Reissner-Nordström black holes and naked singularities

Question

I often read that there is no known mechanism that prevents a charged black hole, if the charge is high enough compared to its irreducible mass, from forming a naked singularity. But I don't get how that is possible.

Imagine a hollow sphere of charge $q$, with an “infinite” radius and “infinitesimal” mass. Imagine we start shrinking it. To do that we need to do work. That work will create mass, which will be added to the mass our sphere.

If we keep shrinking and shrinking, at some point we will reach a radius equal to

$$r = \sqrt{\frac{G q^2}{8 \pi \epsilon_0 c^4}}$$

At that point, the mass that we will have piled up through our shrinking will be equal to

$$m = \sqrt{\frac{q^2}{8 \pi \epsilon_0 G}}$$

Is that not a black hole with an event horizon (at half of its Schwarzschild radius, due to how Reissner-Nordström metric works)? What should I do to create a naked singularity?

EDIT: Maybe my question is not clear enough, so I will try to formulate it better: In which way one could, theoretically, create a naked singularity – or, to put it differently, a black hole “with more charge than mass”? I can't find a way; so to me naked singularities seem mathematically impossible even just theoretically speaking.

Answer 1

The Schwarzschild metric is

$$c^2\mathrm{d}\tau^2=k_Sc^2 \mathrm{d} t^2-k_S^{-1} \mathrm{d} r^2-\mathrm{d} \Omega^2$$

for the metric on a sphere $\mathrm{d}\Omega$ and the “Schwarzschild factor” $k_S=1-\frac{r_s}{r}=1-\frac{2GM}{rc^2}$. There is obviously a horizon at $r=r_s=\frac{2GM}{c^2}$, where the escape velocity is equal to lightspeed, and where the Schwarzschild factor vanishes. Inside this horizon the signs of the timelike and spacelike terms switch (not good for anyone trying to get inside). The Reissner-Nordstrom metric is nearly identical:

$$c^2\mathrm{d}\tau^2=k_{RN}c^2 \mathrm{d} t^2-k_{RN}^{-1} \mathrm{d} r^2-\mathrm{d} \Omega^2$$

where $k_{RN}=1-\frac{r_s}{r}+\frac{r_Q^2}{r^2}=1-\frac{2GM}{rc^2}+\frac{Q^2G}{4\pi\epsilon_0c^4r^2}$. Evidently, for $Q\ll M$, there is still a horizon, but there is also an interior horizon within which the sign of the time and radial terms in the metric are still normal. Increasing $Q$ makes the radius of this interior horizon increase. Specifically, there are two horizons with radii given by

$$r=\frac{1}{2}\left(r_s\pm\sqrt{r_s^2-4r_Q^2}\right).$$

If the interior horizon’s radius is greater than or equal to the radius of the exterior horizon, then there ends up being no event horizon: the “normal” space inside the black hole has been expanded past the event horizon which produces causal connections between points at infinity and the worldline at $r=0$, which is a naked singularity. This also corresponds to the value under the square root up there being negative since event horizons of imaginary radius correspond, physically, to no event horizons.

The reason this happens is because of the sign of the charge term in the Reissner-Nordstrom factor. Unlike the mass term, it is positive, so for large enough $Q$, that term can be made to never actually reach zero, and thus not ever producing a horizon (in this particular case).

To identify one such superextremal naked-singularity case, simply find a $Q$ large enough. Given the above relationship the event horizons coincide for $r_s=2r_Q$ and disappear for $r_s<2r_Q$; that should be easy enough to figure out from there.

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In other words, qualitatively: the charge of the black hole reduces the (exterior) event horizon radius because of how it interacts with spacetime. Under the proper RN metric, increasing $Q$ makes the event horizon shrink and eventually produces a naked singularity for a finite $Q$ and $M$.

In the case of the RN metric, you can't have a superextremal black hole (same with Kerr and KN as far as I know), because adding more charge to a sub-extremal black hole requires doing work which increases the black hole's mass correspondingly so that you can't ever reach extremum. The cosmic censorship hypothesis states that this is generally true, i.e. that no black hole metric can be made to form a naked singularity, but it has not yet been proven.