A flat Maxwell field need not solve the coupled Einstein equations

Given: The action-based field equations and SI stress convention of Einstein–Maxwell equations from compact action stationarity, on a flat background with zero cosmological parameter and no additional matter, under the indexed countable-choice convention [A2].

Einstein–Maxwell equations from compact action stationarity: the smooth fixed-sector field system requires $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, and $G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)$, where $T_{ab}^{\mathrm{E}\mathrm{M}}={\mu }_0^{-1}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})$.

Let $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)$, $x^0=ct$, and Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$. Use the trivial principal bundle $P=M\times U(1)$ with product projection $\pi$ and its global trivialization. For a nonzero constant electric-field component $E_0\in \mathbb{R}$ with SI unit $\mathrm{V}{\mathrm{m}}^{-1}$, defined relative to the observer $n={\mathrm{\partial }}_{x^0}$ in the $x^1$ direction, define the global physical potential $A\in {\mathrm{\Omega }}^1(M)$ by its components $A_0=E_0{x}^1\mathrm{/}c$ and $A_i=0$ for $i=1,2,3$. The normalized local representative is $a=A\mathrm{/}{\kappa }_0$, and the principal connection in this trivialization is $\omega =-i{z}^{-1}dz+{\pi }^{\ast }a$ for $z\in U(1)$. Its physical curvature is $F={\kappa }_{0}da=dA$, so $F_{01}=-E_0\mathrm{/}c$ and all other independent components vanish.

Choose the zero-matter model with configuration space ${\mathcal{C}}_m=\{\ast \}$, no matter-field variations, and differentiable functional $S_m[g,\omega ,\ast ]\equiv 0$. Its matter first variations vanish for every admissible metric and connection variation, hence its current and Hilbert stress are $j=0$ and $T^m=0$. Set $\mathrm{\Lambda }=0$ and $g=\eta$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): every sequence of nonempty sets has a choice function. This is the indexed countable-choice convention used by the compact integration suppliers; no full Axiom of Choice is assumed.