Supported static heat model

Choose positive lengths $L_1,L_2,L_3$, the rectangular box $\mathrm{\Omega }={\prod }_{i=1}^3(0,L_i)$, and an open neighborhood $U\subset {\mathbb{R}}^3$ of $\bar{\mathrm{\Omega }}$. Let $N\in C^{\mathrm{\infty }}(U,(0,\mathrm{\infty }))$ satisfy $0<N_{\min }\le N(x)\le N_{\max }<\mathrm{\infty }$ on $\bar{\mathrm{\Omega }}$. On $M={\mathbb{R}}_{x^0}\times U$, use $x^0=c{t}_K$, where $c>0$ is in $\mathrm{m}{\mathrm{s}}^{-1}$, and the fixed metric

$g=-N(x{)}^2(d{x}^0{)}^2+{\sum }_{i=1}^3(d{x}^i{)}^2\mathrm{.}$

The coordinates $x^i$ and lengths $L_i$ are measured in metres, $t_K$ in seconds, and the lapse $N$ is dimensionless. The metric is independent of $x^0$ and has no time-space cross terms; this is the static background in this model. Give $M$ time orientation $\xi ={\mathrm{\partial }}_{x^0}$. Then $\xi$ is a future timelike Killing field, $N=\sqrt{-g(\xi ,\xi )}$, and $u=N^{-1}\xi$ is the smooth future unit timelike observer congruence in the conventions of Thermodynamics-GR fields, currents, reference normalization and units. Here a frame means this vector field and its pointwise rest spaces, not a coordinate chart or tetrad. The material is held at rest in this congruence; no flow evolution is included. The scalar $T>0$ is proper temperature in kelvins, and $\vartheta :=NT$ is the Killing-normalized (redshifted) temperature of Thermodynamics-GR fields, currents, reference normalization and units; this definition alone does not assert that $\vartheta$ is constant.

Adopt the fixed-composition material branch of Restricted local equilibrium parcel with adopted material state function with constants $C_V,n_0,T_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$, ${\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathbb{R}$, and $e_0\in \mathbb{R}$. The units are $C_V$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, $n_0$ in $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$, $T_{\mathrm{r}\mathrm{e}\mathrm{f}}$ in kelvins, ${\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}$ in $\mathrm{J}{\mathrm{K}}^{-1}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$, and $e_0$ in $\mathrm{J}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$. Put $c_v=C_V\mathrm{/}{n}_0$ in $\mathrm{J}{\mathrm{K}}^{-1}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}{\mathrm{r}}^{-1}$ and choose the state function on $X=(0,\mathrm{\infty })\times \mathbb{R}$ to be

$e(v,{\sigma }_p)=c_v{T}_{\mathrm{r}\mathrm{e}\mathrm{f}}\exp \text{\hspace{0.17em}⁣}\left(\frac{{\sigma }_p-{\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}}{c_v}\right)+e_0,$

independent of the rest volume per marker $v$. For a given positive $T$, set $v=1\mathrm{/}{n}_0$ and ${\sigma }_p={\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}+c_v\log (T\mathrm{/}{T}_{\mathrm{r}\mathrm{e}\mathrm{f}})$. For this branch define $n=v^{-1}$, $s=n{\sigma }_p$, $ϵ=ne$, $T={\mathrm{\partial }}_{{\sigma }_p}e$, and $p=-{\mathrm{\partial }}_{v}e$, using the units of Thermodynamics-GR fields, currents, reference normalization and units and the adopted material convention of Restricted local equilibrium parcel with adopted material state function. These maps give $n=n_0$, $p=0$, $ϵ=C_{V}T+n_0{e}_0$, and $s=n_0{\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}+C_V\log (T\mathrm{/}{T}_{\mathrm{r}\mathrm{e}\mathrm{f}})$. These assignments describe the selected model only; they do not assert a measured equation of state.

Take an admissible initial redshifted-temperature profile ${\vartheta }_0\in C^2(\bar{\mathrm{\Omega }})$ with constants $0<m_0\le M_0<\mathrm{\infty }$ such that $m_0\le {\vartheta }_0\le M_0$, and require a $C^2$ extension ${\tilde{\vartheta }}_0:{\mathbb{R}}^3\to \mathbb{R}$ with the same bounds, period $2L_i$ in each coordinate $x^i$, and evenness under each reflection $x^i\mapsto -x^i$. Its even periodic extension makes the normal derivative vanish at both faces in each coordinate once evolved smoothly. The initial proper temperature is $T(0,x)={\vartheta }_0(x)\mathrm{/}N(x)$.

