Product electrovacuum equations from full factor Ricci

On the two-dimensional base, the inverse metric components are $g^{\tau \tau }=-1\mathrm{/}(L^{2}f)$ and $g^{zz}=f\mathrm{/}{L}^2$. Applying the Christoffel formula [F4] gives, apart from lower-index symmetry, ${\mathrm{\Gamma }}^{\tau }{}_{\tau z}=f^{\mathrm{\prime }}\mathrm{/}(2f)$, ${\mathrm{\Gamma }}^z{}_{\tau \tau }=f{f}^{\mathrm{\prime }}\mathrm{/}2$, and ${\mathrm{\Gamma }}^z{}_{zz}=-f^{\mathrm{\prime }}\mathrm{/}(2f)$. Their traces ${\mathrm{\Gamma }}^a{}_{\tau a}$ and ${\mathrm{\Gamma }}^a{}_{za}$ are zero. Substitution in the coordinate curvature formula [F6] and the Ricci contraction [F5], with convention [F2], gives ${\mathrm{Ric}}_{\tau \tau }={\mathrm{\partial }}_z(f{f}^{\mathrm{\prime }}\mathrm{/}2)-2(f^{\mathrm{\prime }}\mathrm{/}2f)(f{f}^{\mathrm{\prime }}\mathrm{/}2)=f{f}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}2$; similarly ${\mathrm{Ric}}_{zz}={\mathrm{\partial }}_z(-f^{\mathrm{\prime }}\mathrm{/}2f)-\{(f^{\mathrm{\prime }}\mathrm{/}2f{)}^2+(-f^{\mathrm{\prime }}\mathrm{/}2f{)}^2\}=-f^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}(2f)$, and ${\mathrm{Ric}}_{\tau z}=0$. Since $f^{\mathrm{\prime }\mathrm{\prime }}=2$, these are $f$ and $-1\mathrm{/}f$, respectively, which equal $-L^{-2}$ times the corresponding components of the base metric.

On a regular polar chart from [A2], the unit sphere metric $d{\theta }^2+{\sin }^2\theta d{\varphi }^2$ has, by [F4], the only nonzero symbols ${\mathrm{\Gamma }}^{\theta }{}_{\varphi \varphi }=-\sin \theta \cos \theta$ and ${\mathrm{\Gamma }}^{\varphi }{}_{\theta \varphi }={\mathrm{\Gamma }}^{\varphi }{}_{\varphi \theta }=\cot \theta$. The coordinate curvature formula [F6] and contraction [F5] give ${\mathrm{Ric}}_{\theta \theta }=-{\mathrm{\partial }}_{\theta }\cot \theta -{\cot }^2\theta =1$ and ${\mathrm{Ric}}_{\varphi \varphi }={\mathrm{\partial }}_{\theta }(-\sin \theta \cos \theta )-{\cos }^2\theta +2{\cos }^2\theta ={\sin }^2\theta$; the mixed component is zero. Scaling the sphere metric by the constant $L^2$ leaves its connection unchanged, so in the orthonormal coframe of the Statement its Ricci entries are $L^{-2},L^{-2}$. The product metric has no mixed Christoffel symbols because each factor coefficient depends only on its own factor coordinates; hence its mixed Ricci entries vanish.

The field is constant in $(\tau ,z)$ and has only ${\mathcal{F}}_{\tau z}=-L$ and ${\mathcal{F}}_{z\tau }=L$. Thus $d\mathcal{F}=0$. The determinant satisfies $\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}=L^4\sin \theta$, and raising indices gives ${\mathcal{F}}^{\tau z}=L^{-3}$ and ${\mathcal{F}}^{z\tau }=-L^{-3}$. In ${\mathrm{\nabla }}_a{\mathcal{F}}^{ab}={\mathrm{\partial }}_a{\mathcal{F}}^{ab}+{\mathrm{\Gamma }}^a{}_{ac}{\mathcal{F}}^{cb}+{\mathrm{\Gamma }}^b{}_{ac}{\mathcal{F}}^{ac}$, the last term vanishes by symmetry of the connection and antisymmetry of the field. For $b=\tau ,z$, the only derivative terms are ${\mathrm{\partial }}_z{\mathcal{F}}^{z\tau }$ and ${\mathrm{\partial }}_{\tau }{\mathcal{F}}^{\tau z}$, both zero; their trace terms vanish because ${\mathrm{\Gamma }}^a{}_{a\tau }={\mathrm{\Gamma }}^a{}_{az}=0$ from the base Christoffels and the product has no mixed symbols. For $b=\theta ,\varphi$, all ${\mathcal{F}}^{ab}$ vanish. Hence ${\mathrm{\nabla }}_a{\mathcal{F}}^{ab}=0$; [F3] gives $d({\star }_g\mathcal{F})=0$.

In the orthonormal coframe, ${\mathcal{F}}_L=-L^{-1}{e}^0\wedge e^1$, so ${\mathcal{F}}_{AB}{\mathcal{F}}^{AB}=-2\mathrm{/}{L}^2$. Direct contraction in signature $(-,+,+,+)$ gives $Q_{AB}=\frac{1}{2L^2}\mathrm{diag}(1,-1,1,1{)}_{AB}$: the electric pair contributes $+1\mathrm{/}(2L^2)$ to the time entry and $-1\mathrm{/}(2L^2)$ to the radial entry, while the trace term contributes $+1\mathrm{/}(2L^2)$ to each sphere entry. Steps 1.1 and 2.1 give $\mathrm{Ric}=2Q$ on the regular polar locus. The metric and two-form are smooth on the full product by [F1], so $\mathrm{Ric}(g)$ and $Q({\mathcal{F}}_L)$ are smooth tensor fields there; by density in [A2], their equality extends by continuity to both sphere poles. The displayed coframe components are local to those polar charts. Together with the Maxwell equations in step 3.1 this proves the full definition of a geometric electrovac datum on the stated domain.