0$ be lengths, $r^2:={\sum }_{i=1}^3(x^i{)}^2,\mathrm{\Psi }:=\sqrt{r^2+b^2},H:=1+\frac{a}{\mathrm{\Psi }}\mathrm{.}$ Then $\mathrm{\Delta }H=-3a{b}^2\mathrm{/}(r^2+b^2{)}^{5\mathrm{/}2}<0$ and $\rho =3c^{2}a{b}^2\mathrm{/}[4\pi G_N{H}^3(r^2+b^2{)}^{5\mathrm{/}2}]>0$ everywhere. On a slice $\mathrm{\Sigma }=\{x^0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\}$ the induced spatial metric is $h=H^2{\delta }_{ij}d{x}^{i}d{x}^j$ and its volume measure is $d{V}_h=H^{3}d{\lambda }_3$, where ${\lambda }_3$ is Euclidean Lebesgue measure. Define the proper matter mass and signed charge by $M_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}:={\int }_{\mathrm{\Sigma }}\rho d{V}_h$ and $Q_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}:={\int }_{\mathrm{\Sigma }}\sigma d{V}_h$. Both are finite, with $M_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}=\frac{c^{2}a}{G_N},Q_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}=s{Q}_s\frac{c^{2}a}{G_N}\mathrm{.}$ The scalar curvature, meaning the trace of the Ricci tensor in the adopted convention, is $S_g=-2\mathrm{\Delta }H\mathrm{/}{H}^3>0$. These proper integrals are not identified with an ADM mass or a measurement.

For $a=\delta b$ with $0<\delta \le 1$, put $h_0:=b\mathrm{/}\mathrm{\Psi }$ and define the positive linear density profile ${\rho }_{\mathrm{l}\mathrm{i}\mathrm{n}}:=\frac{3c^2\delta b^3}{4\pi G_N(r^2+b^2{)}^{5\mathrm{/}2}}\mathrm{.}$ The same exact coupled solutions satisfy the entrywise Cartesian-coordinate bounds $\mid g_{00}+(1-2\delta h_0)\mid \le 3{\delta }^2,\mid g_{ij}-(1+2\delta h_0){\delta }_{ij}\mid \le {\delta }^2,0\le 1-\frac{\rho }{{\rho }_{\mathrm{l}\mathrm{i}\mathrm{n}}}\le 3\delta \mathrm{.}$ All these are ideal model statements, not observed data. The result does not assert physical realization, stability, formation, or a coupled initial-value theorem.