# 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. :)

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).

• Is it really the case that CCH is alive? I think I recall reading multiple times that there's no problem constructing non-pathological solutions that violate it. Moreover, there doesn't seem any reason to require it to be true (except in the wonderland where people assume GR is all there is to physics and singularity is in any sense physical). Jul 28, 2011 at 19:16
• @Marek: I don't know any more about its current status than what the WP article en.wikipedia.org/wiki/Cosmic_censorship_hypothesis says. There's a weak version and a strong version, and each can be true or false independent of the other. There are issues with how to define the hypothesis correctly. If it does hold, then it has to depend on some energy condition, but we know that basically all energy conditions fail under some circumstances. Re assuming "GR is all there is to physics," I disagree philosophically. If we can prove existence or nonexistence of singularities,[...]
– user4552
Jul 28, 2011 at 19:27
• [...continued...] that really does mean something. For instance, the Hawking singularity theorem doesn't guarantee that the big bang singularity was a physical singularity, but it does suggest that the universe got to somewhere around the Planck density, which is a nontrivial inference. In any case, I strongly doubt that there is such a trivial counterexample as dropping an electron into an extremal R-N black hole.
– user4552
Jul 28, 2011 at 19:28
• I don't quite follow. To get at BB singularity all you need is right FLRW solutions (which follow from the parameters obtained from observations), no need to invoke Hawking. Moreover, BB is one and only special event in this universe completely unrelated to any other BH out there. By the way, I agree that showing existence of singularities is very important -- it shows the theory is breaking down which implies new exciting physics is lurking close by. But there's no need to prevent them from happening. FWIW, I agree with your last sentence though. Jul 28, 2011 at 19:39
• FRW assumes perfect homogeneity and isotropy. The Hawking singularity theorem applies when there is not perfect homogeneity and isotropy.
– user4552
Jul 28, 2011 at 23:02