Electric circulation obstructs a global charged equilibrium scalar

The product metric is smooth, has signature $(-,+,+,+)$, and is invariant under translations of $x^0$; hence $\xi ={\mathrm{\partial }}_{x^0}$ is a future timelike Killing field and $N=\sqrt{-g(\xi ,\xi )}=1$. The form $\vartheta$ is globally defined, closed, and independent of $x^0$, so $A=E_0{x}^0\vartheta$ is a global one-form and differentiation gives $F=dA=E_{0}d{x}^0\wedge \vartheta$. Since ${\mathcal{L}}_{\xi }d{x}^0=d(\xi (x^0))=d1=0$ and ${\mathcal{L}}_{\xi }\vartheta =0$, the product rule gives ${\mathcal{L}}_{\xi }F=0$; direct contraction gives ${\iota }_{\xi }F=E_0\vartheta$.

The product rule gives ${\mathcal{L}}_{\xi }A=E_0\xi (x^0)\vartheta +E_0{x}^0{\mathcal{L}}_{\xi }\vartheta =E_0\vartheta$, whose period around the circle is $E_{0}L\mathrm{\ne }0$. Every exact one-form $d\mathrm{\Lambda }$ for a global smooth real $\mathrm{\Lambda }$ has zero integral around this closed circle, since its line integral is the endpoint difference of $\mathrm{\Lambda }$. Thus no global ${\mathrm{\Lambda }}_{\xi }$ satisfies ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$. Under a gauge change $A\mapsto A+d\chi$, the coordinate formula gives $({\mathcal{L}}_{\xi }d\chi {)}_a={\xi }^b{\mathrm{\partial }}_b{\mathrm{\partial }}_a\chi +({\mathrm{\partial }}_a{\xi }^b){\mathrm{\partial }}_b\chi ={\mathrm{\partial }}_a({\xi }^b{\mathrm{\partial }}_b\chi )$; this is exact and has zero circle period. Hence the nonzero period cannot be removed by a single-valued gauge.

Here $\beta =\xi \mathrm{/}(k_B{T}_K)$ is Killing. If a global smooth $\zeta$ made $f$ from [F1] stationary, [F2] would give $d\zeta =-cq{\iota }_{\beta }F=-\frac{cq{E}_0}{k_B{T}_K}\vartheta \mathrm{.}$ Integrating around the positive circle gives $0=\oint d\zeta =-cq{E}_{0}L\mathrm{/}(k_B{T}_K)$, which contradicts $q{E}_0\mathrm{\ne }0$ and $T_K>0$. Hence no single-valued global $\zeta$ exists.