Selected stationary thermal equilibrium

On a nonempty connected open set $O$ in the smooth, time-oriented Lorentzian four-manifold of Thermodynamics-GR fields, currents, reference normalization and units, select one smooth future timelike Killing field $\xi$ and one smooth scalar $T:O\to (0,\mathrm{\infty })$ in kelvins. Its dimensionless lapse and selected observer congruence are

$N_{\xi }:=\sqrt{-g(\xi ,\xi )}>0,u:=\xi \mathrm{/}{N}_{\xi }\mathrm{.}$

Thus $g(u,u)=-1$ and the physical four-velocity is $cu$ in the length-coordinate convention $x^0=ct$. A reference normalization of $\xi$ is part of the selected model. The local temperature $T$ is assigned to this comoving observer congruence; it is not a coordinate temperature or a value assigned to every observer.

Adopt the inverse-temperature vector

$\beta :=\frac{u}{k_{B}T},$

where $k_B>0$ is Boltzmann's constant in joules per kelvin, and assume that this field is Killing on $O$. In this item, a selected stationary thermal equilibrium means precisely this adopted bath, observer congruence, positive proper-temperature field, and Killing condition on $\beta$. The field $N_{\xi }T$ is not assumed constant here; its constancy is the conclusion of Conditional Tolman–Ehrenfest relation.

This is a model assumption, not a statement that every stationary geometry admits a thermal state or that every material realizes this bath. It asserts no complete Killing flow, KMS state, trace-class stress exponential, collision or thermalization law, Hawking or Unruh temperature, or empirical thermometer reading.

Use the connected stationary thermal carrier of Selected stationary thermal equilibrium on $O$. Let $N_{\xi }=\sqrt{-g(\xi ,\xi )}>0$, $u=\xi \mathrm{/}{N}_{\xi }$, and $\beta =u\mathrm{/}(k_{B}T)$ as there, and assume the postulated Killing condition on $\beta$. Then a positive constant $T_K$ exists such that

$N_{\xi }(x)T(x)=T_K\text{and}\beta =\frac{\xi }{k_B{T}_K}(x\in O)\mathrm{.}$

In the same convention, $\xi (N_{\xi })=0$, the observer acceleration satisfies $a=\mathrm{\nabla }\log N_{\xi }$, the expansion and shear vanish, and

$\mathrm{\nabla }\log T=-a\mathrm{.}$

For any chosen reference event $x_{\mathrm{r}\mathrm{e}\mathrm{f}}\in O$, $T_K=N_{\xi }(x_{\mathrm{r}\mathrm{e}\mathrm{f}})T(x_{\mathrm{r}\mathrm{e}\mathrm{f}})$. If the reference normalization has $N_{\xi }(x_{\mathrm{r}\mathrm{e}\mathrm{f}})=1$, then $T_K$ is the local temperature at that event. Rescaling $\xi$ by a constant $b>0$ rescales $N_{\xi }$ and $T_K$ by $b$ while leaving the local scalar $T$ unchanged. The result is a conditional deduction from the selected physical postulate and the stated mathematical supplier; it is not a thermometer law for arbitrary observers or a KMS-existence theorem.

The mathematical supplier is a conditional local field argument. The cited treatments provide independent context; none supplies an empirical observation or proves that a selected bath describes every material.

For a selected fixed-composition material branch, adopt the local state function and energy convention of Restricted local equilibrium parcel with adopted material state function and Rest fluid state, frame and geometric current definitions. In that state map, $T>0$ is the derivative of the supplied energy per marker with respect to entropy per marker, and the chemical potential is $\mu =H-T{\sigma }_p$ in joules per marker. Rest energy stays in the energy convention used for $\mu$ and for any particle generator; a rest-energy subtraction made in a different convention must be tracked explicitly.

For particle exchange, select and declare one of these physical model protocols:

• Finite-cell exchange. Supply $r\ge 1$ finitely many independent cells with open convex state domains $U_i\subseteq {\mathbb{R}}^2$, coordinates $(E_i,n_i)$, and concave entropy functions $S_i\in C^1(U_i,\mathbb{R})$. Here $E_i$ is in joules, $n_i$ is a continuous dimensionless particle-number coordinate, and $S_i$ is in joules per kelvin. Fix dimensionless lapse values $N_i>0$, a common charge $q$ in coulombs, and potential values ${\mathrm{\Phi }}_i$ in volts in a stated common reference. Put $U={\prod }_i{U}_i$ and $S_{\mathrm{t}\mathrm{o}\mathrm{t}}={\sum }_i{S}_i$. The adopted exchange conserves $\mathcal{E}={\sum }_i(N_i{E}_i+q{\mathrm{\Phi }}_i{n}_i)$ and $\mathcal{N}={\sum }_i{n}_i$. At a supplied feasible interior point $x^{\ast }\in U$, with these totals fixed and ${\mathrm{\partial }}_{E_i}{S}_i(x_i^{\ast })>0$ for every $i$, adopt the following equilibrium criterion: $D{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\ast })[h]=0$ for every $h=(\delta E_1,\delta n_1,\dots ,\delta E_r,\delta n_r)$ satisfying ${\sum }_i(N_i\delta E_i+q{\mathrm{\Phi }}_i\delta n_i)=0$ and ${\sum }_i\delta n_i=0$. All such independent infinitesimal variations are admitted. Define $1\mathrm{/}{T}_i={\mathrm{\partial }}_{E_i}{S}_i(x_i^{\ast })$ and ${\mu }_i\mathrm{/}{T}_i=-{\mathrm{\partial }}_{n_i}{S}_i(x_i^{\ast })$, so $T_i>0$ is in kelvins and ${\mu }_i$ is in joules per particle. This criterion is an adopted condition on any selected equilibrium point, not a consequence of conservation, an existence assertion, or a relaxation law. Independent finite entropy-exchange equilibrium supplies its conditional consequences. Identification of these variables with another material state map is an additional model identification, not inferred here.

