Classification of NearHorizon Geometries of Extremal Black Holes
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Abstract
Any spacetime containing a degenerate Killing horizon, such as an extremal black hole, possesses a welldefined notion of a nearhorizon geometry. We review such nearhorizon geometry solutions in a variety of dimensions and theories in a unified manner. We discuss various general results including horizon topology and nearhorizon symmetry enhancement. We also discuss the status of the classification of nearhorizon geometries in theories ranging from vacuum gravity to EinsteinMaxwell theory and supergravity theories. Finally, we discuss applications to the classification of extremal black holes and various related topics. Several new results are presented and open problems are highlighted throughout.
Keywords
Black holes Extremal black holes Nearhorizon geometry1 Introduction
Equilibrium blackhole solutions to Einstein’s equations have been known since the advent of general relativity. The most obvious reason such solutions are of physical interest is the expectation that they arise as the end state of catastrophic gravitational collapse of some suitably localised matter distribution. A less obvious reason such solutions are important is that they have played a key role in guiding studies of quantum gravity.
The main purpose of this review is to discuss the classification of the nearhorizon geometries of extremal black holes. There are a number of different motivations for considering this, which we will briefly review. As alluded to above, the principle reasons stem from studies in quantum gravity. In Section 1.1 and 1.2 we discuss the various ways extremal black holes and their nearhorizon geometries have appeared in modern studies of quantum gravity. In Section 1.3 we discuss the more general black hole classification problem (which is also partly motivated by quantum gravity), and how nearhorizon geometries provide a systematic tool for investigating certain aspects of this problem for extremal black holes.
1.1 Black holes in string theory
To date, the most promising candidate for a theory of quantum gravity is string theory. Famously, this predicts the existence of extra spatial dimensions. As discussed above, an important test for any candidate theory of quantum gravity is that it is able to explain the semiclassical formulae (1) and (2). A major breakthrough of Strominger and Vafa [200] was to use string theory to supply a microscopic derivation of Eq. (2) for certain five dimensional extremal black holes.
The black holes in question are higherdimensional counterparts of the extremal ReissnerNordström (RN) black holes. These are supersymmetric solutions to a supergravity theory, which can be obtained as a consistent KaluzaKlein (KK) reduction of the ten/eleven dimensional supergravity that describes string theory at low energies. Supersymmetry was crucial for their calculation, since nonrenormalisation results allowed them to perform a weakcoupling string calculation (involving certain Dbrane configurations) to deduce the entropy of the semiclassical black holes that exist in the strong coupling regime (see [56] for a review). This was quickly generalised to supersymmetric black holes with angular momentum [30] and supersymmetric four dimensional black holes [171].
An important assumption in these string theory calculations is that a given black hole is uniquely specified by its conserved charges: its mass/energy, electric charge and angular momentum. For four dimensional EinsteinMaxwell theory this follows from the blackhole uniqueness theorem (see [39] for a review). However, Emparan and Reall demonstrated that black hole uniqueness is violated for fivedimensional asymptoticallyflat vacuum spacetimes [73]. This was via the construction of an explicit counterexample, the black ring, which is a black hole whose spatial horizon topology is S^{1} × S^{2}. Together with the higherdimensional analogues of the Kerr black hole (which have spherical topology) found by Myers and Perry [183], this established that the conserved charges are not sufficient to specify a black hole uniquely and also that other horizon topologies are possible. Indeed, this remarkable result motivated the study of stationary black holes in higher dimensional spacetimes. Subsequently, a supersymmetric black ring was constructed [64, 66] that coexists with the spherical topology black holes used in the original entropy calculations. Although microscopic descriptions for the black rings have been proposed [20, 51], it is fair to say that the description of black hole nonuniqueness within string theory is not properly understood (see [74] for a brief review).
Any supersymmetric black hole is necessarily extremal. Since the StromingerVafa calculation, a substantial amount of work has been directed at removing the assumption of supersymmetry and extremality, with the ultimate goal being a string theory derivation of the thermodynamics of realistic black holes such as the four dimensional Schwarzschild or Kerr black holes. Although little progress has been made in the description of such nonextremal black holes, significant progress has been made for extremal nonsupersymmetric black holes. In particular, this has been via the blackhole attractor mechanism.
The attractor mechanism is the phenomenon that the entropy of certain extremal black holes in string theory does not depend on the moduli of the theory (typically scalar fields in the supergravity theory). This was first observed for supersymmetric static black holes [78, 197, 77] although later it was realised it is valid for generic extremal black holes [100, 194, 12]. The key idea is that extremal black holes have a welldefined nearhorizon geometry that typically possesses an AdS_{2} symmetry. Assuming this symmetry, it was argued that the entropy must be independent of the moduli of the theory. Motivated by this, it was then proved that the nearhorizon geometry of any extremal black hole in this context must in fact possess an AdS_{2} symmetry [162]. This general attractor mechanism thus ensures the black hole entropy is independent of the string coupling, so it can be safely computed at weak coupling. This shows that in fact it is extremality, rather than supersymmetry, that is behind the success of the string theory microscopic calculations [52, 13]. This also explains the success of the entropy calculations for extremal, nonsupersymmetric black holes in four and five dimensions, e.g., [192, 71, 140].
1.2 Gauge/gravity duality
A significant breakthrough in the study of quantum gravity is the Anti de Sitter/Conformal Field Theory (AdS/CFT) duality [169, 206, 207, 109]. In principle, AdS/CFT asserts a fully nonperturbative equivalence of quantum gravity in asymptotically AdS spacetimes with a conformally invariant quantum field theory in one lower spatial dimension. This is an explicit realisation of a ‘holographic principle’ underlying quantum gravity [202, 201].
A crucial feature of the duality is that classical gravity in AdS spacetimes is dual to the stronglycoupled regime in the CFT. This provides a precise framework to analyse the microscopic description of black holes in terms of welldefined quantum field theories. The duality was originally proposed [169] in the context of string theory on AdS_{5} × S^{5}, in which case the CFT is the maximally supersymmetric four dimensional SU(N) YangMills gauge theory. However, the original idea has subsequently been generalised to a number of dimensions and theories, and such gauge/gravity dualities are believed to hold more generally.
Classical nonextremal AdS black holes represent highenergy thermal states in the dual theory at large N and strong coupling [207]. Strong coupling poses the main obstacle to providing a precise entropy counting for such black holes, although excellent qualitative agreement can be found via extrapolating weak coupling calculations [108, 23, 130]. Precise agreement has been achieved [198] for the asymptotically AdS_{3} BañadosTeitelboimZanelli (BTZ) black hole (even for the nonextremal case) [15]. This is because generically any theory of quantum gravity in AdS_{3} must be described by a two dimensional CFT_{2} with a specific central charge [32]. This allows one to compute the entropy from Cardy’s formula, without requiring an understanding of the microscopic degrees of freedom. In fact the string theory calculations described in Section 1.1 can be thought of as applications of this method. This is because the black holes in question can be viewed (from a higherdimensional viewpoint) as black strings with an AdS_{3} factor in the nearhorizon geometry, allowing AdS_{3}/CFT_{2} to be applied.
A major open problem is to successfully account for blackhole entropy using a higher dimensional CFT. The best understood case is when the CFT is four dimensional, in which case the black holes are asymptotically AdS_{5}. As in the original string calculations, a strategy to overcome the strongcoupling problem is to focus on supersymmetric AdS black holes. The dual CFT states then belong to certain Bogomol’nyiPrasadSommerfield (BPS) representations, and so weakcoupling calculations may not receive quantum corrections. It turns out that such blackhole solutions must rotate and hence are difficult to construct. In fact the first examples of supersymmetric AdS_{5} black holes [120, 119] were found via a classification of nearhorizon geometries. Subsequently, a more general fourparameter family of blackhole solutions were found [38, 160]. The problem of classifying all supersymmetric AdS black holes motivated further classifications of nearhorizon geometries, which have ruled out the possibility of other types of black hole such as supersymmetric AdS black rings [154, 161, 106]. Despite significant effort, a microscopic derivation of the entropy from the CFT has not yet been achieved in this context. Due to the low amount of supersymmetry preserved by the black holes, it appears that nonzero coupling effects must be taken into account [149, 22, 148, 21, 36].
The original AdS/CFT duality was established by arguing that there exist two complementary descriptions of the low energy physics of the string theory of a stack of N extremal D3 black branes. Near the horizon of the Dbrane only low energy excitations survive, which are thus described by string theory in the AdS_{5} ×S^{5} nearhorizon geometry. On the other hand, the massless degrees of freedom on a Dbrane arrange themselves into (super) SU(N) YangMills theory. It is natural to extend this idea to extremal black holes. Since extremal black holes typically possess an AdS_{2} factor in their nearhorizon geometry, one may then hope that an AdS_{2}/CFT_{1} duality [199] could provide a microscopic description of such black holes. Unfortunately this duality is not as well understood as the higherdimensional cases. However, it appears that the blackhole entropy can be reproduced from the degeneracy of the ground states of the dual conformal mechanics [170, 195].
Another recentlydeveloped approach is to generalise the AdS_{3}/CFT_{2} derivation of the BTZ entropy to describe more general black holes. This involves finding an asymptotic symmetry group of a given nearhorizon geometry that contains a Virasoro algebra and then applying Cardy’s formula. This was applied to the extremal Kerr black hole and led to the Kerr/CFT correspondence [110], which is a proposal that quantum gravity in the nearhorizon geometry of extremal Kerr is described by a chiral CFT_{2} (see the reviews [31, 48]). This technique has provided a successful counting of the entropy of many black holes. However, as in the AdS_{2}/CFT_{1} case, the duality is poorly understood and it appears that nontrivial excitations of the nearhorizon geometry do not exist [60, 4]. The relation between these various approaches has been investigated in the special case of BTZ [16]. Furthermore, a CFT_{2} description has been proposed for a certain class of nearextremal black holes, which possess a local nearhorizon AdS_{3} factor but a vanishing horizon area in the extremal limit [196, 146].^{3}
Recently, ideas from the gauge/gravity duality have been used to model certain phase transitions that occur in condensedmatter systems, such as superfluids or superconductors [107, 125]. The key motivation to this line of research, in contrast to the above, is to use knowledge of the gravitational system to learn about stronglycoupled field theories. Charged black holes in AdS describe the finite temperature phases. The nonsuperconducting phase is dual to the standard planar RNAdS black hole of EinsteinMaxwell theory, which is stable at high enough temperature. However, at low enough temperatures this solution is unstable to the formation of a charged scalar condensate. The dominant phase at low temperatures is a charged black hole with scalar hair, which describes the superconducting phase. This instability of (near)extremal RNAdS can be understood as occurring due to the violation of the BreitenlohnerFreedmann bound in the AdS_{2} factor of the nearhorizon geometry. A similar result has also been shown for neutral rotating AdS black holes [59]. The nearhorizon AdS_{2} has also been used to provide holographic descriptions of quantum critical points and Fermi surfaces [76].
1.3 Black hole classification
The classification of higherdimensional stationary blackhole solutions to Einstein’s equations is a major open problem in higher dimensional general relativity (see [75, 136] for reviews). As explained above, the main physical motivation stems from studies of quantum gravity and high energy physics. However, its study is also of intrinsic value both physically and mathematically. On the physical side we gain insight into the behaviour of gravity in higherdimensional spacetimes, which in turn often provides renewed perspective for the classic fourdimensional results. On the mathematical side, solutions to Einstein’s equation have also been of interest in differential geometry [24].^{4}
In four dimensions the blackhole uniqueness theorem provides an answer to the classification problem for asymptoticallyflat blackhole solutions of EinsteinMaxwell theory (see [39] for a review).^{5} However, in higher dimensions, uniqueness is violated even for asymptoticallyflat vacuum black holes. To date, the explicit blackhole solutions known are the spherical horizon topology MyersPerry black holes [183] and the black rings [73, 187] that have S^{1} × S^{2} horizon topology (see [75] for a review). If one allows for more complicated boundary conditions, such as KK asymptotics, then uniqueness is violated even for static black holes (see, e.g., [141]). Although of less obvious physical relevance, the investigation of asymptoticallyflat vacuum black holes is the fundamental starting case to consider in higher dimensions, since such solutions can be viewed as limits of black holes with more general asymptotics such as KK, AdS and matter fields.
General results have been derived that constrain the topology of black holes. By generalising Hawking’s horizon topology theorem [127] to higher dimensions, Galloway and Schoen [91] have shown that the spatial topology of the horizon must be such that it admits a positive scalar curvature metric (i.e., positive Yamabe type). Horizon topologies are further constrained by topological censorship [85, 44]. For asymptoticallyflat (and globally AdS) black holes, this implies that there must be a simply connected (oriented) cobordism between cross sections of the horizon and the (D − 2)dimensional sphere at spatial infinity.^{6} In D = 4, this rules out toroidal black holes, although for D = 5 it imposes no constraint. For D > 5 this does provide a logically independent constraint in addition to the positive Yamabe condition [191, 156].
General results have also been derived that constrain the symmetries of black hole spacetimes. Firstly, asymptoticallyflat, static vacuum black holes must be spherically symmetric and hence are uniquely given by the higher dimensional Schwarzschild black hole [98].^{7} By generalising Hawking’s rigidity theorem [127], it was shown that asymptoticallyflat and AdS stationary nonextremal rotating black holes must admit at least ℝ × U(1) isometry [137, 180] (for partial results pertaining to extremal rotating black holes see [134]). This additional isometry can be used to further refine the allowed set of D = 5 black hole horizon topologies [133].
