Proof : We present a streamlined version of the proof in [ ].
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As described in Section 3. The x component of Eq. Substituting this into Eq. Using this, one can write Eq. Observe that this derivation is purely local and does not assume anything about the topology of H unlike the derivation in [ ], which assumed compactness. Now assume H is compact, so by axisymmetry one must have either S 2 or T 2. Integrating Eq. It would be interesting to remove the assumption of axisymmetry in the above theorem. In [ ] it is shown that regular non-axisymmetric linearised solutions of Eq. This supports the conjecture that any smooth solution of Eq.
In this case there are several different symmetry assumptions one could make. Classifications are known for homogeneous horizons and horizons invariant under a U 1 2 -rotational symmetry. It follows that f, h a must also be invariant under the isometry K , showing that our original definition is indeed equivalent to the near-horizon geometry being a homogeneous spacetime. Homogeneous geometries can be straightforwardly classified without assuming compactness of H as follows. Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to.
The proof uses the fact that homogeneity implies h must be a Killing field and then one reduces the problem onto the 2D orbit space. Hence we have:. Corollary 4. If we again have a horizon geometry locally isometric to a homogeneous S 3. This is analogous to a classification first obtained for supersymmetric near-horizon geometries in gauged supergravity, see Proposition 5.
We now consider a weaker symmetry assumption, which allows for inhomogeneous horizons. A U 1 2 -rotational isometry is natural in five dimensions and all known explicit black-hole solutions have this symmetry. The following classification theorem has been derived:. Consider a vacuum non-static near-horizon geometry with a U 1 2 -rotational isometry and a compact cross section H. It must be globally isometric to the near-horizon geometry of one of the following families of black-hole solutions :. The near-horizon geometry of the vacuum extremal black ring [ ] is a 2-parameter subfamily of case 1, corresponding to a Kerr string with vanishing tension [ ].
A perturbative attempt at constructing such a solution is discussed in [ ]. Few general classification results are known, although several large families of vacuum near-horizon geometries have been constructed. An explicit classification of the possible near-horizon geometries for the non-toroidal case was derived in [ ], see their Theorem 1. A generalisation of these metrics with non-zero cosmological constant has been found [ 99 ]. These conditions hold only when the black hole is spinning in all the two planes available, i.
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The near-horizon geometry of the extremal MP black holes can be written in a unified form [ 79 ]:. It is worth noting that if subsets of the angular momentum parameters a i are set equal, the rotational symmetry enhances to a non-Abelian unitary group.
Since these are vacuum solutions one can trivially add flat directions to generate new solutions. Interestingly, for odd dimensions D the resulting geometry has commuting rotational isometries. For this reason, it was conjectured that a special case of this is also the near-horizon geometry of yet-to-be-found asymptotically-flat black rings as is known to be the case in five dimensions [ 79 ]. It is an open problem as to whether there are corresponding black-hole solutions to these near-horizon geometries. All the constructions given below employ the following data.
In even dimensions greater than four, an infinite class of near-horizon geometries is revealed by the following result. Proposition 4.
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The topology of the total space is always a non-trivial S 2 -bundle over K and in fact different m give different topologies, so there are an infinite number of horizon topologies allowed. Furthermore, one can choose K to have no continuous isometries giving examples of near-horizon geometries with a single U 1 -rotational isometry. Hence, if there are black holes corresponding to these horizon geometries they would saturate the lower bound in the rigidity theorem.
Similar constructions of increasing complexity can be made in odd dimensions, again revealing an infinite class of near-horizon geometries. There exists a 1-parameter family of Sasakian solutions to Eq. This leads to an explicit homogeneous near-horizon geometry with. These are deformations of the Sasaki-Einstein Y p,q manifolds [ 94 ].
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For a countably infinite set of non-zero integers m 1 , m 2 , j, k , there exists a two-parameter family of smooth solutions to Eq. Generically these solutions possess two independent angular momenta along the T 2 -fibres. The Sasakian horizon geometries of Proposition 4. The above class of horizon geometries are of the same form as the Einstein metrics found in [ 37 , ].
By definition, a supersymmetric solution of a supergravity theory is a solution that also admits a Killing spinor, i. Hence any supersymmetric horizon is necessarily a degenerate Killing horizon. The simplest supergravity theory in four dimensions admitting supersymmetric black holes is minimal supergravity, whose bosonic sector is simply standard Einstein-Maxwell theory. Using this fact, the following near-horizon uniqueness theorem has been proved:.
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Theorem 5. Notice that staticity here follows from supersymmetry. In Section 7 we will discuss the implications of this result for uniqueness of supersymmetric black holes in four dimensions. For gauged supergravity, whose bosonic sector is Einstein-Maxwell theory with a negative cosmological constant, an analogous classification of supersymmetric near-horizon geometries has not been performed. Nevertheless, one may deduce the following result, from a classification of all near-horizon geometries of this theory under the additional assumption of axisymmetry:.
Proposition 5. Any supersymmetric, axisymmetric, near-horizon geometry in gauged supergravity, is given by the near-horizon limit of the 1-parameter family of supersymmetric Kerr-Newman-AdS 4 black holes [ ]. Note that the above near-horizon geometry is non-static. This is related to the fact that supersymmetric AdS black holes must carry angular momentum. It would be interesting to remove the assumption of axisymmetry. Some related work has been done in the context of supersymmetric isolated horizons [ 29 ].
Supersymmetric black holes are not expected to exist in supergravity.
For the general supergravity the following result, supporting this expectation, has been established. Supersymmetric solutions to this theory were classified in [ 93 ]. This was used to obtain a complete classification of supersymmetric near-horizon geometries in this theory. Note that here supersymmetry implies homogeneity. As discussed in Section 7 , the above theorem can be used to prove a uniqueness theorem for topologically spherical supersymmetric black holes.
The corresponding problem for minimal gauged supergravity has proved to be more difficult. The bosonic sector of this theory is Einstein-Maxwell-Chern-Simons theory with a negative cosmological constant. The following partial results have been shown. Consider a supersymmetric, homogeneous near-horizon geometry of minimal gauged supergravity. The near-horizon geometry of the S 3 case was used to construct the first example of an asymptotically AdS 5 supersymmetric black hole [ ].
Analogous results in gauged supergravity coupled to an arbitrary number of vector multiplets this includes U 1 3 gauged supergravity were obtained in [ ]. Unlike the ungauged theory, homogeneity is not implied by supersymmetry, and indeed there are more general solutions. The most general supersymmetric near-horizon geometry in minimal gauged supergravity, admitting a U 1 2 - rotational symmetry and a compact horizon section, is the near-horizon limit of the topologically-spherical supersymmetric black holes of [ 38 ].
The motivation for assuming this isometry group is that all known black-hole solutions in five dimensions possess this. In fact, recent results allow one to remove all assumptions and obtain a complete classification. This latter result is proved using a Lichnerowicz type identity to establish a correspondence between Killing spinors and solutions to a horizon Dirac equation, and then applying an index theorem.
Therefore, combining the previous three propositions gives a complete classification theorem for near-horizon geometries in minimal gauged supergravity.