Supported-rest effective Cattaneo thermal model

Use the length-coordinate and unit-observer conventions of Thermodynamics-GR fields, currents, reference normalization and units. Fix Minkowski spacetime $M={\mathbb{R}}^4$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$ in coordinates $(x^0,x^1,x^2,x^3)=(ct,x)$, where $t$ is physical time in seconds, $x^i$ are in metres, and $c>0$ is in $\mathrm{m}{\mathrm{s}}^{-1}$. The supported rest observer congruence is the smooth future unit vector field $u={\mathrm{\partial }}_{x^0}$, so $\eta (u,u)=-1$. An external support maintains the modeled material at rest in this congruence; the support is not a dynamical variable, and this subsystem does not claim to include the momentum equations of a full fluid.

Fix constants $T_0>0$ in K, volumetric heat capacity $C>0$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, thermal conductivity $\kappa >0$ in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, and relaxation time $\tau >0$ in seconds. Select an open state neighborhood $D$ of $(0,0)$ in $\mathbb{R}\times {\mathbb{R}}^3$ contained in $\{(\vartheta ,q):T_0+\vartheta >0\}$. The neighborhood is part of the application regime; no universal numerical size is asserted. Let $I\subseteq \mathbb{R}$ be an open interval of physical time, and define ${\iota }_I:I\times {\mathbb{R}}^3\to M$ by ${\iota }_I(t,\mathbf{x})=(ct,\mathbf{x})$; it identifies these coordinates with an open region of Minkowski spacetime. Choose either $\mathrm{\Sigma }={\mathbb{R}}^3$ or, for a fixed $L>0$, $\mathrm{\Sigma }={\mathbb{T}}_L^3:={\mathbb{R}}^3\mathrm{/}(L{\mathbb{Z}}^3)$ with its flat quotient metric; when $\mathrm{\Sigma }={\mathbb{T}}_L^3$, let ${\pi }_L:{\mathbb{R}}^3\to {\mathbb{T}}_L^3$ be the quotient map. The spacetime remains $M={\mathbb{R}}^4$: the torus is only an analytic representation of spatially $L$-periodic fields on its Minkowski rest slices, not a quotient of spacetime. Take smooth fields $\vartheta :I\times \mathrm{\Sigma }\to \mathbb{R}$ and $q:I\times \mathrm{\Sigma }\to {\mathbb{R}}^3$ with $(\vartheta (t,x),q(t,x))\in D$. If $\mathrm{\Sigma }={\mathbb{T}}_L^3$, their physical fields on the rest slices are the smooth periodic lifts $\tilde{\vartheta }(t,\mathbf{x}):=\vartheta (t,{\pi }_L(\mathbf{x}))$ and $\tilde{q}(t,\mathbf{x}):=q(t,{\pi }_L(\mathbf{x}))$ on $I\times {\mathbb{R}}^3$; under ${\iota }_I$, these are coordinate representatives of fields on ${\iota }_I(I\times {\mathbb{R}}^3)\subset M$. If $\mathrm{\Sigma }={\mathbb{R}}^3$, the given fields are already coordinate representatives on that region. Here $\vartheta$ is the temperature departure in K, $T=T_0+\vartheta$ is the local temperature, and $q$ is the physical heat-flux vector in the supported rest space, expressed in its fixed Cartesian rest basis and measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Spatial $\mathrm{\nabla }$ and $\mathrm{div}$ are Euclidean derivatives in metres on the Minkowski rest space, evaluated by periodic lifts when $\mathrm{\Sigma }={\mathbb{T}}_L^3$.

In the source-free effective model, adopt

$C{\mathrm{\partial }}_t\vartheta +\mathrm{div}q=0,\tau {\mathrm{\partial }}_{t}q+q=-\kappa \mathrm{\nabla }\vartheta \mathrm{.}$

The first equation is the subsystem's thermal-energy balance and the second is its Cattaneo constitutive relation. Also adopt the specified effective quadratic entropy density and entropy flux

$s_{\mathrm{e}\mathrm{f}\mathrm{f}}:=s_0+\frac{C\vartheta }{T_0}-\frac{C{\vartheta }^2}{2T_0^2}-\frac{\tau \mathrm{\mid }q{\mathrm{\mid }}^2}{2\kappa T_0^2},j_s:=\frac{q}{T_0}-\frac{\vartheta q}{T_0^2},$

where $s_0$ is constant. Both $s_0$ and $s_{\mathrm{e}\mathrm{f}\mathrm{f}}$ have units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$, and $j_s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. This quadratic function is an adopted effective state function on $D$, not the exact nonlinear material entropy or a particle Gibbs equation of state. The model assigns no measured values to its coefficients, reports no experiment, and derives no microscopic transport law.