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.

Let $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ be nonempty and open, let $e\in C^2(X,\mathbb{R})$, and define $Y:=\{(n,s)\in (0,\mathrm{\infty })\times \mathbb{R}:(1\mathrm{/}n,s\mathrm{/}n)\in X\},\epsilon (n,s):=ne(1\mathrm{/}n,s\mathrm{/}n)\mathrm{.}$ For $(n,s)\in Y$, put $v=1\mathrm{/}n$, ${\sigma }_p=s\mathrm{/}n$, $T=e_{{\sigma }_p}$, $p=-e_v$, $H=e+pv$, $h=\epsilon +p=nH$, and $\mu =H-T{\sigma }_p$, as in Rest fluid state, frame and geometric current definitions. Then the following identities hold as equalities of differentials on $Y$: $d\epsilon =Tds+\mu dn,\epsilon +p=Ts+\mu n,dp=sdT+nd\mu \mathrm{.}$

For parcel totals $\mathsf{S}\in \mathbb{R}$ (entropy), $\mathsf{V}>0$ (proper volume), and $\mathsf{N}>0$ (marker count) with $(\mathsf{V}\mathrm{/}\mathsf{N},\mathsf{S}\mathrm{/}\mathsf{N})\in X$, define $E(\mathsf{S},\mathsf{V},\mathsf{N}):=\mathsf{N}e(\mathsf{V}\mathrm{/}\mathsf{N},\mathsf{S}\mathrm{/}\mathsf{N})\mathrm{.}$ Then $E$ is $C^2$, is positively homogeneous of degree one, and satisfies $dE=Td\mathsf{S}-pd\mathsf{V}+\mu d\mathsf{N},E=T\mathsf{S}-p\mathsf{V}+\mu \mathsf{N}\mathrm{.}$ If $E$ is viewed on its open domain in $(\mathsf{S},\mathsf{V},\mathsf{N})$, its Hessian annihilates the radial vector $(\mathsf{S},\mathsf{V},\mathsf{N})$.

The local response statements are conditional. At a state with $T>0$ and $e_{{\sigma }_p{\sigma }_p}>0$, temperature is a valid local coordinate at fixed $v$, and the per-marker heat capacity $c_v:={\left(\frac{\mathrm{\partial }e}{\mathrm{\partial }T}\right)}_v=\frac{T}{e_{{\sigma }_p{\sigma }_p}}$ is positive. At a state with $e_{vv}>0$, the isentropic compressibility ${\kappa }_{{\sigma }_p}:=-\frac{1}{v}{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }p}\right)}_{{\sigma }_p}=\frac{1}{v{e}_{vv}}$ is positive. At a state with $e_{vv}>0$ and $h>0$, the local pressure-energy response ${\alpha }_{{\sigma }_p}:={\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }\epsilon }\right)}_{{\sigma }_p}=\frac{e_{vv}}{nh}$ is positive. If $X$ is convex and $e$ is convex on $X$, then $D^{2}e$ is positive semidefinite. Positivity of the displayed diagonal responses is not asserted to imply convexity, a sound cone, or an evolution theorem.

Let $\mathcal{V}$ be a four-dimensional real vector space and let $g:\mathcal{V}\times \mathcal{V}\to \mathbb{R}$ be a symmetric nondegenerate bilinear form of signature $(-,+,+,+)$, meaning that in some basis its matrix is $\mathrm{diag}(-1,1,1,1)$. A vector $X$ is timelike when $g(X,X)<0$, and null when $X\mathrm{\ne }0$ and $g(X,X)=0$. Choose one connected component $C^{+}$ of the timelike set $\{X:g(X,X)<0\}$ as the future cone; a timelike vector is future-directed when it belongs to $C^{+}$. Components and connectedness use the basis-independent finite-dimensional real vector-space topology. Lower indices with $g$ and raise them with its inverse, summing repeated upper/lower indices over $0,1,2,3$. Let $u\in C^{+}$ be unit timelike, $g(u,u)=-1$, and define ${\mathrm{\Delta }}^a{}_b={\delta }^a{}_b+u^a{u}_b$ as in Rest fluid state, frame and geometric current definitions. For every vector $J\in \mathcal{V}$ and symmetric contravariant rank-two tensor $\mathsf{T}$ on $\mathcal{V}$, the following statements hold.

