Gauge-compensated charged Killing energy

Let $r\ge 1$ be finite. For each cell $i=1,\dots ,r$, let $U_i\subseteq {\mathbb{R}}^2$ be open and convex, with coordinates $(E_i,n_i)$, and let $S_i\in C^1(U_i,\mathbb{R})$ be concave. The coordinates may be assigned SI dimensions $[E_i]=\mathrm{J}$, $[n_i]=1$, and $[S_i]=\mathrm{J}{\mathrm{K}}^{-1}$; the particle-number coordinate is a continuous thermodynamic variable. Fix constants $N_i>0$ (dimensionless), ${\mathrm{\Phi }}_i\in \mathbb{R}$ (volts), and a common charge $q\in \mathbb{R}$ (coulombs). Define $\mathcal{E}(x):={\sum }_{i=1}^r(N_i{E}_i+q{\mathrm{\Phi }}_i{n}_i),\mathcal{N}(x):={\sum }_{i=1}^r{n}_i,S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x):={\sum }_{i=1}^r{S}_i(E_i,n_i)\mathrm{.}$ For a supplied feasible interior point $x^{\ast }\in U:={\prod }_i{U}_i$, let $A:{\mathbb{R}}^{2r}\to {\mathbb{R}}^2$ be the linear map whose rows are the coefficients of $(\mathcal{E},\mathcal{N})$. Call $x^{\ast }$ stationary under independent exchange variations when $D{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\ast })[h]=0$ for every $h\in \ker A$. Assume ${\mathrm{\partial }}_{E_i}{S}_i(x_i^{\ast })>0$ and define $\frac{1}{T_i}:={\mathrm{\partial }}_{E_i}{S}_i(x_i^{\ast }),\frac{{\mu }_i}{T_i}:=-{\mathrm{\partial }}_{n_i}{S}_i(x_i^{\ast })\mathrm{.}$ Then $x^{\ast }$ is stationary if and only if there are constants $T_K>0$ and ${\mu }_K\in \mathbb{R}$ such that, for every cell, $N_i{T}_i=T_K,N_i{\mu }_i+q{\mathrm{\Phi }}_i={\mu }_K\mathrm{.}$ If each $S_i$ is concave, every such stationary point is a global maximizer of $S_{\mathrm{t}\mathrm{o}\mathrm{t}}$ on the feasible set in $U$. If each $S_i$ is strictly concave, that maximizer is unique whenever it exists. No existence or boundary attainment of a stationary point is asserted. These are finite-dimensional conditional deductions; the coefficients $N_i,{\mathrm{\Phi }}_i$ are inputs, not predictions about a physical system.

The proof uses only the finite affine constraint map and differentiable concave functions. It does not import a statistical sampling rule, a Gibbs state, an equation of state, or a gravitational field equation. In the neutral case $q=0$, the particle relation reduces to $N_i{\mu }_i={\mu }_K$ without division by $q$.

Use SI dimensions and the signature $(-,+,+,+)$. Take the geometric data of Stationary thermal geometric data: $O$ is a nonempty connected open domain in a smooth oriented, time-oriented Lorentzian four-manifold $(M,g)$, $\mathrm{\nabla }$ is the Levi–Civita connection, and $\xi$ is a smooth future-directed timelike Killing field with dimensionless components in physical length coordinates $x^0=ct$ and $x^i$ measured in metres. Put $N:=\sqrt{-g(\xi ,\xi )}>0$ and $u:=\xi \mathrm{/}N$, so $g(u,u)=-1$. Its smooth tangent-bundle data carry the supplier's explicit foundational assumption The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) (${\mathrm{A}\mathrm{C}}_{\omega }$); no stronger choice principle is added here.