An important class of spacetimes, for which substantial progress towards classification has been made, are the generalised Weyl solutions [72, 124]. By definition these possess an ℝ × U(1)^{D−3} symmetry group and generalise D = 4 stationary and axisymmetric spacetimes. As in the D = 4 case, it turns out that the vacuum Einstein equations for spacetimes with these symmetries are integrable. For D = 5 this structure has allowed one to prove certain uniqueness theorems for asymptoticallyflat black holes with ℝ × U(1)^{2} symmetry, using the same methods as for D = 4 [138]. Furthermore, this has led to the explicit construction of several novel asymptoticallyflat, stationary, multiblackhole vacuum solutions, the first example being a (nonlinear) superposition of a black ring and a spherical black hole [67]. For D > 5, the symmetry of these spacetimes is not compatible with asymptotic flatness, that would require the number of commuting rotational symmetries to not exceed \(\left\lfloor {{{D  1} \over 2}} \right\rfloor\), the rank of the rotation group SO(D − 1). In this case, Weyl solutions are compatible with KK asymptotics and this has been used to prove uniqueness theorems for (uniform) KK black holes/strings [139].
The general topology and symmetry constraints discussed above become increasingly weak as one increases the number of dimensions. Furthermore, there is evidence that black hole uniqueness will be violated much more severely as one increases the dimensions. For example, by an analysis of gravitational perturbations of the MyersPerry black hole, evidence for a large new family of black holes was found [193]. Furthermore, the investigation of “blackfolds”, where the longrange effective dynamics of certain types of black holes can be analysed, suggests that many new types of black holes should exist, see [70] for a review. In the absence of new ideas, it appears that the general classification problem for asymptoticallyflat black holes is hopelessly out of reach.
As discussed in Section 1.2, the blackholeclassification problem for asymptotically AdS black holes is of interest in the context of gauge/gravity dualities. The presence of a (negative) cosmological constant renders the problem even more complicated. Even in four dimensions, there is no analogue of the uniqueness theorems. One reason for this comes from the fact that Einstein’s equations with a cosmological constant for stationary and axisymmetric metrics are not integrable. Hence the standard method used to prove uniqueness of Kerr cannot be generalised. This also means that constructing charged generalisations, in four and higher dimensions, from a neutral seed can not be accomplished using standard solution generating methods. In fact, perturbation analyses of known solutions (e.g., KerrAdS and higher dimensional generalisations), reveal that if the black holes rotate sufficiently fast, superradiant instabilities exist [130, 34, 159, 35]. It has been suggested that the endpoint of these instabilities are new types of nonaxisymmetric blackhole solutions that are not stationary in the usual sense, but instead invariant under a single Killing field corotating with the horizon [159]. (Examples of such solutions have been constructed in a scalar gravity theory [58]). A further complication in AdS comes from the choice of asymptotic boundary conditions. In AdS there is the option of replacing the sphere on the conformal boundary with more general manifolds, in which case topological censorship permits more blackhole topologies [90].
It is clear that supersymmetry provides a technicallysimplifying assumption to classifying spacetimes, since it reduces the problem to solving firstorder Killing spinor equations rather than the full Einstein equations. A great deal of work has been devoted to developing systematic techniques for constructing supersymmetric solutions, most notably in fivedimensional ungauged supergravity [93] and gauged supergravity [92]. These have been used to construct new fivedimensional supersymmetric blackhole solutions, which are asymptotically flat [66, 64] and AdS [120, 119, 160], respectively. Furthermore, the first uniqueness theorem for asymptoticallyflat supersymmetric black holes was proved using these methods [191].
Less obviously, it turns out that the weaker assumption of extremality can also be used as a simplifying assumption, as follows. The event horizon of all known extremal black holes is a degenerate Killing horizon with compact spatial cross sections H. It turns out that restricting Einstein’s equations for a Ddimensional spacetime to a degenerate horizon gives a set of geometric equations for the induced metric on such (D − 2)dimensional cross sections H, that depend only on quantities intrinsic to H. By studying solutions to this problem of Riemannian geometry on a compact manifold H, one can thus consider the possible horizon geometries (and topologies) independently of the full parent spacetime. This strategy often also works in cases where the standard black hole uniqueness/classification techniques do not apply (e.g., AdS, higher dimensions etc.).
One can understand this feature of degenerate horizons in terms of the nearhorizon limit, which, as we explain in Section 2, exists for any spacetime containing a degenerate horizon. This allows one to define an associated nearhorizon geometry, which must also satisfy the full Einstein equations [191, 162], so classifying nearhorizon geometries is then equivalent to classifying possible horizon geometries (and topologies). Indeed, the topic of this review is the classification of nearhorizon geometries in diverse dimensions and theories.
The classification of nearhorizon geometries allows one to explore in a simplified setup the main issues that appear in the general blackhole classification problem, such as the horizon topology, spacetime symmetry and the “number” of solutions. The main drawback of this approach is that the existence of a nearhorizon geometry solution does not guarantee the existence of a corresponding blackhole solution (let alone its uniqueness).^{8} Hence, one must keep this in mind when interpreting nearhorizon classifications in the context of black holes, although definite statements can be learned. In particular, one can use this method to rule out possible blackhole horizon topologies, for if one can classify nearhorizon geometries completely and a certain horizon topology does not appear, this implies there can be no extremal black hole with that horizon topology either. A notable example of this method has been a proof of the nonexistence of supersymmetric AdS black rings in D = 5 minimal gauged supergravity [161, 106].
1.4 This review
1.4.1 Scope
In this review we will consider nearhorizon geometries of solutions to Einstein’s equations, in all dimensions D ≥ 3, that contain smooth degenerate horizons. Our aim is to provide a unified treatment of such nearhorizon solutions in diverse theories with matter content ranging from vacuum gravity, to EinsteinMaxwell theories and various (minimal) supergravity theories.
We do not assume the nearhorizon geometry arises as a nearhorizon limit of a blackhole solution. However, due to the application to extremal black holes we will mostly consider horizons that admit a spatial cross section that is compact. As we will see in various setups, compactness often allows one to avoid explicitly solving the full Einstein equations and instead use global arguments to constrain the space of solutions. As a result, the classification of nearhorizon geometries with noncompact horizon cross sections is a much more difficult problem about which less is known. This is relevant to the classification of extremal black branes and therefore lies outside the scope of this article. Nevertheless, along the way, we will point out cases in which classification has been achieved without the assumption of compactness, and in Section 7.5 we briefly discuss extremal branes in this context.
Although this is a review article, we streamline some of the known proofs and we also present several new results that fill in various gaps in the literature. Most notably, we fully classify three dimensional nearhorizon geometries in vacuum gravity and EinsteinMaxwellChernSimons theories, in Section 4.2 and 6.1 respectively, and classify homogeneous nearhorizon geometries in five dimensional EinsteinMaxwellChernSimons theories in Section 6.3.2.
1.4.2 Organisation
In Section 2 we provide key definitions, introduce a suitably general notion of a nearhorizon geometry and set up the Einstein equations for such nearhorizon geometries.
In Section 3 we review various general results that constrain the topology and symmetry of nearhorizon geometries. This includes the horizon topology theorem and various nearhorizon symmetry enhancement theorems. We also discuss the physical charges one can calculate from a nearhorizon geometry.
In Section 4 we discuss the classification of nearhorizon geometries in vacuum gravity, including a cosmological constant, organised by dimension. In cases where classification results are not known, we describe the known solutions.
In Section 5 we discuss the classification of supersymmetric nearhorizon geometries in various supergravity theories, organised by dimension.
In Section 6 we discuss the classification of general nearhorizon geometries coupled to gauge fields. This includes D = 3, 4, 5 EinsteinMaxwell theories, allowing for ChernSimons terms where appropriate, and D = 4 EinsteinYangMills theory.
In Section 7 we discuss various applications of nearhorizon geometries and related topics. This includes uniqueness/classification theorems of the corresponding extremal blackhole solutions, stability of nearhorizon geometries and extremal black holes, geometric inequalities, analytic continuation of nearhorizon geometries, and extremal branes and their nearhorizon geometries.
2 Degenerate Horizons and NearHorizon Geometry
2.1 Coordinate systems and nearhorizon limit
In this section we will introduce a general notion of a nearhorizon geometry. This requires us to first introduce some preliminary constructions. Let \({\mathcal N}\) be a smooth^{9} codimension1 null hypersurface in a D dimensional spacetime (M, g). In a neighbourhood of any such hypersurface there exists an adapted coordinate chart called Gaussian null coordinates that we now recall [179, 86].
Let N be a vector field normal to \({\mathcal N}\) whose integral curves are futuredirected null geodesic generators of \({\mathcal N}\). In general these will be nonaffinely parameterised so on \({\mathcal N}\) we have ∇_{N}N = κN for some function κ. Now let H denote a smooth (D − 2)dimensional spacelike submanifold of \({\mathcal N}\), such that each integral curve of N crosses H exactly once: we term H a cross section of \({\mathcal N}\) and assume such submanifolds exist. On H choose arbitrary local coordinates (x^{a}), for a = 1, …, D − 2, containing some point p ∈ H. Starting from p ∈ H, consider the point in \({\mathcal N}\) a parameter value v along the integral curve of N. Now assign coordinates (v, x^{a}) to such a point, i.e., we extend the functions x^{a} into \({\mathcal N}\) by keeping them constant along such a curve. This defines a set of coordinates (v, x^{a}) in a tubular neighbourhood of the integral curves of N through p ∈ H, such that N = ∂/∂v. Since N is normal to \({\mathcal N}\) we have N · N = g_{vv} = 0 and N · ∂/∂x^{a} = g_{va} = 0 on \({\mathcal N}\).
We now extend these coordinates into a neighbourhood of \({\mathcal N}\) in M as follows. For any point \(q \in {\mathcal N}\) contained in the above coordinates (v, x^{a}), let L be the unique pastdirected null vector satisfying L · N = 1 and L · ∂/∂x^{a} = 0. Now starting at q, consider the point in M an affine parameter value r along the null geodesic with tangent vector L. Define the coordinates of such a point in M by (v,r,x^{a}), i.e., the functions v, x^{a} are extended into M by requiring them to be constant along such null geodesics. This provides coordinates (v, r, x^{a}) defined in a neighbourhood of \({\mathcal N}\) in M, as required.
The coordinates developed above are valid in the neighbourhood of any smooth null hypersurface \({\mathcal N}\). In this work we will in fact be concerned with smooth Killing horizons. These are null hypersurfaces that possess a normal that is a Killing field K in M. Hence we may set N = K in the above construction. Since K = ∂/∂v we deduce that in the neighbourhood of a Killing horizon \({\mathcal N}\), the metric can be written as Eq. (5) where the functions f, h_{a}, γ_{ab} are all independent of the coordinate v. Using the Killing property one can rewrite ∇_{K K} = κK as d(K · K) = −2κK on \({\mathcal N}\), where κ is now the usual surface gravity of a Killing horizon.
Intuitively, the nearhorizon limit is a scaling limit that focuses on the spacetime near the horizon \({\mathcal N}\). We emphasise that the degenerate assumption g_{vv} = O(r^{2}) is crucial for defining this limit and such a general notion of a nearhorizon limit does not exist for a nondegenerate Killing horizon.
2.2 Curvature of nearhorizon geometry
As we will see, geometric equations (such as Einstein’s equations) for a nearhorizon geometry can be equivalently written as geometric equations defined purely on a (D − 2)dimensional cross section manifold H of the horizon. In this section we will write down general formulae relating the curvature of a nearhorizon geometry to the curvature of the horizon H. For convenience we will denote the dimension of H by n = D − 2.
It is worth noting that the following components of the Weyl tensor automatically vanish: C_{−a−b} = 0 and C_{−abc} = 0. This means that e_{−} = ∂_{r} is a multiple Weyl aligned null direction and hence any nearhorizon geometry is at least algebraically special of type II within the classification of [47]. In fact, it can be checked that the null geodesic vector field ∂_{r} has vanishing expansion, shear and twist and therefore any nearhorizon geometry is a Kundt spacetime.^{11} Indeed, by inspection of Eq. (7) it is clear that nearhorizon geometries are a subclass of the degenerate Kundt spacetimes,^{12} which are all algebraically special of at least type II [184].
Henceforth, we will drop the “hats” on all horizon quantities, so R_{ab} and ∇_{a} refer to the Ricci tensor and LeviCivita connection of γ_{ab} on H.
2.3 Einstein equations and energy conditions
An important fact is that if a spacetime containing a degenerate horizon satisfies Einstein’s equations then so does its nearhorizon geometry. This is easy to see as follows. If the metric g in Eq. (6) satisfies Einstein’s equations, then so will the 1parameter family of diffeomorphic metrics g_{ϵ} for any ϵ > 0. Hence the limiting metric ϵ → 0, which by definition is the nearhorizon geometry, must also satisfy the Einstein equations.
