Rotation and charged equilibrium obstruction

Let $O$ be a nonempty connected open subset of a smooth time-oriented Lorentzian four-manifold $(M,g,N)$ of signature $(-,+,+,+)$, with Levi-Civita connection $\mathrm{\nabla }$ and time-orientation field $N$. Let $\beta$ be any smooth vector field on $O$ satisfying $g(\beta ,\beta )<0$ and $g(\beta ,N)<0$ everywhere; it is not assumed Killing, and Killing means ${\mathcal{L}}_{\beta }g=0$. In length coordinates $x^0=ct$, take $\beta$ to have components in ${\mathrm{J}}^{-1}$, fix $k_B>0$ in ${\mathrm{J}\mathrm{K}}^{-1}$, and let $\mu :O\to \mathbb{R}$ be smooth in joules. Define $T:=\frac{1}{k_B\sqrt{-g(\beta ,\beta )}},u:=k_{B}T\beta ,{\varpi }_{ab}:=-\frac{1}{2}({\mathrm{\nabla }}_a{\beta }_b-{\mathrm{\nabla }}_b{\beta }_a),$ and $h_{ab}:=g_{ab}+u_a{u}_b,D:=u^a{\mathrm{\nabla }}_a,a_b:=u^a{\mathrm{\nabla }}_a{u}_b,\theta :={\mathrm{\nabla }}_a{u}^a,{\sigma }_{ab}:=h_a{}^c{h}_b{}^d{\mathrm{\nabla }}_{(c}{u}_{d)}-\frac{1}{3}\theta h_{ab},\alpha :=-\frac{\mu }{k_{B}T}\mathrm{.}$ Then $\beta$ is Killing if and only if $DT=0$, $\theta =0$, ${\sigma }_{ab}=0$, and $h_a{}^b{\mathrm{\nabla }}_b\log T=-a_a$.

In the Killing subcase, let $A$ be a global smooth real one-form with SI length-coordinate components in $\mathrm{T}\mathrm{m}$, set $F:=dA$ (with components in teslas), and define $\omega :={\iota }_{\beta }F$ by $({\iota }_{\beta }F{)}_a={\beta }^b{F}_{ba}$. The components of $\omega$ have units ${\mathrm{T}\mathrm{J}}^{-1}$. Assume $d\omega =0$. Here exact means $\omega =d\psi$ for a globally defined smooth scalar $\psi$ with units ${\mathrm{T}\mathrm{m}\mathrm{J}}^{-1}$; for a piecewise-$C^1$ closed curve $\gamma :[0,1]\to O$, its period is the oriented integral ${\mathrm{Per}}_{\gamma }(\omega ):={\sum }_j{\int }_{t_j}^{t_{j+1}}{\omega }_{\gamma (t)}(\dot{\gamma }(t))dt$, with the curve regularity and endpoint convention in the period definition. Let $m>0$ be a particle rest mass in kilograms, let $q_{\mathrm{S}\mathrm{I}}\mathrm{\ne }0$ be a constant charge in coulombs, and take $c>0$ in metres per second. Set $q_{\mathrm{e}\mathrm{f}\mathrm{f}}:=c{q}_{\mathrm{S}\mathrm{I}}$, with units ${\mathrm{C}\mathrm{m}\mathrm{s}}^{-1}$. Choose any constant $T_{\ast }>0$ kelvins and put $\xi :=k_B{T}_{\ast }\beta$ and $N_{\xi }:=\sqrt{-g(\xi ,\xi )}$. Then $\xi$ is a future-timelike Killing field with dimensionless components, $u=\xi \mathrm{/}{N}_{\xi }$ is future unit timelike, $N_{\xi }=T_{\ast }\mathrm{/}T$, and $\beta =u\mathrm{/}(k_{B}T)$. These are the stationary geometric data required by the charged stationarity theorem. For the ansatz $f=\exp (\zeta +g(\beta ,P))$ on the future massive shell defined by the charged particle data, with $\zeta :=\mu \mathrm{/}(k_{B}T)$ and $\alpha :=-\zeta$, stationarity is equivalent, in this already-Killing subcase, to $d\alpha =q_{\mathrm{e}\mathrm{f}\mathrm{f}}\omega$; both sides have inverse-length units. Thus the equation for an unknown global smooth dimensionless scalar $\alpha$ has a solution exactly when $\omega$ is exact; any nonzero period obstructs such a solution. The scalar $\alpha$ corresponds to chemical potential $\mu =-k_{B}T\alpha$ in joules.

If the same normalized field $\xi =k_B{T}_{\ast }\beta$ has a global potential compensator ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$, then $\omega$ is exact and all its periods vanish. For the rotating model, take global Cartesian length coordinates $(x^0,x,y,z)$ on Minkowski space $M={\mathbb{R}}^4$, with $x^0=ct$, metric $-(d{x}^0{)}^2+d{x}^2+d{y}^2+d{z}^2$, and future direction selected by ${\mathrm{\partial }}_{x^0}$. Put $r:=\sqrt{x^2+y^2}$ and, away from the axis $r=0$, let $\varphi$ be the polar angle in the $x,y$-plane. The rotation field ${\mathrm{\partial }}_{\varphi }:=-y{\mathrm{\partial }}_x+x{\mathrm{\partial }}_y$ is defined by this Cartesian formula on all of $M$, including the axis; away from the axis the metric is $-(d{x}^0{)}^2+d{r}^2+r^{2}d{\varphi }^2+d{z}^2$. For $b>0$ in ${\mathrm{J}}^{-1}$ and $\mathrm{\Omega }\in \mathbb{R}$ in ${\mathrm{m}}^{-1}$, set $\beta :=b({\mathrm{\partial }}_{x^0}+\mathrm{\Omega }{\mathrm{\partial }}_{\varphi })$. This field is future timelike exactly where $1-{\mathrm{\Omega }}^2{r}^2>0$. When $\mathrm{\Omega }\mathrm{\ne }0$ this is $r<1\mathrm{/}\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$ and $T=1\mathrm{/}(k_{B}b\sqrt{1-{\mathrm{\Omega }}^2{r}^2})$; when $\mathrm{\Omega }=0$ every $r$ is allowed and $T=1\mathrm{/}(k_{B}b)$. These rotating formulas use the same SI length-coordinate convention $x^0=ct$ as the preceding clauses; no natural-unit substitution is made.

Thermal vorticity and equilibrium domain

Period of a smooth one-form along a closed curve

An exact one-form has zero period on every closed piecewise-C¹ curve

Charged particle and Liouville data on a stationary Lorentzian domain

A global potential compensator makes the contracted field exact

Exact thermal Killing kinematic equivalence

Global electrochemical redshift from charged Liouville stationarity

Closed and exact differential forms

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The field $\beta$ is the selected thermal flow in this definition. Stationary non-static geometries can admit distinct timelike Killing flows and therefore distinct temperature profiles; no universal heat-bath frame or global thermal state is asserted. The charged condition is a conditional mathematical consequence of the specified Liouville model, not evidence that a material equilibrium is realized.