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  as well as in [156, 54]. The result is the NHEK geometry, which is written as (25) without matter fields and with3 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).
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
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
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
The near-horizon geometry was obtained in [159, 71] (except the coefficient given here). The result is
Living Rev. Relativity 15, (2012), 11
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