The Einstein equations for a nearhorizon geometry can also be interpreted as geometrical equations arising from the restriction of the Einstein equations for the full spacetime to a degenerate horizon, without taking the nearhorizon limit, as follows. The nearhorizon limit can be thought of as the ϵ → 0 limit of the “boost” transformation (K, L) → (ϵK, ϵ^{−1}L). This implies that restricting the boostinvariant components of the Einstein equations for the full spacetime to a degenerate horizon is equivalent to the boost invariant components of the Einstein equations for the nearhorizon geometry. The boostinvariant components are +− and ab and hence we see that Eqs. (17) and (18) are also valid for the full spacetime quantities restricted to the horizon. We deduce that the restriction of these components of the Einstein equations depends only on data intrinsic to H: this special feature only arises for degenerate horizons.^{13} It is worth noting that the horizon equations (17) and (18) remain valid in the more general context of extremal isolated horizons [163, 209, 28] and Kundt metrics [144].
In this review, we describe the current understanding of the space of solutions to the basic horizon equation (17), together with the appropriate horizon matter field equations, in a variety of dimensions and theories.
2.4 Physical charges
Note that in general the gauge field A will not be globally defined on H, so care must be taken to evaluate expressions such as (35) and (38), see [123, 155].
3 General Results
In this section we describe a number of general results concerning the topology and symmetry of nearhorizon geometries under various assumptions.
3.1 Horizon topology theorem
Hawking’s horizon topology theorem is one of the fundamental ingredients of the classic fourdimensional blackhole uniqueness theorems [127]. It states that cross sections of the event horizon of an asymptoticallyflat, stationary, blackhole solution to Einstein’s equations, satisfying the dominant energy condition, must be homeomorphic to S^{2}. The proof is an elegant variational argument that shows that any cross section with negative Euler characteristic can be deformed outside the event horizon such that its outward null geodesics converge. This means one has an outer trapped surface outside the event horizon, which is not allowed by general results on black holes.^{15}
Galloway and Schoen have shown how to generalise Hawking’s horizon topology theorem to higher dimensional spacetimes [91]. Their theorem states if the dominant energy condition holds, a cross section H of the horizon of a black hole, or more generally a marginally outer trapped surface, must have positive Yamabe invariant.^{16} The positivity of the Yamabe invariant, which we define below, is equivalent to the existence of a positive scalar curvature metric and is well known to impose restrictions on the topology, see, e.g., [89]. For example, when H is three dimensional, the only possibilities are connected sums of S^{3} (and their quotients) and S^{1} × S^{2}, consistent with the known examples of blackhole solutions.
In the special case of degenerate horizons a simple proof of this topology theorem can be given directly from the nearhorizon geometry [166]. This is essentially a specialisation of the simplified proof of the GallowaySchoen theorem given in [189]. However, we note that since we only use properties of the nearhorizon geometry, in particular only the horizon equation (17), we do not require the existence of a black hole.
For four dimensional spacetimes, so dim H = 2, the proof is immediate, see, e.g., [144, 152].
Theorem 3.1. Consider a spacetime containing a degenerate horizon with a compact cross section H and assume the dominant energy condition holds. If Λ ≥ 0 then H ≅ S^{2}, except for the special case where the nearhorizon geometry is flat (so Λ = 0) and H ≅ T^{2}. If Λ < 0 and χ(H) < 0 the area of H satisfies A_{H} ≥ 2πΛ^{−1}χ(H) with equality if and only if the nearhorizon geometry is AdS_{2} × Σ_{g}, where Σ_{g} is a compact quotient of hyperbolic space of genus g.
We are now ready to present the degenerate horizon topology theorem.
A simple proof exploits the solution to the Yamabe problem mentioned above [189, 166]. First observe that if there exists a conformal class of metrics [7] for which the Yamabe constant Y (H, [γ]) > 0 then it follows that σ(H) > 0. Therefore, to establish that H has positive Yamabe invariant, it is sufficient to show that for some γ_{ab} the functional E_{γ} [ϕ] > 0 for all ϕ > 0, since the solution to the Yamabe problem then tells us that Y (H, [γ]) = E_{γ} [ϕ_{0}] > 0 for some ϕ_{0} > 0.
We note that the above topology theorems in fact only employ the scalar curvature of the horizon metric and not the full horizon equation (17). It would be interesting if one could use the nontrace part of the horizon equation to derive further topological restrictions.
3.2 AdS_{2}structure theorems
It is clear that a general nearhorizon geometry, Eq. (7), possesses enhanced symmetry: in addition to the translation symmetry v → v + c one also has a dilation symmetry (v, r) → (λv, λ^{−1}r) where λ ≠ 0 and together these form a twodimensional nonAbelian isometry group. In this section we will discuss various nearhorizon symmetry theorems that guarantee further enhanced symmetry.
3.2.1 Static nearhorizon geometries
A static nearhorizon geometry is one for which the normal Killing field K is hypersurface orthogonal, i.e., K ∧ dK ≡ 0 everywhere.
Theorem 3.3 ([162]). Any static nearhorizon geometry is locally a warped product of AdS_{2}, dS_{2} or ℝ^{1,1} and H. If H is simply connected this statement is global. In this case if H is compact and the strong energy conditions holds it must be the AdS_{2} case or the direct product ℝ^{1,1} × H.
3.2.2 Nearhorizon geometries with rotational symmetries
We begin by considering nearhorizon geometries with a U(1)^{D−3} rotational symmetry, whose orbits are generically cohomogeneity1 on cross sections of the horizon H. The orbit spaces H/U(1)^{D−3} have been classified and are homeomorphic to either the closed interval or a circle, see, e.g., [139]. The former corresponds to H of topology S^{2}, S^{3}, L(p, q) times an appropriate dimensional torus, whereas the latter corresponds to H ≅ T^{D−3}. Unless otherwise stated we will assume nontoroidal topology.
We are now ready to state the simplest of the AdS_{2} nearhorizon symmetry enhancement theorems:
Theorem 3.4 ([162]). Consider a Ddimensional spacetime containing a degenerate horizon, invariant under an ℝ × U(1)^{D−3} isometry group, and satisfying the Einstein equations R_{μν} = Λg_{μν}. Then the nearhorizon geometry has a global G × U(1)^{D−3} symmetry, where G is either O(2, 1) or the 2D Poincaré group. Furthermore, if Λ ≤ 0 and the nearhorizon geometry is nonstatic the Poincaré case is excluded.
The toroidal case can in fact be excluded [131], although as remarked above the coordinate system needs to be developed differently.
In fact as we will see in Section 4 one can completely solve for the nearhorizon geometries of the above form in the Λ = 0 case.
Theorem 3.5 ([162]). Consider an extremalblackhole solution of the above D = 4, 5 theory with ℝ × U(1)^{D−3} symmetry. The nearhorizon limit of this solution has a global G × U(1)^{D−3} symmetry, where G is either SO(2, 1) or (the orientationpreserving subgroup of) the 2D Poincaré group. The Poincarésymmetric case is excluded if f_{AB} (Φ) and g_{IJ}(Φ) are positive definite, the scalar potential is nonpositive, and the horizon topology is not T^{D−2}.

In the original statement of this theorem asymptoticallyflat or AdS boundary conditions were assumed [162]. These were only used at one point in the proof, where the property that the generator of each rotational symmetry must vanish somewhere in the asymptotic region (on the “axis” of the symmetry) was used to constrain the Maxwell fields. In fact, using the general form for the nearhorizon limit of a Maxwell field (23) and the fact that for nontoroidal topology at least one of the rotational Killing fields must vanish somewhere, allows one to remove any assumptions on the asymptotics of the blackhole spacetime.

In the context of black holes, toroidal topology is excluded for Λ = 0 when the dominant energy condition holds by the blackholetopology theorems.
Corollary 3.1. Consider a D ≥ 5 spacetime with a degenerate horizon invariant under a ℝ × U(1)^{D−3} symmetry as in Theorem 3.4 and 3.5. The nearhorizon geometry is static if it is either a warped product of AdS_{2} and H, or it is a warped product of locally AdS_{3} and a (D − 3)manifold.
Proposition 3.1 ([162]). Consider an extremal blackhole solution of the above higherderivative theory, obeying the same assumptions as in Theorem 3.5. Assume there is a regular horizon when λ = 0 with SO(2, 1) × U(1)^{D−3} nearhorizon symmetry, and the nearhorizon solution is analytic in λ. Then the nearhorizon solution has SO(2, 1) × U(1)^{D−3} symmetry to all orders in λ.
Hence Theorem 3.5 is stable with respect to higherderivative corrections. However, it does not apply to “small” black holes (i.e., if there is no regular black hole for λ = 0).
So far, the results described all assume D − 3 commuting rotational Killing fields. For D = 4, 5 this is the same number as the rank of the rotation group SO(D − 1), so the above results are applicable to asymptoticallyflat or globallyAdS black holes. For D > 5 the rank of this rotation group is \(\left\lfloor {{{D  1} \over 2}} \right\rfloor\), which is smaller than D − 3, so the above theorems do not apply to asymptoticallyflat or AdS black holes. An important open question is whether the above theorems generalise when fewer than D − 3 commuting rotational isometries are assumed, in particular the case with \(\left\lfloor {{{D  1} \over 2}} \right\rfloor\) commuting rotational symmetries. To this end, partial results have been obtained assuming a certain nonAbelian cohomogeneity1 rotational isometry.
Proposition 3.2 ([79]). Consider a nearhorizon geometry with a rotational isometry group U(1)^{m} × K, whose generic orbit on H is a cohomogeneity1 T^{m}bundle over a Kinvariant homogeneous space B. Furthermore, assume B does not admit any Kinvariant oneforms. If Einstein’s equations R_{μν} = Λg_{μν} hold, then the nearhorizon geometry possesses a G × U(1)^{m} × K isometry group, where G is either 0(2, 1) or the 2D Poincaré group. Furthermore, if Λ ≤ 0 and the nearhorizon geometry is nonstatic then the Poincaré group is excluded.
The assumptions in the above result reduce the Einstein equations for the nearhorizon geometry to ODEs, which can be solved in the same way as in Theorem 3.4. The special case K = SU(q), B = ℂℙ^{q−1} with m =1 and m = 2, gives a nearhorizon geometry of the type that occurs for a MyersPerry black hole with all the angular momenta of set equal in 2q + 2 dimensions, or all but one set equal in 2q + 3 dimensions, respectively.
The preceding results apply only to cohomogeneity1 nearhorizon geometries. As discussed above, this is too restrictive to capture the generic case for D > 5. The following result for highercohomogeneity nearhorizon geometries has been shown.
Theorem 3.6 ([166]). Consider a spacetime containing a degenerate horizon invariant under orthogonally transitive^{17} isometry group ℝ × U(1)^{N}, where 1 ≤ N ≤ D − 3, such that the surfaces orthogonal to the surfaces of transitivity are simply connected. Then the nearhorizon geometry has an isometry group G × U(1)^{N}, where G is either SO(2, 1) or the 2D Poincaré group. Furthermore, if the strong energy condition holds and the nearhorizon geometry is nonstatic, the Poincaré case is excluded.
4 Vacuum Solutions
4.1 Static: all dimensions
A complete classification is possible for Λ ≤ 0. Recall from Section 3.2.1, the staticity conditions for a nearhorizon geometry are dh = 0 and dF = hF.
Theorem 4.1 ([42]). The only vacuum static nearhorizon geometry for Λ ≤ 0 and compact H is given by h_{a} ≡ 0, F = Λ and R_{ab} = Λγ_{ab}. For D = 4 this result is also valid for Λ > 0.
4.2 Three dimensions
The classification of nearhorizon geometries in D = 3 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 H are onedimensional, so the horizon equations are automatically ODEs. Furthermore, there is no intrinsic geometry on H and so the only choice concerns its global topology, which must be either H ≅ S^{1} or H ≅ ℝ.
Theorem 4.2. Consider a nearhorizon geometry with compact cross section H ℝ S^{1}, which satisfies the vacuum Einstein equations including cosmological constant Λ. If Λ < 0 the nearhorizon geometry is given by the quotient of AdS_{3} in Eq. (73) . For Λ = 0 the only solution is the trivial flat geometry ℝ^{1,1} × S^{1}. There are no solutions for Λ > 0.
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 Λ = 0 and in [154] for Λ < 0.
Theorem 4.3 ([122, 163, 154]). Consider a spacetime containing a degenerate horizon, invariant under an ℝ × U(1) isometry, satisfying the vacuum Einstein equations including a cosmological constant. Any nonstatic nearhorizon geometry, with compact cross section, is given by the nearhorizon limit of the extremal Kerr or Kerr(A)dS black hole.
Now assume H is compact, so by axisymmetry one must have either S^{2} or T^{2}. Integrating Eq. (77) over H then shows that if Λ ≤ 0 then A_{0} < 0 and so the metric in square brackets is AdS_{2}. The horizon metric extends to a smooth metric on H ≅ S^{2} if and only if c_{1} = 0. It can then be checked the nearhorizon geometry is isometric to that of extremal Kerr for Λ = 0 or KerrAdS for Λ < 0 [154]. It is also easy to check that for Λ > 0 it corresponds to extremal KerrdS. In the nonstatic case, the horizon topology theorem excludes the possibility of H ≅ T^{2} for Λ ≥ 0. If Λ < 0 the nonstatic possibility with H ≅ T^{2} 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 nonaxisymmetric linearised solutions of Eq. (66) about the extremal Kerr nearhorizon geometry do not exist. This supports the conjecture that any smooth solution of Eq. (66) on H ≅ S^{2} 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 U(1)^{2}rotational symmetry.
