### 2.4 Explicit near-horizon geometries

Let us now present explicit examples of near-horizon geometries of interest. We will discuss the cases of the extremal Kerr and Reissner–Nordström black holes as well as the extremal Kerr–Newman and Kerr–Newman–AdS black holes. Other near-horizon geometries of interest can be found, e.g., in [88, 121, 203].

#### 2.4.1 Near-horizon geometry of extremal Kerr

The near-horizon geometry of extremal Kerr with angular momentum can be obtained by the above procedure, starting from the extremal Kerr metric written in usual Boyer–Lindquist coordinates; see the original derivation in [33] as well as in [156, 54]. The result is the NHEK geometry, which is written as (25) without matter fields and with

The angular momentum only affects the overall scale of the geometry. There is a value degrees for which becomes null. For , is spacelike. This feature is a consequence of the presence of the ergoregion in the original Kerr geometry. Near the equator we have a “stretched” AdS3 self-dual orbifold (as the fiber is streched), while near the poles we have a “squashed” AdS3 self-dual orbifold (as the fiber is squashed).

#### 2.4.2 Near-horizon geometry of extremal Reissner–Nordström

The extremal Reissner–Nordström black hole is determined by only one parameter: the electric charge . The mass is and the horizon radius is . This black hole is static and, therefore, its near-horizon geometry takes the form (21). We have explicitly

#### 2.4.3 Near-horizon geometry of extremal Kerr–Newman

It is useful to collect the different functions characterizing the near-horizon limit of the extremal Kerr–Newman black hole. We use the normalization of the gauge field such that the Lagrangian is proportional to . The black hole has mass . The horizon radius is given by . One finds

In the limit , the NHEK functions (37) are recovered. The near-horizon geometry of extremal Kerr–Newman is therefore smoothly connected to the near-horizon geometry of Kerr. In the limit one finds the near-horizon geometry of the Reissner–Nordström black hole (38). The limiting procedure is again smooth.

#### 2.4.4 Near-horizon geometry of extremal Kerr–Newman–AdS

As a last example of near-horizon geometry, let us discuss the extremal spinning charged black hole in AdS or Kerr–Newman–AdS black hole in short. The Lagrangian is given by where . It is useful for the following to start by describing a few properties of the non-extremal Kerr–Newman–AdS black hole. The physical mass, angular momentum, electric and magnetic charges at extremality are expressed in terms of the parameters of the solution as

where and . The horizon radius is defined as the largest (smallest) root, respectively, of
Hence, one can trade the parameter for . If one expands up to quadratic order around , one finds
where and are defined by
In AdS, the parameter obeys , and coincides with and only at extremality. In the flat limit , we have and . The Hawking temperature is given by
The extremality condition is then or, more explicitly, the following constraint on the four parameters ,

The near-horizon geometry was obtained in [159, 71] (except the coefficient given here). The result is

where we defined
The near-horizon geometry of the extremal Kerr–Newman black hole is recovered in the limit .