Extremal RN Black Hole

In the chapter on black holes we already discussed that when the charge to mass ratio is big enough the RN solution is a naked singularity. For astrophysical or macroscopic black holes this seems to be an unlikely situation (first of all, astrophysical black holes are neutral, and secondly, black holes formed through gravitational collapse are very massive). But we also mentioned that black holes emit thermal Hawking radiation. Assuming this radiation consists of particles with $M \gg Q$, the black hole will lose mass, but it's charge will stay (approximately) the same. Eventually (the rate of Hawking evaporation is very small for very massive black holes) the black hole could turn into a naked singularity, violating the Cosmic Censorship Conjecture.

The extremal state is a solution where the black hole is on the verge of becoming a naked singularity. In the case of SS (neutral) this happens when $M=0$, so space-time becomes flat and the black hole has evaporated completely.

In the case of RN, we find that the black hole is extremal when $r_+ =r_-$ with $Q^2=4M^2$ (see the section on RN black holes). For charge to mass ratio's bigger than that the solution will be a naked singularity because the metric function $\lambda (r)$ will have no roots (horizons) in that case.

Let's use this to write the complete extremal solution in a useful, simple way. Making a shift in r, $r \rightarrow r'=r- r_+$, where $r_+ =r_- =M={\textstyle{1\over 2}}\sqrt{Q^2}=\Sigma$, we can write the line element as

$$ds^2=\frac{{r'}^2}{(r'+\Sigma)^2}dt^2 - \frac{(r'+\Sigma)^2}{{r'}^2} {dr'}^2 - (r'+\Sigma)^2 d\Omega ^2$$

Identifying

$$H=\frac{r'+\Sigma}{r'} =1+\frac{\Sigma}{r'}$$

, which means $H$ is a harmonic function ($\Box H=0$, where the second derivatives are with respect to space-like coordinates), we can write this as (dropping primes)

$$ds^2 = H^{-2} dt^2 - H^2 d{\vec x}^2$$ (6.1)

,where $d{\vec x}^2=dr^2 +r^2 d\Omega^2$, the space-like part of the metric. Also, if the charge is purely electric, the electric vector field can be expressed as $A_t=2H$. As we will see later on, all extremal solutions can be written with the help of harmonic functions.

The structure of space-time is the same as SS now, one event horizon and one curvature singularity at $r=-\Sigma$ (this is negative because of the shift we performed). When you fall into the extremal black hole you will end up at the singularity, as opposed to the non-extremal case. Calculating the Hawking temperature we find

$$T_H=\frac{r_+ -r_-}{4\pi r_+^2}$$ (6.2)

, meaning that in the extremal case the Hawking temperature is zero and therefore the Hawking evaporation stops, in agreement with the Cosmic Censorship Conjecture.

So the extremal solution is stable against Hawking radiation [5]. This made people suggest that extremal solutions could be stable quantum states, this was also motivated by the fact that the mass condition could be explained through supersymmetry. In supersymmetry theories stable quantum states should obey a so-called Bogomol'nyi bound, saying that the mass of a state should be bigger or equal than the charges of that state. It turns out that the mass condition, arising from demanding extremality, satisfies this Bogomol'nyi bound (we will never proof this, it is just to give you some information on why these extremal states are interesting). So you could say supersymmetry acts as a Cosmic Censor, meaning supersymmetry forbids the existence of naked singularities [12] [24].

Because the entropy of a black hole is equal to the horizon's area, the entropy can be written as

$$S=\pi M^2$$ (6.3)

, because $r_+=r_-=M$. The number of (degenerate) states of the extremal black hole is thus $e^{\pi M^2}$. From Kaluza-Klein and String theories it is known that the number of degenerate states of massive modes behave in a similar matter. This opened up the possibility to identify black holes with massive Kaluza-Klein or String modes [9] [10]. This is still under investigation and other ways of counting states (e.g., black holes as bound states) have been conjectured.

Another important feature of extremal states is that the mass condition implies that the forces between extremal black holes vanish (considering multi black hole solutions). The gravitational pull exactly balances the electromagnetic repulsion between extremal black holes with same masses and charges. Because of this, multi black hole solutions can be obtained just by adding them [5].

We mentioned some properties of extremal black holes and showed why they are interesting to study. Off course, we only considered the RN extremal solution here, but when we go on we will see that other extremal solutions have similar properties making them interesting.