## 4 Vacuum Solutions

The Einstein equations for a near-horizon geometry (7) in the absence of matter fields are equivalent to the following equation on with the function determined by see Eqs. (17), (18). In this section we will explore solutions to Eq. (66) in various dimensions. It is useful to note that the contracted Bianchi identity for the horizon metric is equivalent to

### 4.1 Static: all dimensions

A complete classification is possible for . Recall from Section 3.2.1, the staticity conditions for a near-horizon geometry are and .

Theorem 4.1 ([42]). The only vacuum static near-horizon geometry for and compact is given by , and . For this result is also valid for .

Proof: A simple proof of the first statement is as follows. Substituting the staticity conditions (50) into (67) gives

where . Irrespective of the topology of one can argue that is a globally-defined function (for simply connected this is automatic, otherwise it can be shown by working in patches on and exploiting the fact that in each patch is only defined up to an additive constant). For it is then clear that compactness implies must be a constant. For , if one assumes is non-constant one can easily derive a contradiction by evaluating the above equation at the maximum and minimum of (which must exist by compactness). Hence in either case , which gives the claimed near-horizon data.In four dimensions one can solve Eq. (66) in general without assuming compactness of . For non-constant and any one obtains the near-horizon geometry (sending )

where . (This is an analytically-continued Schwarzschild with a cosmological constant.) This local form of the metric can be used to show that for and non-constant, there are also no smooth horizon metrics on a compact .

### 4.2 Three dimensions

The classification of near-horizon geometries in vacuum gravity with a cosmological constant can be completely solved. Although very simple, to the best of our knowledge this has not been presented before, so for completeness we include it here.

The main simplification comes from the fact that cross sections of the horizon are one-dimensional, so the horizon equations are automatically ODEs. Furthermore, there is no intrinsic geometry on and so the only choice concerns its global topology, which must be either or .

Theorem 4.2. Consider a near-horizon geometry with compact cross section , which
satisfies the vacuum Einstein equations including cosmological constant . If the
near-horizon geometry is given by the quotient of AdS_{3} in Eq. (73). For the only solution
is the trivial flat geometry . There are no solutions for .

Proof: We may choose a periodic coordinate on so the horizon metric is simply and the 1-form . Observe that must be a globally-defined function and hence must be a periodic function of . Since the curvature and metric connection trivially vanish, the horizon equations (66) and (67) simplify to

This system of ODEs can be explicitly integrated as we explain below. Instead, we will avoid this and employ a global argument on . If , integrate Eq. (71) over to deduce that and , which gives the trivial flat near-horizon geometry . For we argue as follows. Multiply Eq. (71) by and integrate over to find . Hence must be a constant and substituting into the horizon equations gives and (without loss of generality we have chosen a sign for ). The near-horizon geometry is then This metric is locally AdS_{3}and in the second equality we have written it as a fibration over AdS

_{2}.

It is worth remarking that the ODE (71) can be completely integrated without assuming compactness. For this reveals a second solution and , where we have set the integration constant to zero by translating the coordinate . Upon changing the resulting near-horizon geometry is:

Again, this metric is locally AdS_{3}. Unlike the previous case though, cannot be taken to be compact and hence we must have . For there is also a second solution given by and , although this is singular. For there is a unique solution given by and , although this is also singular.

^{18}

### 4.3 Four dimensions

The general solution to Eq. (66) is not known in this case. In view of the rigidity theorem it is natural to assume axisymmetry. If one assumes such a symmetry, the problem becomes of ODE type and it is possible to completely solve it. The result is summarised by the following theorem, first proved in [122, 163] for and in [154] for .

Theorem 4.3 ([122, 163, 154]). Consider a spacetime containing a degenerate horizon, invariant under an isometry, satisfying the vacuum Einstein equations including a cosmological constant. Any non-static near-horizon geometry, with compact cross section, is given by the near-horizon limit of the extremal Kerr or Kerr-(A)dS black hole.

Proof: We present a streamlined version of the proof in [154]. As described in Section 3.2.2, axisymmetry implies one can introduce coordinates on so that

The component of Eq. (66) implies is a constant. The component of Eq. (68) then implies where is a constant. Substituting this into Eq. (67) gives Now subtracting the component from the component of Eq. (66) gives A non-static near-horizon geometry must have and therefore from the above equation is non-constant. Using this, one can write Eq. (77) as The solution to the ODE for is given by where is a positive constant, which can then be used to solve the ODE for : where and is a constant. Changing affine parameter in the full near-horizon geometry finally gives with determined above. Observe that this derivation is purely local and does not assume anything about the topology of (unlike the derivation in [154], which assumed compactness). If we recover the general static solution (70), hence let us now assume . Now assume is compact, so by axisymmetry one must have either or . Integrating
Eq. (77) over then shows that if then and so the metric in square brackets is AdS_{2}.
The horizon metric extends to a smooth metric on if and only if . It can then be checked
the near-horizon geometry is isometric to that of extremal Kerr for or Kerr-AdS for [154].
It is also easy to check that for it corresponds to extremal Kerr-dS. In the non-static case, the
horizon topology theorem excludes the possibility of for . If the non-static
possibility with can also be excluded [164].