We may define a homogeneous nearhorizon geometry to be one for which the Riemannian manifold (H, γ_{ab}) is a homogeneous space whose transitive isometry group K also leaves the rest of the nearhorizon data (f, h_{a}) invariant. Since any nearhorizon geometry (7) possesses the 2D symmetry generated by v → v + c and (v, r) → (λv, λ^{−1}r) where λ ≠ 0, it is clear that this definition guarantees the nearhorizon geometry itself is a homogeneous spacetime. Conversely, if the nearhorizon geometry is a homogeneous spacetime, then any cross section (H, γ_{ab}) must be a homogeneous space under a subgroup K of the spacetime isometry group, which commutes with the 2D symmetry in the (v,r) plane (since H is a constant (v, r) submanifold). It follows that (f, h_{a}) must also be invariant under the isometry K, showing that our original definition is indeed equivalent to the nearhorizon geometry being a homogeneous spacetime.
Homogeneous geometries can be straightforwardly classified without assuming compactness of H as follows.
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. Observe that for k → 0 one recovers the static nearhorizon geometries. For k ≠ 0 and Λ ≥ 0 we see that \(\hat \lambda > 0\) so that the 2D metric ĝ is a round S^{2} and the horizon metric is locally isometric to a homogeneously squashed S^{3}. Hence we have:
Corollary 4.1. Any vacuum, homogeneous, nonstatic nearhorizon geometry is locally isometric to the nearhorizon limit of the extremal MyersPerry black hole with SU(2) × U(1) rotational symmetry (i.e., equal angular momenta). For Λ > 0 one gets the same result with the MyersPerry black hole replaced by its generalisation with a cosmological constant [129].
For Λ < 0 we see that there are more possibilities depending on the sign of \({\hat \lambda}\). If \(\hat \lambda > 0\) we again have a horizon geometry locally isometric to a homogeneous S^{3}. If \(\hat \lambda > 0\), we can write ĝ = dx^{2} + dy^{2} and the U(1)connection \(\hat \omega = \sqrt {2\left\vert \Lambda \right.\left\vert {(x{\rm{d}}y  y{\rm{d}}x)} \right.}\) is nontrivial, so the cross sections H are the Nil group manifold with its standard homogeneous metric. For \(\hat \lambda > 0\), we can write \(\hat g = {\rm{d}}{x^2} + {\rm{d}}{y^2}\) and the connection \(\left\vert {\hat \lambda} \right.\left\vert {\hat \omega} \right. = \sqrt {{k^2} + 2\Lambda} \,{\rm{d}}x/y\), so the cross sections H are the SL(2, ℝ) group manifold with its standard homogeneous metric, unless k^{2} = −2Λ, which gives H = ℝ × ℍ^{2}. Hence we have:
Corollary 4.2. For Λ < 0 any vacuum, homogeneous, nonstatic nearhorizon geometry is locally isometric to either the nearhorizon limit of the extremal rotating black hole [129] with SU(2) × U(1) rotational symmetry, or a nearhorizon geometry with: (i) H = Nil and its standard homogenous metric, (ii) H = SL(2, ℝ) and its standard homogeneous metric or (iii) H = ℝ × ℍ^{2}.
This is analogous to a classification first obtained for supersymmetric nearhorizon geometries in gauged supergravity, see Proposition (5.3).
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 blackhole solutions have this symmetry. The following classification theorem has been derived:
 1.
H ≅ S^{1} × S^{2}: the 3parameter boosted extremal Kerr string.
 2.
H ≅ S^{3}: the 2parameter extremal MyersPerry black hole or the 3parameter ‘fast’ rotating extremal KK black hole [190].
 3.
H ≅ L(p, q): the Lens space quotients of the above H ≅ S^{3} solutions.

The nearhorizon geometry of the vacuum extremal black ring [187] is a 2parameter subfamily of case 1, corresponding to a Kerr string with vanishing tension [162].

The nearhorizon geometry of the ‘slowly’ rotating extremal KK black hole [190] is identical to that of the 2parameter extremal MyersPerry in case 2.

The H ≅ S^{3} cases can be written as a single 3parameter family of nearhorizon geometries [135].

The H ≅ T^{3} case has been ruled out [131].
4.5 Higher dimensions
For spacetime dimension D ≥ 6, so the horizon cross section dim H ≥ 4, the horizon equation (66) is far less constraining than in lower dimensions. Few general classification results are known, although several large families of vacuum nearhorizon geometries have been constructed.
4.5.1 Weyl solutions
The only known classification result for vacuum D > 5 nearhorizon geometries is for Λ = 0 solutions with U(1)^{D−3}rotational symmetry. These generalise the D = 4 axisymmetric solutions and D = 5 solutions with U(1)^{2}symmetry discussed in Sections 4.3 and 4.4 respectively. By performing a detailed study of the orbit spaces H/U(1)^{D−3} it has been shown that the only possible topologies for H are: S^{2} × T^{D−4}, S^{3} × T^{D−5}, L(p,q) × T^{D−5}, and T^{D−2} [139].
An explicit classification of the possible nearhorizon geometries (for the nontoroidal case) was derived in [135], see their Theorem 1. Using their theorem it is easy to show that the most general solution with H ≅ S^{2} × T^{D−4} is in fact isometric to the nearhorizon geometry of a boosted extremal Kerrmembrane (i.e., perform a general boost of Kerr×ℝ^{D−4} along the {t} × ℝ^{D−4} coordinates and then compactify ℝ^{D−4} → T^{D−4}). Nonstatic nearhorizon geometries with H ≅ T^{D−2} have been ruled out [131] (including a cosmological constant).
4.5.2 MyersPerry metrics
The MyersPerry (MP) blackhole solutions [183] generically have isometry groups ℝ × U(1)^{s} where \(s = \left\lfloor {{{D  1} \over 2}} \right\rfloor\). Observe that if D > 5 then s < D − 3 and hence these solutions fall outside the classification discussed in Section 4.5.1. They are parameterised by their mass parameter μ and angular momentum parameters a_{i} for i = 1, … s. The topology of the horizon cross section H ≅ S^{D−2}. A generalisation of these metrics with nonzero cosmological constant has been found [99]. We will focus on the Λ = 0 case, although analogous results hold for the Λ ≠ 0 solutions.
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 nearhorizon geometries have H ≅ S^{1} × S^{D−3} topology. Interestingly, for odd dimensions D the resulting geometry has \(\left\lfloor {{{D  1} \over 2}} \right\rfloor\) commuting rotational isometries. For this reason, it was conjectured that a special case of this is also the nearhorizon geometry of yettobefound asymptoticallyflat black rings (as is known to be the case in five dimensions) [79].
4.5.3 Exotic topology horizons
Despite the absence of explicit D > 5 blackhole solutions, a number of solutions to Eq. (66) are known. It is an open problem as to whether there are corresponding blackhole solutions to these nearhorizon geometries.
All the constructions given below employ the following data. Let K be a compact Fano KählerEinstein manifold^{19} of complex dimension q − 1 and a ∈ H^{2}(K, ℤ) is the indivisible class given by c_{1}(K) = Ia with I ∈ ℕ (the Fano index I and satisfies I ≤ q with equality iff K = ℂℙ^{q−1}). The KählerEinstein metric ḡ on K is normalised as Ric(ḡ) = 2qḡ and we denote its isometry group by G. The simplest example occurs for q = 2, in which case K = ℂℙ^{1} ≅ S^{2} with ḡ = ¼ (dθ^{2} + sin^{2} θdχ^{2}).
In even dimensions greater than four, an infinite class of nearhorizon geometries is revealed by the following result.
Proposition 4.1 ([156]). Let m ∈ ℤ and P_{m} be the principal S^{1}bundle over any Fano KählerEinstein manifold K, specified by the characteristic class ma. For each m > I there exists a 1parameter family of smooth solutions to Eq. (66) on the associated S^{2}bundles \(H \cong {P_m}{\times _{{S^1}}}{S^2}\).
For q > 2 the Fano base K is higher dimensional and there are more choices available. The topology of the total space is always a nontrivial 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 nearhorizon 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.
It is worth noting that the local form of the above class of nearhorizon metrics includes as a special case that of the extremal MP metrics H ≅ S^{2q} with equal angular momenta (for m = I). 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 Λ ≤ 0.
Similar constructions of increasing complexity can be made in odd dimensions, again revealing an infinite class of nearhorizon geometries.
Proposition 4.2. Let m ∈ ℤ and be the principal S^{1}bundle over K specified by the characteristic class ma. There exists a 1parameter family of Sasakian solutions to Eq. (66) on H ≅ P_{m}.
The above proposition can be generalised as follows.
Proposition 4.3 ([158]). Given any Fano KählerEinstein manifold K of complex dimension q − 1 and coprime p_{1},p_{2} ∈ ℕ satisfying 1 < Ip_{1}/p_{2} < 2, there exists a 1parameter family of solutions to Eq. (66) where H is a compact Sasakian (2q + 1)manifold.
These examples have U(1)^{2} × G symmetry, although possess only one independent angular momentum along the T^{2}fibres. These are deformations of the SasakiEinstein Y^{p,q} manifolds [94].
There also exist a more general class of nonSasakian horizons in odd dimensions.
Proposition 4.4 ([157]). Let Pm_{1},m_{2} be the principal T^{2}bundle over any Fano KählerEinstein manifold K, specified by the characteristic classes (m_{1}a,m_{2}a,) where m_{1},m_{2} ∈ ℤ. For a countably infinite set of nonzero integers (m_{1}, m_{2}, j, k), there exists a twoparameter family of smooth solutions to Eq. (66) on the associated Lens space bundles H ≅ Pm_{1},m_{2} × T^{2} L(j,k).
It is worth noting that the local form of this class of nearhorizon metrics includes as a special case that of the extremal MP metrics H ≅ S^{2q+1} 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].
5 Supersymmetric Solutions
By definition, a supersymmetric solution of a supergravity theory is a solution that also admits a Killing spinor, i.e., a spinor field ψ that satisfies D_{μ}ψ = 0, where D_{μ} is a spinorial covariant derivative that depends on the matter fields of the theory. Given such a Killing spinor ψ, the bilinear \({K^\mu} = \bar \psi {\Gamma ^\mu}\psi\) is a nonspacelike Killing field. By definition, a supersymmetric horizon is invariant under the Killing field K^{μ} and thus K^{μ} must be tangent to the horizon. Hence K^{μ} must be null on the horizon; in other words the horizon is a Killing horizon of K^{μ}. Furthermore, since K^{2} ≤ 0 both outside and inside the horizon, it follows that on the horizon dK^{2} = 0, i.e., it must be a degenerate horizon. Hence any supersymmetric horizon is necessarily a degenerate Killing horizon.
5.1 Four dimensions
The simplest supergravity theory in four dimensions admitting supersymmetric black holes is minimal \({\mathcal N} = 2\) supergravity, whose bosonic sector is simply standard EinsteinMaxwell theory. The general supersymmetric solution in this theory is given by the IsraelWilsonPerjés metrics. Using this fact, the following nearhorizon uniqueness theorem has been proved:
Theorem 5.1 ([41]). Any supersymmetric nearhorizon geometry in \({\mathcal N} = 2\) minimal supergravity is one of the maximally supersymmetric solutions ℝ^{1,1} × T^{2} or AdS_{2} × S^{2}.
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 \({\mathcal N} = 2\) gauged supergravity, whose bosonic sector is EinsteinMaxwell theory with a negative cosmological constant, an analogous classification of supersymmetric nearhorizon geometries has not been performed. Nevertheless, one may deduce the following result, from a classification of all nearhorizon geometries of this theory under the additional assumption of axisymmetry:
Proposition 5.1 ([154]). Any supersymmetric, axisymmetric, nearhorizon geometry in \({\mathcal N} = 2\) gauged supergravity, is given by the nearhorizon limit of the 1parameter family of supersymmetric KerrNewmanAdS_{4} black holes [150].
Note that the above nearhorizon geometry is nonstatic. 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 \({\mathcal N} = 1\) supergravity. For the general \({\mathcal N} = 1\) supergravity the following result, supporting this expectation, has been established.
Proposition 5.2 ([115]). A supersymmetric nearhorizon geometry of \({\mathcal N} = 1\) supergravity is either the trivial solution ℝ^{1,1} × T^{2}, or ℝ^{1,1} × S^{2} where S^{2} may be a nonround sphere.
An example of a supersymmetric nearhorizon geometry of the form ℝ^{1,1} × S^{2}, with a round S^{2}, was given in [176].
5.2 Five dimensions
The simplest D = 5 supergravity theory admitting supersymmetric black holes is \({\mathcal N} = 1\) minimal supergravity. The bosonic sector is EinsteinMaxwell theory with a ChernSimons term given by Eq. (112) with a specific coupling ξ = 1. Supersymmetric solutions to this theory were classified in [93]. This was used to obtain a complete classification of supersymmetric nearhorizon geometries in this theory.
Theorem 5.2 ([191]). Any supersymmetric nearhorizon geometry of minimal supergravity is locally isometric to one of the following maximally supersymmetric solutions: AdS_{3} × S^{2}, ℝ^{1,1} × T^{3}, or the nearhorizon geometry of the BreckenridgeMyersPeetVafa (BMPV) black hole (of which AdS_{2} × S^{3} is a special case).
