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.

Use the smooth time-oriented Lorentzian geometry, metric-selected Levi-Civita connection $\mathrm{\nabla }$, and length-coordinate conventions of Thermodynamics-GR fields, currents, reference normalization and units, with signature $(-,+,+,+)$. On an open region $O$ of a smooth four-manifold without boundary, take a smooth positive temperature field $T:O\to (0,\mathrm{\infty })$ and a smooth future unit observer-congruence field $u$ satisfying $g(u,u)=-1$. Adopt the separate entropy-only state maps of Separate entropy-only radiation state maps with a fixed parameter $a_R>0$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$:

$\epsilon =a_R{T}^4,p=\frac{1}{3}{a}_R{T}^4,s=\frac{4}{3}{a}_R{T}^3,h:=\epsilon +p\mathrm{.}$

The fields $\epsilon ,p,h$ have units $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{P}\mathrm{a}$ and $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$. Adopt the perfect stress tensor and its source-free balance

${\mathsf{T}}^{ab}:=h{u}^a{u}^b+p{g}^{ab},{\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$

on the specified smooth background. In the proper-time velocity convention $U^a:=c{u}^a$, this is equivalently ${\mathsf{T}}^{ab}=(h\mathrm{/}{c}^2)U^a{U}^b+p{g}^{ab}$ with $g(U,U)=-c^2$. This branch has no conserved particle-number coordinate, particle-number current, or chemical-potential field. It adopts an effective equation of state and perfect-stress balance; it does not determine $a_R$ from quantum statistics, assert that a physical system realizes these maps, report a measurement, or derive an Einstein-field equation.

For the global-evolution and forward-time clauses below, assume that the postulate's open time interval $I$ contains $0$. The local identities and telegrapher equation hold on any open interval on which a smooth solution is defined.

Let $C,\kappa ,\tau ,T_0,c,L>0$ and let $\mathrm{\Sigma }$ be ${\mathbb{R}}^3$ or ${\mathbb{T}}_L^3$. If $\mathrm{\Sigma }={\mathbb{T}}_L^3$, use the periodic-lift convention of Supported-rest effective Cattaneo thermal model: torus fields represent smooth spatially $L$-periodic fields on the rest slices of $M={\mathbb{R}}^4$. Adopt the supported-rest Cattaneo equations and effective current of Supported-rest effective Cattaneo thermal model. The equations are $C{\vartheta }_t+\mathrm{div}q=0$ and $\tau q_t+q=-\kappa \mathrm{\nabla }\vartheta$, with Euclidean spatial derivatives. For any smooth solution on an open time interval, the exact effective entropy-production and quadratic Lyapunov identities are

${\mathrm{\partial }}_t{s}_{\mathrm{e}\mathrm{f}\mathrm{f}}+\mathrm{div}{j}_s=\frac{\mathrm{\mid }q{\mathrm{\mid }}^2}{\kappa T_0^2}\ge 0,{\mathrm{\partial }}_t{e}_L+\mathrm{div}(\vartheta q)=-\frac{\mathrm{\mid }q{\mathrm{\mid }}^2}{\kappa },e_L:=\frac{1}{2}(C{\vartheta }^2+\frac{\tau }{\kappa }\mathrm{\mid }q{\mathrm{\mid }}^2)\mathrm{.}$

The associated telegrapher equation is $\tau {\vartheta }_{tt}+{\vartheta }_t-(\kappa \mathrm{/}C)\mathrm{\Delta }\vartheta =0$, where $\mathrm{\Delta }:=\mathrm{div}\mathrm{\nabla }={\sum }_{i=1}^3{\mathrm{\partial }}_{x^i}^2$ on the Euclidean rest space, and its characteristic propagation speed is $v_h:=\sqrt{\kappa \mathrm{/}(C\tau )}$. Put $U_0=({\vartheta }_0,q_0):=(\vartheta (0,\cdot ),q(0,\cdot ))$. For Schwartz data on ${\mathbb{R}}^3$ or smooth periodic data on ${\mathbb{T}}_L^3$, the unique global smooth evolution, weighted-norm contraction, and finite propagation at $v_h$ follow conditionally from the exact mathematical interface in [F1]. Here Schwartz data mean componentwise $C^{\mathrm{\infty }}$ data for which every $x^{\alpha }{\mathrm{\partial }}^{\beta }{U}_0$ is bounded for multi-indices $\alpha ,\beta \in {\mathbb{N}}^3$, as specified in [F2]. Since the spacetime model uses light speed $c$, the heat subsystem is relativistically causal exactly when

$v_h\le c⟺\kappa \le C\tau c^2\mathrm{.}$

On ${\mathbb{T}}_L^3$, for each integer $r>3\mathrm{/}2$, the periodic solution supplied conditionally by [F1] obeys $T_0+\vartheta (t,x)>0$ for all $t\ge 0$ whenever $K_{r,L,C}\mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H_M^r({\mathbb{T}}_L^3)}<T_0$, with $K_{r,L,C}$ and $H_M^r$ exactly as defined there. This bound ensures positive temperature but does not by itself ensure that the solution stays in the selected domain $D$ of the postulate. The global linear PDE statement concerns the mathematical extension of the subsystem; its physical interpretation is restricted to times when $(\vartheta ,q)\in D$. No empirical coefficient or material-specific validity is asserted.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. On an open region $O$ of a smooth four-manifold without boundary, let $(g,u,T)$ and the entropy-only equation of state and source-free perfect-stress balance be as in Separate no-conserved-number radiation fluid branch. For every such smooth solution,

${\mathrm{\nabla }}_a(s{u}^a)=0.$

This is a conditional local model deduction; it assumes no particle-number current. In the additional submodel $M={\mathbb{R}}^4$ with Minkowski coordinates $(x^0,x^1,x^2,x^3)=(ct,x)$, constant reference temperature $T_{\ast }>0$, and uniform rest observer $u_{\ast }=(1,0,0,0)$, the frozen linearized system for $r:=\delta \epsilon$ and $w^i:=\delta u^i$ has two transverse zero characteristic speeds and longitudinal speeds $\pm c\mathrm{/}\sqrt{3}$. Thus its linear acoustic speed is $c\mathrm{/}\sqrt{3}<c$. This is a conditional prediction of the adopted model, not an observed speed or a claim of global nonlinear evolution. The parameter $a_R$ remains an adopted positive coefficient; no quantum blackbody calibration is supplied.