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8 General Results and Open Problems

8.1 Introduction

In four dimensions, the black-hole uniqueness theorem states that there is at most one stationary, asymptotically-flat, vacuum black-hole solution with given mass and angular momentum: the Kerr black hole. The coexistence of Myers–Perry black holes and black rings shows explicitly that black-hole uniqueness is violated in five dimensions. By now there is strong evidence that this is even more dramatically true in more than five dimensions. However, even if higher-dimensional black holes are not uniquely characterized by their conserved charges, we can still hope to classify them. A major goal of research in higher-dimensional general relativity is to solve the classification problem: determine all stationary, asymptotically-flat black-hole solutions of the higher-dimensional vacuum Einstein equation (or Einstein equation coupled to appropriate matter). We are still a long way from this goal, but partial progress has been made, as we shall review below.

8.2 Black-hole topology

Logically, the first step in the proof of the d = 4 black-hole uniqueness theorem is Hawking’s black-hole topology theorem [132Jump To The Next Citation Point], which states that (a spatial cross-section of) the event horizon must be topologically S2. The strongest version of this theorem makes use of topological censorship. Loosely, this is the statement that, in an asymptotically-flat and globally-hyperbolic spacetime obeying the null-energy condition, every causal curve beginning and ending at infinity can be homotopically deformed to infinity. This can be used to prove the topology theorem for stationary [48], and even nonstationary [15395], black holes.

The existence of black rings demonstrates that topologically nonspherical horizons are possible for d > 4. But there are still restrictions on the topology of the event horizon. It has been shown [9896] that a stationary, asymptotically-flat, black-hole spacetime, obeying the dominant energy condition, must have a horizon that is “positive Yamabe”, i.e., it must admit a metric of positive Ricci scalar. This restricts the allowed topologies. After d = 4, the strongest restriction is for d = 5, in which case the topology must be a connected sum of spherical spaces (3-spheres with identifications) and 1 2 S × S’s.

Another topological restriction arises from cobordism theory. Consider surrounding the black hole with a large sphere. Let Σ denote a spacelike hypersurface that runs from this sphere down to the event horizon, which it intersects in some compact surface H. Then Σ is a cobordism between H and a sphere. The existence of such a cobordism imposes topological restrictions on H. These have been discussed in [136]. The results are much less restrictive than those just mentioned.

It may be that horizon topology is not very useful for classifying black holes in d ≥ 6. As we discussed in the introduction 1, it is the combination of extended horizons with the novel possibilities for rotation that give higher-dimensional black holes much of their richness. For a black ring with horizon topology S1 × S2, the two factors differ in that the S1, being contractible, needs rotation to be stabilized, whereas the S2 is already a minimal surface. In d ≥ 6 we can envisage even more complicated situations arising from the bending into different surfaces of the worldvolume directions of a variety of black p-branes. Not only horizon topology, but also, and more importantly, extrinsic geometry and dynamical considerations seem to be relevant to the existence of these black holes.

8.3 Uniqueness of static black holes

In d = 4 dimensions, the Schwarzschild solution is the unique, static, asymptotically-flat, vacuum black-hole solution. The strongest version of this theorem allows for a possibly-disconnected event horizon, and the proof uses the positive energy theorem [19]. This proof can be extended to d > 4-dimensions to establish uniqueness of the d > 4-dimensional Schwarzschild solution amongst static vacuum solutions [110]. The method can also be generalized to prove a uniqueness theorem for static, asymptotically-flat, black-hole solutions of d > 4-dimensional Einstein–Maxwell-dilaton theory; such black holes are uniquely characterized by their mass and charge and are described by generalized Reissner–Nordström solutions [109].

These theorems assume that there are no degenerate components of the horizon. This assumption can be eliminated for d = 4 vacuum gravity [4446]. In Einstein–Maxwell theory, one can show that the only solutions with degenerate horizons are the Majumdar–Papapetrou multi-Reissner–Nordström solutions [47]. These results have been generalized to d > 4 Einstein–Maxwell theory [218220].

In conclusion, the classification problem for static black holes has been solved, at least for the class of theories mentioned.

It must be noted, though, that the assumption of staticity is stronger than requiring vanishing total angular momentum. The existence of black Saturns (Section 5.3) shows that there exists an infinite number of solutions (with disconnected event horizons) characterized by a given mass and vanishing angular momentum.

8.4 Stationary black holes

In d = 4, the uniqueness theorem for stationary black holes relies on Hawking’s result that a stationary black hole must be axisymmetric [132]. This result has been generalized to higher dimensions [138Jump To The Next Citation Point]. More precisely, it can be shown that a stationary, nonextremal, asymptotically-flat, rotating, black-hole solution of d > 4 dimensional Einstein–Maxwell theory must admit a spacelike Killing vector field that generates rotations. Here “rotating” means that the Killing field that generates time translations is not null on the event horizon, i.e., the angular velocity is nonvanishing. However, it can be shown that a nonrotating black hole must be static for Einstein–Maxwell theory in d = 4 [230] and d > 4 [219], so this assumption can be eliminated. The result of [138] also applies in the presence of a cosmological constant (e.g., asymptotically anti-de Sitter black holes).

This theorem guarantees the existence of a single rotational symmetry. However, the known higher-dimensional black-hole solutions (i.e., Myers–Perry and black rings) admit multiple rotational symmetries; in d dimensions there are ⌊(d − 1)∕2⌋ rotational Killing fields. Is there some underlying reason that this must be true, or is this simply a reflection of the fact that we are only able to find solutions with a lot of symmetry?