Note that here supersymmetry implies homogeneity. The AdS_{3} × S^{2} nearhorizon geometry has cross sections H ≅ S^{1} × S^{2} and arises as the nearhorizon limit of supersymmetric black rings [64, 65] and supersymmetric black strings [93, 19]. Analogous results have been obtained in D = 5 minimal supergravity coupled to an arbitrary number of vector multiplets [111]. 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 EinsteinMaxwellChernSimons theory with a negative cosmological constant. The theory admits asymptotically AdS_{5} blackhole solutions that are relevant in the context of the AdS/CFT correspondence [120, 38]. The following partial results have been shown.
Proposition 5.3 ([120]). Consider a supersymmetric, homogeneous nearhorizon geometry of minimal gauged supergravity. Cross sections of the horizon must be one of the following: a homogeneously squashed S^{3}, Nil or SL(2, ℝ) manifold.
The nearhorizon geometry of the S^{3} case was used to construct the first example of an asymptotically AdS_{5} supersymmetric black hole [120]. Analogous results in gauged supergravity coupled to an arbitrary number of vector multiplets (this includes U(1)^{3} gauged supergravity) were obtained in [151]. Unlike the ungauged theory, homogeneity is not implied by supersymmetry, and indeed there are more general solutions.
Proposition 5.4 ([161]). The most general supersymmetric nearhorizon geometry in minimal gauged supergravity, admitting a U(1)^{2}rotational symmetry and a compact horizon section, is the nearhorizon limit of the topologicallyspherical supersymmetric black holes of [38].
The motivation for assuming this isometry group is that all known blackhole solutions in five dimensions possess this. Interestingly, this result implies the nonexistence of supersymmetric AdS5 black rings with ℝ × U(1)^{2} isometry.
In fact, recent results allow one to remove all assumptions and obtain a complete classification. Generic supersymmetric solutions of minimal gauged supergravity preserve ¼supersymmetry.
Proposition 5.5 ([121, 106]). Any ½supersymmetric nearhorizon geometry in minimal gauged supergravity must be invariant under a local U(1)^{2}rotational isometry.
Furthermore, the following has also been shown.
Proposition 5.6 ([105]). Any supersymmetric nearhorizon geometry in minimal gauged supergravity with a compact horizon section must preserve ½supersymmetry.
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 nearhorizon geometries in minimal gauged supergravity.
Theorem 5.3 ([161, 105, 106]). A supersymmetric nearhorizon geometry in minimal gauged supergravity, with a compact horizon section, must be locally isometric to the nearhorizon limit of the topologicallyspherical supersymmetric black holes [38], or the homogeneous nearhorizon geometries with the Nil or SL(2, ℝ) horizons.
This theorem establishes a striking corollary for the corresponding black hole classification theorem.
Corollary 5.1. Supersymmetric black rings in minimal gauged supergravity do not exist.
We emphasise that the absence of supersymmetric AdS_{5} black rings is rather suprising, given asymptoticallyflat counterparts are known to exist [64].
Parts of the above analysis have been generalised by U(1)^{n} gauged supergravity, although the results are slightly different.
Proposition 5.7 ([151]). Consider a supersymmetric nearhorizon geometry in U(1)^{n} minimal gauged supergravity with U(1)^{2}rotational symmetry and a compact horizon section. It must be either: (i) the nearhorizon limit of the topologically spherical black holes of [160]; or (ii) AdS_{3} × S^{2} with H ≅ S^{1} × S^{2} or (iii) AdS_{3} × T^{2} with H ≅ T^{3}. These latter two cases have constant scalars and only exist in certain regions of the scalar moduli space (not including the minimal theory).
Therefore, in this theory one cannot rule out the existence of supersymmetric AdS_{5} black rings (although as argued in [151] they would not be connected to the asymptoticallyflat black rings [66]). It would be interesting to complete the classification of nearhorizon geometries in this more general theory, along the lines of the minimal theory.
5.3 Six dimensions
The simplest supergravity in six dimensions is minimal supergravity. The bosonic field context of this theory is a metric and a 2form potential with selfdual field strength. The classification of supersymmetric solutions to this theory was given in [112]. This was used to work out a complete classification of supersymmetric nearhorizon geometries.
Theorem 5.4 ([112]). Any supersymmetric nearhorizon geometry of D = 6 minimal supergravity, with a compact horizon cross section, is either ℝ^{1,1} × T^{4}, ℝ^{1,1} × K_{3} or locally AdS_{3} × S^{3}.
The AdS_{3} × S^{3} solution has H ≅ S^{1} × S^{3} and arises as the nearhorizon limit of a supersymmetric rotating black string.
Analogous results have been obtained for minimal supergravity coupled to an arbitrary number of scalar and tensor multiplets [3].
5.4 Ten dimensions
Various results have been derived for heterotic supergravity and type IIB supergravity.
The bosonic field content of D = 10 heterotic supergravity consist of the metric, a 2form gauge potential and a scalar field (dilaton). The full theory is invariant under 16 supersymmetries. There are two classes of supersymmetric nearhorizon geometries [113]. One is the direct product ℝ^{1,1} × H, with vanishing flux and constant dilaton, where H is Spin(7) holonomy manifold, which generically preserves one supersymmetry (there are solutions in this class which preserve more supersymmetry provided H has certain special holonomy). In the second class, the nearhorizon geometry is a fibration of AdS_{3} over a base B_{7} (with a U(1)connection) with a G_{2} structure, which must preserve 2 supersymmetries. This class may preserve 4, 6, 8 supersymmetries if B_{7} is further restricted. In particular, an explicit classification for ½supersymmetric nearhorizon geometries is possible.
Proposition 5.8 ([113]). Any supersymmetric nearhorizon geometry of heterotic supergravity invariant under 8 supersymmetries, with a compact horizon cross section, must be locally isometric to one of AdS_{3} × S^{3} × T^{4}, AdS_{3} × S^{3} × K_{3} or ℝ^{1,1} × T^{4} × K_{3} (with constant dilaton). A large number of heterotic horizons preserving 4 supersymmetries have been constructed, including explicit examples where H is SU(3) and S^{3} × S^{3} × T^{2} [114].
The bosonic field content of type IIB supergravity consists of a metric, a complex scalar, a complex 2form potential and a selfdual 5form field strength. The theory is invariant under 32 supersymmetries. A variety of results concerning the classification of supersymmetric nearhorizon geometries in this theory have been derived [102, 101, 103]. Certain explicit classification results are known for nearhorizon geometries with just a 5form flux preserving more than two supersymmetries, albeit under certain restrictive assumptions [101]. More generally, the existence of one supersymmetry places rather weak geometric constraints on the horizon cross sections H: generically H may be any almost Hermitian spin_{c} manifold [103]. There are also special cases for which H has a Spin(7) structure and those for which it has an SU(4) structure (where the Killing spinor is pure). More recently, it has been shown that any supersymmetric nearhorizon geometry in type IIB supergravity must preserve an even number of supersymmetries, and furthermore, if a certain horizon Dirac operator has nontrivial kernel the bosonic symmetry group must contain SO(22, 1) [104].
5.5 Eleven dimensions
Various classification results have been derived for supersymmetric nearhorizon geometry solutions under the assumption that cross sections of the horizon are compact. Static supersymmetric nearhorizon geometries are warped products of either ℝ^{1,1} or AdS_{2} with M_{9}, where M_{9} admits a particular Gstructure [117]. We note that these warped product forms are guaranteed by the general analysis of static nearhorizon geometries in Section 3.2.1. Supersymmetric nearhorizon geometries have been studied more generally in [116]. Most interestingly, a nearhorizon (super)symmetry enhancement theorem has been established.
Theorem 5.5 ([118]). Any supersymmetric nearhorizon geometry solution to eleven dimensional supergravity, with compact horizon cross sections, must preserve an even number of supersymmetries. Furthermore, the bosonic symmetry group must contain SO(2, 1).
The proof of this follows by first establishing a Lichnerowicz type identity for certain horizon Dirac operators and then application of an index theorem. As far as bosonic symmetry is concerned, the above result is a direct analogue of the various nearhorizon symmetry theorems discussed in Section 3.2, which are instead established under various assumptions of rotational symmetry.
6 Solutions with Gauge Fields
In this section we will consider general nearhorizon geometries coupled to nontrivial gauge fields. We will mostly focus on theories that are in the bosonic sector of minimal supergravity theories (since these are the best understood cases). Extremal, nonsupersymmetric, nearhorizon geometries may be thought of as interpolating between vacuum and supersymmetric solutions. They consist of a much larger class of solutions, which, at least in higher dimensions, are much more difficult to classify. In particular, we consider D = 3, 4, 5 EinsteinMaxwell theory, possibly coupled to a ChernSimons term in odd dimensions, and D = 4 EinsteinYangMills theory.
6.1 Three dimensional EinsteinMaxwellChernSimons theory
The classification of nearhorizon geometries in D = 3 EinsteinMaxwell theory with a cosmological constant can be completely solved. To the best of our knowledge this has not been presented before, so for completeness we include it here. It should be noted though that partial results which capture the main result were previously shown in [175].
Theorem 6.1. Consider a nearhorizon geometry with a compact horizon cross section H ≅ S^{1} in EinsteinMaxwellΛ theory. If Λ < 0 the nearhorizon geometry must be either AdS_{2} × S^{1} with a constant AdS_{2} Maxwell field, or the quotient of AdS_{3} Eq. (73) with a vanishing Maxwell field. If Λ = 0 the only solution is the trivial flat nearhorizon geometry ℝ^{1,1} × S^{1}. If Λ > 0 there are no solutions.
This result implies that the nearhorizon limit of any charged rotating blackhole solution to 3D EinsteinMaxwellΛ theory either has vanishing charge or angular momentum. The AdS_{2} × S^{1} solution is the nearhorizon limit of the nonrotating extremal charged BTZ black hole, whereas the AdS_{3} solution is the nearhorizon limit of the vacuum rotating extremal BTZ [15]. Charged rotating black holes were first obtained within a wide class of stationary and axisymmetric solutions to EinsteinMaxwellΛ theory [45], and later by applying a solution generating technique to the charged nonrotating black hole [46, 174]. We have checked that in the extremal limit their nearhorizon geometry is the AdS_{2} × S^{1} solution, so the angular momentum is lost in the nearhorizon limit, in agreement with the above analysis. It would be interesting to investigate whether charged rotating black holes exist that instead possess a locally AdS_{3} nearhorizon geometry. In this case, charge would not be captured by the nearhorizon geometry, a phenomenon that is known to occur for fivedimensional supersymmetric black rings whose nearhorizon geometry is locally AdS_{3} × S^{2}.
Theorem 6.2. Consider a nearhorizon geometry with a cross section H ≅ S^{1} in EinsteinMaxwellΛ theory with a ChernSimons term and mass μ. If Λ < 0 the functions Δ and h are constant and the nearhorizon geometry is the homogeneous S^{1}bundle over AdS2 (106). If Λ = 0 the only solution is the trivial flat nearhorizon geometry ℝ^{1,1} × S^{1}. If Λ > 0 there are no solutions.
This implies that the nearhorizon geometry of any charged rotating blackhole solution to this theory is either the vacuum AdS_{3} solution, or the nontrivial solution (106), which is sometimes referred to as “warped AdS3”. Examples of charged rotating blackhole solutions in this theory have been found [2].
6.2 Four dimensional EinsteinMaxwell theory
Static nearhorizon geometries have been completely classified. For Λ = 0 this was first derived in [43], and generalised to Λ ≠ 0 in [154].
Theorem 6.3 ([43, 154]). Consider a static nearhorizon geometry in D = 4 EinsteinMaxwellΛ theory, with compact horizon cross section H. For Λ ≥ 0 it must be AdS_{2} × S^{2}. For Λ < 0 it must be AdS_{2} × H where H is one of the constant curvature surfaces S^{2}, T^{2}, Σ_{g}.
Nonstatic nearhorizon geometries are not fully classified, except under the additional assumption of axisymmetry.
Theorem 6.4 ([163, 154]). Any axisymmetric, nonstatic nearhorizon geometry in D = 4 EinsteinMaxwellΛ theory, with a compact horizon cross section, must be given by the nearhorizon geometry of an extremal KerrNewmanΛ black hole.
[163] solved the Λ = 0 in the context of isolated degenerate horizons, where the same equations on H arise. [154] solved the case with Λ ≠ 0. Note that the horizon topology theorem excludes the possibility of toroidal horizon cross sections for Λ ≥ 0. [164] also excluded the possibility of a toroidal horizon cross section if Λ < 0 under the assumptions of the above theorem.
It is worth noting that the results presented in this section, as well as the techniques used to establish them, are entirely analogous to the vacuum case presented in Section 4.3.
6.3 Five dimensional EinsteinMaxwellChernSimons theory
A number of new complications arise that render the classification problem more difficult, most obviously the lack of electromagnetic duality. Therefore, purely electric solutions, which correspond to Δ ≠ 0 and B ≡ 0, are qualitatively different to purely magnetic solutions, which correspond to Δ ≡ 0 and B ≠ 0.
6.3.1 Static
Perhaps somewhat surprisingly a complete classification of static nearhorizon geometries in this theory has not yet been achieved. Nevertheless, a number of results have been proved under various extra assumptions. All the results summarised in this section were proved in [153].
As in other five dimensional nearhorizon geometry classifications, the assumption of U(1)^{2} rotational symmetry proves to be useful. Static nearhorizon geometries in this class in general are either warped products of AdS_{2} and H, or AdS_{3} and a 2D manifold, see the Corollary 3.1. The AdS_{3} nearhorizon geometries are necessarily purely magnetic and can be classified for any ξ.