• Charged kinetic exchange. On the selected stationary domain, supply a species of rest mass $m>0$, charge $q$, smooth local fields $T>0$ and $\mu$, a global smooth real potential $A$ with $F=dA$, and a smooth compensator ${\mathrm{\Lambda }}_{\xi }$ satisfying ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$. Use the future massive shell, charged transport, and dimensionless exponential ansatz exactly as defined in Charged particle and Liouville data on a stationary Lorentzian domain. Adopt whole-shell stationarity ${\mathcal{L}}_{F,q}f=0$ as the exchange criterion. This is a model assumption on every future mass-shell momentum, not an assertion that collisions drive a gas to this ansatz.

The two protocols are separate model choices. This postulate does not derive either protocol from gravity, assert a global electrostatic potential when no single-valued symmetry compensator exists, establish a Gibbs state or material realization, or identify finite entropy variables with kinetic fields without an explicit model assumption.

Use the selected stationary carrier of Selected stationary thermal equilibrium and the declared exchange protocol of Specified equilibrium particle exchange. The following two conclusions are conditional on their respective model inputs.

Charged kinetic protocol. Let $O$ be connected, let $\xi$ be the selected future timelike Killing field, and set $N_{\xi }=\sqrt{-g(\xi ,\xi )}>0$. For a species with $m>0$ and charge $q$, assume a global smooth potential symmetry ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$ and define ${\mathrm{\Phi }}_{\xi }=-c({\iota }_{\xi }A-{\mathrm{\Lambda }}_{\xi })$ in volts. Suppose the smooth fields $T>0$ and $\mu$ define $\beta =u\mathrm{/}(k_{B}T)$ and $\zeta =\mu \mathrm{/}(k_{B}T)$, and the ansatz $f=\exp (\zeta +g(\beta ,P))$ is stationary under the charged transport of Charged particle and Liouville data on a stationary Lorentzian domain at every future massive-shell momentum. Then there are constants $T_K>0$ and ${\mu }_K$ such that

$N_{\xi }T=T_K,\beta =\frac{\xi }{k_B{T}_K},N_{\xi }\mu +q{\mathrm{\Phi }}_{\xi }={\mu }_K\mathrm{.}$

Here $\mu$ and ${\mu }_K$ use the same full-energy convention as the conserved generator, including rest energy when that generator includes it. The gauge-compensated potential reference may be shifted by ${\mathrm{\Phi }}_{\xi }\mapsto {\mathrm{\Phi }}_{\xi }+C$ only with ${\mu }_K\mapsto {\mu }_K+qC$.

Finite-cell exchange protocol. For $r\ge 1$ finitely many independent cells with $C^1$ concave entropies $S_i(E_i,n_i)$, positive lapse values $N_i$, common charge $q$, and the supplied potential values ${\mathrm{\Phi }}_i$, define $1\mathrm{/}{T}_i={\mathrm{\partial }}_{E_i}{S}_i>0$ and ${\mu }_i\mathrm{/}{T}_i=-{\mathrm{\partial }}_{n_i}{S}_i$ at a supplied feasible interior point. If independent variations preserving ${\sum }_i(N_i{E}_i+q{\mathrm{\Phi }}_i{n}_i)$ and ${\sum }_i{n}_i$ leave the total entropy stationary, then constants $T_K>0,{\mu }_K$ exist with

$N_i{T}_i=T_K,N_i{\mu }_i+q{\mathrm{\Phi }}_i={\mu }_K(1\le i\le r)\mathrm{.}$

That stationary point is a global entropy maximum on the feasible set when the cell entropies are concave, and it is unique if they are strictly concave. No existence or boundary attainment is asserted. In either branch, $q=0$ gives the neutral relation $N\mu ={\mu }_K$ (or $N_i{\mu }_i={\mu }_K$). The finite-cell symbols are not identified with the kinetic fields unless the model explicitly supplies that identification.

The finite-cell maximum-entropy argument is finite-dimensional and assumes independent variations. The charged branch uses the exact global gauge-compensated energy and full-shell stationarity. Neither branch imports a mean-field stellar equation, a chemical-reaction law, a Gibbs/Euler equation of state, or a continuum equilibrium state.