It would be interesting to remove the assumption of axisymmetry in the above theorem. In [145] it is shown that regular non-axisymmetric linearised solutions of Eq. (66) about the extremal Kerr near-horizon geometry do not exist. This supports the conjecture that any smooth solution of Eq. (66) on must be axisymmetric and hence given by the above theorem.

### 4.4 Five dimensions

In this case there are several different symmetry assumptions one could make. Classifications are known for homogeneous horizons and horizons invariant under a -rotational symmetry.

We may define a homogeneous near-horizon geometry to be one for which the Riemannian manifold is a homogeneous space whose transitive isometry group also leaves the rest of the near-horizon data invariant. Since any near-horizon geometry (7) possesses the 2D symmetry generated by and where , it is clear that this definition guarantees the near-horizon geometry itself is a homogeneous spacetime. Conversely, if the near-horizon geometry is a homogeneous spacetime, then any cross section must be a homogeneous space under a subgroup of the spacetime isometry group, which commutes with the 2D symmetry in the plane (since is a constant submanifold). It follows that must also be invariant under the isometry , 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 as follows.

Theorem 4.4 ([158]). Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to

where is a -connection over a 2D base space satisfying with . The curvature of the connection is , where is the volume form of the 2D base, and .The proof uses the fact that homogeneity implies must be a Killing field and then one reduces the problem onto the 2D orbit space. Observe that for one recovers the static near-horizon geometries. For and we see that so that the 2D metric is a round and the horizon metric is locally isometric to a homogeneously squashed . Hence we have:

Corollary 4.1. Any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to the near-horizon limit of the extremal Myers–Perry black hole with rotational symmetry (i.e., equal angular momenta). For one gets the same result with the Myers–Perry black hole replaced by its generalisation with a cosmological constant [129].

For we see that there are more possibilities depending on the sign of . If we again have a horizon geometry locally isometric to a homogeneous . If , we can write and the -connection is non-trivial, so the cross sections are the Nil group manifold with its standard homogeneous metric. For , we can write and the connection , so the cross sections are the group manifold with its standard homogeneous metric, unless , which gives . Hence we have:

Corollary 4.2. For any vacuum, homogeneous, non-static near-horizon geometry is locally isometric to either the near-horizon limit of the extremal rotating black hole [129] with rotational symmetry, or a near-horizon geometry with: (i) and its standard homogenous metric, (ii) and its standard homogeneous metric or (iii) .

This is analogous to a classification first obtained for supersymmetric near-horizon geometries in gauged supergravity, see Proposition (5.3).

We now consider a weaker symmetry assumption, which allows for inhomogeneous horizons. A -rotational isometry is natural in five dimensions and all known explicit black-hole solutions have this symmetry. The following classification theorem has been derived:

Theorem 4.5 ([152]). Consider a vacuum non-static near-horizon geometry with a -rotational isometry and a compact cross section . It must be globally isometric to the near-horizon geometry of one of the following families of black-hole solutions:

- : the 3-parameter boosted extremal Kerr string.
- : the 2-parameter extremal Myers–Perry black hole or the 3-parameter ‘fast’ rotating extremal KK black hole [190].
- : the Lens space quotients of the above solutions.

Remarks:

- The near-horizon geometry of the vacuum extremal black ring [187] is a 2-parameter subfamily of case 1, corresponding to a Kerr string with vanishing tension [162].
- The near-horizon geometry of the ‘slowly’ rotating extremal KK black hole [190] is identical to that of the 2-parameter extremal Myers–Perry in case 2.
- The cases can be written as a single 3-parameter family of near-horizon geometries [135].
- The case has been ruled out [131].

For , the analogous problem has not been solved. The only known solution in this case is the
near-horizon geometry of the rotating black hole with a cosmological constant [129], which generalises
the Myers–Perry black hole. It would be interesting to classify near-horizon geometries with
in this case since this would capture the near-horizon geometry of the yet-to-be-found
asymptotically-AdS_{5} black ring. A perturbative attempt at constructing such a solution is discussed in
[152].

### 4.5 Higher dimensions

For spacetime dimension , so the horizon cross section dim, the horizon equation (66) is far less constraining than in lower dimensions. Few general classification results are known, although several large families of vacuum near-horizon geometries have been constructed.

#### 4.5.1 Weyl solutions

The only known classification result for vacuum near-horizon geometries is for solutions with -rotational symmetry. These generalise the axisymmetric solutions and solutions with -symmetry discussed in Sections 4.3 and 4.4 respectively. By performing a detailed study of the orbit spaces it has been shown that the only possible topologies for are: , , , and [139].

An explicit classification of the possible near-horizon geometries (for the non-toroidal case) was derived in [135], see their Theorem 1. Using their theorem it is easy to show that the most general solution with is in fact isometric to the near-horizon geometry of a boosted extremal Kerr-membrane (i.e., perform a general boost of Kerr along the coordinates and then compactify ). Non-static near-horizon geometries with have been ruled out [131] (including a cosmological constant).

