Nonzero parallel Maxwell fields satisfy torus constraints with isotropic initial curvature

The flat torus has $R(h)=0$ and zero Christoffel symbols. Constant $e,b$ therefore have $D_i{e}^i=D_i{b}^i=0$; their collinearity gives $e{\times }_{h}b=0$. Since $K=Hh$ with constant $H$, $DK=0$, ${\mathrm{tr}}_{h}K=3H$, and $\mathrm{\mid }K{\mathrm{\mid }}_h^2=3H^2$. Hence the Hamiltonian left side is $0+9H^2-3H^2=6H^2$, which by the definition of $H$ equals $\kappa {\mu }_0^{-1}(\mathrm{\mid }e{\mathrm{\mid }}^2+\mathrm{\mid }b{\mathrm{\mid }}^2)$, exactly the right side of [L1] for $\mathrm{\Lambda }=0$. The momentum left side and right side both vanish. Thus all four constraints hold. The scalar under the square root is positive since both fields are nonzero, and $H$ is finite and real.

The data are smooth and $h$ is uniformly positive on the compact torus. For any integer $s\ge 5$ they belong to the Sobolev classes in [F1]; their verified constraints meet its hypotheses. Applying [F1] yields a smooth local full source-free solution. The reconstructed initial two-form has nonzero electric and magnetic components because $e,b\mathrm{\ne }0$, so the solution is nontrivial. The argument is a model deduction, not an empirical claim.

(A nonzero magnetic period.) The initial spatial two-form is ${\omega }_b=\frac{1}{2}{ϵ}_{ijk}{b}_{k}d{X}^i\wedge d{X}^j=\frac{1}{2}{L}_0^2{ϵ}_{ijk}{b}_{k}d{\theta }^i\wedge d{\theta }^j$. Write the development from step 2.1 as $M=(-T,T)\times {\mathbb{T}}_{2\pi }^3$ with initial embedding ${\iota }_{\mathrm{\Sigma }}(\theta )=(0,\theta )$. For each $i$, choose the cyclic pair $(j,k)$ with ${ϵ}_{ijk}=1$ and orient the coordinate two-torus ${\mathbb{T}}_{jk}^2=\{{\theta }^i=0\}$ by $d{\theta }^j\wedge d{\theta }^k$. Its magnetic period, the integral of the restricted two-form, is ${\int }_{{\mathbb{T}}_{jk}^2}{\omega }_b=(2\pi L_0{)}^2{b}_i$. Since $b\mathrm{\ne }0$ and $L_0>0$, some such period is nonzero. If $F=dA$ for a globally defined smooth one-form on a neighborhood of the full initial torus, then ${\omega }_b=d({\iota }_{\mathrm{\Sigma }}^{\ast }A)$. Write ${\iota }_{\mathrm{\Sigma }}^{\ast }A={\sum }_{\mathrm{\ell }}{A}_{\mathrm{\ell }}(\theta )d{\theta }^{\mathrm{\ell }}$ with smooth periodic coefficients. Integrating ${\mathrm{\partial }}_j{A}_k-{\mathrm{\partial }}_k{A}_j$ over the coordinate two-torus gives zero by periodicity in each variable, contradicting its nonzero period. Thus no such global potential exists.