Perfect stress and particle balances with entropy identity

Given: A nonempty open set $O$ in a smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of signature $(-,+,+,+)$; a smooth future unit field $u$ on $O$; a $C^2$ state map into a nonempty open material-state domain; and the fields and balances specified in the Statement.

The geometric and tensor-bundle construction carries Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no stronger choice principle is assumed.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: $M$ is a smooth four-manifold without boundary, $g$ is a smooth symmetric nondegenerate metric of signature $(-,+,+,+)$, and $N$ selects the future cone. Its stated tangent-bundle construction assumes only ${\mathrm{A}\mathrm{C}}_{\omega }$.

Rest fluid state, frame and geometric current definitions: on a nonempty open set $O$, the observer field is smooth and future unit timelike; a local state has a nonempty open domain $X\subset (0,\mathrm{\infty })\times \mathbb{R}$, a specified $e\in C^2(X)$, and a $C^2$ state map $(v,{\sigma }_p):O\to X$. It defines $n=v^{-1}$, $s=n{\sigma }_p$, $ϵ=ne$, $T=e_{{\sigma }_p}$, $p=-e_v$, $H=e+pv$, $h=ϵ+p=nH$, and $\mu =H-T{\sigma }_p$. The convention is signature $(-,+,+,+)$ with $x^0=ct$; $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $u$ is dimensionless, $n$ has units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, $T$ has units K, $\mu$ has units $\mathrm{J}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$, and $ϵ,p,h$ have units Pa.

Local Gibbs, Euler, parcel first law and response identities (current batch-53 draft): for this state, $dϵ=Tds+\mu dn$ and $ϵ+p=Ts+\mu n$ as identities of differentials and state functions. The identities are used conditionally at the current carrier's stated $C^2$ hypotheses.

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: for a smooth nondegenerate metric, the metric-selected connection is torsion-free and metric-compatible, and its induced tensor derivative obeys the product and contraction rules. This is the published mathematical supplier actually inspected here.

Define ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a^b=u^a{\mathrm{\nabla }}_a{u}^b$. The perfect model data are $T^{ab}=ϵu^a{u}^b+p{\mathrm{\Delta }}^{ab}$, $J^a=n{u}^a$, and ${\mathrm{\nabla }}_a{T}^{ab}=f^b$, ${\mathrm{\nabla }}_a{J}^a=0$. In SI, $f^a$ has units $\mathrm{N}{\mathrm{m}}^{-3}$, $ϵ,p,h$ have units Pa, $n$ has units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, and $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$.

Given: A nonempty open set $O$ in a smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of signature $(-,+,+,+)$, a smooth future unit field $u$ on $O$, a $C^2$ state map into a nonempty open material-state domain, an isolated perfect-fluid stress and particle balances ${\mathrm{\nabla }}_a{T}^{ab}=0$ and ${\mathrm{\nabla }}_a(n{u}^a)=0$, a compact closed curve in $O$, and $c>0$.

The geometric and tensor-bundle constructions use only Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Rest fluid state, frame and geometric current definitions (current batch-53 draft): on a nonempty open set $O$, the observer field is smooth and future unit timelike; a local state has a nonempty open domain $X\subset (0,\mathrm{\infty })\times \mathbb{R}$, a specified $e\in C^2(X)$, and a $C^2$ state map $(v,{\sigma }_p):O\to X$. It defines $n=v^{-1}$, $s=n{\sigma }_p$, $ϵ=ne$, $T=e_{{\sigma }_p}$, $p=-e_v$, $H=e+pv$, $h=ϵ+p=nH$, and $\mu =H-T{\sigma }_p$. The SI convention has signature $(-,+,+,+)$ and $x^0=ct$; $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $u$ is dimensionless, $H$ has units $\mathrm{J}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$, and the components of $(H\mathrm{/}c)u^{\mathrm{♭}}$ have units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$. The circulation $\mathcal{C}$ has units $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$. Here $H$ is enthalpy per marker and $h$ is an energy density.

