2.1 Properties of extremal black holes

For simplicity, we will strictly concentrate our analysis on stationary black holes. Since the Kerr/CFT correspondence and its extensions are only concerned with the region close to the horizon, one could only require that the near-horizon region is stationary, while radiation would be allowed far enough from the horizon. Such a situation could be treated in the framework of isolated horizons [11, 10] (see [14] for a review). However, for our purposes, it will be sufficient and much simpler to assume stationarity everywhere. We expect that all results derived in this review could be generalized for isolated horizons (see [268] for results along these lines).

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 axisymmetric3 – and their event horizon is a Killing horizon4. In asymptotically-flat spacetimes, black holes have spherical topology [163].

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

TH = 0. (8 )
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.


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