general relativity - Charged particle close to a charged black hole - what happens?

Let's assume the Reissner–Nordström metric (charged black hole, non-rotating), for simplicity. The black hole is charged with a powerful electric charge. There's a particle nearby, of non-zero mass, let's say an electron, its charge being the same sign like the black hole's charge. The particle's initial speed, relative to the BH, is zero. The particle is close to the event horizon, but still outside of it.

The question is - what happens? Are there any combinations of parameters where the particle starts falling in, but stops before being swallowed? Or even repulsed outright? If so, any quantitative, intuitive examples?

I'm asking because I know how easily intuition can get deceived by General Relativity, and doing the math involving the R-N metric is probably not feasible for me now. :)

Answer

Let's say the black hole has mass and charge $Q$ and $M$, and the electron has $m$ and $q$. An extremal black hole has $|Q|=2M$ (in the appropriate units). An electron has $|q| \gg 2m$ in the same units. If the electron fell into a negatively charged extremal R-N black hole, then the black hole would have $|Q| > 2M$, which would make it more than extremal. This would cause it to be a naked singularity, which would be exciting, since it would be a counterexample to the cosmic censorship hypothesis; but I'm pretty sure there is no such trivial counterexample, since cosmic censorship is still alive and kicking, decades after being conjectured. Another way of seeing that it will be repelled is that extremal R-N black holes with like charges do not interact; their gravitational attraction exactly cancels their electrostatic repulsion. Since the electron has a greater $|q|/m$, it will definitely be repelled.

One thing to watch out for in this kind of situation is that different observers can disagree on the direction of the forces. For example, a light ray that falls radially into a black hole experiences a repulsion at certain points as measured in Schwarzschild coordinates. This is the famous "Hilbert repulsion" beloved of kooks like Angelo Loinger. The light ray nevertheless passes through the horizon, and local observers never see a repulsion -- McGruder, Gravitational repulsion in the Schwarzschild field, Phys. Rev. D 25, 3191–3194 (1982).