#### 4.5.2 Myers–Perry metrics

The Myers–Perry (MP) black-hole solutions [183] generically have isometry groups where . Observe that if then and hence these solutions fall outside the classification discussed in Section 4.5.1. They are parameterised by their mass parameter and angular momentum parameters for . The topology of the horizon cross section . A generalisation of these metrics with non-zero cosmological constant has been found [99]. We will focus on the case, although analogous results hold for the solutions.

The location of the horizon is determined by the largest positive number such that in odd and even dimensions and , respectively, where

The extremal limit of these black holes in odd and even dimensions is given by and , respectively. These conditions hold only when the black hole is spinning in all the two planes available, i.e., we need for all . Without loss of generality we will henceforth assume and use the extremality condition to eliminate the mass parameter . The near-horizon geometry of the extremal MP black holes can be written in a unified form [79]: where and in odd and even dimensions respectively. The direction cosines and take values in the range with and in odd and even dimensions satisfy respectively. The generalisation of these near-horizon geometries for was given in [165]. It is worth noting that if subsets of the angular momentum parameters 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. For example, by adding one flat direction one can generate a boosted MP string, whose near-horizon geometries have topology. Interestingly, for odd dimensions 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].

#### 4.5.3 Exotic topology horizons

Despite the absence of explicit black-hole solutions, a number of solutions to Eq. (66) are known. 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. Let be a compact Fano Kähler–Einstein
manifold^{19}
of complex dimension and is the indivisible class given by
with (the Fano index and satisfies with equality iff ). The
Kähler–Einstein metric on is normalised as and we denote its isometry
group by . The simplest example occurs for , in which case with
.

In even dimensions greater than four, an infinite class of near-horizon geometries is revealed by the following result.

Proposition 4.1 ([156]). Let and be the principal -bundle over any Fano Kähler–Einstein manifold , specified by the characteristic class . For each there exists a 1-parameter family of smooth solutions to Eq. (66) on the associated -bundles .

The dim ansatz used for the near-horizon data is the invariant form

where is a -connection over with curvature . The solutions depend on one continuous parameter and the integer . The continuous parameter corresponds to the angular momentum where generates the -isometry in the -fibre. The various functions are given by , , and where is a polynomial in and smoothness fixes to be a function of .The simplest example is the , solution, for which the near-horizon data takes the explicit form

where and . The coordinate ranges are , , , . Cross sections of the horizon , are homeomorphic to if is even, or the non-trivial bundle if is odd. For the solutions are analogous.For the Fano base is higher dimensional and there are more choices available. The topology of the total space is always a non-trivial -bundle over and in fact different give different topologies, so there are an infinite number of horizon topologies allowed. Furthermore, one can choose to have no continuous isometries giving examples of near-horizon geometries with a single -rotational isometry. Hence, if there are black holes corresponding to these horizon geometries they would saturate the lower bound in the rigidity theorem.

It is worth noting that the local form of the above class of near-horizon metrics includes as a special case that of the extremal MP metrics with equal angular momenta (for ). The above class of horizon geometries are of the same form as the Einstein metrics on complex line bundles [186], which in four dimensions corresponds to the Page metric [185], although we may of course set .

Similar constructions of increasing complexity can be made in odd dimensions, again revealing an infinite class of near-horizon geometries.

Proposition 4.2. Let and be the principal -bundle over specified by the characteristic class . There exists a 1-parameter family of Sasakian solutions to Eq. (66) on .

As a simple example consider . This leads to an explicit homogeneous near-horizon geometry with

where for convenience we have written is a vector field, is a constant and . Regularity requires that the Chern number of the -fibration over each to be the same and the period . The total space is a Lens space -bundle over and is topologically . For and this corresponds to a Sasaki–Einstein metric on sometimes known as .The above proposition can be generalised as follows.

Proposition 4.3 ([158]). Given any Fano Kähler–Einstein manifold of complex dimension and coprime satisfying , there exists a 1-parameter family of solutions to Eq. (66) where is a compact Sasakian -manifold.

These examples have symmetry, although possess only one independent angular momentum along the -fibres. These are deformations of the Sasaki–Einstein manifolds [94].

There also exist a more general class of non-Sasakian horizons in odd dimensions.

Proposition 4.4 ([157]). Let be the principal -bundle over any Fano Kähler–Einstein manifold , specified by the characteristic classes where . For a countably infinite set of non-zero integers , there exists a two-parameter family of smooth solutions to Eq. (66) on the associated Lens space bundles .

The dim form of the near-horizon data in the previous two propositions is the invariant form

where is a principal -connection over whose curvature is . The explicit functions are linear in and are ratios of various polynomials in . Generically these solutions possess two independent angular momenta along the -fibres. The Sasakian horizon geometries of Proposition 4.3 arise as a special case with and possess only one independent angular momentum. The base gives horizon topologies or depending on whether is even or odd respectively.It is worth noting that the local form of this class of near-horizon metrics includes as a special case that of the extremal MP metrics with all but one equal angular momenta. The above class of horizon geometries are of the same form as the Einstein metrics found in [37, 157].