7 Applications and Related Topics

7.1 Black-hole uniqueness theorems

One of the main motivations for classifying near-horizon geometries is to prove uniqueness theorems for the corresponding extremal black-hole 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 near-horizon geometry is to provide the correct boundary conditions near the horizon for the global–black-hole solution.

7.1.1 Supersymmetric black holes

Uniqueness theorems for supersymmetric black holes have been proved in the simplest four and five-dimensional supergravity theories, by employing the associated near-horizon classifications described in Section 5.

In four dimensions, the simplest supergravity theory that admits supersymmetric black holes is minimal supergravity; its bosonic sector is simply Einstein–Maxwell theory.

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 minimal supergravity. Using general properties of supersymmetric solutions in this theory, as well as the near-horizon classification discussed in Section 5, the following uniqueness theorem has been obtained.

Remarks:

• The BMPV solution is a stationary, non-static, non-rotating black hole with angular momentum and electric charge . For 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 , 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 yet-to-be-obtained and this remains a very interesting open problem. This is particularly significant due to the lack of black-hole uniqueness theorems even for pure gravity in AdS. However, it is worth mentioning that the analysis of [120] used the homogeneous near-horizon geometry of Theorem 5.3 with , together with supersymmetry, to explicitly integrate for the full cohomogeneity-1 AdS₅ black hole solution. It would be interesting to prove a uniqueness theorem for supersymmetric AdS₅ black holes assuming symmetry.

7.1.2 Extremal vacuum black holes

The classic black-hole uniqueness theorem of general relativity roughly states that any stationary, asymptotically-flat black-hole solution to the vacuum Einstein equations must be given by the Kerr solution. Traditionally this theorem assumed that the event horizon is non-degenerate, 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 non-degeneracy 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 two-dimensional domain (the orbit space), just as in the case of a non-degenerate 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 near-horizon geometry. Curiously, this has only been realised rather recently. This has allowed one to extend the uniqueness theorem for Kerr to the degenerate case.

We note that [178] employs methods from integrability and the inverse scattering method. The remaining proofs employ the near-horizon geometry classification theorem discussed in Section 4.3 together with the standard method used to prove uniqueness of non-extremal Kerr. The above uniqueness theorem has also been established for the extremal Kerr–Newman black hole in Einstein–Maxwell theory [5, 40, 177].

The assumption of a non-toroidal horizon is justified by the black-hole–horizon 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 non-rotating black hole must be static has only been established for non-degenerate horizons, so the classification of non-rotating degenerate black holes remains an open problem.

Of course, in higher dimensions, there is no such simple general uniqueness theorem. For spacetimes with symmetry though, one has a mathematical structure analogous to stationary and axisymmetric spacetimes. Namely, one can reduce the problem to an integrable boundary-value problem on a 2D orbit space. However in this case the boundary data is more complicated. It was first shown that non-degenerate black-hole solutions in this class are uniquely specified by certain topological data, which specifies the -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 near-horizon geometry to determine the correct boundary conditions.

It is worth emphasising that although there is no near-horizon uniqueness theorem in this case, see Section 4.4, this result actually only requires the general form for the near-horizon geometry and not its detailed functional form.

It seems likely that the results of this section could be extended to invariant extremal black holes in Einstein–Maxwell type theories in higher dimensions. In 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 non-linear 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 near-horizon geometries and extremal black holes

All known near-horizon geometries are fibrations of the horizon cross section over an AdS₂ base (see Section 3.2). By writing the AdS₂ in global coordinates one obtains examples of smooth complete spacetimes that solve the Einstein equations. Explicitly, by converting the near-horizon geometry (65) to AdS₂ global coordinates, such spacetimes take the general form

where we have written the constant in terms of the radius of AdS ₂. These spacetimes possess two timelike boundaries, and of course do not contain a horizon. It is of interest to consider the stability of such “global” near-horizon geometries, as spacetimes in their own right. It turns out that this problem is rather subtle.

In fact, general arguments suggest that any near-horizon geometry must be unstable when backreaction is taken into account and the nearby solution must be singular [18]. This follows from the fact that 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 near-horizon geometry, then this need not be an issue.²¹ For , heuristic arguments also indicating its instability have been obtained by dimensional reduction to an AdS₂ theory of gravity [170]. In particular, this suggests that the backreaction of matter in , is not consistent with a fall-off preserving both of the AdS ₂ boundaries.

