The barotropic fluid equations in enthalpy form are symmetric hyperbolic with positive principal symbol

(Equations and conservation.) By [L1], $Dr=c_{s}Dϵ\mathrm{/}(ϵ+p)$ and $c_s{\mathrm{\partial }}_{a}r=p^{\mathrm{\prime }}{\mathrm{\partial }}_aϵ\mathrm{/}(ϵ+p)$. Hence division by $ϵ+p>0$ transforms the energy and Euler equations into the displayed enthalpy system, and the inverse transformation recovers them. Applying [L2] proves equivalence to stress conservation for every metric and normalized velocity, without requiring Einstein's equation.

(Actual independent velocity coordinates.) Restrict a chart to the open metric states with $g_{00}<0$ and positive spatial metric block, as holds near its initially orthogonal surface. Use independent components $y^i=u^i$ and set $b=g_{0i}{y}^i$, $c=g_{ij}{y}^i{y}^j$. The future branch of the normalization equation is $u^0=\frac{-b-\sqrt{b^2-g_{00}(1+c)}}{g_{00}}\mathrm{.}$ The discriminant is positive and $u^0>0$. It is smooth, and differentiating normalization gives $g(u,e_A)=0$ for $e_A=\mathrm{\partial }u\mathrm{/}\mathrm{\partial }{y}^A$ at fixed metric. The spatial components of $e_A$ are the standard basis, so these three vectors are a smooth basis of $u^{\perp }$. This parametrizes all future unit velocities in the local chart. Terms from differentiating $u(g,y)$ with respect to $g$ multiply $\mathrm{\partial }g$; the connection terms do likewise. When metric first-order variables are appended, all are zeroth-order sources in the full unknowns.

(Smooth symmetric coefficient matrices.) In the leading part of step 1.1, ${\mathrm{\partial }}_{\mu }u=e_B{\mathrm{\partial }}_{\mu }{y}^B$ modulo the metric-derivative terms of step 1.2. Multiply the Euler vector equation by $g(e_A,\cdot )$. Its components together with the energy equation have coefficient matrices at ${\mathrm{\partial }}_{\mu }$ with entries $A_{rr}^{\mu }=u^{\mu },A_{rB}^{\mu }=c_s{e}_B^{\mu }=A_{Br}^{\mu },A_{AB}^{\mu }=u^{\mu }g(e_A,e_B)\mathrm{.}$ They are smooth and symmetric, and their covector contraction is exactly $Q_{\zeta }$ in the Statement. The Euler equation is tangent to $u^{\perp }$ because unit normalization is enforced, so taking its three $e_A$ components is equivalent to its full vector equation. Thus this is an actual equivalent coordinate system, not only a symbol computed in a different rest frame at each point.

(Positivity and eigenvalues.) For $a=\delta r$ and $v=\delta u\in u^{\perp }$, [L4] and Cauchy-Schwarz give $\mathrm{\mid }\zeta (v)\mathrm{\mid }\le \mathrm{\mid }{\zeta }_{\perp }\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }$ and therefore $Q_{\zeta }((a,v),(a,v))\ge (\zeta (u)-c_s\mathrm{\mid }{\zeta }_{\perp }\mathrm{\mid })(a^2+\mathrm{\mid }v{\mathrm{\mid }}^2)>0.$ The strict last inequality uses $c_s<1$ and $\mathrm{\mid }{\zeta }_{\perp }\mathrm{\mid }<\zeta (u)$. In a rest orthonormal basis the matrix has diagonal blocks $\zeta (u)$ and $\zeta (u)I_3$ and off-diagonal $c_s{\zeta }_{\perp }$. The two directions perpendicular to ${\zeta }_{\perp }$ give eigenvalue $\zeta (u)$, and the remaining two-dimensional block gives $\zeta (u)\pm c_s\mathrm{\mid }{\zeta }_{\perp }\mathrm{\mid }$. Thus the actual time matrix for the positively evaluated timelike time conormal is positive; after relatively compact shrinking its lower bound is uniform.

(Coupling, complete initialization and recovery.) Append [L3]'s metric block. Stress depends algebraically on $(g,r,y)$; the fluid metric derivatives from step 1.2 depend algebraically on the metric first-order variables, so the combined principal matrices remain block diagonal and symmetric. The time block is positive by step 3.1 and [L3]. At the initial orthogonal metric, the prescribed $v_0$ gives the normalized bundle velocity $U_0=W_0(N+v_0)$ and has coordinate components $u^0=W_0\mathrm{/}{\alpha }_0$, $u^i=W_0{v}_0^i$, since $N={\alpha }_0^{-1}{\mathrm{\partial }}_t$. Indeed $g^0(U_0,U_0)=W_0^2(-1+h(v_0,v_0))=-1$. This initializes every independent fluid variable as stated. Reconstruction keeps $u$ unit and future; the smooth inverse enthalpy map and step 2.1 recover the full fluid equations on any reduced solution. Step 1.1 then gives conserved stress on that reduced metric before imposing Einstein recovery. Conversely every physical fluid solution transforms to these same equations and variables. All asserted conclusions follow.