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 [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
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 = ![]() ![]() ![]() ![]() ![]() |
(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 [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 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 [100
].9
The geometries enjoy a global timelike Killing vector (which can be identified as ) if and only if
One 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
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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