So far we have been talking about the non-linear stability of near-horizon geometries. Of course, linearised perturbations in these backgrounds can be analysed in more detail. A massless scalar field in the near-horizon geometry of an extremal Kerr black hole (NHEK) reduces to a massive charged scalar field in AdS₂ 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 non-linear 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 near-horizon 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 near-horizon geometry is insufficient to establish stability of the black hole, but it has been argued that certain instabilities of the near-horizon geometry imply instability of the black hole [62]. For higher dimensional vacuum near-horizon geometries, one can construct gauge-invariant quantities (Weyl scalars), whose perturbation equations decouple, generalising the Teukolsky equation [62].²² One can then perform a KK reduction on to find that linearised gravitational (and electromagnetic) perturbations reduce to an equation for a massive charged scalar field in AdS₂ with a homogeneous electric field (as for the NHEK case above). The authors of [62] define the near-horizon geometry to be unstable if the effective Breitenlohner–Freedman bound for this charged scalar field is violated. They propose the following conjecture: an instability of the near-horizon 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 cohomogeneity-1 MP black holes [57]. It is also supported by the known stability results for the five-dimensional MP black hole [181] and the Kerr-AdS₄ black hole [61]. This conjecture suggests that even-dimensional near-extremal 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 Kerr–black-hole 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 th-transverse derivative blows up as along the horizon. These results follow from the existence of a non-vanishing 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 near-horizon geometry. From the point of view of the near-horizon geometry, it is merely a coordinate artefact corresponding to the fact that the Poincaré horizon of AdS₂ 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 near-horizon geometry satisfies a certain assumption [168]. This assumption in fact follows from the AdS₂-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 higher-dimensional 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 Einstein–Maxwell theory also exhibit such a horizon instability [168]. An interesting open question is what is the fate of the non-linear evolution of such horizon instabilities. To this end, an analogous instability has been established for certain non-linear wave equations [10].

7.3 Geometric inequalities

Interestingly, near-horizon 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 four-dimensional dynamical axially-symmetric spacetimes, the area of black-hole 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 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 area-angular momentum-charge inequality has been derived in Einstein–Maxwell theory, which is saturated by the extreme Kerr–Newman black hole [87].

The above result can be generalised to higher dimensions, albeit under stronger symmetry assumptions.

In particular, for , the horizon is topologically and and one recovers (130).

Other generalisations of such inequalities have been obtained in Einstein–Maxwell-dilaton theories [210, 211].

The proof of the above results involve demonstrating that axisymmetric near-horizon 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 near-horizon 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 near-horizon geometries have an isometry. Generically, the orbits of this isometry are three-dimensional line or circle bundles over AdS₂. One can often analytically continue these geometries so and (or ) with orbits that are circle bundles over . It is most natural to work with the near-horizon geometry written in global AdS ₂ coordinates (129). Such analytic continuations are then obtained by setting and .

First we discuss four dimensions. One can perform an analytic continuation of the near-horizon geometry of extremal Kerr to obtain the zero mass Lorentzian Taub–NUT solution [162]. More generally, there is an analytic continuation of the near-horizon geometry of extremal Kerr–Newman- to the zero mass Lorentzian RN–Taub–NUT- solution [154]. In fact, Page constructed a smooth compact Riemannian metric on the non-trivial -bundle over with isometry, by taking a certain limit of the Euclidean Kerr-dS metric [185]. He showed that locally his metric is the Euclidean Taub–NUT- with zero mass. Hence, it follows that there exists an analytic continuation of the near-horizon geometry of extremal Kerr- to Page’s Einstein manifold. (Also we deduce that Page’s limit is a Riemannian version of a combined extremal and near-horizon limit).

Explicitly, the analytic continuation of the near-horizon extremal Kerr- metric (83) to the Page metric, can be obtained as follows. First write the near-horizon geometry in global coordinates (129), then analytically continue and and redefine the coordinates appropriately, to find

with By an appropriate choice of parameters, this metric extends to a smooth global, inhomogeneous Einstein metric on the non-trivial bundle over , as follows. Compactness and positive definiteness requires one to take and where are two adjacent roots of such that . The part of the metric has conical singularities at , which are removed by imposing resulting in an fibre. This fibration is globally defined if , where , so combining these results implies As Page showed, the only solution is , which implies the -bundle is non-trivial. This manifold is diffeomorphic to the first del Pezzo surface .

Similarly, there is an analytic continuation of the near-horizon geometry of the extremal Kerr–Newman- that gives a family of smooth Riemannian metrics on -bundles over that satisfy the Riemannian Einstein–Maxwell equations (and hence have constant scalar curvature). Interestingly, this leads to an infinite class of metrics (i.e., there are an infinite number of possibilities for the integer ). The local solution can be derived directly by classifying solutions of the Riemannian Einstein–Maxwell equations with isometry group acting on three-dimensional orbits (see, e.g., [173]). One finds the geometry is given by (131) but with now given by

where is a constant related to the electric and magnetic charges of the analytically continued extremal Kerr–Newmann black hole. The regularity condition now becomes which for reduces to Eq. (134). One can check the 2nd term above is monotonically increasing in the range and unbounded as . If then there is a unique solution for every integer . For the only allowed solution is , and in fact there exist values of such that there are two solutions with . For even, the metric extends to a global metric on the trivial bundle over , whereas for odd, globally the space is . Hence one obtains a generalisation of the Page metric.