On $O=(0,\mathrm{\infty })\times \mathrm{\Omega }$, identified with the subset of $M$ by $x^0=c{t}_K$, the admissible temperature field is a positive map $T:[0,\mathrm{\infty })\times \bar{\mathrm{\Omega }}\to (0,\mathrm{\infty })$ continuous with initial trace $T(0,x)={\vartheta }_0(x)\mathrm{/}N(x)$ and $C^{\mathrm{\infty }}$ on $O$. For every $0<a<b<\mathrm{\infty }$, it extends smoothly in all space-time variables to a neighborhood of $[a,b]\times \bar{\mathrm{\Omega }}$; no smoothness across $t_K=0$ is required. The state map on $O$ is

$v=1\mathrm{/}{n}_0,{\sigma }_p={\sigma }_{\mathrm{r}\mathrm{e}\mathrm{f}}+c_v\log (T\mathrm{/}{T}_{\mathrm{r}\mathrm{e}\mathrm{f}})\mathrm{.}$

With the definition $\mu =e+pv-T{\sigma }_p$ in Rest fluid state, frame and geometric current definitions, this branch has $\mu \mathrm{/}T=c_v+e_0\mathrm{/}T-{\sigma }_p$ and ${\mathrm{\nabla }}_a(\mu \mathrm{/}T)=-(e_0\mathrm{/}{T}^2+c_v\mathrm{/}T){\mathrm{\nabla }}_{a}T$. The state map and these derived fields are smooth on $O$.

For Restricted gradient constitutive dissipative model, declare the regime $\mathcal{R}$ as follows. Its tuples record $(v,{\sigma }_p,u,T,\mu \mathrm{/}T)$ and the first covariant derivatives $\mathrm{\nabla }u$, $\mathrm{\nabla }T$, and $\mathrm{\nabla }(\mu \mathrm{/}T)$ generated by first derivatives of the displayed state maps and of a future unit timelike field $u$; thus $u_b{\mathrm{\nabla }}_a{u}^b=0$. Require $1\mathrm{/}(2n_0)<v<3\mathrm{/}(2n_0)$, with arbitrary ${\sigma }_p\in \mathbb{R}$ and no bounds on the derivatives. This is a nonempty open regime in the relative topology of the state/first-jet space after the normalization and state-map constraints are imposed. It contains the fixed-density heat branch pointwise because $v=1\mathrm{/}{n}_0$, $T>0$, and $u=N^{-1}\xi$ is future unit timelike. In this application set the diffusion coefficient $L=0$, so $V=0$ and the supplier's number current is $J=nu$; it is future timelike throughout $\mathcal{R}$ since $n=1\mathrm{/}v>0$, and on the fixed-density branch $J=n_{0}u$.

Adopt from Restricted gradient constitutive dissipative model the first-order heat closure and current/stress/entropy conventions on this regime. The physical heat-flux field $q$ is spatial, has units $\mathrm{W}{\mathrm{m}}^{-2}$, and has constitutive coefficient ${\kappa }_{\mathrm{t}\mathrm{h}}=A_0\mathrm{/}N$ for a constant $A_0>0$ in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. With ${\mathrm{\nabla }}_x$ the Euclidean spatial gradient, its specified static heat law, local energy balance, and insulated-face condition are

$q=-{\kappa }_{\mathrm{t}\mathrm{h}}({\mathrm{\nabla }}_{x}T+T{\mathrm{\nabla }}_x\log N),{\kappa }_{\mathrm{t}\mathrm{h}}=\frac{A_0}{N},N{C}_V{\mathrm{\partial }}_{t_K}T+{\mathrm{div}}_x(N^{2}q)=0,N^{2}q\cdot \nu =0\text{ on each face},$

where ${\mathrm{div}}_x$ is Euclidean divergence and $\nu$ is the outward Euclidean unit normal to a box face. The adopted first-order closure specializes to this lapse-gradient form in the selected static frame; it is not inferred from an experiment. Since $N$ is time independent, ${\mathrm{\nabla }}_{x}T+T{\mathrm{\nabla }}_x\log N=N^{-1}{\mathrm{\nabla }}_x\vartheta$, so this coefficient choice gives $N^{2}q=-A_0{\mathrm{\nabla }}_x\vartheta$ and diffusivity $A_0\mathrm{/}{C}_V$ in ${\mathrm{m}}^2{\mathrm{s}}^{-1}$.