Proposition 6.1. Any static AdS_{3} nearhorizon geometry with a U(1)^{2}rotational symmetry, in EinsteinMaxwellChernSimons theory, with a compact horizon cross section, is the direct product of a quotient of AdS_{3} and a round S^{2}.
This classifies a subset of purely magnetic geometries. By combining the results of [153] together with Proposition 6.4 of [155], it can be deduced that for ξ = 1 there are no purely magnetic AdS_{2} geometries; therefore, with these symmetries, one has a complete classification of purely magnetic geometries.
Corollary 6.1. Any static, purely magnetic, nearhorizon geometry in minimal supergravity, possessing U(1)^{2}rotational symmetry and compact cross sections H, must be locally isometric to AdS_{3} × S^{2} with H = S^{1} × S^{2}.
We now turn to purely electric geometries.
Proposition 6.2. Consider a static, purely electric, nearhorizon geometry in EinsteinMaxwellChernSimons theory, with a U(1)^{2}rotational symmetry and compact cross section. It must be given by either AdS_{2} × S^{3}, or a warped product of AdS_{2} and an inhomogeneous S^{3}.
The latter nontrivial solution is in fact the nearhorizon limit of an extremal RN black hole immersed in a background electric field (this can be generated via a Harrison type transformation).
Finally, we turn to the case where the nearhorizon geometry possess both electric and magnetic fields. In fact one can prove a general result in this case, i.e., without the assumption of rotational symmetries.
Proposition 6.3. Any static nearhorizon geometry with compact cross sections H, in EinsteinMaxwellChernSimons theory with coupling ξ = 0, with nontrivial electric and magnetic fields, is a direct product of AdS_{2} × H, where the metric on H is not Einstein.
Explicit examples for 0 < ξ^{2} < 1/4 were also found, which all have H = S^{3} with U(1)^{2}rotational symmetry. However, we should emphasise that no examples are known for minimal supergravity (ξ = 1). Hence there is the possibility that in this case static nearhorizon geometries with nontrivial electric and magnetic fields do not exist, although this has not yet been shown. If this is the case, then the above results fully classify static nearhorizon geometries with U(1)^{2}rotational symmetry. For ξ = 0 the analysis of electromagnetic geometries is analogous to the purely magnetic case above; in fact there exists a dyonic AdS_{2} geometry that is a direct product AdS_{2} × S^{2} × S^{1} and it is conjectured there are no others.
6.3.2 Homogeneous
The classification of homogeneous nearhorizon geometries can be completely solved, even including a cosmological constant Λ. This does not appear to have been presented explicitly before, so for completeness we include it here with a brief derivation. We may define a homogeneous nearhorizon geometry as follows. The Riemannian manifold (H, γ_{ab}) is a homogeneous space, i.e., admits a transitive isometry group K, such that the rest of the nearhorizon data (f, h_{a}, Δ, B_{ab}) are invariant under K. As discussed in Section (4.4), this is equivalent to the nearhorizon geometry being a homogeneous spacetime with a Maxwell field invariant under its isometry group.
An immediate consequence of homogeneity is that an invariant function must be a constant and any invariant 1form must be a Killing field. Hence the 1forms h and j ≡ ★_{3}B are Killing and the functions F, Δ, h^{2}, j^{2} must be constants. Thus, the horizon Einstein equations (17), (18), (113), (114) and Maxwell equation (115) simplify.
Firstly, note that if h and j vanish identically then H is Einstein R_{ab} = ½ (Δ^{2} + 2Λ)γ_{ab}, so H is a constant curvature space S^{3}, ℝ^{3}, ℍ^{3} (the latter two can only occur if Λ < 0). This family includes the static nearhorizon geometry AdS_{2} × H.
The proof of this proceeds as in the vacuum case, by reducing the horizon equations to the 2D orbit space defined by the Killing field u. The above result contains a number of special cases of interest, which we now elaborate upon. Firstly, note that q = Δ = 0 reduces to the vacuum case, see Theorem 4.4.
Before discussing the general case consider k = 0, so the nearhorizon geometry is static, which connects to Section 6.3.1. The constraint on the parameters is \((1  4{\xi ^2}){\Delta ^2} = {3 \over 4}{q^2} + 2\Lambda\) and hence, if Λ ≤ 0 one must have 0 ≤ ξ^{2} < ¼. For ξ = 0 the connection is trivial and hence the nearhorizon geometry is locally isometric to the dyonic AdS_{2} × S^{1} × S^{2} solution. For 0 < ξ^{2} < ¼ we get examples of the geometries in Proposition (6.3), where H ≅ S^{3} with its standard homogeneous metric.
Now we consider Λ = 0 in generality so at least one of k, q, Δ is nonzero, in which case \(\hat \lambda = 0\) and so ĝ is the round S^{2}. The horizon is then either H ≅ S^{3} with its homogeneous metric or H ≅ S^{1} × S^{2}, depending on whether the connection ῶ is nontrivial or not, respectively. Notably we have:
Corollary 6.2. Any homogeneous nearhorizon geometry of minimal supergravity is locally isometric to AdS_{3} × S^{2}, or the nearhorizon limit of either (i) the BMPV black hole (including AdS_{2} × S^{3}), or (ii) an extremal nonsupersymmetric charged black hole with SU(2) × U(1) rotational symmetry.
The proof of this follows immediately from Theorem 6.5 with Λ = 0 and ξ =1. In this case the constraint on the parameters factorises to give two branches of possible solutions (a) \(k =  2q/\sqrt 3\) or (b) \({\Delta ^2}(k + 2\sqrt {3q}) = 4{q^2}(k  2q/\sqrt 3)/3\). Case (a) gives two solutions. If Δ ≠ 0 it must have H ≅ S^{3} and corresponds to the BMPV solution (i) (for q → 0 this reduces to AdS_{2} × S^{3}), whereas if Δ = 0 it is the AdS_{3} × S^{2} solution with H ≅ S^{1} × S^{2}. Case (b) also gives two solutions. If \(k = 2q/\sqrt 3\) then Δ = 0, which gives AdS_{3} × S^{2} with H ≅ S^{1} × S^{2}, otherwise we get solution (ii) with H ≅ S^{3}. Note that solution (ii) reduces to the vacuum case as Δ,q → 0.
The blackhole solution (ii) may be constructed as follows. A charged generalisation of the MP black hole can be generated in minimal supergravity [50]. Generically, the extremal limit depends on 3parameters with two independent angular momenta J_{1}, J_{2} and ℝ × U(1)^{2} symmetry. Setting J_{1} = J_{2} gives two distinct branches of 2parameter extremal blackhole solutions with enhanced SU(2) × U(1) rotational symmetry corresponding to the BMPV solution (i) (which reduces to the RN solution if J = 0) or solution (ii) (which reduces to the vacuum extremal MP black hole in the neutral limit).
It is interesting to note the analogous result for pure EinsteinMaxwell theory:
This also follows from application of Theorem 6.5 with Λ = 0 and ξ = 0. Solution (i) for q → 0 reduces to the vacuum solution of Theorem 4.4, whereas for k = 0, it gives the static dyonic AdS_{2} × S^{1} × S^{2} solution. Solution (ii) for Δ = 0 gives AdS_{3} × S^{2}, whereas for Δ ≠ 0 it gives a nearhorizon geometry with H ≅ S^{3}, which for q → 0 is AdS_{2} × S^{3}. A charged rotating blackhole solution to EinsteinMaxwell theory generalising MP is not known explicitly. Hence this corollary could be of use for constructing such an extremal charged rotating black hole with ℝ × SU(2) × U(1) symmetry.
For Λ = −4/ℓ^{2} < 0 there are even more possibilities, since \({\hat \lambda}\) may be positive, zero, or negative. One then gets nearhorizon geometries that generically have H ≅ S^{3}, Nil, SL(2, ℝ), respectively, equipped with their standard homogeneous metrics. Each of these may have a degenerate limit with H ≅ S^{1} × S^{2}, T^{3}, S^{1} × ℍ^{2}, as occurs in the vacuum and supersymmetric cases. The full space of solutions interpolates between the vacuum case given in Corollary 4.2, and the supersymmetric nearhorizon geometries of gauged supergravity of Proposition 5.3. For example, the supersymmetric horizons [120] correspond to \(k =  2\sqrt 3 q\) and k^{2} = 9/ℓ^{2} with \(\hat \lambda = {\Delta ^2}  3{\ell ^{ 2}}\). We will not investigate the full space of solutions in detail here.
6.3.3 U(1)^{2}rotational symmetry
The classification of nearhorizon geometries in D = 5 EinsteinMaxwellCS theory, under the assumption of U(1)^{2} symmetry, turns out to be significantly more complicated than the vacuum case. This is unsurprising; solutions may carry two independent angular momenta, electric charge and dipole/magnetic charge (depending on the spacetime asymptotics). As a result, there are several ways for a black hole to achieve extremality. Furthermore, such horizons may be deformed by background electric fields [153].
In the special case of minimal supergravity one can show:
Proposition 6.4 ([155]). Any nearhorizon geometry of minimal supergravity with U(1)^{2}rotational symmetry takes the form of Eqs. (58) and (62) , where the functional form of Γ(x), B_{ij}(x), b_{i}(x) can be fully determined in terms of rational functions of x. In particular, Γ(x) is a quadratic function.
The method of proof is discussed in Section 6.4. Although this solves the problem in principle, it turns out that in practice it is very complicated to disentangle the constraints on the constants that specify the solution. Hence an explicit classification of all possible solutions has not yet been obtained, although in principle it is contained in the above result.
We now summarise all known examples of five dimensional nonstatic nearhorizon geometries with nontrivial gauge fields, which arise as nearhorizon limits of known black hole or black string solutions. All these examples possess U(1)^{2} rotational symmetry and hence fall into the class of solutions covered by Theorem 3.5, so the nearhorizon metric and field strength \((g,{\mathcal F})\) take the form of Eqs. (58) and (62) respectively. We will divide them by horizon topology.
6.3.3.1 Spherical topology
Charged MyersPerry black holes: This asymptoticallyflat solution is known explicitly only for minimal supergravity ξ = 1 (in particular it is not known in pure EinsteinMaxwell ξ = 0), since it can be constructed by a solutiongenerating procedure starting with the vacuum MP solution. It depends on four parameters M, J_{1},J_{2}, and Q corresponding to the ADM mass, two independent angular momenta and an electric charge. The extremal limit generically depends on three parameters and gives a nearhorizon geometry with H ≅ S^{3}.
Charged KaluzaKlein black holes: The most general known solution to date was found in [204] (see references therein for a list of previously known solutions) and carries a mass, two independent angular momenta, a KK monopole charge, an electric charge and a ‘magnetic charge’^{20}. The extremal limit will generically depend on fiveparameters, however, as for the vacuum case the extremal locus must have more than one connected component. These solutions give a large family of nearhorizon geometries with H ≅ S^{3}.
6.3.3.2 S^{1} × S^{2} topology
Supersymmetric black rings and strings: The asymptoticallyflat supersymmetric black ring [64] and the supersymmetric TaubNUT black ring [65] both have a nearhorizon geometry that is locally AdS_{3} × S^{2}. There are also supersymmetric black string solutions with such nearhorizon geometries [93, 19].
Charged nonsupersymmetric black rings: The dipole ring solution admits a threeparameter charged generalisation with one independent angular momentum and electric and dipole charges [63] (the removal of DiracMisner string singularities imposes an additional constraint, so this solution has the same number of parameters as that of [69]). The charged black ring has a twoparameter extremal limit with a corresponding twoparameter nearhorizon geometry. As in the above case, at the level of nearhorizon geometries there is an additional independent parameter corresponding to the arbitrary size of the radius of the S^{1}.
Electromagnetic Kerr black strings: Black string solutions have been constructed carrying five independent charges: mass M, linear momentum P along the S^{1} of the string, angular momentum J along the internal S^{2}, as well as electric Q_{e} and magnetic charge Q_{m} [49]. These solutions admit a four parameter extremal limit, which in turn give rise to a fiveparameter family of nonstatic near horizon geometries (the additional parameter is the radius of the S^{1} at spatial infinity) [155].
Although the two solutions (124) and (125) share many features, it is important to emphasise that only the former is known to correspond to the nearhorizon geometry of an asymptoticallyflat black ring. It is conjectured that there exists a general blackring solution to minimal supergravity that carries mass, two angular momenta, electric and dipole charges all independently. Hence there should exist corresponding 4parameter families of extremal black rings. [155] discusses the possibility that the tensionless Kerrstring solution is the nearhorizon geometry of these yettobeconstructed black rings.
6.3.3.3 Theories with hidden symmetry
The most notable example which can be treated in the above formalism is vacuum Ddimensional gravity for which G/K = SL(D − 2, ℝ)/SO(D − 2). The classification problem for nearhorizon geometries has been completely solved using this approach [135], as discussed earlier in Section 4.5.1.
Fourdimensional EinsteinMaxwell also possesses such a structure where the coset is now G/K = SU(2, 1)/SU(2), although the nearhorizon classification discussed earlier in Section 6.2 does not exploit this fact.
It turns out that D = 5 minimal supergravity also has a nonlinear sigma model structure with G/K = G_{2,2}/SO(4) (G_{2,2} refers to the split real form of the exceptional Lie group G_{2}). The classification of nearhorizon geometries in this case was analysed in [155] using the hidden symmetry and some partial results were obtained, see Proposition 6.4.
