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 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
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
![]() This work is licensed under a Creative Commons License. E-mail us: |