One is free to redefine as for any and, therefore, the near-horizon geometry admits the enhanced symmetry generator
Now, contrary to the static case, the existence of a third Killing vector is not guaranteed by geometric considerations. Nevertheless, it turns out that Einstein’s equations derived from the action (1) imply that there is an additional Killing vector in the near-horizon geometry [194, 19] (see also  for a geometrical derivation). The vectors turn out to obey the algebra. This dynamical enhancement is at the origin of many simplifications in the near-horizon limit. More precisely, one can prove  that any stationary and axisymmetric asymptotically-flat or anti-de Sitter extremal black-hole solution of the theory described by the Lagrangian (1) admits a near-horizon geometry with isometry. The result also holds in the presence of higher-derivative corrections in the Lagrangian provided that the black hole is big, in the technical sense that the curvature at the horizon remains finite in the limit where the higher-derivative corrections vanish. The general near-horizon geometry of extremal spinning black holes consistent with these symmetries is given by8
The term in (25) is physical since it cannot be gauged away by an allowed gauge transformation. For example, one can check that the near-horizon energy would be infinite in the Kerr–Newman near-horizon geometry if this term would be omitted. One can alternatively redefine and the gauge field takes the form
The static near-horizon geometry (21) is recovered upon choosing only covariant quantities with a well-defined static limit. This requires and it requires the form
Going back to the spinning case, the symmetry is generated by
|ζ−1 = ∂t, ζ0 = t∂t − r∂r,|
|ζ1 = + ∂t − tr∂r −∂ϕ, L0 = ∂ϕ.||(29)|
The geometry (25) is a warped and twisted product of . The coordinates are analogous to Poincaré coordinates on AdS2 with an horizon at . One can find global coordinates in the same way that the global coordinates of AdS2 are related to the Poincaré coordinates . Let
Geodesic completeness of these geometries has not been shown in general, even though it is expected that they are geodesically complete. For the case of the near-horizon geometry of Kerr, geodesic completeness has been proven explicitly in  after working out the geodesic equations.
At fixed polar angle , the geometry can be described in terms of warped anti-de Sitter geometries; see  for a relevant description and [226, 158, 238, 127, 223, 175, 174, 6, 119, 43, 26, 93, 222] for earlier work on these three-dimensional geometries. Warped anti-de Sitter spacetimes are deformations of AdS3, where the fiber is twisted around the AdS 2 base. Because of the identification , the geometries at fixed are quotients of the warped AdS geometries, which are characterized by the presence of a Killing vector of constant norm (namely ). These quotients are often called self-dual orbifolds by analogy to similar quotients in AdS3 .9
The geometries enjoy a global timelike Killing vector (which can be identified as ) if and only if3 spacetime and acquires a local isometry. At all other values of , one is broken to . Note that there is still a global time function for each near-horizon geometry. Constant global time in the global coordinates (33) are spacelike surfaces because their normal is timelike,
One can show the existence of an attractor mechanism for extremal spinning black holes, which are solutions of the action (1) . According to , the complete near-horizon solution is generically independent of the asymptotic data and depends only on the electric charges , magnetic charges and angular momentum carried by the black hole, but in special cases there may be some dependence of the near horizon background on this asymptotic data. In all cases, the entropy only depends on the conserved electromagnetic charges and the angular momentum of the black hole and might only jump discontinuously upon changing the asymptotic values of the scalar fields, as it does for static charged black holes [227, 118].
One can generalize the construction of near-horizon extremal geometries to higher dimensions. In five dimensions, there are two independent planes of rotation since the rotation group is a direct product . Assuming the presence of two axial symmetries , (with fixed points at the poles), one can prove  that the near-horizon geometry of a stationary, extremal black-hole solution of the five-dimensional action (1) possibly supplemented by Chern–Simons terms (4) is given by.
Living Rev. Relativity 15, (2012), 11
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