Let $A$ be a smooth real one-form on $O$: for each $x\in O$, $A_x:T_{x}O\to \mathbb{R}$ is linear, and in every smooth chart $(x^a)$ its coefficients $A_a$ are smooth and obey the covector transformation law $A_i^{\mathrm{\prime }}=A_a\mathrm{\partial }{x}^a\mathrm{/}\mathrm{\partial }{y}^i$ on an overlapping chart $(y^i)$. The coefficients have units tesla metres. Define $F$ in each chart by $F_{ab}:=(dA{)}_{ab}:={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a\mathrm{.}$ This prescription defines a global smooth antisymmetric covariant two-tensor: under a change of chart $x=x(y)$ the chain rule gives $F_{ij}^{\mathrm{\prime }}=F_{ab}\frac{\mathrm{\partial }{x}^a}{\mathrm{\partial }{y}^i}\frac{\mathrm{\partial }{x}^b}{\mathrm{\partial }{y}^j},$ since the only additional term is $A_a({\mathrm{\partial }}_i{\mathrm{\partial }}_j{x}^a-{\mathrm{\partial }}_j{\mathrm{\partial }}_i{x}^a)=0$. Thus $F=dA$ is a two-form; its components have units tesla. A global gauge compensator is a smooth real function ${\mathrm{\Lambda }}_{\xi }$ on $O$ satisfying ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi },({\mathcal{L}}_{\xi }A{)}_a:={\xi }^b{\mathrm{\partial }}_b{A}_a+A_b{\mathrm{\partial }}_a{\xi }^b$ in every smooth chart. This defines the Lie-derivative coefficients globally: under $x=x(y)$, put $J_i^a:=\mathrm{\partial }{x}^a\mathrm{/}\mathrm{\partial }{y}^i$ and $K_a^i:=\mathrm{\partial }{y}^i\mathrm{/}\mathrm{\partial }{x}^a$. Substitution of $A_i^{\mathrm{\prime }}=A_a{J}_i^a$ and ${\xi }^{\mathrm{\prime }i}=K_a^i{\xi }^a$ gives $({\mathcal{L}}_{\xi }A{)}_i^{\mathrm{\prime }}={\xi }^b({\mathrm{\partial }}_b{A}_a)J_i^a+{\xi }^b{A}_a{K}_b^j{\mathrm{\partial }}_j{J}_i^a+A_c{J}_j^c({\mathrm{\partial }}_i{K}_b^j){\xi }^b+A_b{J}_i^a{\mathrm{\partial }}_a{\xi }^b\mathrm{.}$ Differentiating $J_j^c{K}_b^j={\delta }_b^c$ and using ${\mathrm{\partial }}_j{J}_i^a={\mathrm{\partial }}_i{J}_j^a$ cancels the two middle terms, leaving $({\mathcal{L}}_{\xi }A{)}_i^{\mathrm{\prime }}=({\mathcal{L}}_{\xi }A{)}_a{J}_i^a$. The units of ${\mathrm{\Lambda }}_{\xi }$ are tesla metres. No global compensator is assumed to exist for an arbitrary smooth $A$. Define contraction by $({\iota }_{\xi }F{)}_a:={\xi }^b{F}_{ba}$. When a global smooth compensator ${\mathrm{\Lambda }}_{\xi }:O\to \mathbb{R}$ satisfying ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$ is supplied as additional data, set ${\mathrm{\Psi }}_{\xi }:={\xi }^a{A}_a-{\mathrm{\Lambda }}_{\xi },{\mathrm{\Phi }}_{\xi }:=-c{\mathrm{\Psi }}_{\xi }\mathrm{.}$ Thus ${\mathrm{\Psi }}_{\xi }$ has units tesla metres and ${\mathrm{\Phi }}_{\xi }$ has units volts. These two scalars are defined only for such supplied compensated data; the trajectory, mass shell, transport and ansatz below require no compensator.

Let $c>0$ be the speed of light, let $m>0$ be a particle rest mass in kilograms, let $q\in \mathbb{R}$ be its charge in coulombs, and put $M:=m{c}^2>0$ joules. A charged massive Lorentz trajectory is a $C^2$ curve $x:J\to O$, where $J$ is an open real interval, parametrized by physical length $s_m=c\tau$ in metres, with future unit tangent $w:=dx\mathrm{/}d{s}_m$, $g(w,w)=-1$, energy four-vector $P:=Mw$, and equation ${\mathrm{\nabla }}_w{P}^a=cq{F}^a{}_b{w}^b,F^a{}_b:=g^{ac}{F}_{cb}\mathrm{.}$ In particular $g(P,P)=-M^2$ and $P$ is future directed. This is a conditional definition of curves satisfying the equation; it makes no existence claim.

The future massive shell for this definition is the subset ${\mathcal{P}}_M^{+}(O):=\{(x,P)\in TO:g_x(P,P)=-M^2,P\text{ future directed}\}\mathrm{.}$ On the full tangent bundle $TO$, define charged transport as the vector field whose integral curves, with parameter $\lambda =s_m\mathrm{/}M$, satisfy $\frac{d{x}^a}{d\lambda }=P^a,{\mathrm{\nabla }}_{dx\mathrm{/}d\lambda }{P}^a=cq{F}^a{}_b{P}^b\mathrm{.}$ In a smooth chart its expression is ${\mathcal{L}}_{F,q}:=P^a{\mathrm{\partial }}_{x^a}+(-{\mathrm{\Gamma }}^a{}_{bc}{P}^b{P}^c+cq{F}^a{}_b{P}^b){\mathrm{\partial }}_{P^a}\mathrm{.}$ The restriction of this field to ${\mathcal{P}}_M^{+}(O)$ is justified by Charged Liouville transport preserves the future massive shell ↗. For any smooth real scalar $\zeta$ and smooth vector field $\beta$ with components in inverse joules, define the dimensionless charged exponential ansatz on the shell by $f(x,P):=\exp \text{\hspace{0.17em}⁣}(\zeta (x)+g_x(\beta (x),P)),{\beta }_b:=g_{bc}{\beta }^c\mathrm{.}$ The equation ${\mathcal{L}}_{F,q}f=0$ means that the derivative of this function along the restricted charged transport vector field vanishes.

Under a smooth gauge change $A\mapsto A+d\chi$, with $\chi$ in tesla square metres, a supplied compensator changes by ${\mathrm{\Lambda }}_{\xi }\mapsto {\mathrm{\Lambda }}_{\xi }+\xi (\chi )$. These definitions fix the sign convention for $F$, the Lorentz force, and the potential terms used on this page.

