|\vec{k}| < 4\alpha E,
\label{inequality}$$ where $E$ is the background electric field and $\vec{k}$ is a projection of the spatial momentum onto the plane orthogonal the electric field. Let us describe the instability mode found in [@Nakamura:2009tf]. If the electric field is in the $x^1$ direction, it is convenient to decompose the 5-dimensional momentum as $(p_{\mu}, k_i)$, $\mu=0,1$ and $i=2,3,4$. Consider a linear fluctuation of the Maxwell field of the form, $$a_i = c_i^{(\pm)} e^{i p_{\mu} x^{\mu} + i k_i x^i},
\label{ansatz}$$ where $c_i^{(\pm)}$ are an eigenvectors of $\epsilon_{ijk} k_j$, $$\epsilon_{ijk} k_j c_k^{(\pm)} = \pm |\vec k| c_i^{(\pm)},$$ obeying the transverse gauge condition $k_i c_i^{(\pm)}=0$. Substituting (\[ansatz\]) into the equations of motion (\[eom\]), we find the dispersion relation for this mode as $$(p_0)^2-(p_1)^2 = k^2 \mp 4\alpha E k\ ,~~~k =|\vec{k}|.$$ This means that it is tachyonic for the range (\[inequality\]) if we choose $c_i^{(+)}$.

Non-linear Solutions
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We can find a non-linear solution triggered by the perturbation $a_i = c_i^{(+)} e^{i p_{\mu} x^{\mu} + i k_i x^i}$ as follows. Although the unstable mode breaks the translational invariance along the direction of the momentum $\vec{k}$, it is invariant under some combination of translation along $\vec{k}$ and rotation in the transverse 