Perfect stress and particle balances with entropy identity (current batch-55 draft): for smooth state fields, perfect stress and particle current, ${\mathrm{\nabla }}_a{T}^{ab}=f^b$ and ${\mathrm{\nabla }}_a{J}^a=0$ are equivalent to the stated parallel/spatial stress balances and $Dn+n\theta =0$. With both balances, it states $T{\mathrm{\nabla }}_a(s{u}^a)=-u_b{f}^b$ and $nTD{\sigma }_p=-u_b{f}^b$. Here $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, $a^b=u^a{\mathrm{\nabla }}_a{u}^b$, and $(\mathrm{\Delta }\alpha {)}_b={\alpha }_b+u_b{u}^a{\alpha }_a$ for a covector $\alpha$. The supplier's smooth-state hypothesis is stronger than [F1]; step 1.2 proves the needed isolated identities directly at [F1]'s regularity.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: the metric is smooth, nondegenerate, time-oriented, and has signature $(-,+,+,+)$.

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: $\mathrm{\nabla }g=0$, the connection is torsion-free, and it obeys the tensor product rule.

Local flows by a contraction argument supplies unique local flows for $C^1$ coordinate vector fields, and Variational equations and commuting local flows supplies their joint $C^2$ dependence when the field is $C^2$. Both are published mathematical items; the coordinate hypotheses are checked in the proof.

The given closed curve is smooth, ${\gamma }_0(0)={\gamma }_0(1)$, and its image is compact. The local flow is parameterized by ${\mathrm{\partial }}_{\lambda }{\mathrm{\Phi }}_{\lambda }=u$. For a one-form $\alpha$, $d\alpha (X,Y)=X(\alpha (Y))-Y(\alpha (X))-\alpha ([X,Y])$; ${\iota }_u\omega =\omega (u,\cdot )$ and $u^{\mathrm{♭}}=g(u,\cdot )$.

Given: A nonempty open set $O$ and the smooth Lorentzian data of Rest fluid state, frame and geometric current definitions, a $C^2$ state map into a nonempty open state domain, the thermodynamic identities of Local Gibbs, Euler, parcel first law and response identities, and the stress and particle balances displayed in the Statement.

The spacetime and tensor construction assumes Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no stronger choice principle is used.

Rest fluid state, frame and geometric current definitions (current batch-53 draft): on a nonempty open set $O$, the observer field is smooth and future unit timelike; a local state has a nonempty open domain $X\subset (0,\mathrm{\infty })\times \mathbb{R}$, a specified $e\in C^2(X)$, and a $C^2$ state map $(v,{\sigma }_p):O\to X$. It defines $n=v^{-1}>0$, $s=n{\sigma }_p$, $ϵ=ne$, $T=e_{{\sigma }_p}$, $p=-e_v$, $H=e+pv$, $h=ϵ+p=nH$, and $\mu =H-T{\sigma }_p$.

Local Gibbs, Euler, parcel first law and response identities (current batch-53 draft): under the stated nonempty open-domain and $C^2$ hypotheses, it gives $dϵ=Tds+\mu dn$ and $ϵ+p=Ts+\mu n$. Applying the first identity to the state map along $D$ gives the directional identity used below. This use is conditional on the current draft carrier.

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: the induced tensor derivative satisfies the product rule, $\mathrm{\nabla }g=0$, and contraction commutes with differentiation.

In the SI convention of [F1], $u$ and $\mathrm{\Delta }$ are dimensionless, $D,\theta ,a,\sigma$ have inverse-length components, $ϵ,p,\mathrm{\Pi },Q,\pi$ have units Pa, $n,V$ have units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, $T$ has units K, and $\mu$ has units J per marker. The force density $f$ has units $\mathrm{N}{\mathrm{m}}^{-3}$, the physical heat flux is $q=cQ$, and $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$.

Define ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a^b=u^a{\mathrm{\nabla }}_a{u}^b$. The stress decomposition, spatial symmetries, and balance equations are exactly those in the Statement.

Given: Countable Choice; a nonempty open spacetime domain with the smooth Lorentzian frame and $C^2$ local state specified by Rest fluid state, frame and geometric current definitions, with $n,T>0$; and the model equations and constants stated above.

Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ is assumed where the heat-kernel and Lorentzian suppliers require it (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Frame-qualified dissipative entropy cancellation: for the displayed stress and particle balances, ${\mathrm{\nabla }}_a{S}^a=-\frac{\mathrm{\Pi }\theta +{\pi }^{ab}{\sigma }_{ab}}{T}-\frac{Q^a({\mathrm{\nabla }}_{a}T+T{a}_a)}{T^2}-V^a{\mathrm{\nabla }}_a(\mu \mathrm{/}T)-\frac{u_b{f}^b}{T}\mathrm{.}$ This is the current batch-55 conditional draft; its tgr-mc01 input hashes are stale, so its use remains conditional.

