Dust continuity, transport and Lorentz acceleration

Let ${\mathrm{\Omega }}_{+}\subseteq \mathrm{\Omega }$ be an open positive-density region of the charged-dust model in Selected transported charged-dust Einstein–Maxwell model. Let $\rho ,\sigma \in C^1({\mathrm{\Omega }}_{+},\mathbb{R})$ with $\rho >0$, let $U\in C^2({\mathrm{\Omega }}_{+},T{\mathrm{\Omega }}_{+})$ be future timelike with $g(U,U)=-c^2$, and let the smooth electromagnetic two-form and metric be those of that model. Set $a^b:=U^a{\mathrm{\nabla }}_a{U}^b$, $T_m^{ab}:=\rho U^a{U}^b$, and $j^a:=\sigma U^a$.

On ${\mathrm{\Omega }}_{+}$ the adopted charged-dust matter-exchange equation ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$ is equivalent to

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

Under the mass-continuity equation, ${\mathrm{\nabla }}_a{j}^a=0$ is equivalent to transport of the charge-to-mass ratio $\alpha :=\sigma \mathrm{/}\rho$ along the flow, $U^a{\mathrm{\nabla }}_a\alpha =0$. These are model deductions on ${\mathrm{\Omega }}_{+}$; the ratio and the divided force law are not defined at zero density. The result is local to its stated smooth positive-density domain and asserts neither a global smooth continuation nor validity of the single-velocity model beyond that domain. It supplies no Maxwell or Einstein solution or coupled-field existence theorem.