A neutral, constraint-compatible linear-EOS sector

On $J=(0,\mathrm{\infty })$, $p^{\mathrm{\prime }}=w\in (0,1)$ and $ϵ+p=(1+w)ϵ>0$. By [F1] and [F3], $N(ϵ)=N_{\ast }\exp \text{\hspace{0.17em}⁣}\left(\frac{1}{1+w}{\int }_{{ϵ}_{\ast }}^{ϵ}\frac{dz}{z}\right)=N_{\ast }\exp \text{\hspace{0.17em}⁣}\left(\frac{\log (ϵ\mathrm{/}{ϵ}_{\ast })}{1+w}\right)=N_{\ast }{\left(\frac{ϵ}{{ϵ}_{\ast }}\right)}^{1\mathrm{/}(1+w)}\mathrm{.}$ The ratio in the logarithm is dimensionless.

Since $\sqrt{p^{\mathrm{\prime }}(z)}=\sqrt{w}$, [F2] and [F3] give $r(ϵ)=\frac{\sqrt{w}}{1+w}\log \text{\hspace{0.17em}⁣}\left(\frac{ϵ}{{ϵ}_{\ast }}\right),r^{\mathrm{\prime }}(ϵ)=\frac{\sqrt{w}}{(1+w)ϵ}>0,$ so this is the strictly increasing enthalpy coordinate of [F2]. By [F7], the adopted physical characteristic speed is $c_{\mathrm{s}}=c\sqrt{w}\in (0,c)$.

The definition [F4] identifies the torus as the quotient by translations in $(2\pi \mathbb{Z}{)}^3$. For any point of ${\mathbb{R}}^3$, an open ball of radius less than $\pi$ is disjoint from each of its nontrivial lattice translates; the quotient map therefore gives local smooth charts with transition maps that are translations, so the quotient is a smooth three-manifold without boundary. The translations preserve both the Euclidean orientation and metric, so these descend to an orientation and the flat metric $h=\delta$. The continuous quotient map sends the compact cube $[0,2\pi {]}^3$ onto the torus, hence the torus is compact. In these local quotient charts $h$ has constant Euclidean coefficients, so its Levi-Civita derivative vanishes and its curvature is zero.

For the data in the Example, $j_0=0$ because $Z_0=0$. Also $T_{m,0}(\mathfrak{n},\mathfrak{n})=({ϵ}_{\ast }+p({ϵ}_{\ast }))(-1{)}^2+p({ϵ}_{\ast })g_0(\mathfrak{n},\mathfrak{n})={ϵ}_{\ast }$, and $T_{m,0}(\mathfrak{n},X)=0$ for every tangent $X$. The electromagnetic stress vanishes because $e=b=0$.

By step 1.3 the flat torus has $R(h)=0$ and $Dh=0$. In three dimensions, $\tau ={\mathrm{tr}}_h(Hh)=3H$ and $\mathrm{\mid }K{\mathrm{\mid }}_h^2=3H^2$, so the Hamiltonian left side is $9H^2-3H^2=6H^2$. Its right side is $2\mathrm{\Lambda }+2\kappa T_0(\mathfrak{n},\mathfrak{n})=2\kappa {ϵ}_{\ast }=6H^2$ by the choice of ${ϵ}_{\ast }$. The density units are $[3H^2\mathrm{/}\kappa ]={\mathrm{m}}^{-2}\mathrm{/}({\mathrm{m}\mathrm{J}}^{-1})={\mathrm{J}\mathrm{m}}^{-3}$ by the stated units and [F8]. Both sides of the momentum constraint vanish because $H$ is constant and $T_0(\mathfrak{n},X)=0$.

The two Maxwell constraints reduce to ${\mathrm{div}}_{h}0=0$ and $-{\mu }_0{g}_0(j_0,\mathfrak{n})=0$, so both hold. The fields are smooth, and their density range $\{{ϵ}_{\ast }\}$ is contained in the compact interval $K_{ϵ}=[{ϵ}_{\ast }\mathrm{/}2,3{ϵ}_{\ast }\mathrm{/}2]⋐J$. All strict EOS inequalities hold on $K_{ϵ}$ by step 1.1. Thus the complete smooth-data hypotheses of [F5] hold.

Applying [F5] under [A1] and [A2] gives the smooth local development for these exact constrained data. The conclusion is conditional on the adopted fluid model and the stated mathematical hypotheses of [L1] and [L2]. This example reports no observation or empirical validation of the model.