is large, for a solution with finite $\tilde A$, the energy-momentum tensor is of the order $O(1/\alpha^2)$ and the coupling of the Maxwell field to the metric is suppressed. This limit is analogous to the infinite charge limit considered in the holographic description of superconductivity [@Hartnoll:2008vx].

The Large $\alpha$ Limit
------------------------

To keep $\tilde A$ finite, we have to scale the background gauge field as well. This means that the chemical potential $\mu$ of the black hole should also be scaled in such a way that the combination $\alpha \mu$ remains finite. Let us examine what this limit means to the black hole solution. The Reissner-Nordström solution has the metric $$\label{E:RNblackhole}
ds^2=- H(r) dt^2 + \frac 1 {H(r)} dr^2 + r^2 d{\vec x}^2\;, \qquad \vec x = 
(x^2, x^3, x^4) \;,$$ where the function $H(r)$ is given by $$\label{E:RNblackhole2}
H(r)=r^2\left[1-\left(\frac {r_+} r\right)^4\right]\;,$$ The temperature in this limit of $\mu \rightarrow 0$ becomes $$T=\frac{r_+}{2\pi} \left(2- \frac{\mu^2}{3r_+^2}\right)\rightarrow \frac{r_+}{\pi}\;.$$ The background geometry in this limit is simply the (neutral) Schwarzschild $AdS_5$ solution. In terms of the rescaled finite gauge field $\tilde{A}$, the background field strength is given by $$\tilde F=\frac{\tilde E}{r^3} dt\wedge dr\;, \qquad \text{where $\tilde E=-2\alpha \mu r_+^2=-\frac{2 r_+^3}{\pi}\frac 1 {\tau}$,}$$ where 