Extremal Black Holes and Cosmic Censorship

The Cosmic Censorship Hypothesis (CSH) was put forward by Roger Penrose in 1969, and (roughly) states

“there are no naked singularities”.

The hypothesis proposes that whenever a singularity occurs, such as in the center of a black hole, it must occur behind an event horizon. A singularity outside of an event horizon is termed a naked singularity, and the CSH says that such singularites do not exist. Cosmic Censorship has pretty profound implications for fundamental physics. For instance, a failure of the Cosmic Censorship Hypothesis leads to a failure of determinism in classical physics, since one can’t predict the behaviour of spacetime in the causal future of a singularity. On the other hand, if the Cosmic Censorship Hypothesis holds then (outside the event horizon), the singularity does not affect determinism. In 1991, Stephen Hawking made a wager with physicists Kip Thorne and John Preskill that the Cosmic Censorship Hypothesis is true. Six years later in 1997, Hawking conceded the bet “on a technicality”, after computer calculations showed that naked singularities could exist, albeit only in exceptional (not physically realistic) circumstances. In this post, we’re going to be looking at extremal black holes, and their relationship to the Cosmic Censorship Hypothesis.

Recall that all black holes can be described by three quantities: mass (M), charge (Q) and angular momentum (J). The charge and angular momentum partition the possible black holes into four categories:

  • Uncharged, non-rotating: Schwarzchild
  • Uncharged, rotating: Kerr
  • Charged, non-rotating: Reissner–Nordström
  • Charged, rotating: Kerr-Newman

There is also a classification of black holes based on their mass, but since the properties of these black holes are qualitatively the same, we’re not really interested in it. The two types of black holes we are interested in at the moment are the Kerr black hole and the Reissner-Nordström black hole. We’ve already seen the Kerr black hole, so let’s have a look at the Reissner-Nordström black hole. It’s actually reasonable to combine the analysis for both rotating and charged black holes by considering the Kerr-Newmann metric, but for notational simplicity we’ll stick with the Reissner-Nordström for now.

The metric for the Reissner-Nordström black hole in spherical coordinates (t, r, \theta, \phi):

ds^2 = \frac{1}{r^2} \left ( r^2 - 2Mr + Q^2 \right ) dt^2 - \frac{r^2}{ r^2 - 2Mr + Q^2} dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta \, d \phi^2

Remember that Q is the charge of the black hole. Assume, for simplicity, that Q > 0.

The event horizon of the Kerr black hole is determined by the value of r which makes the dr^2 term singular, i.e. the value of r for which r^2 - 2Mr + Q^2= 0. Solving the quadratic equation gives r = M \pm \sqrt{M^2 - Q^2}. Already we see some issues with the allowed values of Q. If the charge is larger than the mass (in appropriate units), there is no real solution to the previous equation – there is no event horizon! As we’ve already mentioned, black holes with no event horizons can cause us some issues, so we’d like to know if this is a physically realistic solution. Let’s suppose for the moment that we have a so-called ‘Extremal’ charged black hole. This is a Reissner-Nordström black hole with the critical value of charge, Q = M. (Note that the sign of Q isn’t important, as it would just correspond to charge of the opposite sign). In this case, we have a single solution to r^2 - 2Mr + Q^2 = 0, namely r = M.

Can we increase the charge of this black hole? If we can, then we are able to create naked singularities simply by ‘overcharging’ a charged black hole. Let’s try and do this by adding a small test charge with mass m and charge q (where the charge is the same sign as Q). We don’t want to appreciably affect the mass of the black hole, so we want q > m.

Naively computing the radial (Newtonian) gravitational force on the particle and the radial (Coulomb) electric force on the particle, we get (recalling that we are using natural units and Q = M for an extremal black hole):

F_Q - F_G = \frac{Qq}{r^2} - \frac{Mm}{r^2} = \frac{Q}{r^2} (q-m) > 0

That is, the electrostatic repulsion is greater than the gravitational attraction! The charged particle is repelled, and you don’t get a supercharged black hole. You might think, “sure, at rest the particle won’t fall in, but what happens if I shoot the particle in using a particle accelerator?”. That’s a perfectly reasonable question, and it turns out that if you increase the energy of the particle enough to overcome the electrostatic repulsion, then (remembering mass-energy equivalence) you increase the mass of the particle by just the right amount to preserve the inequality M+m \leq Q+q. It seems to be impossible to supercharge a black hole just by adding mass.

You can do a similar analysis with rotating black holes – in that case it is the angular momentum, J, which is required to be less than or equal to the mass. For extremal rotating black holes, you can show that trying to add angular momentum by ‘spinning’ the black hole won’t work – any particle you send at the black hole which would increase it’s angular momentum more than it’s mass ends up ‘missing’ the black hole, precisely because it has too much angular momentum. Although the math works out fine, this is a bit harder to picture, which is why I stuck with the charged black hole example.

There are two final comments I want to make before I finish up. One is that charged black holes admit a natural generalisation to include magnetic charge, as well as electric charge. In that case, it is the total charge Q = \sqrt{Q_e^2 + Q_m^2} which is required to be less than the mass.

Secondly, general relativity has a natural extension to higher dimensional spacetimes. Solving the Einstein Field Equations for higher dimensional spacetimes is significantly harder than in four dimensions, but there are natural extensions of the rotating black hole solutions (which I will probably dedicate an entire blog post to at some stage) called Myers-Perry black holes. Interestingly, these black holes (in dimensions \geq 6) can have arbitrarily high angular momentum per unit mass without introducing naked singularities.

I’m currently at the Institut Henri Poincaré in Paris for a trimester on General Relativity, so expect a few more blog posts on black holes in the near future!


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