Selected transported charged-dust Einstein–Maxwell model

On an open region $\mathrm{\Omega }$ of the smooth oriented, time-oriented Lorentzian spacetime and SI conventions of Smooth matter quantities, current and observer observables, adopt the equation-based Einstein and smooth curved Maxwell systems of Equation-based Einstein–Maxwell model and Smooth curved vacuum Maxwell constitutive adoption, with the smooth matter quantities and electromagnetic stress of Smooth matter quantities, current and observer observables. Thus the coupled branch includes the adopted Einstein equation $G_{ab}+\mathrm{\Lambda }{g}_{ab}=(8\pi G_N\mathrm{/}{c}^4)(T_{m,ab}+T_{ab}^{\mathrm{E}\mathrm{M}})$ and Maxwell equations $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, using the fixed local SI signs, units and curvature convention of those suppliers. Their adopted equations remain physical assumptions; neither is deduced here from flat spacetime, variational stationarity, or mathematical structure.

Choose smooth $\rho ,\sigma ,U$ on $\mathrm{\Omega }$, with $\rho \ge 0$, $U$ future timelike, $g(U,U)=-c^2$, and $\sigma =0$ wherever $\rho =0$. Select the pressureless dust closure

$T_m^{ab}=\rho U^a{U}^b,j^a=\sigma U^a\mathrm{.}$

As the matter model, adopt the undivided smooth mass and Lorentz-force equations

${\mathrm{\nabla }}_a(\rho U^a)=0,\rho U^a{\mathrm{\nabla }}_a{U}^b=\sigma F^b{}_c{U}^c,$

and charge-current conservation ${\mathrm{\nabla }}_a{j}^a=0$. On each open region ${\mathrm{\Omega }}_{+}:=\{p\in \mathrm{\Omega }:\rho (p)>0\}$ define $\alpha :=\sigma \mathrm{/}\rho$. Under the adopted mass continuity and charged-dust forms, Charged dust balance equivalence and advected charge ratio gives the exact equivalence between charge-current conservation and transport $U^a{\mathrm{\nabla }}_a\alpha =0$. The ratio is not defined and no division is made on $\rho =0$. This ratio statement is restricted to ${\mathrm{\Omega }}_{+}$; no global coupled evolution theorem is asserted.

This is a selected classical pressureless continuum model with one four-velocity field $U$ on the stated smooth domain. It excludes viscosity, conduction, multistream averaging, spin, point sources, self-field products and a general dust action. The finely fixed charge-to-mass ratio used in the specific ECD solutions of Meinel and Hütten is not a law of this general model. The postulate asserts neither a global smooth continuation nor validity of a single-velocity description beyond the stated domain; it makes no coupled-solution existence, uniqueness, stability, or empirical claim.