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1 Introduction

Classical general relativity in more than four spacetime dimensions has been the subject of increasing attention in recent years. Among the reasons it should be interesting to study this extension of Einstein’s theory, and in particular its black-hole solutions, we may mention that

These, however, refer to applications of the subject – important though they are – but we believe that higher-dimensional gravity is also of intrinsic interest. Just as the study of quantum field theories, with a field content very different than any conceivable extension of the Standard Model, has been a very useful endeavor, throwing light on general features of quantum fields, we believe that endowing general relativity with a tunable parameter – namely the spacetime dimensionality d – should also lead to valuable insights into the nature of the theory, in particular into its most basic objects: black holes. For instance, four-dimensional black holes are known to have a number of remarkable features, such as uniqueness, spherical topology, dynamical stability, and to satisfy a set of simple laws — the laws of black hole mechanics. One would like to know which of these are peculiar to four-dimensions, and which hold more generally. At the very least, this study will lead to a deeper understanding of classical black holes and of what spacetime can do at its most extreme.

There is a growing awareness that the physics of higher-dimensional black holes can be markedly different, and much richer, than in four dimensions. Arguably, two advances are largely responsible for this perception: the discovery of dynamical instabilities in extended black-hole horizons [118Jump To The Next Citation Point] and the discovery of black-hole solutions with horizons of nonspherical topology that are not fully characterized by their conserved charges [83Jump To The Next Citation Point].

At the risk of anticipating results and concepts that will be developed only later in this review, in the following we try to give simple answers to two frequently asked questions: 1) why should one expect any interesting new dynamics in higher-dimensional general relativity, and 2) what are the main obstacles to a direct generalization of the four-dimensional techniques and results. A straightforward answer to both questions is to simply say that as the number of dimensions grows, the number of degrees of freedom of the gravitational field also increases, but more specific, yet intuitive, answers are possible.

1.1 Why gravity is richer in d > 4

The novel features of higher-dimensional black holes that have been identified so far can be understood in physical terms as due to the combination of two main ingredients: different rotation dynamics and the appearance of extended black objects.

There are two aspects of rotation that change significantly when spacetime has more than four dimensions. First, there is the possibility of rotation in several independent rotation planes [200Jump To The Next Citation Point]. The rotation group SO (d − 1) has Cartan subgroup U(1)N, with

⌊ ⌋ d-−-1- N ≡ 2 ; (1 )
hence, there is the possibility of N independent angular momenta. In simpler and more explicit terms, one can group the d − 1 spatial dimensions (say, at asymptotically-flat infinity) into pairs (x1,x2), (x3,x4),…, each pair defining a plane, and choose polar coordinates in each plane, (r1,φ1), (r2,φ2 ),…. Here we see the possibility of having N independent (commuting) rotations associated to the vectors ∂φ1, ∂φ2 …. To each of these rotations we associate an angular momentum component Ji.

The other aspect of rotation that changes qualitatively as the number of dimensions increases is the relative competition between the gravitational and centrifugal potentials. The radial falloff of the Newtonian potential

− GM--- (2 ) rd−3
depends on the number of dimensions, whereas the centrifugal barrier
2 --J--- (3 ) M 2r2
does not, since rotation is confined to a plane. We see that the competition between (2View Equation) and (3View Equation) is different in d = 4, d = 5, and d ≥ 6. In Newtonian physics this is well known to result in a different stability of Keplerian orbits, but this precise effect is not directly relevant to the black-hole dynamics we are interested in. Still, the same kind of dimension dependence will have rather dramatic consequences for the behavior of black holes.

The other novel ingredient that appears in d > 4 but is absent in lower dimensions (at least in vacuum gravity) is the presence of black objects with extended horizons, i.e., black strings and, in general, black p-branes. Although these are not asymptotically-flat solutions, they provide the basic intuition for understanding novel kinds of asymptotically-flat black holes.

Let us begin with the simple observation that, given a black-hole solution of the vacuum Einstein equations in d dimensions with horizon geometry Σ H, we can immediately construct a vacuum solution in d + 1 dimensions by simply adding a flat spatial direction1. The new horizon geometry is then a black string with horizon ΣH × ℝ. Since the Schwarzschild solution is easily generalized to any d ≥ 4, it follows that black strings exist in any d ≥ 5. In general, adding p flat directions we find that black p-branes with horizon Sq × ℝp (with q ≥ 2) exist in any d ≥ 6 + p − q.

How are these related to new kinds of asymptotically-flat black holes? Heuristically, take a piece of black string with Sq × ℝ horizon, and curve it to form a black ring with horizon topology Sq × S1. Since the black string has a tension, the S1, being contractible, will tend to collapse. But we may try to set the ring into rotation and in this way provide a centrifugal repulsion that balances the tension. This turns out to be possible in any d ≥ 5, so we expect that nonspherical horizon topologies are a generic feature of higher-dimensional general relativity.

It is also natural to try to apply this heuristic construction to black p-branes with p > 1, namely, to bend the worldvolume spatial directions into a compact manifold and balance the tension by introducing suitable rotations. The possibilities are still under investigation, but it is clear that an increasing variety of black holes should be expected as d grows. Observe again that the underlying reason is a combination of extended horizons with rotation.

