Adopted strict charged barotropic fluid model

Use the smooth oriented, time-oriented Lorentzian four-manifold, metric-selected connection, signature $(-,+,+,+)$, two-form field, and SI conventions stated in Einstein–Maxwell primitive quantities and SI conventions and Strict charged barotropic tensor system. On an open set $U$ of that spacetime, let $ϵ:U\to J$ be a smooth rest-energy-density scalar, let $u$ be a smooth future-directed vector field with $g(u,u)=-1$, and let $Z:U\to \mathbb{R}$ be a smooth charge-per-particle scalar. Here $J\subseteq \mathbb{R}$ is an open interval, $ϵ$ and $p(ϵ)$ have units J m${}^{-3}$ = Pa, $u$ is dimensionless, and $Z$ has units C. In this adopted model, the word barotropic means that pressure is a function of rest-energy density alone.

Adopt a smooth equation of state $p:J\to \mathbb{R}$ satisfying $ϵ+p(ϵ)>0,0<p^{\mathrm{\prime }}(ϵ)<1$ for every $ϵ\in J$. The dimensionless derivative gives the model's barotropic characteristic speed $c_{\mathrm{s}}:=c\sqrt{p^{\mathrm{\prime }}(ϵ)}$, so this selection requires $0<c_{\mathrm{s}}<c$. Fix ${ϵ}_{\ast }\in J$ and a positive normalization $N_{\ast }>0$ with units m${}^{-3}$, and define $N(ϵ):=N_{\ast }\exp \text{\hspace{0.17em}⁣}\left({\int }_{{ϵ}_{\ast }}^{ϵ}\frac{dz}{z+p(z)}\right)\mathrm{.}$ For each $ϵ\in J$, the closed segment between ${ϵ}_{\ast }$ and $ϵ$ is compact and contained in $J$; the smooth integrand is finite there. Its integral is dimensionless, so $N(ϵ)>0$ has units m${}^{-3}$. Within this model, $N$ is the EOS-defined number-density variable and $N_{\ast }$ is its chosen normalization. This definition makes no claim that a microscopic particle theory derives $N$ for every EOS.

Adopt the charged-fluid current and matter stress tensor $j^a:=cN(ϵ)Z{u}^a,T_{m,ab}:=(ϵ+p(ϵ))u_a{u}_b+p(ϵ)g_{ab},$ and the material balance and advection equations ${\mathrm{\nabla }}^a{T}_{m,ab}=F_{bc}{j}^c,u^a{\mathrm{\nabla }}_{a}Z=0.$ Thus $j$ is a smooth four-current with components in A m${}^{-2}$, and $T_m$ has components in Pa in length coordinates $x^0=ct$. The balance law adopts the Lorentz-force density with the same field and current conventions as the chosen Einstein-Maxwell formulation. Charge-per-particle is carried along the fluid flow. Current conservation is not an additional adopted material law.

This is a physical postulate for smooth classical matter in the strict interior regime only. It excludes shocks, vacuum interfaces, free boundaries, degenerate or superluminal sound speed, finite-conductivity and dissipative currents, point-charge sources, and arbitrary currents chosen independently of the fluid. The cited treatments motivate the perfect-fluid and charged-current structures in their own conventions; they do not establish this exact EOS normalization, the $Z$ transport choice, or the physical applicability of this model. No matter action, experiment, or universal material law is asserted.