Rest fluid state, frame and geometric current definitions

Use a smooth time-oriented Lorentz four-manifold $(M,g,N)$ of signature $(-,+,+,+)$, in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, with the unique torsion-free metric-compatible connection $\mathrm{\nabla }$ of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Use length coordinates $x^0=ct$ when assigning physical units: $g$ and the components of $u$ are dimensionless, and $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$. The physical reading of the geometry is not asserted here.

Let $O\subseteq M$ be a nonempty open set and let $u$ be an observer congruence on $O$ in the sense of Observer congruences, their observer curves and adapted charts: $u$ is a smooth vector field whose values are future-directed unit timelike observers, so $g(u,u)=-1$. Its maximal integral curves have open interval parameter domains and are smooth regular future timelike curves; no completeness, geodesic equation, or orthogonal rest-space foliation is assumed. The frame in this definition means this congruence field and its associated pointwise rest spaces. It is not an adapted chart or a tetrad field; those are distinct objects in Observer congruences, their observer curves and adapted charts and Local frame fields, coordinate frames and tetrad fields. The physical four-velocity associated with $u$ is $U=cu$, with $g(U,U)=-c^2$.

At $x\in O$, the rest space is $u^{\perp }$ as in Observers, rest spaces and the spatial projection at an event. Put $u_b=g_{bc}{u}^c$ and define the mixed rest-space endomorphism ${\mathrm{\Delta }}^a{}_b:={\delta }^a{}_b+u^a{u}_b,{\mathrm{\Delta }}^{ab}:=g^{ac}{\mathrm{\Delta }}^b{}_c\mathrm{.}$ For any vector $X^a$, write $\mathrm{\Delta }X$ for ${\mathrm{\Delta }}^a{}_b{X}^b$. This defines a tensor field; its projection properties are proved in the accompanying fluid-frame lemma.

A local material state consists of a nonempty open set $X\subset (0,\mathrm{\infty })\times \mathbb{R}$, coordinates $(v,{\sigma }_p)$ on $X$, and a specified $e\in C^2(X,\mathbb{R})$. A marker is a formal counted constituent used to normalize these state quantities; calling it conserved does not impose a spacetime conservation equation. Here $v$ is the positive rest-volume-per-marker state coordinate, interpreted as volume measured in the observer's local rest space; the model selects no spatial region whose volume is being integrated. It has units ${\mathrm{m}}^3\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$. The coordinate ${\sigma }_p$ is entropy per marker, with units $\mathrm{J}{\mathrm{K}}^{-1}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$, and $e$ is energy per marker, with units $\mathrm{J}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$. For a $C^2$ state map $(v,{\sigma }_p):O\to X$, define $n:=v^{-1},s:=n{\sigma }_p,\epsilon :=ne,T:={\mathrm{\partial }}_{{\sigma }_p}e,p:=-{\mathrm{\partial }}_{v}e,$ $H:=e+pv,h:=\epsilon +p=nH,\mu :=H-T{\sigma }_p\mathrm{.}$ Thus $n$ has units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, $\epsilon ,p,h$ have units $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{P}\mathrm{a}$, $T$ has units $\mathrm{K}$, and $H,\mu$ have units $\mathrm{J}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$. These are mathematical state maps from the supplied function $e$; no constitutive law is asserted to describe an actual material.

A geometric number current is a $C^1$ vector field $J^a$ with components of units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$; its physical flux is $cJ$, with units $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. Define its frame density and spatial part by ${\nu }_u:=-u_a{J}^a,V^a:={\mathrm{\Delta }}^a{}_b{J}^b\mathrm{.}$ A geometric entropy current is a $C^1$ vector field $S^a$ with components of units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$; its physical flux is $cS$, with units $\mathrm{W}{\mathrm{K}}^{-1}{\mathrm{m}}^{-2}$. Define $s_u:=-u_a{S}^a,{\mathrm{\Sigma }}^a:={\mathrm{\Delta }}^a{}_b{S}^b\mathrm{.}$ A geometric stress-energy current is a $C^1$ symmetric contravariant tensor field ${\mathsf{T}}^{ab}$ with components of units $\mathrm{P}\mathrm{a}$. Define its frame projections ${\epsilon }_u:={\mathsf{T}}^{ab}{u}_a{u}_b,Q^a:=-{\mathrm{\Delta }}^a{}_c{\mathsf{T}}^{cd}{u}_d,P^{ab}:={\mathrm{\Delta }}^a{}_c{\mathrm{\Delta }}^b{}_d{\mathsf{T}}^{cd},$ $p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}:=\frac{1}{3}{g}_{ab}{P}^{ab},{\pi }^{ab}:=P^{ab}-p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}{\mathrm{\Delta }}^{ab}\mathrm{.}$ Here $Q$ has units $\mathrm{P}\mathrm{a}$ and $cQ$ is physical energy flux with units $\mathrm{W}{\mathrm{m}}^{-2}$; $P,p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}},\pi$ have units $\mathrm{P}\mathrm{a}$. The thermodynamic quantities $(\epsilon ,p)$ and the tensor projections $({\epsilon }_u,p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}})$ are not identified by this definition.

For these $C^1$ fields, ${\mathrm{\nabla }}_a{J}^a$ and ${\mathrm{\nabla }}_a{S}^a$ denote scalar covariant divergences, and ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}$ denotes the vector covariant divergence, using the connection fixed above. No conservation equation, perfect-fluid stress form, equality ${\nu }_u=n$, equality $s_u=s$, or physical interpretation is imposed.