Smooth perfect local material balance model

Let $O$ be an open region of a smooth, oriented, time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$, with its metric-compatible torsion-free connection $\mathrm{\nabla }$. Use the local material-state and unit conventions of Rest fluid state, frame and geometric current definitions and Thermodynamics-GR fields, currents, reference normalization and units: a smooth future-directed unit timelike observer-congruence field $u$ has $g(u,u)=-1$; a $C^2$ state map $(v,{\sigma }_p):O\to X$ takes values in a nonempty open $X\subset (0,\mathrm{\infty })\times \mathbb{R}$; and a supplied $e\in C^2(X,\mathbb{R})$ defines $n=1\mathrm{/}v>0$, $s=n{\sigma }_p$, $\epsilon =ne$, $T={\mathrm{\partial }}_{{\sigma }_p}e$, and $p=-{\mathrm{\partial }}_{v}e$. Adopt the selected fixed-composition material branch of Restricted local equilibrium parcel with adopted material state function, including its stipulated local-equilibrium regime and $T>0$.

For this branch, adopt the perfect-fluid matter model $J^a:=n{u}^a,{\mathsf{T}}^{ab}:=\epsilon u^a{u}^b+p{\mathrm{\Delta }}^{ab},{\mathrm{\Delta }}^{ab}:=g^{ab}+u^a{u}^b,$ with local balances ${\mathrm{\nabla }}_a{J}^a=0$ and ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$ on $O$. Here $J^a$ is the selected material-marker current and ${\mathsf{T}}^{ab}$ is the symmetric stress-energy tensor; the perfect branch adopts zero spatial marker diffusion and isotropic rest-space stress. The balances are physical model assumptions in the stated smooth regime, not conclusions from the state function or from geometry. They may be used on a prescribed background or together with an explicitly specified coupled Einstein model; this postulate does not assert a solution exists for arbitrary initial data or an arbitrary equation of state.

The marker receives its material-particle interpretation from the fixed-composition premise. No microscopic equation of state, universal material applicability, nonzero external force, dissipative flux, global self-gravitating energy additivity, or empirical observation is asserted.