Assuming a spherically symmetric line element of the form 1) \begin{equation} ds^2 = -f\,dt^2 + \frac{dr^2}{f} + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2 \end{equation} where $f$ is an arbitrary function of $r$, Einstein's equation with an electromagnetic source, representing a point charge, can be solved for $f$, yielding \begin{equation} f = 1 - \frac{2m}{r} + \frac{q^2}{r^2} \end{equation} The resulting spacetime is known as the Reissner-Nordström geometry, and represents a black hole with mass $m$ and charge $q$.

The global geometry of a Reissner-Nordström black hole differs considerably from that of a Schwarzschild black hole. One way to see this is to notice that (assuming $|q|<m$) there are now two horizons, since \begin{equation} 1 - \frac{2m}{r} + \frac{q^2}{r^2} = 0 \Longrightarrow r=r_\pm=m\pm\sqrt{m^2-q^2} \end{equation} where we have assumed that $m>|q|>0$. The geometry can be extended across one horizon at a time using a procedure analogous to the Kruskal extension of the Schwarzschild geometry, and the resulting Penrose diagram has many asymptotic regions, not just two. The cases $m=|q|>0$ and $|q|>m>0$ can be handled similarly, noting that $r=m$ is a double root of $f$ in the first case, and that there are no roots in the second case.

Further discussion, including the corresponding Penrose diagrams, can be found in Chapter 18 of d'Inverno.