Local evolution for a strict barotropic perfect fluid: smooth constrained data admit a development, unique up to a data-fixing isometry

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Fix $\mathrm{\Lambda }\in \mathbb{R}$, $\kappa >0$ and one smooth EOS $p=p(ϵ)$ on an open interval $J$ with $ϵ+p>0$ and $0<p^{\mathrm{\prime }}(ϵ)<1$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy). Let $\mathrm{\Sigma }$ be a smooth three-manifold with smooth metric $h$, symmetric tensor $K_2$, density ${ϵ}_0:\mathrm{\Sigma }\to J$ and tangent field $v_0$ with $h(v_0,v_0)<1$. These specify the future unit initial velocity $U_0=W_0(N+v_0),W_0=(1-h(v_0,v_0){)}^{-1\mathrm{/}2}$ in $\mathcal{V}=\mathbb{R}N\oplus T\mathrm{\Sigma }$, with metric $g_0(aN+X,bN+Y)=-ab+h(X,Y)$. Require the constraints of Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form for $T_0=({ϵ}_0+p({ϵ}_0))U_0^{{\mathrm{♭}}_{g_0}}\otimes U_0^{{\mathrm{♭}}_{g_0}}+p({ϵ}_0)g_0\mathrm{.}$ Then a smooth time-oriented local development has metric $g$, density $ϵ$ and future unit fluid velocity $u$ satisfying $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa [(ϵ+p)u_a{u}_b+p{g}_{ab}],Dϵ+(ϵ+p)\vartheta =0,(ϵ+p)a^b+{\mathrm{\Pi }}^{ba}{\mathrm{\partial }}_{a}p=0.$ Here $D={\mathrm{\nabla }}_u$, $\vartheta ={\mathrm{\nabla }}_a{u}^a$, $a={\mathrm{\nabla }}_{u}u$ and ${\mathrm{\Pi }}^{ab}=g^{ab}+u^a{u}^b$. Initial geometric data are $(h,K_2)$ and initial matter data are $({ϵ}_0,U_0)$ under $N\mapsto n$, $X\mapsto i_{\ast }X$. Arbitrary smooth subluminal tilt is allowed. The initial surface lies in the interior of its intrinsic domain of dependence; its interior Cauchy neighborhood is globally hyperbolic. Any two developments of this same EOS and complete data agree near the entire initial surface by a data-fixing future isometry matching density and velocity. The neighborhood may have position-dependent time width; shocks, vacuum boundaries and vanishing sound speed are outside this strict interior-fluid statement (Local geometric development for a fixed covariant Einstein-matter model with complete constrained data).

A timelike initial velocity is specified in the abstract Lorentz bundle over the Riemannian surface; it is not an intrinsic timelike vector of its tangent bundle. No small-tilt restriction or uniform noncompact width is required.