It is clear that this method has wider applicability. It would be interesting to use it to classify nearhorizon geometries in other theories possessing such hidden symmetry.
6.4 NonAbelian gauge fields
Much less work has been done on classifying extremal black holes and their nearhorizon geometries coupled to nonAbelian gauge fields.
The simplest setup for this is fourdimensional EinsteinYangMills theory. As is well known, the standard four dimensional blackhole uniqueness theorems fail in this case (at least in the nonextremal case), for a review, see [205]. Nevertheless, nearhorizon geometry uniqueness theorems analogous to the EinsteinMaxwell case have recently been established for this theory.
Static nearhorizon geometries in this theory have been completely classified.
Theorem 6.6 ([164]). Consider D = 4 EinsteinYangMillsΛ theory with a compact semisimple gauge group G. Any static nearhorizon geometry with compact horizon cross section is given by: AdS_{2} × S^{2} if Λ > 0; AdS_{2} × H where H is one of S^{2}, T^{2}, Σg if Λ < 0.
The proof employs the same method as in EinsteinMaxwell theory. However, it should be noted that the YangMills field need not be that of the Abelian embedded solution. The horizon gauge field may be any YangMills connection on S^{2}, or on the higher genus surface as appropriate, with a gauge group G (if there is a nonzero electric field E the gauge group G is broken to the centraliser of E). The moduli space of such connections have been previously classified [14]. Hence, unless the gauge group is SU(2), one may have genuinely nonAbelian solutions. It should be noted that static nearhorizon geometries have been previously considered in EinsteinYangMillsHiggs under certain restrictive assumptions [25].
Nonstatic nearhorizon geometries have also been classified under the assumption of axisymmetry.
Theorem 6.7 ([164]). Any axisymmetric nonstatic nearhorizon geometry with compact horizon cross section, in D = 4 EinsteinYangMillsΛ with a compact semisimple gauge group, must be given by the nearhorizon geometry of an Abelian embedded extreme KerrNewmanΛ black hole.
The proof of this actually requires new ingredients as compared to the EinsteinMaxwell theory. The AdS_{2}symmetry enhancement theorems discussed in Section 3.2 do not apply in the presence of a nonAbelian gauge field. Nevertheless, assuming the horizon cross sections are of S^{2} topology allows one to use a global argument to show the symmetry enhancement phenomenon still occurs. This implies the solution is effectively Abelian and allows one to avoid finding the general solution to the ODEs that result from the reduction of the EinsteinYangMills equations. One can also rule out toroidal horizon cross sections, hence giving a complete classification of horizons with a U(1)symmetry.
Interestingly, EinsteinYangMills theory with a negative cosmological constant is a consistent truncation of the bosonic sector of D = 11 supergravity on a squashed S^{7} [188]. It would be interesting if nearhorizon classification results could be obtained in more general theories with nonAbelian YangMills fields, such as the full \({\mathcal N} = 8\), D = 4, SO(8)gauged supergravity that arises as a truncation of D = 11 supergravity on S^{7}.
7 Applications and Related Topics
7.1 Blackhole uniqueness theorems
One of the main motivations for classifying nearhorizon geometries is to prove uniqueness theorems for the corresponding extremal blackhole solutions. This turns out to be a much harder problem and has only been achieved when extra structure is present that constrains certain global aspects of the spacetime. The role of the nearhorizon geometry is to provide the correct boundary conditions near the horizon for the globalblackhole solution.
7.1.1 Supersymmetric black holes
Uniqueness theorems for supersymmetric black holes have been proved in the simplest four and fivedimensional supergravity theories, by employing the associated nearhorizon classifications described in Section 5.
In four dimensions, the simplest supergravity theory that admits supersymmetric black holes is \({\mathcal N} = 2\) minimal supergravity; its bosonic sector is simply EinsteinMaxwell theory.
Theorem 7.1 ([41]). Consider an asymptoticallyflat, supersymmetric blackhole solution to \({\mathcal N} = 2\), D = 4 minimal supergravity. Assume that the supersymmetric Killing vector field is timelike everywhere outside the horizon. Then it must belong to the MajumdarPapapetrou family of black holes. If the horizon is connected it must be the extremal RN black hole.
It would be interesting to remove the assumption on the supersymmetric Killing vector field to provide a complete classification of supersymmetric black holes in this case.
In five dimensions, the simplest supergravity theory that admits supersymmetric black holes is \({\mathcal N} = 2\) minimal supergravity. Using general properties of supersymmetric solutions in this theory, as well as the nearhorizon classification discussed in Section 5, the following uniqueness theorem has been obtained.
Theorem 7.2 ([191]). Consider an asymptoticallyflat, supersymmetric blackhole solution to \({\mathcal N} = 2\), D = 5 minimal supergravity, with horizon cross section H ≅ S^{3}. Assume that the supersymmetric Killing vector field is timelike everywhere outside the horizon. Then it must belong to the BMPV family of black holes [30].

The BMPV solution is a stationary, nonstatic, nonrotating black hole with angular momentum J and electric charge Q. For J = 0 it reduces to the extremal RN black hole.

It would be interesting to investigate the classification of supersymmetric black holes without the assumption on the supersymmetric Killing vector field.

An analogous uniqueness theorem for supersymmetric black rings [66], i.e., for H ≅ S^{1} × S^{2}, remains an open problem.

An analogous result has been obtained for minimal supergravity theory coupled to an arbitrary number of vector multiplets [111].
Analogous results for asymptotically AdS black holes in gauged supergravity have yettobeobtained and this remains a very interesting open problem. This is particularly significant due to the lack of blackhole uniqueness theorems even for pure gravity in AdS. However, it is worth mentioning that the analysis of [120] used the homogeneous nearhorizon geometry of Theorem 5.3 with H ≅ S^{3}, together with supersymmetry, to explicitly integrate for the full cohomogeneity1 AdS5 black hole solution. It would be interesting to prove a uniqueness theorem for supersymmetric AdS_{5} black holes assuming ℝ × U(1)^{2} symmetry.
7.1.2 Extremal vacuum black holes
The classic blackhole uniqueness theorem of general relativity roughly states that any stationary, asymptoticallyflat blackhole solution to the vacuum Einstein equations must be given by the Kerr solution. Traditionally this theorem assumed that the event horizon is nondegenerate, at a number of key steps. Most notably, the rigidity theorem, which states that a stationary rotating black hole must be axisymmetric, is proved by first showing that the event horizon is a Killing horizon. Although the original arguments [127] assumed nondegeneracy of the horizon, in four dimensions this assumption can be removed [179, 180, 134].
This allows one to reduce the problem to a boundary value problem on a twodimensional domain (the orbit space), just as in the case of a nondegenerate horizon. However, the boundary conditions near the boundary corresponding to the horizon depend on whether the surface gravity vanishes or not. Unsurprisingly, the boundary conditions near a degenerate horizon can be deduced from the nearhorizon geometry. Curiously, this has only been realised rather recently. This has allowed one to extend the uniqueness theorem for Kerr to the degenerate case.
Theorem 7.3 ([178, 5, 80, 40]). The only fourdimensional, asymptoticallyflat, stationary and axisymmetric, rotating, blackhole solution of the Einstein vacuum equations, with a connected degenerate horizon with nontoroidal horizon sections, is the extremal Kerr solution.
We note that [178] employs methods from integrability and the inverse scattering method. The remaining proofs employ the nearhorizon geometry classification theorem discussed in Section 4.3 together with the standard method used to prove uniqueness of nonextremal Kerr. The above uniqueness theorem has also been established for the extremal KerrNewman black hole in EinsteinMaxwell theory [5, 40, 177].
The assumption of a nontoroidal horizon is justified by the blackholehorizon topology theorems. Similarly, as discussed above, axisymmetry is justified by the rigidity theorem under the assumption of analyticity. Together with these results, the above theorem provides a complete classification of rotating vacuum black holes with a single degenerate horizon, under the stated assumptions. The proof that a nonrotating black hole must be static has only been established for nondegenerate horizons, so the classification of nonrotating degenerate black holes remains an open problem.
Of course, in higher dimensions, there is no such simple general uniqueness theorem. For spacetimes with ℝ × U(1)^{D−3} symmetry though, one has a mathematical structure analogous to D = 4 stationary and axisymmetric spacetimes. Namely, one can reduce the problem to an integrable boundaryvalue problem on a 2D orbit space. However in this case the boundary data is more complicated. It was first shown that nondegenerate blackhole solutions in this class are uniquely specified by certain topological data, which specifies the U(1)^{D−3}action on the manifold, referred to as interval data (i.e., rod structure) [138, 139]. The proof is entirely analogous to that for uniqueness of Kerr amongst stationary and axisymmetric black holes. This has been extended to cover the degenerate case, again by employing the nearhorizon geometry to determine the correct boundary conditions.
Proposition 7.1 ([80]). Consider a fivedimensional, asymptoticallyflat, stationary blackhole solution of the vacuum Einstein equations, with ℝ × U(1)^{2} isometry group and a connected degenerate horizon (with nontoroidal sections). There exists at most one such solution with given angular momenta and a given interval structure.
It is worth emphasising that although there is no nearhorizon uniqueness theorem in this case, see Section 4.4, this result actually only requires the general SO(2, 1) × U(1)^{2} form for the nearhorizon geometry and not its detailed functional form.
It seems likely that the results of this section could be extended to ℝ × U(1)^{D−3} invariant extremal black holes in EinsteinMaxwell type theories in higher dimensions. In D = 5 it is known that coupling a CS term in such a way to give the bosonic sector of minimal supergravity, implies such solutions are determined by a nonlinear sigma model analogous to the pure vacuum case. Hence it should be straightforward to generalise the vacuum uniqueness theorems to this theory.
7.2 Stability of nearhorizon geometries and extremal black holes
In fact, general arguments suggest that any nearhorizon geometry must be unstable when backreaction is taken into account and the nearby solution must be singular [18]. This follows from the fact that H is marginally trapped, so there exist perturbations that create a trapped surface and by the singularity theorems the resulting spacetime must be geodesically incomplete. If the perturbed solution is a black hole sitting inside the nearhorizon geometry, then this need not be an issue.^{21} For AdS_{2} × S^{2}, heuristic arguments also indicating its instability have been obtained by dimensional reduction to an AdS_{2} theory of gravity [170]. In particular, this suggests that the backreaction of matter in AdS_{2} × S^{2}, is not consistent with a falloff preserving both of the AdS_{2} boundaries.
So far we have been talking about the nonlinear stability of nearhorizon geometries. Of course, linearised perturbations in these backgrounds can be analysed in more detail. A massless scalar field in the nearhorizon geometry of an extremal Kerr black hole (NHEK) reduces to a massive charged scalar field in AdS_{2} with a homogeneous electric field [18]. This turns out to capture the main features of the Teukolsky equation for NHEK, which describes linearised gravitational perturbations of NHEK [4, 60]. In contrast to the above instability, these works revealed the stability of NHEK against linearised gravitational perturbations. In fact, one can prove a nonlinear uniqueness theorem in this context: any stationary and axisymmetric solution that is asymptotic to NHEK, possibly containing a smooth horizon, must in fact be NHEK [4].
So far we have discussed the stability of nearhorizon geometries as spacetimes in their own right. A natural question is what information about the stability of an extremal black hole can be deduced from the stability properties of its near horizon geometry. Clearly, stability of the nearhorizon geometry is insufficient to establish stability of the black hole, but it has been argued that certain instabilities of the nearhorizon geometry imply instability of the black hole [62]. For higher dimensional vacuum nearhorizon geometries, one can construct gaugeinvariant quantities (Weyl scalars), whose perturbation equations decouple, generalising the Teukolsky equation [62].^{22} One can then perform a KK reduction on H to find that linearised gravitational (and electromagnetic) perturbations reduce to an equation for a massive charged scalar field in AdS_{2} with a homogeneous electric field (as for the NHEK case above). The authors of [62] define the nearhorizon geometry to be unstable if the effective BreitenlohnerFreedman bound for this charged scalar field is violated. They propose the following conjecture: an instability of the nearhorizon geometry (in the above sense), implies an instability of the associated extreme black hole, provided the unstable mode satisfies a certain symmetry requirement. This conjecture is supported by the linear stability of NHEK and was verified for the known instabilities of odd dimensional cohomogeneity1 MP black holes [57]. It is also supported by the known stability results for the fivedimensional MP black hole [181] and the KerrAdS_{4} black hole [61]. This conjecture suggests that evendimensional nearextremal MP black holes, which are more difficult to analyse directly, are also unstable [203].
Recently, it has been shown that extremal black holes exhibit linearised instabilities at the horizon. This was first observed for a massless scalar field in an extremal RN and extremal Kerrblackhole background [8, 7, 6, 9]. The instability is somewhat subtle. While the scalar decays everywhere on and outside the horizon, the first transverse derivative of the scalar does not generically decay on the horizon, and furthermore the kthtransverse derivative blows up as v^{k−1} along the horizon. These results follow from the existence of a nonvanishing conserved quantity on the horizon linear in the scalar field. If the conserved quantity vanishes it has been shown that a similar, albeit milder, instability still occurs on the horizon [54, 26, 11, 167]. It should be noted that this instability is not in contradiction with the above linear stability of the nearhorizon geometry. From the point of view of the nearhorizon geometry, it is merely a coordinate artefact corresponding to the fact that the Poincaré horizon of AdS_{2} is not invariantly defined [167].
