Selected charged perfect-fluid exchange/Ohmic branch

On a smooth open state domain $\mathrm{\Omega }$ in the spacetime and SI conventions of Smooth matter quantities, current and observer observables, choose $C^1$ rest-energy density $ϵ>0$, pressure $p$, rest-charge density $\sigma$, and a $C^2$ future-timelike field $U$ with $g(U,U)=-c^2$. The equation-based coupled branch uses Equation-based Einstein–Maxwell model and Smooth curved vacuum Maxwell constitutive adoption with a smooth symmetric matter tensor and current supplied by this selected fluid model. Adopt the perfect-fluid tensor

$T_m^{ab}=\frac{ϵ+p}{c^2}{U}^a{U}^b+p{g}^{ab}$

and either the convective current $j^a=\sigma U^a$ or a separately specified conduction current $j^a=\sigma U^a+q^a$ with $q^a{U}_a=0$. Adopt charge continuity ${\mathrm{\nabla }}_a{j}^a=0$ and the local matter-exchange balance ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$. The fields and derivatives are smooth enough for these equations to hold pointwise.

For an optional Ohmic branch, choose a smooth scalar conductivity $\chi \ge 0$ in $\mathrm{S}{\mathrm{m}}^{-1}$ and impose $q^a=\chi E_U^a$, where $E_U^a=F^a{}_b{U}^b$ is the rest-frame electric field in $\mathrm{V}{\mathrm{m}}^{-1}$. This is a separate constitutive adoption, not a consequence of covariance or of the Maxwell equations. A smooth equation of state may be supplied on the chosen local state domain; no particular function, relativistic entropy law, causality assertion, or thermodynamic interpretation of $ϵ,p$ is included here. The stress-projection and comoving work identities follow conditionally from Perfect-fluid exchange projections and conduction heating, but no global fluid or conducting-fluid solution theory is asserted.

This postulate adopts a local classical matter branch. It does not presume an unspecified matter action or infer empirical validity from the source literature.