How do electric charges interact in general relativity? In solutions to Einsteins equations one normally look at how a mass curves spacetime and then how that effect the geodesics of a test particle. So one only consider free movement of the particle. I know there is a solution with a black hole that has an electric charge and one can then describe how a neutrally charged testparticle will be attracted and repelled and so on. But what if the testparticle has an electric charge? Are there any solutions which describe how electric charges interact in general relativity or is it too difficult to figure out? 
 A: Well, of course General Relativity can handle charged particles in electromagnetic fields! The equations of motion for such a particle are the generalization to a curved spacetime of the Lorentz equation,
$$
\frac{d^{2}x^{\mu}}{ds^{2}}=-\Gamma^{\mu}_{\nu\sigma}\ \frac{d x^{\nu}}{ds} \frac{d x^{\sigma}}{ds}+\frac{q}{m}\ F^{\mu}_{\ \ \nu}\ \frac{d x^{\nu}}{ds}\qquad\qquad (\star)
$$
$(c=1)$, where $s$ is the proper time along the particle's world line, $F_{\mu\nu}$ is the electromagnetic field strength tensor and $F^{\mu}_{\ \ \nu}=g^{\mu\sigma}\ F_{\sigma \nu}$. The equation is just the geodesic equation plus an electromagnetic force term and can be derived from the action
$$
S=-m\int ds\ \bigg\{\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{d x^{\nu}}{ds}}+\frac{q}{m}\ A_{\mu}\,\frac{d x^{\mu}}{ds}\bigg\}
$$
by defining $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Equation $(\star)$ differs from the Lorentz equation in flat spacetime in that gravity enters both through the Christoffel symbols $\Gamma^{\mu}_{\nu\sigma}$ and through the inverse metric used to raise the index of $F^{\mu}_{\ \ \nu}$. Moreover, the Maxwell equations too are modified on a curved spacetime, as follows:
$$
\nabla_{\mu}F^{\mu\nu}=J^{\nu}
$$
Here $J^{\mu}$ is the four-current source for the electromagnetic field and $\nabla_{\mu}$ is the covariant derivative associated to the metric. Since gravity enters the Maxwell equations through the covariant derivative and raising of the indices of $F^{\mu\nu}$, its solutions are different from their analogue on a flat spacetime. Therefore, also the $F_{\mu\nu}$ itself in the Lorentz equation is in general different from that on flat spacetime.
