result should be contrasted with the corresponding instability of the charged black hole in $AdS_5$, which we will study in the next section. There, we do not expect that the background electric field to disappear since the electric charge of the black hole is fixed by the chemical potential at the boundary. Indeed, we will find stable solutions with non-zero momentum $k$ in this case.

Maxwell-Chern-Simons Theory in the $AdS_5$ Black Hole Geometry {#sec:MCSSBHAdS}
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In this section, we will construct non-linear solutions which describe the end-point of the instability of the charged black hole in $AdS_5$. Since the Maxwell field contributes to the energy momentum tensor, in general we need to analyze the coupled Einstein and Maxwell equations. Here we will simplify the problem by taking a limit where we can ignore the back-reaction of the Maxwell field to the metric.

The Lagrangian density for the Maxwell-Chern-Simons theory is given by $$\mathcal{L} = -\frac 1 4 F_{IJ}F^{IJ} + \frac{\alpha}{3!\sqrt{-g}} \epsilon^{IJKLM} A_I F_{JK} F_{LM}\;.$$ Rescaling the gauge field as $\tilde A = \alpha A$, the Lagrangian density becomes $$\label{E:rescaledLagrangian}
\mathcal L = \frac{1}{\alpha^2} \left(-\frac 1 4 \tilde F_{IJ}\tilde F^{IJ} + \frac{1}{3!\sqrt{-g}} \epsilon^{IJKLM} \tilde A_I \tilde F_{JK} \tilde F_{LM}\right)\;.$$ When $\alpha$ 