Strict charged barotropic tensor system

Use the oriented, time-oriented smooth Lorentzian four-manifold, metric-selected Levi-Civita connection, signature $(-,+,+,+)$, two-form $F$, Einstein tensor $G$, and SI tensor conventions of Electrovacuum tensor Cauchy system. In particular, $F$ is a smooth real section of ${\mathrm{\Lambda }}^2{T}^{\ast }M$, $x^0=ct$ is a length coordinate, the electromagnetic stress is

$T_{\mathrm{E}\mathrm{M},ab}={\mu }_0^{-1}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd}),$

and the sourced Maxwell convention is $dF=0$ and $d{\star }_{g}F={\mu }_0{\iota }_j{\mathrm{vol}}_g$, equivalently ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$. These are mathematical model equations, not claims that the model describes nature.

Choose an open real interval $J$ and a smooth equation of state $p:J\to \mathbb{R}$ such that $ϵ+p(ϵ)>0$ and $0<p^{\mathrm{\prime }}(ϵ)<1$ for every $ϵ\in J$. The rest-energy density $ϵ:M\to J$ and pressure $p(ϵ)$ have SI units J m${}^{-3}$=Pa; $p$ depends only on $ϵ$, which is the meaning of barotropic here. Fix ${ϵ}_{\ast }\in J$ and a constant $N_{\ast }>0$ with units m${}^{-3}$ and define the positive EOS scalar

$N(ϵ)=N_{\ast }\exp \text{\hspace{0.17em}⁣}\left({\int }_{{ϵ}_{\ast }}^{ϵ}\frac{dz}{z+p(z)}\right)\mathrm{.}$

For each $ϵ\in J$, the closed interval between ${ϵ}_{\ast }$ and $ϵ$ is a compact subset of $J$, so the smooth integrand has a finite ordinary integral there. The exponent is dimensionless and $N(ϵ)>0$. This is an EOS-defined number-density variable and normalization, not a microscopic particle derivation.

Let $u$ be a smooth future-directed vector field with $g(u,u)=-1$, and let $Z:M\to \mathbb{R}$ be a smooth scalar with units coulomb, interpreted within this model as charge per number-density unit. Define the current and matter stress by

$j^a=cN(ϵ)Z{u}^a,T_{m,ab}=(ϵ+p(ϵ))u_a{u}_b+p(ϵ)g_{ab}\mathrm{.}$

Here $u$ is dimensionless, $j$ has SI component units A m${}^{-2}$, and $T_m$ has units Pa in the $x^0=ct$ convention. The strict charged barotropic system is the smooth tensor system

${\mathrm{\nabla }}^a{T}_{m,ab}=F_{bc}{j}^c,u^a{\mathrm{\nabla }}_{a}Z=0,$

coupled to the stated sourced Maxwell equations and

$G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa (T_{m,ab}+T_{\mathrm{E}\mathrm{M},ab}),\kappa =8\pi G_N\mathrm{/}{c}^4>0,\mathrm{\Lambda }\in \mathbb{R}\mathrm{.}$

The first equation is the charged-fluid energy-momentum balance; its Lorentz-force term is orthogonal to $u$ by antisymmetry of $F$. The second transports the charge-per-particle scalar along the fluid flow. No independent current continuity law is added: it is derived in Charged-fluid current and total stress conservation. No dust, shock, vacuum interface, free boundary, degenerate sound speed, superluminal sound speed, finite-conductivity law, or point-charge source is included.

The same displayed tensor formulas also define the classical finite-regularity branch on a smooth Hausdorff, second-countable underlying oriented four-manifold with a time-oriented $C^3$ Lorentzian metric and $C^2$ fields $F,ϵ,u,Z$. Retain $ϵ(M)\subset J$, $g(u,u)=-1$, future direction, the positive constants, the strict EOS inequalities and all tensor/sign conventions above. In this branch the equations are pointwise classical equations, not a smoothness assertion or a distributional extension. The original smooth sector is retained.

These expressions are well defined at the declared orders: the connection is $C^2$ and the curvature is $C^1$; $N(ϵ)$, $j$ and both stresses are $C^2$ by the ordinary chain and product rules, since the scalar EOS maps are smooth on $J$. Their first covariant derivatives and $dF$ are continuous, so every displayed equation compares classical tensors. This domain extension asserts no solution existence, shock continuation or regularity below these orders.