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.

Adopt the equation-based Einstein-Maxwell field postulate Equation-based Einstein–Maxwell model and the smooth material closure of Adopted strict charged barotropic fluid model, with the SI, signature, and coordinate conventions of Einstein–Maxwell primitive quantities and SI conventions. Thus the spacetime field equations are $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, and $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa (T_{m,ab}+T_{\mathrm{E}\mathrm{M},ab}),\kappa =8\pi G_N\mathrm{/}{c}^4,$ while the fluid contributes exactly its adopted $T_m,j$ and material balance equations. No matter action is assumed.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ and Dependent Choice. Let $p=p(ϵ)$ be smooth on an open interval $J$, and let $K_{ϵ}⋐J$ be a compact interval containing the initial density values, on which $ϵ+p(ϵ)>0$ and $0<p^{\mathrm{\prime }}(ϵ)<1$. Take $\kappa >0$, ${\mu }_0>0$, $c>0$, and $\mathrm{\Lambda }\in \mathbb{R}$.

Let $\mathrm{\Sigma }$ be a smooth oriented three-manifold without boundary, and let $\mathfrak{n}$ be its future unit normal. Let $h$ be a Riemannian metric and let $K\in \mathrm{\Gamma }({\mathrm{Sym}}^2{T}^{\ast }\mathrm{\Sigma })$ use the second fundamental form convention $K(X,Y):=g({\mathrm{\nabla }}_X\mathfrak{n},Y)\mathrm{.}$ The initial normal bundle is $\mathcal{V}:=\mathbb{R}\mathfrak{n}\oplus T\mathrm{\Sigma }$ with $g_0(a\mathfrak{n}+X,b\mathfrak{n}+Y):=-ab+h(X,Y)\mathrm{.}$ Let ${ϵ}_0,Z_0$ be smooth scalar fields, $u_0$ a smooth future unit timelike section of $\mathcal{V}$ satisfying $g_0(u_0,u_0)=-1$, and $e,b$ smooth spatial vector fields representing $E\mathrm{/}c$ and $B$ in an oriented orthonormal frame. Their units are J m${}^{-3}$, C, dimensionless, and tesla, respectively; $K$ has units m${}^{-1}$. The components of the smooth initial two-form $F_0$ are defined in every positively oriented $h$-orthonormal frame $({\mathsf{e}}_i)$ by $e_i=F_0({\mathsf{e}}_i,\mathfrak{n}),b_i=\frac{1}{2}{\epsilon }_i{}^{jk}{F}_0({\mathsf{e}}_j,{\mathsf{e}}_k),{\epsilon }_{123}=1.$ This fixes all six independent components of $F_0$. Set $j_0=cN({ϵ}_0)Z_0{u}_0$ and $T_0=T_{m,0}+T_{\mathrm{E}\mathrm{M},0}$ from the adopted model.

Write $D$ for the Levi-Civita connection of $h$, $\tau :={\mathrm{tr}}_{h}K=h^{ij}{K}_{ij}$, $\mathrm{\mid }K{\mathrm{\mid }}_h^2:=h^{ik}{h}^{jl}{K}_{ij}{K}_{kl}$, $({\mathrm{div}}_{h}K{)}_j:=h^{ik}(D_{i}K{)}_{kj}$, and $R(h)$ for the scalar curvature of $h$. Assume the complete Einstein and sourced Maxwell initial constraints $R(h)+{\tau }^2-\mathrm{\mid }K{\mathrm{\mid }}_h^2=2\mathrm{\Lambda }+2\kappa T_0(\mathfrak{n},\mathfrak{n}),({\mathrm{div}}_{h}K)(X)-D_X\tau =\kappa T_0(\mathfrak{n},X)$ for every tangent vector $X$, and ${\mathrm{div}}_{h}e=-{\mu }_0{g}_0(j_0,\mathfrak{n}),{\mathrm{div}}_{h}b=0.$

Then the adopted equation-based fluid model has the following conditional local developments.

(a) Torus Sobolev branch. If $\mathrm{\Sigma }={\mathbb{T}}_{2\pi }^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$ with the periodic Sobolev spaces of The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s, assigning each spatial period length $2\pi \mathrm{m}$ in the length-coordinate convention of [F6]. Assume $s\ge 5$ is an integer, $h\in H^{s+1}$ and $K,e,b,{ϵ}_0,u_0,Z_0\in H^s$, with the same strict compact admissibility. Then there is $T>0$ and a solution on $(-T,T)\times {\mathbb{T}}^3$ in the harmonic gauge of The harmonic gauge vector, the reduced Einstein equation and the Einstein residual, relative to the fixed background used by Local charged Einstein–Maxwell–barotropic development. In periodic charts, $g\in L^{\mathrm{\infty }}{H}^{s+1}\cap C^0{H}^s$, ${\mathrm{\partial }}_{t}g\in L^{\mathrm{\infty }}{H}^s\cap C^0{H}^{s-1}$, and $F,ϵ,u,Z\in L^{\mathrm{\infty }}{H}^s\cap C^0{H}^{s-1}$. The equations hold classically, and the reduced solution is unique on common intervals with common compact admissible bounds.

(b) Smooth geometric branch. For smooth data on any $\mathrm{\Sigma }$ above, there is a smooth time-oriented solution on an open variable-width neighborhood of $\{0\}\times \mathrm{\Sigma }$, inducing exactly $(h,K,F_0,{ϵ}_0,u_0,Z_0)$. It is unique near the initial surface up to a data-fixing time-orientation-preserving isometry matching $F,ϵ,u,Z$. No uniform time width for noncompact $\mathrm{\Sigma }$, global existence, shock continuation, vacuum/free-boundary solution, or empirical validity is asserted.

This is a model deduction. It does not assert that the postulates describe nature.