The heat kernel on ${\mathbb{R}}^n$ and its causal extension: for $\tau >0$, $\mathrm{\Gamma }(\mathbf{x},\tau )=(4\pi \tau {)}^{-3\mathrm{/}2}\exp (-\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}^2\mathrm{/}(4\tau ))$, which is strictly positive.

Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel: assuming Countable Choice, $\mathrm{\Gamma }(\cdot ,\tau )>0$ and has unit integral for every $\tau >0$.

The heat Cauchy problem for bounded uniformly continuous data: assuming Countable Choice, for bounded uniformly continuous ${\vartheta }_0:{\mathbb{R}}^3\to \mathbb{R}$, the convolution $\int \mathrm{\Gamma }(\mathbf{x}-\mathbf{y},\tau ){\vartheta }_0(\mathbf{y})d\mathbf{y}$ is smooth for $\tau >0$, solves $U_{\tau }=\mathrm{\Delta }U$, is bounded, and recovers ${\vartheta }_0$ locally uniformly at $\tau =0$; it also obeys $\mathrm{\parallel }U(\cdot ,\tau ){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{\vartheta }_0{\mathrm{\parallel }}_{\mathrm{\infty }}$.

For the supported fixed-rest branch, $M={\mathbb{R}}^4$ has Minkowski metric $g=\mathrm{diag}(-1,1,1,1)$, $x^0=ct$, the field domain is $t>0$, $\mathbf{x}\in {\mathbb{R}}^3$, and $u=(1,0,0,0)$. With $V=\mathrm{\Pi }=\pi =0$ and $\eta =\zeta =L=0$, the external support is specified by $f^0=0$ and $f^i={\mathrm{\partial }}_{i}p+c^{-1}{\mathrm{\partial }}_t{Q}^i$. On a fixed-volume state slice $v=v_0$, $T=e_{{\sigma }_p}$ is a valid coordinate and $(\mathrm{\partial }ϵ\mathrm{/}\mathrm{\partial }T{)}_{v_0}=C>0$ is constant on an interval $[T_0,T_0+\delta ]$ inside the material-state domain, so $ϵ={ϵ}_0+C(T-T_0)$ there. Here $T_0,\delta ,\kappa >0$ and ${\vartheta }_0\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ satisfies $0\le {\vartheta }_0\le \delta$ and ${\vartheta }_0\equiv ̸0$.

In the entropy-sign branch assume $u_b{f}^b=0$. This includes the isolated case $f^b=0$; in the supported fixed-rest branch of [A2], $u_b{f}^b=-f^0=0$ although the spatial support force may be nonzero.

In the homogeneous Eckart linearization, $f=0$, the background is constant with $n_0,T_0>0$, $h_0={ϵ}_0+p_0>0$, and $\kappa >0$; $\delta n=\delta T=0$, $V=0$, and all spatial gradients vanish. In SI, $C$ has units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, $\kappa$ has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, $q$ has units $\mathrm{W}{\mathrm{m}}^{-2}$, $h_0$ has units Pa, $\lambda$ has units ${\mathrm{s}}^{-1}$, $\eta ,\zeta$ have units $\mathrm{P}\mathrm{a}\mathrm{s}$, $L$ has units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^2\mathrm{K}{\mathrm{J}}^{-1}{\mathrm{m}}^{-2}$, and $Q,\mathrm{\Pi },\pi$ have units Pa; $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: in signature $(-,+,+,+)$, the orthogonal complement of a unit timelike vector has positive-definite metric; this supplies the sign of the projected quadratic terms.

Rest fluid state, frame and geometric current definitions (current batch-53 draft): on a nonempty open set $O$, a local state is specified by $e\in C^2(X)$ on nonempty open $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ and a $C^2$ state map $(v,{\sigma }_p):O\to X$. It defines $n=v^{-1}$, $s=n{\sigma }_p$, $ϵ=ne$, $T=e_{{\sigma }_p}$, $p=-e_v$, $H=e+pv$, and $\mu =H-T{\sigma }_p$. The convention has signature $(-,+,+,+)$ and $x^0=ct$, with $c$ in $\mathrm{m}{\mathrm{s}}^{-1}$.