Horizon topologies other than spherical are forbidden in d = 4 by well-known theorems [132Jump To The Next Citation Point]. These are rigorous, but also rather technical and formal results. Can we find a simple, intuitive explanation for the absence of vacuum black rings in d = 4? The previous argument would trace this fact back to the absence of asymptotically-flat vacuum black holes in d = 3. This is often attributed to the absence of propagating degrees of freedom for the three-dimensional graviton (or one of its paraphrases: 2 + 1-gravity is topological, the Weyl tensor vanishes identically, etc), but here we shall use the simple observation that the quantity GM is dimensionless in d = 3. Hence, given any amount of mass, there is no length scale to tell us where the black-hole horizon should be2. So we attribute the absence of black strings in d = 4 to the lack of such a scale. This observation goes some way towards understanding the absence of vacuum black rings with horizon topology S1 × S1 in four dimensions; it implies that there cannot exist black-ring solutions with different scales for each of the two circles, and, in particular one can not make one radius arbitrarily larger than the other. This argument, though, could still allow for black rings, where the radii of the two S1 are set by the same scale, i.e., the black rings should be plump. The horizon-topology theorems then tell us that plump black rings do not exist; they would actually be within a spherical horizon.

Extended horizons also introduce a feature absent in d = 4: dynamical horizon instabilities [118Jump To The Next Citation Point]. Again, this is to some extent an issue of scales. Black brane horizons can be much larger in some of their directions than in others, and so perturbations with wavelengths on the order of the ‘short’ horizon length can fit several times along the ‘long’ extended directions. Since the horizon area tends to increase by dividing up the extended horizon into black holes of roughly the same size in all its dimensions, this provides grounds to expect an instability of the extended horizon (however, when other scales are present, as in charged solutions, the situation can become quite a bit more complicated). It turns out that higher-dimensional rotation can extend the horizon much more in some directions than in others, which is expected to trigger this kind of instability [81Jump To The Next Citation Point]. At the threshold of the instability, a zero-mode deformation of the horizon has been conjectured to lead to new ‘pinched’ black holes that do not have four-dimensional counterparts.

Finally, an important question raised in higher dimensions refers to the rigidity of the horizon. In four dimensions, stationarity implies the existence of a U(1) rotational isometry [132Jump To The Next Citation Point]. In higher dimensions stationarity has been proven to imply one rigid rotation symmetry too [138Jump To The Next Citation Point], but not (yet?) more than one. However, all known higher-dimensional black holes have multiple rotational symmetries. Are there stationary black holes with less symmetry, for example just the single U(1) isometry guaranteed in general? Or are black holes always as rigid as can be? This is, in our opinion, the main unsolved problem on the way to a complete classification of five-dimensional black holes and an important issue in understanding the possibilities for black holes in higher dimensions.

1.2 Why gravity is more difficult in d > 4

Again, the simple answer to this question is the larger number of degrees of freedom. However, this cannot be an entirely satisfactory reply, since one often restricts oneself to solutions with a large degree of symmetry for which the number of actual degrees of freedom may not depend on the dimensionality of spacetime. A more satisfying answer should explain why the methods that are so successful in d = 4 become harder, less useful, or even inapplicable, in higher dimensions.

Still, the larger number of metric components, and of equations determining them, is the main reason for the failure so far to find a useful extension of the Newman–Penrose (NP) formalism to d > 4. This formalism, in which all the Einstein equations and Bianchi identities are written out explicitly, was instrumental in deriving the Kerr solution and analyzing its perturbations. The formalism is tailored to deal with algebraically-special solutions, but even if algebraic classifications have been developed for higher dimensions [51Jump To The Next Citation Point] and applied to known black-hole solutions, no practical extension of the NP formalism has appeared yet that can be used to derive the solutions, nor to study their perturbations.

Then, it seems natural to restrict oneself to solutions with a high degree of symmetry. Spherical symmetry yields easily by force of Birkhoff’s theorem. The next simplest possibility is to impose stationarity and axial symmetry. In four dimensions this implies the existence of two commuting Abelian isometries, time translation and axial rotation, which are extremely powerful; by integrating out the two isometries from the theory, we obtain an integrable two-dimensional GL (2,ℝ ) sigma-model. The literature on these theories is enormous and many solution-generating techniques are available, which provide a variety of derivations of the Kerr solution.

There are two natural ways of extending axial symmetry to higher dimensions. We may look for solutions invariant under the group O (d − 2) of spatial rotations around a given line axis, where the orbits of O (d − 2) are (d − 3)-spheres. However, in more than four dimensions these orbits have nonzero curvature. As a consequence, after dimensional reduction of these orbits, the sigma model acquires terms (of exponential type) that prevent a straightforward integration of the equations (see [34Jump To The Next Citation Point35Jump To The Next Citation Point] for an investigation of these equations).

This suggests that one should look for a different higher-dimensional extension of the four-dimensional axial symmetry. Instead of rotations around a line, consider rotations around (spatial) codimension-2 hypersurfaces. These are U (1) symmetries. If we assume d − 3 commuting U(1) symmetries, so that we have a spatial d−3 U(1) symmetry in addition to the timelike symmetry ℝ, then the vacuum Einstein equations again reduce to an integrable two-dimensional GL (d − 2,ℝ ) sigma model with powerful solution-generating techniques.

However, there is an important limitation: only in d = 4,5 can these geometries be globally asymptotically flat. Global asymptotic flatness implies an asymptotic factor d− 2 S in the spatial geometry, whose isometry group O (d − 1) has a Cartan subgroup N U (1 ). If, as above, we demand d − 3 axial isometries, then, asymptotically, these symmetries must approach elements of O (d − 1 ), so we need U (1)d−3 ⊂ U (1)N, i.e.,

⌊ ⌋ d-−-1- d − 3 ≤ N = 2 , (4 )
which is only possible in d = 4,5. This is the main reason for the recent great progress in the construction of exact five-dimensional black holes, and the failure to extend it to d > 5.

Finally, the classification of possible horizon topologies becomes increasingly complicated in higher dimensions [98Jump To The Next Citation Point]. In four spacetime dimensions the (spatial section of the) horizon is a two-dimensional surface, so the possible topologies can be easily characterized and restricted. Much less restriction is possible as d is increased.

All these aspects will be discussed in more detail below.

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