The horizon instability has been generalised to a massless scalar in an arbitrary extremal black hole in any dimension, provided the nearhorizon geometry satisfies a certain assumption [168]. This assumption in fact follows from the AdS_{2}symmetry theorems and hence is satisfied by all known extremal black holes. Furthermore, by considering the Teukolsky equation, it was shown that a similar horizon instability occurs for linearised gravitational perturbations of extremal Kerr [168]. This was generalised to certain higherdimensional vacuum extremal black holes [182]. Similarly, using Moncrief’s perturbation formalism for RN, it was shown that coupled gravitational and electromagnetic perturbations of extremal RN within EinsteinMaxwell theory also exhibit such a horizon instability [168]. An interesting open question is what is the fate of the nonlinear evolution of such horizon instabilities. To this end, an analogous instability has been established for certain nonlinear wave equations [10].
7.3 Geometric inequalities
Interestingly, nearhorizon geometries saturate certain geometric bounds relating the area and conserved angular momentum and charge of dynamical axisymmetric horizons, see [53] for a review.
In particular, for fourdimensional dynamical axiallysymmetric spacetimes, the area of blackhole horizons with a given angular momentum is minimised by the extremal Kerr black hole. The precise statement is:
Furthermore, it has been shown that if S is a section of an isolated horizon the above equality is saturated if and only if the surface gravity vanishes [142] (see also [172]). An analogous areaangular momentumcharge inequality has been derived in EinsteinMaxwell theory, which is saturated by the extreme KerrNewman black hole [87].
The above result can be generalised to higher dimensions, albeit under stronger symmetry assumptions.
Theorem 7.5 ([132]). Consider a Ddimensional spacetime satisfying the vacuum Einstein equations with nonnegative cosmological constant Λ that admits a U(1)^{D−3}rotational isometry. Then the area of any (stably outer) marginallyoutertrapped surface satisfies A ≥ 8πJ_{+}J_{−}^{1/2} where J± are distinguished components of the angular momenta associated to the rotational Killing fields, which have fixed points on the horizon. Further, if Λ = 0 then equality is achieved if the spacetime is a nearhorizon geometry and conversely, if the bound is saturated, the induced geometry on spatial cross sections of the horizon is that of a nearhorizon geometry.
In particular, for D = 4, the horizon is topologically S^{2} and J_{+} = J_{−} and one recovers (130).
Other generalisations of such inequalities have been obtained in D = 4, 5 EinsteinMaxwelldilaton theories [210, 211].
The proof of the above results involve demonstrating that axisymmetric nearhorizon geometries are global minimisers of a functional of the form (128) that is essentially the energy of a harmonic map, as discussed in Section 6.4.
7.4 Analytic continuation
In this section we discuss analytic continuation of nearhorizon geometries to obtain other Lorentzian or Riemannian metrics. As we will see, this uncovers a number of surprising connections between seemingly different spacetimes and geometries.
As discussed in Section 3.2, typically nearhorizon geometries have an SO(2, 1) isometry. Generically, the orbits of this isometry are threedimensional line or circle bundles over AdS_{2}. One can often analytically continue these geometries so AdS_{2} → S^{2} and SO(2, 1) → SU(2) (or SO(3)) with orbits that are circle bundles over S^{2}. It is most natural to work with the nearhorizon geometry written in global AdS_{2} coordinates (129). Such analytic continuations are then obtained by setting \(\rho \rightarrow i(\theta  {\pi \over 2})\) and k^{I} → ik^{I}.
First we discuss four dimensions. One can perform an analytic continuation of the nearhorizon geometry of extremal Kerr to obtain the zero mass Lorentzian TaubNUT solution [162]. More generally, there is an analytic continuation of the nearhorizon geometry of extremal KerrNewmanΛ to the zero mass Lorentzian RNTaubNUTΛ solution [154]. In fact, Page constructed a smooth compact Riemannian metric on the nontrivial S^{2}bundle over S^{2} with SU(2) × U(1) isometry, by taking a certain limit of the Euclidean KerrdS metric [185]. He showed that locally his metric is the Euclidean TaubNUTΛ with zero mass. Hence, it follows that there exists an analytic continuation of the nearhorizon geometry of extremal KerrΛ to Page’s Einstein manifold. (Also we deduce that Page’s limit is a Riemannian version of a combined extremal and nearhorizon limit).
Curiously, in five dimensions there exist analytic continuations of nearhorizon geometries to stationary blackhole solutions [162]. For example, one can perform an analytic continuation of the nearhorizon geometry of an extremal MP black hole with J_{1} ≠ J_{2} to obtain the full cohomogeneity1 MP black hole with J_{1} = J_{2} (which need not be extremal). In this case the S^{1} bundle over S^{2} that results after analytic continuation is the homogenous S^{3} of the resulting black hole. This generalises straightforwardly with the addition of a cosmological constant and/or charge. Interestingly, this also works with the nearhorizon geometries of extremal black rings. For example, there is an analytic continuation of the nearhorizon geometry of the extremal dipole black ring that gives a static charged squashed KK black hole with S^{3} horizon topology. In these fivedimensional cases, the isometry of the nearhorizon geometries is SO(2, 1) × U(1)^{2}, which has 4D orbits; hence one can arrange the new time coordinate to lie in these orbits in such a way it is not acted upon by the SU(2). This avoids NUT charge, which is inevitable in the fourdimensional case discussed above. As in the fourdimensional case, there are analytic continuations which result in Einstein metrics on compact manifolds. For example, there is an analytic continuation of the nearhorizon geometry of extremal MPΛ that gives an infinite class of Einstein metrics on S^{3}bundles over S^{2}, which were first found by taking a Page limit of the MPdS black hole [126].
In higher than five dimensions one can similarly perform analytic continuations of Einstein nearhorizon geometries to obtain examples of compact Einstein manifolds. The nearhorizon geometries of MPΛ give the Einstein manifolds that have been constructed by a Page limit of the MPdS metrics [99]. On the other hand, the new families of nearhorizon geometries [156, 157], discussed in Section 4.5.3, analytically continue to new examples of Einstein metrics on compact manifolds that have yet to be explored.
So far we have discussed analytic continuations in which the AdS_{2} is replaced by S^{2}. Another possibility is to replace the AdS_{2} with hyperbolic space ℍ^{2}. For simplicity let us focus on static nearhorizon geometries. Such an analytic continuation is then easily achieved by replacing the global AdS_{2} time with imaginary time, i.e., t → it in Eq. (129). In this case, instead of a horizon, one gets a new asymptotic region corresponding to ρ → ∞. General static Riemannian manifolds possessing an end that is asymptotically extremal in this sense were introduced in [81]. Essentially, they are defined as static manifolds possessing an end in which the metric can be written as an Euclideanised static spacetime containing a smooth degenerate horizon. It was shown that Ricci flow preserves this class of manifolds, and furthermore asymptoticallyextremal Ricci solitons must be Einstein spaces [81]. These results were used to numerically simulate Ricci flow to find a new Einstein metric that has an interesting interpretation in the AdS/CFT correspondence [81], which we briefly discuss in Section 7.5. It would be interesting to investigate nonstatic nearhorizon geometries in this context.
7.5 Extremal branes
Due to the applications to blackhole solutions, we have mostly focused on the nearhorizon geometries of degenerate horizons with cross sections H that are compact. However, as we discussed in Section 2, the concept of a nearhorizon geometry exists for any spacetime containing a degenerate horizon, independent of the topology of H. In particular, extremal black branes possess horizons with noncompact cross sections H. Hence the general techniques discussed in this review may be used to investigate the classification of the nearhorizon geometries of extremal black branes. In general, this is a more difficult problem, since as we have seen, compactness of H can often be used to avoid solving for the general local metric by employing global arguments. Since it is outside the scope of this review, we will not give a comprehensive overview of this topic; instead we shall select a few noteworthy examples.
The BPS extremal D3, M2, M5 black branes of 10,11dimensional supergravity are well known to have a nearhorizon geometry given by AdS_{5} × S^{5}, AdS_{4} × S^{7} and AdS_{7} × S^{4} respectively with their horizons corresponding to a Poincaré horizon in the AdS factor. However, as discussed above, the standard Poincaré coordinates are not valid on the horizon and hence unsuitable if one wants to extend the brane geometry onto and beyond the horizon. To this end, it is straightforward to construct Gaussian null coordinates adapted to the horizon of these black branes and check their nearhorizon limit is indeed given by Eq. (140) plus the appropriate sphere.^{23}
Of course, extremal branes occur in other contexts. For example, the RandallSundrum model posits that we live on a 3 + 1 dimensional brane in a 4 + 1dimensional bulk spacetime with a negative cosmological constant. A longstanding open problem has been to construct solutions to the fivedimensional Einstein equations with a black hole localised on such a brane, the braneworld black hole.^{24} In the fivedimensional spacetime, the horizons of such black holes extend out into the bulk. [147] constructed the nearhorizon geometry of an extremal charged black hole on a brane. This involved constructing (numerically) the most general five dimensional static nearhorizon geometry with SO(3) rotational symmetry, which turns out to be a 1parameter family generalisation of Eq. (140) (this is the 5D analogue of the 4D general static nearhorizon geometry with a noncompact horizon, see Section 4.1).
Notably, [83] numerically constructed the first example of a braneworld blackhole solution. This corresponds to a Schwarzschildlike black hole on a brane suspended above the Poincaré horizon in AdS_{5}. An important step towards this solution was the construction of a novel asymptotically AdS Einstein metric with a Schwarzschild conformal boundary metric and an extremal Poincaré horizon in the bulk (sometimes called a black droplet) [81]. This was found by numerically simulating Ricci flow on a suitable class of stationary and axisymmetric metrics. This solution is particularly interesting since by the AdS/CFT duality it is the gravity dual to a strongly coupled CFT in the Schwarzschild black hole, thus allowing one to investigate strongly coupled QFT in blackhole backgrounds. Recently, generalisations in which the boundary black hole is rotating have been constructed, in which case there is also the possibility of making the black hole on the boundary extremal [82, 84].
Footnotes
 1.
We work in geometrised units throughout.
 2.
Of course, a charged extremal black hole can always discharge in the presence of charged matter.
 3.
 4.
Although spacetimes correspond to Lorentzian metrics, one can often analytically continue these to complete Riemannian metrics. Indeed, the first example of an inhomogeneous Einstein metric on a compact manifold was found by Page, by taking a certain limit of the Kerrde Sitter metrics [185], giving a metric on \(\mathbb C{\mathbb P^2}\# {\overline {\mathbb C \mathbb P} ^2}\).
 5.
Albeit, under some technical assumptions such as analyticity of the metric.
 6.
Two oriented manifolds are said to be orientedcobordant if there exists some other oriented manifold whose boundary (with the induced orientation) is their disjoint union.
 7.
 8.
Indeed counterexamples are known in both senses.
 9.
In fact our constructions only assume the metric is C^{2} in a neighbourhood of the horizon. This encompasses certain examples of multiblackhole spacetimes with nonsmooth horizons [33].
 10.
To avoid proliferation of indices we will denote both coordinate and vielbein indices on H by lower case latin letters a, b, ….
 11.
A Kundt spacetime is one that admits a null geodesic vector field with vanishing expansion, shear and twist.
 12.
A Kundt spacetime is said to be degenerate if the Riemann tensor and all its covariant derivatives are type II with respect to the defining null vector field [184].
 13.
The remaining components of the Einstein equations for the full spacetime restricted to \({\mathcal N}\) give equations for extrinsic data (i.e., rderivatives of F, h_{a}, γ_{ab} that do not appear in the nearhorizon geometry). On the other hand, the rest of the Einstein equations for the nearhorizon geometry restricted to the horizon vanish.
 14.
The Komar integral associated with the null generator of the horizon ∂/∂v vanishes identically. In fact, one can show that for a general nonextremal Killing horizon, this integral is merely proportional to κ.
 15.
 16.
A borderline case also arises in this proof, corresponding to the induced metric on H being Ricci flat. This was in fact later excluded [88].
 17.
An isometry group whose surfaces of transitivity are p < D dimensional is said to be orthogonally transitive if there exists D − p dimensional surfaces orthogonal to the surfaces of transitivity at every point.
 18.
This can be obtained by analytically continuing Eq. (74) by ℓ → iℓ.
 19.
A complex manifold is Fano if its first Chern class is positive, i.e., c_{1}(K) > 0. It follows that any KählerEinstein metric on such a manifold must have positive Einstein constant.
 20.
This is a conserved charge for such asymptotically KK spacetimes.
 21.
Similarly, higherdimensional pureAdS spacetime is unstable to the formation of small black holes [27].
 22.
This stems from the fact that such spacetimes admit null geodesic congruences with vanishing expansion, rotation, and shear (i.e., they are Kundt spacetimes and hence algebraically special).
 23.
We would like to thank Carmen Li for verifying this.
 24.
There is a vast literature on this problem, which we will not attempt to review here.
Notes
Acknowledgements
HKK is supported by an NSERC Discovery Grant. JL is supported by an ESPRC Career Acceleration Fellowship. We would especially like to thank Harvey Reall for several fruitful collaborations on this topic and numerous discussions over the years. We would also like to thank Pau Figueras and Mukund Rangamani for a collaboration on this topic. JL would also like to thank Pau Figueras, Keiju Murata, Norihiro Tanahashi and Toby Wiseman for collaborations on related topics. —
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