1. The operator $\mathrm{\Delta }$ is the $g$-orthogonal projection onto $u^{\perp }$, and $J^a={\nu }_u{u}^a+V^a,{\nu }_u=-u_a{J}^a,V^a={\mathrm{\Delta }}^a{}_b{J}^b,$ is the unique decomposition with $V\in u^{\perp }$.

2. With ${\epsilon }_u={\mathsf{T}}^{ab}{u}_a{u}_b,Q^a=-{\mathrm{\Delta }}^a{}_c{\mathsf{T}}^{cd}{u}_d,P^{ab}={\mathrm{\Delta }}^a{}_c{\mathrm{\Delta }}^b{}_d{\mathsf{T}}^{cd},$ one has the unique decomposition ${\mathsf{T}}^{ab}={\epsilon }_u{u}^a{u}^b+u^a{Q}^b+Q^a{u}^b+P^{ab}\mathrm{.}$ The scalar $p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=\frac{1}{3}{g}_{ab}{P}^{ab}$ and tensor ${\pi }^{ab}=P^{ab}-p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}{\mathrm{\Delta }}^{ab}$ satisfy $u_a{Q}^a=u_a{P}^{ab}=0$, $u_a{\pi }^{ab}=0$, and $g_{ab}{\pi }^{ab}=0$. Also, $Q=0$ if and only if $u$ is a timelike eigenvector of ${\mathsf{T}}^a{}_b$.

3. If $J$ is future timelike, then $u_E:=\frac{J}{\sqrt{-g(J,J)}}$ is the unique future unit timelike observer vector for which the spatial number current vanishes, ${\mathrm{\Delta }}_{u_E}J=0$, where ${\mathrm{\Delta }}_{u_E}{}^a{}_b:={\delta }^a{}_b+u_E^a(u_E{)}_b$ is the clause-1 projection computed using $u_E$. Call $u_E$ the pointwise Eckart observer vector associated with $J$; it is a vector, not a chart or a tetrad.

4. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, let $O\subset M$ be open, and let $\mathsf{T}$ be a $C^1$ symmetric tensor field on $O$. At $x_0\in O$, choose an open neighborhood $O_0\subset O$ and a smooth future tetrad $(E_0,E_1,E_2,E_3)$ on $O_0$ as supplied by Existence of a smooth future-directed tetrad field near every event, and put $u_0:=E_0(x_0)$ and ${\mathrm{\Delta }}_0{}^a{}_b:={\delta }^a{}_b+u_0^a(u_0{)}_b$. Define ${\mathbb{B}}^3:=\{w\in {\mathbb{R}}^3:{\sum }_{i=1}^3(w^i{)}^2<1\}$ and, for $x\in O_0$ and $w\in {\mathbb{B}}^3$, define $u(x,w):=(1-\mathrm{\mid }w{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}(E_0(x)+{\sum }_{i=1}^3{w}^i{E}_i(x))\in T_{x}M\mathrm{.}$ The tetrad equations make $u(x,w)$ future unit timelike. If ${\mathsf{T}}^{ab}(x_0)={\epsilon }_0{u}_0^a{u}_0^b+p_0{\mathrm{\Delta }}_0^{ab},h_0:={\epsilon }_0+p_0>0,$ then there are a neighborhood $U\subset O_0$ of $x_0$ and an open neighborhood $W\subset {\mathbb{B}}^3$ of $0$ such that there is a unique $C^1$ map $w_L:U\to W$ for which $u_L(x):=u(x,w_L(x))$ satisfies $Q_{u_L(x)}(x)=0$ for every $x\in U$. Thus $u_L$ is a $C^1$ observer field, meaning a $C^1$ vector field whose values are future unit timelike; it is distinct from the chosen tetrad field and from any coordinate chart. This is a local result; no far-from-equilibrium or global Landau-field existence is asserted.

5. In a Lorentz vector space, for every nonzero null vector $\mathrm{\ell }$ and $\rho >0$, the pointwise tensor ${\mathsf{T}}^{ab}=\rho {\mathrm{\ell }}^a{\mathrm{\ell }}^b$ has no timelike eigenvector. Hence no future unit timelike observer vector has $Q=0$ for this tensor, so no pointwise Landau observer vector exists.