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.

In the adopted fixed-composition local-equilibrium material model of Restricted local equilibrium parcel with adopted material state function, let $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ be its open state domain, let $e\in C^2(X,\mathbb{R})$ include the rest-energy contribution, and set $T=e_{{\sigma }_p}$, $p=-e_v$, $\epsilon =ne$, and $\mu =e+pv-T{\sigma }_p$, with $n=1\mathrm{/}v$ and $s=n{\sigma }_p$ as in Rest fluid state, frame and geometric current definitions. The model's local state identities are $d\epsilon =Tds+\mu dn,\epsilon +p=Ts+\mu n,dp=sdT+nd\mu \mathrm{.}$ For a local continuum parcel with real extensive marker variable $\mathsf{N}>0$, proper volume $\mathsf{V}>0$, entropy $\mathsf{S}\in \mathbb{R}$, and $(\mathsf{V}\mathrm{/}\mathsf{N},\mathsf{S}\mathrm{/}\mathsf{N})\in X$, its adopted energy is $E=\mathsf{N}e(\mathsf{V}\mathrm{/}\mathsf{N},\mathsf{S}\mathrm{/}\mathsf{N})$. On this open parcel domain it obeys $dE=Td\mathsf{S}-pd\mathsf{V}+\mu d\mathsf{N},E=T\mathsf{S}-p\mathsf{V}+\mu \mathsf{N}\mathrm{.}$ The count variable is treated continuously as part of the local-equilibrium continuum model; these differentials are not claims about discrete microscopic transitions. The result is conditional model calculus and does not establish that a chosen state function describes a particular material.

Assume the smooth isolated perfect material model of Smooth perfect local material balance model on an open region $O$ of a time-oriented Lorentzian four-manifold $(M,g)$ with signature $(-,+,+,+)$. Thus $u$ is a smooth future-directed unit timelike observer-congruence field, $g(u,u)=-1$, the local state is the $C^2$ state of Rest fluid state, frame and geometric current definitions with $n>0$ and $T>0$, $J^a=n{u}^a$, ${\mathsf{T}}^{ab}=\epsilon u^a{u}^b+p(g^{ab}+u^a{u}^b)$, and ${\mathrm{\nabla }}_a{J}^a=0$, ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$. Write $D=u^a{\mathrm{\nabla }}_a$ and $u^{\mathrm{♭}}=g(u,\cdot )$. For every smooth closed curve ${\gamma }_0:[0,1]\to O$ whose image is compact and for which a local flow ${\mathrm{\Phi }}_{\lambda }$ of $u$ is defined on a neighborhood of that image for $\lambda$ in an interval about $0$, set ${\gamma }_{\lambda }={\mathrm{\Phi }}_{\lambda }\circ {\gamma }_0$, $H=(\epsilon +p)\mathrm{/}n$, $P=(H\mathrm{/}c)u^{\mathrm{♭}}$, and $\mathcal{C}(\lambda )={\int }_{{\gamma }_{\lambda }}P$, with $c$ and the length-coordinate convention of Thermodynamics-GR fields, currents, reference normalization and units. Conditional on the exact current mathematical supplier interfaces, the entropy identities and loop law are ${\mathrm{\nabla }}_a(s{u}^a)=0,D{\sigma }_p=0,\frac{d\mathcal{C}}{d\lambda }=\frac{1}{c}{\int }_{{\gamma }_{\lambda }}Td{\sigma }_p\mathrm{.}$ In particular, $\mathcal{C}$ is locally constant if ${\sigma }_p$ is constant on the whole region swept out by the loop. Material advection alone, $D{\sigma }_p=0$, does not imply $d{\sigma }_p=0$ or circulation conservation. These are conditional model deductions, not an existence theorem or empirical claim.