Let $(M,g,N_0)$ be the smooth time-oriented Lorentzian geometry of Thermodynamics-GR fields, currents, reference normalization and units, with signature $(-,+,+,+)$ and physical length coordinates $x^0=ct$. On an open region $O\subseteq M$, take a smooth observer congruence $u$ whose value at each event is a future unit timelike observer in the sense of Observers, rest spaces and the spatial projection at an event and Observer congruences, their observer curves and adapted charts. The congruence is a field of observer directions, not an adapted coordinate chart or a tetrad. Let $n:O\to (0,\mathrm{\infty })$ be a smooth proper number density in ${\mathrm{m}}^{-3}$ and $T:O\to (0,\mathrm{\infty })$ a smooth proper-temperature scalar in kelvins. These fields are prescribed model data.

Choose one species with rest mass $m>0$ in kilograms, put $M_{\ast }:=m{c}^2>0$ in joules, choose $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$, and choose a reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ in $\mathrm{J}\mathrm{s}$. At an event $x\in O$, the future massive shell ${\mathcal{H}}_{M_{\ast },x}^{+}$ and its tetrad-independent measure $d{\mu }_{M_{\ast }}$ are those of Canonical future massive mass-shell measure. For each $P\in {\mathcal{H}}_{M_{\ast },x}^{+}$, define the positive observer-measured energy

$E_u(P):=-g_x(P,u_x)>0,\alpha (x):=\frac{1}{k_{B}T(x)}\mathrm{.}$

In any future orthonormal tetrad whose time axis is $u_x$, write the spatial energy-momentum coordinates as $\mathbf{p}\in {\mathbb{R}}^3$, so $E_u(P)=\sqrt{M_{\ast }^2+\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2}$ and $d^3\mathbf{p}=E_u(P)d{\mu }_{M_{\ast }}(P)$. The spatial mechanical momentum is $\mathbf{p}\mathrm{/}c$; $\mathbf{p}$ itself is the spatial part of the energy four-vector $P$.

Define

$Z_{M_{\ast }}(\alpha ):=\frac{4\pi M_{\ast }^2}{\alpha }{K}_2(\alpha M_{\ast }),K_j(z):={\int }_0^{\mathrm{\infty }}{e}^{-z\cosh t}\cosh (jt)dt(j=1,2)\mathrm{.}$

and the observer-rest momentum probability law

$d{\nu }_{u,x}(\mathbf{p}):=Z_{M_{\ast }}(\alpha (x){)}^{-1}{e}^{-\alpha (x)E_u(P)}{d}^3\mathbf{p}\mathrm{.}$

Set

$\beta :=\frac{u}{k_{B}T},\zeta :=\log \text{\hspace{0.17em}⁣}\left(\frac{n{h}_{\mathrm{r}\mathrm{e}\mathrm{f}}^3{c}^3}{Z_{M_{\ast }}(\alpha )}\right),f(x,P):=e^{\zeta (x)+g_x(\beta (x),P)}\mathrm{.}$

Finally define the model number current and stress-energy tensor by the convergent shell moments

$J^a(x):=(h_{\mathrm{r}\mathrm{e}\mathrm{f}}^3{c}^3{)}^{-1}{\int }_{{\mathcal{H}}_{M_{\ast },x}^{+}}{P}^{a}f(x,P)d{\mu }_{M_{\ast }}(P),{\mathsf{T}}^{ab}(x):=(h_{\mathrm{r}\mathrm{e}\mathrm{f}}^3{c}^3{)}^{-1}{\int }_{{\mathcal{H}}_{M_{\ast },x}^{+}}{P}^a{P}^{b}f(x,P)d{\mu }_{M_{\ast }}(P)\mathrm{.}$

The mathematical suppliers Jüttner partition, probability, and perfect-fluid moments and Probability measures and probability spaces give $0<Z_{M_{\ast }}<\mathrm{\infty }$, show that ${\nu }_{u,x}$ has total mass one, and evaluate the current and stress moments:

$J^a=n{u}^a,{\mathsf{T}}^{ab}=(\epsilon +p)u^a{u}^b+p{g}^{ab},\epsilon =n(M_{\ast }\frac{K_1(\alpha M_{\ast })}{K_2(\alpha M_{\ast })}+\frac{3}{\alpha }),p=\frac{n}{\alpha }=n{k}_{B}T\mathrm{.}$

Here $\epsilon$ and $p$ are the exact moments of this one-species kinetic model. They are not automatically identified with an independently adopted material state function or stress tensor. The suppliers use the stated countable-choice strength ${\mathrm{A}\mathrm{C}}_{\omega }$ only; this definition adds no arbitrary-index choice principle. The scale $h_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is a declared phase-space unit convention, not Planck's constant or a measured parameter.

This is an exact classical model definition. It does not claim that an actual material realizes this distribution, that collisions drive it to equilibrium, that the full field $f$ solves a transport equation, or that any quantum-statistical or empirical law follows.