Does the divergence of the electric potential in the event horizon describe a singularity? A possible way to analyze the presence of spacetime singularities deals with divergent invariants of the curvature tensor. For example, in the Schwarzschild black hole the scalar quantity defined from the square of the curvature tensor diverges in $r=0$, and it indicates the presence of a spacetime singularity (which can also be pointed out with the analysis of geodesics).
If we consider now the Reissner-Nordstrom black hole with an electric charge described by the potential $A_\mu=[q/r,0,0,0]$, the norm of this vector is $A_\mu A^\mu = q^2/(r^2-2mr+q^2)$. But this scalar quantity diverges in the event horizons, i.e. if $r^2-2mr+q^2=0$.
On the other hand, the Faraday tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ provides the electric field and is totally regular in the horizon, but if we study for example the Dirac equation of an electron the quantity that appears in the equation is not $F_{\mu\nu}$ but $A_\mu$.
Does the divergence of $A_\mu A^\mu$ in the horizon have a physical meaning? Why don't textbooks or other references say anything about this topic?
 A: @hulsey Yes, it is not gauge invariant but the potential appears for example in the Dirac equation (contracted with gamma-matrices). Furthermore, now I'm thinking that in fact this contraction $\gamma^{a}e^\mu{}_a A_{\mu}$ appearing in the Dirac equation is singular in the event horizons too:
\begin{align}
    e^\mu{}_a=
    \left(
    \begin{array}{cccc}
    \frac{1}{\sqrt{1-2m/r+q^2/r^2}} & 0 & 0 & 0\\
    0 & \sqrt{1-2m/r+q^2/r^2} & 0 & 0\\
    0 & 0 & \frac{1}{r} & 0\\
    0 & 0 & 0 & \frac{1}{r\sin\theta}\\
    \end{array}
    \right)
\end{align}
\begin{equation}A_{\mu}=\left[\frac{q}{r},0,0,0\right].\end{equation}
If we calculate this term that appears in the Dirac equation describing the interaction with charged fermions the result is:
\begin{equation}\gamma^{a}e^\mu{}_a A_{\mu}=\gamma^{0}\frac{q}{\sqrt{r^2-2m+q^2}}.\end{equation}
This term goes to infinity in the event horizons and the gamma matrix $\gamma^{0}$ takes values in the Lorentz algebra, i.e. it does not depend on $m$ and $q$.
According to this, the divergence at the event horizons is a problem, isn't it?