For the smooth Lorentzian data, $m>0$, $q\in \mathbb{R}$, $F=dA$, and charged transport vector field defined in Charged particle and Liouville data on a stationary Lorentzian domain, the set ${\mathcal{P}}_M^{+}(O)$ is a smooth hypersurface of $TO$, and ${\mathcal{L}}_{F,q}$ is tangent to it. Hence its restriction to the future massive shell is a well-defined smooth vector field. No mass-shell measure or particle normalization is asserted.

Use the Lorentzian geometry, units, charge sign, potential $A$, field $F=dA$, compensator ${\mathrm{\Lambda }}_{\xi }$, and $C^2$ charged trajectory $x(s_m)$ of Charged particle and Liouville data on a stationary Lorentzian domain. Assume the global potential symmetry ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$. Define $H_{\xi }:=-g(P,\xi )+q{\mathrm{\Phi }}_{\xi }=-g(P,\xi )-cq{\mathrm{\Psi }}_{\xi }\mathrm{.}$ Then ${\iota }_{\xi }F=-d{\mathrm{\Psi }}_{\xi }$, and $H_{\xi }$ is constant along every interval on which the trajectory obeys the stated Lorentz equation. It is invariant under $A\mapsto A+d\chi$, ${\mathrm{\Lambda }}_{\xi }\mapsto {\mathrm{\Lambda }}_{\xi }+\xi (\chi )$. Replacing the potential reference by ${\mathrm{\Phi }}_{\xi }+C_0$ changes $H_{\xi }$ by $q{C}_0$.

This is a conditional mathematical statement about curves satisfying the displayed equation. It asserts neither existence of such curves for arbitrary initial data nor that the model describes nature.

For a global grand-canonical multiplier ${\mu }_K$, the reference shift ${\mathrm{\Phi }}_{\xi }\mapsto {\mathrm{\Phi }}_{\xi }+C_0$ is accompanied by ${\mu }_K\mapsto {\mu }_K+q{C}_0$; hence $H_{\xi }-{\mu }_K$ is reference independent. The source treatments are context only: the proof above fixes its own Lorentz-force and potential conventions and does not assume a fluid model.

Let $O$ be a connected domain with the stationary Lorentzian data, potential $A$, field $F=dA$, Killing field $\xi$, lapse $N=\sqrt{-g(\xi ,\xi )}$, future unit stationary observer $u=\xi \mathrm{/}N$, and charged transport on the future mass shell from Charged particle and Liouville data on a stationary Lorentzian domain. Let $m>0$, $q\in \mathbb{R}$, $\beta :=u\mathrm{/}(k_{B}T)$ for a smooth proper-temperature scalar $T:O\to (0,\mathrm{\infty })$, and $\zeta :=\mu \mathrm{/}(k_{B}T)$ for a smooth chemical-potential scalar $\mu :O\to \mathbb{R}$. Define $f(x,P)=\exp (\zeta +g(\beta ,P))$. Without any global potential-symmetry assumption, ${\mathcal{L}}_{F,q}f=0⟺\beta \text{ is Killing and }d\zeta =-cq{\iota }_{\beta }F\mathrm{.}$

If, in addition, a global smooth compensator satisfies ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$, define ${\mathrm{\Phi }}_{\xi }=-c({\iota }_{\xi }A-{\mathrm{\Lambda }}_{\xi })$. Then the stationarity conditions above are equivalent to the existence of constants $T_K>0$ and ${\mu }_K\in \mathbb{R}$ such that $NT=T_K,\beta =\frac{\xi }{k_B{T}_K},N\mu +q{\mathrm{\Phi }}_{\xi }={\mu }_K\mathrm{.}$ With $E_{\mathrm{l}\mathrm{o}\mathrm{c}}:=-g(P,u)$ and $H_{\xi }:=-g(P,\xi )+q{\mathrm{\Phi }}_{\xi }$, the same exponent then has both forms $\zeta +g(\beta ,P)=-\frac{E_{\mathrm{l}\mathrm{o}\mathrm{c}}-\mu }{k_{B}T}=-\frac{H_{\xi }-{\mu }_K}{k_B{T}_K}\mathrm{.}$ Here $N$ is dimensionless, $T,T_K$ are kelvins, $\mu ,{\mu }_K,E_{\mathrm{l}\mathrm{o}\mathrm{c}},H_{\xi }$ are joules per particle, ${\mathrm{\Phi }}_{\xi }$ is volts, $q$ is coulombs, and $\beta \cdot P$ and $\zeta$ are dimensionless. All claims are mathematical conditional deductions from the displayed smooth data; no equilibrium state, particle ensemble, or physical realization is asserted to exist.

The source treatments cover narrower fluid or symmetry-invariant-potential settings. The local proof keeps the exact whole-mass-shell condition and the global potential symmetry explicit. The TGR-ET01 thermal-normalization and massive-shell suppliers provide the exact connected-domain and positive-mass interfaces used above.