For the material stress/current in this selected branch put $Q^0=0$, $Q^i=q^i\mathrm{/}c$, $J^a=n_0{u}^a$, and

${\mathsf{T}}^{ab}=ϵu^a{u}^b+u^a{Q}^b+Q^a{u}^b\mathrm{.}$

Here $Q=q\mathrm{/}c$ has units $\mathrm{P}\mathrm{a}$ and ${\mathsf{T}}^{ab}$ has components in $\mathrm{P}\mathrm{a}$. In the first-order notation set the bulk, shear, and diffusion coefficients to zero, so $\mathrm{\Pi }=0$, ${\pi }^{ab}=0$, and $V^a=0$; the thermal coefficient is $\kappa ={\kappa }_{\mathrm{t}\mathrm{h}}$. These are nonnegative $C^1(O)$ coefficient fields. The source-free number balance ${\mathrm{\nabla }}_a{J}^a=0$ remains part of this branch. The external support is the force-density field $f^b:={\mathrm{\nabla }}_a{\mathsf{T}}^{ab}$, in $\mathrm{N}{\mathrm{m}}^{-3}$, required to hold the medium on the fixed background. This model adopts the first-order heat closure but replaces that supplier's source-free stress balance by this supported balance. Require $u_b{f}^b=0$: the support may supply spatial force but no local power. Thus the state maps, constitutive fields, currents, and stress are smooth on $O$, and $f^b$ is a classical covariant divergence; this is a supported material model, not source-free full matter dynamics. Its entropy current is $S^a=s{u}^a+Q^a\mathrm{/}T$, with units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$; this postulate alone does not assert an entropy inequality.

This is an adopted effective parabolic model. It asserts no Einstein equation, no general relativistic-fluid initial-value theorem, no finite propagation speed, and no universal material law. It supplies no experiment, measurement, empirical premise, or evidence that a physical medium realizes the model.

Use the fixed static box, positive bounded lapse $N$, constants, material state function, observer field, heat flux, support, initial-data class, and SI conventions of Supported static heat model. Let ${\tilde{\vartheta }}_0$ be its positive $C^2$ even periodic extension, and define $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=L_1{L}_2{L}_3$, $\bar{\vartheta }=\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{-1}{\int }_{\mathrm{\Omega }}{\vartheta }_{0}dx$, and $C_{\mathrm{\Omega }}=\frac{3}{2}({\max }_i{L}_i{)}^2$. Use Euclidean Lebesgue volume $dx$ and the associated $L^2(\mathrm{\Omega })$ norm. Then the model has a unique solution

$T\in C([0,\mathrm{\infty })\times \bar{\mathrm{\Omega }})\cap C^{2,1}((0,\mathrm{\infty })\times \bar{\mathrm{\Omega }}),$

where $C^{2,1}$ means two continuous spatial derivatives and one continuous time derivative up to the box faces at positive times. With $\vartheta =NT$ and $\mathrm{\Gamma }(z,\tau )=(4\pi \tau {)}^{-3\mathrm{/}2}\exp (-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}(4\tau ))$ for $z\in {\mathbb{R}}^3$ and $\tau >0$, the solution is

$\vartheta (t_K,x)={\int }_{{\mathbb{R}}^3}\mathrm{\Gamma }\text{\hspace{0.17em}⁣}(x-y,\frac{A_0}{C_V}{t}_K){\tilde{\vartheta }}_0(y)dy(t_K>0),\vartheta (0,x)={\vartheta }_0(x),T=\frac{\vartheta }{N}\mathrm{.}$

It satisfies $C_V{\mathrm{\partial }}_{t_K}\vartheta =A_0{\mathrm{\Delta }}_x\vartheta$, the six insulated-face conditions ${\mathrm{\partial }}_{\nu }\vartheta =0$, and $m_0\le \vartheta \le M_0$. The lapse-weighted material energy

${\mathcal{E}}_K(t_K)={\int }_{\mathrm{\Omega }}N(x)ϵ(t_K,x)dx$

is constant. For the entropy current $S^a=s{u}^a+Q^a\mathrm{/}T$ and total entropy $\mathcal{S}(t_K)={\int }_{\mathrm{\Omega }}s(t_K,x)dx$, the exact model rate is

$\frac{d\mathcal{S}}{d{t}_K}=A_0{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }{\mathrm{\nabla }}_x\vartheta {\mathrm{\mid }}^2}{{\vartheta }^2}dx\ge 0.$

Moreover,

${\parallel \vartheta (t_K,\cdot )-\bar{\vartheta }\parallel }_{L^2(\mathrm{\Omega })}\le \exp \text{\hspace{0.17em}⁣}(-\frac{A_0{t}_K}{C_V{C}_{\mathrm{\Omega }}}){\parallel {\vartheta }_0-\bar{\vartheta }\parallel }_{L^2(\mathrm{\Omega })}\mathrm{.}$

Thus $T$ converges in $L^2(\mathrm{\Omega })$ to the positive proper-temperature profile $T_{\mathrm{e}\mathrm{q}}(x)=\bar{\vartheta }\mathrm{/}N(x)$, while the redshifted temperature $NT$ converges to the constant $\bar{\vartheta }$. These are conditional predictions of the adopted model. They do not establish that a real material obeys the postulates, and they do not assert a full Einstein-fluid solution, causal or stable relativistic-fluid evolution, or finite propagation speed. No empirical data are used.