Many theorems have been derived that characterize the generic properties of four-dimensional stationary black holes
that admit an asymptotically-timelike Killing vector. First, they have one additional axial Killing vector – they are
axisymmetric^{3} – and their event
horizon is a Killing horizon^{4}.
In asymptotically-flat spacetimes, black holes have spherical topology [163].

Extremal black holes are defined as stationary black holes with vanishing Hawking temperature,

Equivalently, extremal black holes are defined as stationary black holes whose inner and outer horizons coincide. No physical process is known that would make an extremal black hole out of a non-extremal black
hole.^{5}
If one attempts to send finely-tuned particles or waves into a near-extremal black hole in order to further
approach extremality, one realizes that there is a smaller and smaller window of parameters that
allows one to do so when approaching extremality. In effect, a near-extremal black hole has
a potential barrier close to the horizon, which prevents it from reaching extremality. Also, if
one artificially continues the parameters of the black holes beyond the extremality bound in a
given solution, one typically obtains a naked singularity instead of a black hole. Such naked
singularities are thought not to be reachable, which is known as the cosmic censorship hypothesis.
Extremal black holes can then be thought of as asymptotic or limiting black holes of physical black
holes. The other way around, if one starts with an extremal black hole, one can simply throw
in a massive particle to make the black hole non-extremal. Therefore, extremal black holes
are finely tuned black holes. Nevertheless, as we will discuss, studying the extremal limit is
very interesting because many simplifications occur and powerful specialized methods can be
used.

Extremal spinning or charged rotating black holes enjoy several interesting properties that we will summarize below. In order to be self-contained, we will also first provide some properties of generic (extremal or non-extremal) black holes. We refer the reader to the excellent lecture notes [259] for the derivation of most of these properties.

- Angular velocity. Spinning black holes are characterized by a chemical potential – the angular velocity – conjugate to the angular momentum. The angular velocity can be defined in geometrical terms as the coefficient of the black-hole–horizon generator proportional to the axial Killing vector The net effect of the angular velocity is a frame-dragging effect around the black hole. This gravitational kinematics might be the clue of an underlying microscopic dynamics. Part of the intuition behind the extremal spinning black hole/CFT correspondence is that the degrees of freedom responsible for the black hole entropy are rotating at the speed of light at the horizon.
- Electrostatic potential. Electrically-charged black holes are characterized by a chemical potential – the electrostatic potential – conjugated to the electric charge. It is defined on the horizon as where is the horizon generator defined in (9). Similarly, one can associate a magnetic potential to the magnetic monopole charge. The form of the magnetic potential can be obtained by electromagnetic duality, or reads as the explicit formula derived in [99] (see also [91] for a covariant expression). Part of the intuition behind the extremal charged black hole/CFT correspondence is that this kinematics is the sign of microscopic degrees of freedom “moving along the gauge direction”. We will make that statement more precise in Section 4.1.
- Ergoregion. Although the Killing generator associated with the mass of the black hole, , is timelike at infinity, it does not need to be timelike everywhere outside the horizon. The region where is spacelike is called the ergoregion and the boundary of that region where is lightlike is the ergosphere. If there is no ergoregion, is a global timelike Killing vector outside the horizon. However, it should be noted that the presence of an ergoregion does not preclude the existence of a global timelike Killing vector. For example, the extremal spinning Kerr–AdS black hole has an ergoregion. When the horizon radius is smaller than the AdS length, the horizon generator becomes spacelike at large enough distances and there is no global timelike Killing vector, as for the Kerr black hole. On the contrary, when the horizon radius is larger than the AdS length, the horizon generator is timelike everywhere outside the horizon and is therefore a global timelike Killing vector.
- Superradiance. One of the most fascinating properties of some rotating black holes is that neutral particles or waves sent towards the black hole with a frequency and angular momentum inside a specific band come back to the exterior region with a higher amplitude. This amplification effect or Penrose effect allows the extraction of energy very efficiently from the black hole. Superradiance occurs for the Kerr and Kerr–Newman black hole and is related to the presence of the ergoregion and the lack of a global timelike Killing vector. Because of the presence of a global timelike Killing vector, there is no superradiance for large Kerr–AdS black holes (when reflective boundary conditions for incident massless waves are imposed) [165, 264].
- Electromagnetic analogue to superradiance. Charged black holes contain electrostatic energy that can
also be extracted by sending charged particles or waves with frequency and charge inside a
specific band [84] (see [177] for a review)
There is no ergoregion in the four-dimensional spacetime. However, for asymptotically-flat black holes,
there is a five-dimensional ergoregion when considering the uplift (2). For the Reissner–Nordström
black hole, the five-dimensional ergoregion lies in the range , where is the mass
and the standard Boyer–Lindquist radius.
The combined effect of rotation and charge allows one to extract energy in the range

When considering a wave scattering off a black hole, one can define the absorption probability or macroscopic greybody factor as the ratio between the absorbed flux of energy at the horizon and the incoming flux of energy from infinity, In the superradiant range (13), the absorption probability is negative because the outgoing flux of energy is higher than the incoming flux. - No thermal radiation but spontaneous emission. Taking quantum mechanical effects into account, non-extremal black holes radiate with a perfect black-body spectrum at the horizon at the Hawking temperature [162]. The decay rate of a black hole as observed from the asymptotic region is the product of the black-body spectrum decay rate with the greybody factor , The greybody factor accounts for the fact that waves (of frequency , angular momentum and electric charge ) need to travel from the horizon to the asymptotic region in the curved geometry. In the extremal limit, the thermal factor becomes a step function. The decay rate then becomes As a consequence, ordinary Hawking emission with and vanishes while quantum superradiant emission persists. Therefore, extremal black holes that exhibit superradiance, spontaneously decay to non-extremal black holes by emitting superradiant waves.
- Innermost stable orbit approaching the horizon in the extremal limit. Near-extremal black holes have
an innermost stable circular orbit (ISCO) very close to the horizon. (In Boyer–Lindquist coordinates,
the radius of such an orbit coincides with the radius of the horizon. However, since the
horizon is a null surface, while the ISCO is timelike, the orbit necessarily lies outside
the horizon, which can be seen explicitly in more appropriate coordinates. See Figure 2
of [34]
^{6}). As a consequence, the region of the black hole close to the horizon can support accretion disks of matter and, therefore, measurements of electromagnetic waves originating from the accretion disk of near-extremal rotating black holes contain (at least some marginal) information from the near-horizon region. For a careful analysis of the physical processes around rotating black holes, see [34]. See also [154] for a recent discussion. - Classical singularities approaching the horizon in the extremal limit. Stationary axisymmetric non-extremal black holes admit a smooth inner and outer horizon, where curvatures are small. However, numerical results [52, 50, 51, 112] and the identification of unstable linear modes using perturbation theory [220, 125, 124] showed that the inner horizon is unstable and develops a curvature singularity when the black hole is slightly perturbed. The instability is triggered by tiny bits of gravitational radiation that are blueshifted at the inner Cauchy horizon and which create a null singularity. In the near-extremality limit, the inner horizon approaches the outer horizon and it can be argued that test particles encounter a curvature singularity immediately after they enter the horizon of a near-extremal black hole [212].

Living Rev. Relativity 15, (2012), 11
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