Abstract
This chapter explains how evaporating black holes with electrical charge or angular momentum might, at least at first sight, violate the cosmic censorship conjecture, i.e., lead to naked singularities. The asymptotically flat Reissner–Nordström black hole spacetime is then reviewed. The main focus of this chapter is to study the evolution of such black holes as they undergo Hawking evaporation.
“But he has nothing at all on!” at last cried out all the people. The Emperor was vexed, for he knew that the people were right; but he thought the procession must go on now!
Kejserens nye Klæder, Hans Christian Andersen
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Notes
- 1.
In spacetime dimensions 6 and above, singly rotating (there can be more than one angular momenta in higher dimensions) black holes do not have an extremal limit! They can be ultraspinning—i.e., rotating arbitrarily fast—although these black holes may be unstable [1].
- 2.
Stephen Hawking had a bet with Kip Thorne and John Preskill in 1991, taking the position that naked singularities could not exist. The stakes were 100 pounds sterling and a clothing “embroidered with a suitable concessionary message.” Hawking admitted defeat in 1997 after it was found that there exist (nongeneric) conditions under which naked singularities may exist (e.g., [3]), yet he presented a T-shirt with the now famous words “Nature abhors a naked singularity,” displaying his belief that cosmic censorship is generically true. See also [4] for some evidence that the censorship holds.
- 3.
The fact that black holes emit Hawking radiation stochastically also implies that they undergo random walk motion. That is to say, the center of mass of an evaporating black hole does not simply stay put at one fixed location, instead it drifts around due to the momentum recoil of the particles emitted via Hawking radiation [11, 12]. Consider an asymptotically flat Schwarzschild black hole with initial mass M. Suppose the black hole has emitted a fraction f of its total mass M. (For simplicity we will use the Planck units here.) This means that it would emit about \(N \sim fM^2\) particles in the time period \(\Delta t \sim fM^3\). Each of these particles would have root-mean-square (rms) momentum of \(\left\langle p^2\right\rangle ^{1/2} \sim 1/M\). The total rms momentum is therefore \(\Delta P \sim \left\langle Np^2\right\rangle ^{1/2} \sim f^{1/2}\). The rms uncertainty in the position induced by momentum recoil would therefore be \(\Delta x \sim \Delta P \Delta t/M \sim f^{3/2} M^2.\) This distance can be astronomically large for sufficiently massive black holes.
- 4.
Wheeler eventually abandoned the view that geometry is fundamental, in favor instead of the idea that information is fundamental—it from bit. The important role of information (recovery from black hole) is of course an important theme of this thesis.
- 5.
Part of the analysis in this chapter was published in [13].
- 6.
One could also include a hypothetical magnetic charge P in addition to the electric charge Q, then the metric is simply obtained by \(Q^2 \mapsto Q^2 + P^2\). This is called a dyonic black hole.
- 7.
We cannot directly generalize the magnetic Reissner–Nordtsröm solution to a spacetime with dimensions \(d>4\). This is related to the fact that the magnetic field in d-dimensional spacetime is described by a \((d-3)\)-form potential. For \(d>4\), the magnetic charge will therefore not be carried by a point particle, but by \((d-4)\)-dimensional objects.
- 8.
- 9.
This behavior is not universal for all models of Hawking radiation, see [20], in which asymptotically flat charged black holes have different final fates due to the different physical assumptions made.
- 10.
The model has limitations. For example, Schwinger emission is of course not thermal.
- 11.
Note that this series diverges for all \(x > 0\), but if a fixed number of terms is taken, then for large enough x, the approximation is good. However, the divergence means that, for any fixed x, increasing the number of terms in the series does not help to increase the accuracy of the approximation. Such a series is called an asymptotic series.
- 12.
This is 4 / c times the Stefan–Boltzmann constant, although HW refer to a as simply the “Stefan–Boltzmann constant”.
- 13.
Let \(V(\lambda )=\dot{t}\partial _t + \dot{r}\partial _r + \dot{\theta }\partial _\theta + \dot{\phi }\partial _\phi \), where dot denotes the derivative with respect to the affine parameter \(\lambda \). The conserved quantities \(E_\infty \) and J arise from the fact that \(\partial _t\) and \(\partial _\phi \) are both Killing vector fields. Specifically, \(E=-\left\langle \partial _t, V \right\rangle \), and \(J=\left\langle \partial _\phi , V\right\rangle \). Note that since r is an area radius, J is not, strictly speaking, an angular momentum in the usual sense of classical mechanics.
- 14.
This is calculated using the chain rule: \(\frac{\text {d}T(M,Q)}{\text {d}t}=\frac{\partial T}{\partial M}\frac{\text {d}M}{\text {d}t}+\frac{\partial T}{\partial Q}\frac{\text {d}Q}{\text {d}t}\). Note that Hiscock and Weems missed a power of \(\pi \) in this expression (their Eq. (23))—they wrote \(\frac{\alpha \hbar ^2}{3840\pi }\) instead of \(\frac{\alpha \hbar ^2}{3840\pi ^2}\).
- 15.
There is a large literature on whether a semi-classical extremal black hole exists (see e.g., [27–29]); even at the classical level it is not clear what the final state of an extremal black hole would be since it is actually unstable [30–35]. Here we are neither concerned about the actual physical existence nor the stability of such a solution—we are merely interested in the mathematical solution as it provides insight into the more complicated non-extremal case.
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Ong, Y.C. (2016). Hiscock and Weems: Modeling the Hawking Evaporation of Asymptotically Flat Charged Black Holes. In: Evolution of Black Holes in Anti-de Sitter Spacetime and the Firewall Controversy. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48270-4_4
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