One is free to redefine as for any and, therefore, the near-horizon geometry admits the enhanced symmetry generator

in addition to and . Together and form a non-commutative algebra under the Lie bracket.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 [64] 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 [194] 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 by

where , , and are fixed by the equations of motion. By inverting and redefining , we can always set , . The function can be removed by redefining but it is left for convenience because some near-horizon geometries are then more easily described.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

where are some pure numbers, which are the magnetic charges.Going back to the spinning case, the symmetry is generated by

ζ_{−1} = ∂_{t}, ζ_{0} = t∂_{t} − r∂_{r}, |
|||

ζ_{1} = + ∂_{t} − tr∂_{r} −∂_{ϕ}, L_{0} = ∂_{ϕ}. |
(29) |

The geometry (25) is a warped and twisted product of . The coordinates are
analogous to Poincaré coordinates on AdS_{2} with an horizon at . One can find global coordinates in
the same way that the global coordinates of AdS_{2} are related to the Poincaré coordinates [33]. 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 [33] after working out the geodesic equations.

At fixed polar angle , the geometry can be described in terms of warped anti-de Sitter geometries;
see [8] 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 AdS_{3}, 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
AdS_{3} [100].^{9}

The geometries enjoy a global timelike Killing vector (which can be identified as ) if and only if

If there is no global timelike Killing vector, there is at least one special value of the polar angle , where . At that special value, the slice is locally an ordinary AdSOne can show the existence of an attractor mechanism for extremal spinning black holes, which are solutions of the action (1) [17]. According to [17], 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 [194] 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

In particular, the solutions obtained from the uplift (2) – (3) fall into this class. In general, these solutions can be obtained starting from both black holes (with horizon topology) and black rings (with horizon topology) [133].
Living Rev. Relativity 15, (2012), 11
http://www.livingreviews.org/lrr-2012-11 |
This work is licensed under a Creative Commons License. E-mail us: |