Curiously, in five dimensions there exist analytic continuations of near-horizon geometries to stationary black-hole solutions [162]. For example, one can perform an analytic continuation of the near-horizon geometry of an extremal MP black hole with to obtain the full cohomogeneity-1 MP black hole with (which need not be extremal). In this case the bundle over that results after analytic continuation is the homogenous of the resulting black hole. This generalises straightforwardly with the addition of a cosmological constant and/or charge. Interestingly, this also works with the near-horizon geometries of extremal black rings. For example, there is an analytic continuation of the near-horizon geometry of the extremal dipole black ring that gives a static charged squashed KK black hole with horizon topology. In these five-dimensional cases, the isometry of the near-horizon geometries is , 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 . This avoids NUT charge, which is inevitable in the four-dimensional case discussed above. As in the four-dimensional case, there are analytic continuations which result in Einstein metrics on compact manifolds. For example, there is an analytic continuation of the near-horizon geometry of extremal MP- that gives an infinite class of Einstein metrics on -bundles over , which were first found by taking a Page limit of the MP-dS black hole [126].

In higher than five dimensions one can similarly perform analytic continuations of Einstein near-horizon geometries to obtain examples of compact Einstein manifolds. The near-horizon geometries of MP- give the Einstein manifolds that have been constructed by a Page limit of the MP-dS metrics [99]. On the other hand, the new families of near-horizon 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 AdS₂ is replaced by . Another possibility is to replace the AdS₂ with hyperbolic space . For simplicity let us focus on static near-horizon geometries. Such an analytic continuation is then easily achieved by replacing the global AdS₂ time with imaginary time, i.e., in Eq. (129). In this case, instead of a horizon, one gets a new asymptotic region corresponding to . 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 asymptotically-extremal 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 non-static near-horizon geometries in this context.

7.5 Extremal branes

Due to the applications to black-hole solutions, we have mostly focused on the near-horizon geometries of degenerate horizons with cross sections that are compact. However, as we discussed in Section 2, the concept of a near-horizon geometry exists for any spacetime containing a degenerate horizon, independent of the topology of . In particular, extremal black branes possess horizons with non-compact cross sections . Hence the general techniques discussed in this review may be used to investigate the classification of the near-horizon geometries of extremal black branes. In general, this is a more difficult problem, since as we have seen, compactness of 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.

First consider AdS_D space written in Poincaré coordinates

where . The surface , often called the Poincaré horizon, is a degenerate Killing horizon of the Killing field . However, these coordinates are not valid at (the induced metric is singular) and hence are unsuitable for extracting the geometry of the Poincaré horizon. One may introduce coordinates adapted to the Poincaré horizons by constructing Gaussian null coordinates as described in Section 2. We need to find null geodesics that in particular satisfy . Explicitly, this condition is simply . Now, since are Killing fields, along any geodesic the quantities must be constant; this gives , where are constant along the geodesics. The null constraint now simplifies to and so . This latter equation is easily integrated to give and using the above we may now integrate for and . The result is where is an integration constant and we have set the other integration constants to zero to ensure the horizon is at . This gives a family of null geodesics parameterised by , which shoot out from every point on the horizon; hence we may take the as coordinates on the horizon. We can then change from Poincaré coordinates to the desired Gaussian null coordinates . Indeed, one can check that in the coordinates , the Killing field and the metric takes the form (6) (with ). In fact, as is often the case, it is convenient to use a different affine parameter . Also, since we may write , where parameterise a unit -sphere. Now the coordinate transformation becomes and the AdS_D metric in these coordinates is It is now manifest that the surface is a smooth degenerate Killing horizon of , corresponding to the Poincaré horizon, which we may now extend onto and through by taking . Cross sections of the Poincaré horizon are non-compact and of topology with a (non-flat) induced metric given by the standard Einstein metric on hyperbolic space . Observe that the above expresses AdS_D as a warped product of AdS₂ and hyperbolic space , i.e., as a static near-horizon geometry (with no need to take a near-horizon limit!), as observed in [81].

The BPS extremal black branes of 10,11-dimensional supergravity are well known to have a near-horizon geometry given by , and 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 near-horizon limit is indeed given by Eq. (140) plus the appropriate sphere.²³

Of course, extremal branes occur in other contexts. For example, the Randall–Sundrum model posits that we live on a 3 + 1 dimensional brane in a 4 + 1-dimensional bulk spacetime with a negative cosmological constant. A longstanding open problem has been to construct solutions to the five-dimensional Einstein equations with a black hole localised on such a brane, the brane-world black hole.²⁴ In the five-dimensional spacetime, the horizons of such black holes extend out into the bulk. [147] constructed the near-horizon geometry of an extremal charged black hole on a brane. This involved constructing (numerically) the most general five dimensional static near-horizon geometry with rotational symmetry, which turns out to be a 1-parameter family generalisation of Eq. (140) (this is the 5D analogue of the 4D general static near-horizon geometry with a non-compact horizon, see Section 4.1).

Notably, [83] numerically constructed the first example of a brane-world black-hole solution. This corresponds to a Schwarzschild-like black hole on a brane suspended above the Poincaré horizon in AdS₅. 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 black-hole 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].