Naked singularity of a charged black hole

Question

Consider the Reissner-Nordstrom metric for a black hole:

$$ds^{2} = - f(r)dt^{2} + \frac{dr^{2}}{f(r)} + r^{2}d\Omega_{2}^{2},$$

where

$$f(r) = 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$$

We can write

$$f(r) = \frac{1}{r^{2}}(r-r_{+})(r-r_{-}), \qquad r_{\pm} = M \pm \sqrt{M^{2}-Q^{2}}.$$

Then $r_{+}$ is called the event horizon and $r_{-}$ is called the Cauchy horizon.

There is a curvature singularity at $r = 0$.

If $|Q|>M$, then $r_{+} < 0$, so the curvature singularity is not hidden behind the horizon.

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I do not understand the final sentence.

Firstly, for $|Q|>M$, I find that $r_{\pm}$ is imaginary.

Secondly, even if $r_{+} < 0$, this does not make sense: $r$ is a radial coordinate. How can it be less than $0$?

Answer 1

It can't. You are right, it is simply imaginary, i.e. It does not exist. And neither does $r_-$. It means there is no horizon, but you still have the singularity. So, if it existed as a physical case, it would be a naked singularity.

The conjecture is that there are no naked singularities. That the physics doesn't let them form. The same issue holds is true for almost all calculated BH solutions (exceptions have been found with certain quantum fields, eg, a scalar quantum field instead of an electromagnetic charge). Calculations and findings that back up the conjecture include those taking a BH near the critical values (say your solution with |Q| approaching M in value), and then trying to add more charge. The charge in the BH tends to repel the incoming charge, and not let it fall in beyond the critical value. Similarly for trying to rotate the BH faster, there is also a similar limit.

Those BHs where, in your example of |Q| = M are called extremal BHs. But Q being high is extremely unlikely in large astrophysical objects because positive charges tend to attract negative ones and viceversa and it is thought very unlikely to have those highly charged astrophysical bodies, except under very strange conditions. The case that is of most interest astrophysically is for rotating bodies, i.e. the Kerr solution. The limit on angular momentum so far has been true in the detected BHs, i.e., if it rotates too fast any new matter coming in with the same direction of angular momentum that would make it hyper-extremal will instead tend to fly around the BH and escape.

Clearly there is still some unknowns. And what happens for certain quantum fields, and whether there could be some naked singularities, is still unknown. It is worthily to point out that a naked singularity, if it existed, would create a breakdown of causality in the spacetime. Then there is also the hope that somehow some valid (of which we still don't have one that is considered to be it) theory of quantum gravity will eliminate singularities.

See the Wikipedia article on extremal BHs at https://en.m.wikipedia.org/wiki/Extremal_black_hole