If there do exist solutions with less symmetry then they must be nonstatic (because of the uniqueness theorem for d > 4 static black holes). One could look for evidence that such solutions exist by considering perturbations of known solutions [216Jump To The Next Citation Point]. For example, the existence of nonuniform black strings was first conjectured on the evidence that uniform strings exhibit a static zeromode that breaks translational symmetry. If a Myers–Perry black hole had a stationary zero-mode that breaks some of its rotational symmetry then that would be evidence in favor of the existence of a new branch of solutions with less symmetry [216Jump To The Next Citation Point]. Alternatively, bifurcations could occur without breaking any rotational symmetry. As we have discussed, [80Jump To The Next Citation Point81Jump To The Next Citation Point] have conjectured that such bifurcations will happen in d ≥ 6; see Figure 12View Image. Either case could lead to nonuniqueness within solutions of spherical topology.

The formalism of [151Jump To The Next Citation Point] allows one to show that the only regular stationary perturbations of a Schwarzschild black hole lead into the MP family of black holes [162Jump To The Next Citation Point]. Thus, the MP solutions are the only stationary black holes that have a regular static limit.

The issue of how much symmetry a general stationary black hole must possess is probably the main impediment to progress with the classification problem. At present, the only uniqueness results for stationary higher-dimensional black holes assume the existence of multiple rotational symmetries. These results are for d = 5; if one assumes the existence of two rotational symmetries then it can be shown that the Myers–Perry solution is the unique, stationary, nonextremal, asymptotically-flat, vacuum black-hole solution of spherical topology [194]. More generally, it has been shown that stationary, nonextremal, asymptotically-flat, vacuum black-hole solutions with two rotational symmetries are uniquely characterized by their mass, angular momenta, and rod structure (see Section 5.2.2.1[139]. The remaining step in a full classification of all d = 5 stationary vacuum black holes with two rotational symmetries is to prove that the only rod structures giving rise to regular black-hole solutions are those associated with the known (Myers–Perry and black ring) solutions.

The situation in d ≥ 6 is much further away from a complete classification, even for the class of solutions with the maximal number of rotational symmetries. For instance, the tools to classify the (yet to be found) infinite number of families of solutions with ‘pinched horizons’ proposed in [8081] are still to be developed.

It is clear that the notion of black-hole uniqueness that holds in four dimensions, namely that conserved charges serve to fully specify a black hole, does not admit any simple extension to higher dimensions. This leaves us with two open questions: (a) what is the simplest and most convenient set of parameters that fully specify a black hole; (b) how many black-hole solutions with given conserved charges are relevant in a given physical situation.

Concerning the first question, we note that while the rod structure may provide the additional data to determine five-dimensional vacuum black holes, one may still desire a characterization in terms of physical parameters. In other words, since the dimensionless angular momenta ja are insufficient to specify the solutions, an adequate physical parameterization of the phase space of higher-dimensional black holes is still missing. [225241] have studied whether higher multipole moments may serve this purpose, but the results appear to be inconclusive.

The second question is more vague, as it hinges not only on the answer to the previous question, but also on the specification of the problem one is studying. It has been speculated that the conserved charges may still suffice to select a unique classical stationary configuration, if supplied with additional conditions, such as the specification of horizon topology or requiring classical stability [166]. In five dimensions we already know that horizon topology alone is not enough, since there are both thin and fat black rings with the same j. We have seen that, very likely, an even larger nonuniqueness occurs in all d ≥ 6. Classical dynamical stability to linearized perturbations, which is not at all an issue in the four-dimensional classification, is presumably a much more restrictive condition, but even in five dimensions it is unclear whether it always picks out only one solution for a given (j ,j ) 1 2. One must also bear in mind that the classical instability of a black hole does not per se rule it out as a physically relevant solution; the time-scale of the instability must be compared to the time scale of the situation at hand. Some classical instabilities (e.g., ergoregion instabilities [25]) may be very slow.

It seems possible, although so far we are nowhere near having any compelling argument, that the requirement of classical, linearized stability, plus, possibly, horizon topology, suffices to fully specify a unique vacuum black-hole solution with given conserved charges. However, in the presence of gauge fields, this seems less likely, since dipoles not only introduce larger degeneracies but also tend to enhance the classical stability of the solutions.

8.5 Supersymmetric black holes

As mentioned earlier, the study of BPS black holes is of special importance in string theory and it is natural to ask whether one can classify BPS black holes. Asymptotically-flat BPS black-hole solutions are known only for d = 4,5.

In addition to rendering microscopic computations tractable, the additional ingredient of supersymmetry makes the classification of black holes easier. A supersymmetric solution admits a globally-defined super-covariantly constant spinor (see, e.g., [107]). This is such a strong constraint that it is often possible to determine the general solution with this property. This was first done for minimal d = 4, N = 2 supergravity, whose bosonic sector is Einstein–Maxwell theory. It can be shown [237] that any supersymmetric solution of this theory must be either a pp-wave or a member of the well-known Israel-Wilson-Perjes family of solutions (see, e.g., [228]). The only black-hole solutions in the IWP family are believed to be the multi-Reissner–Nordström solutions [131], and this can be proved subject to one assumption9 [45]. Hence, this is a uniqueness theorem for supersymmetric black-hole solutions of minimal d = 4, N = 2 supergravity.

This success has been partially extended to minimal d = 5 supergravity. It can be shown that any supersymmetric black hole must have near-horizon geometry locally isometric to either (i) the near-horizon geometry of the BMPV black hole, (ii) AdS3 × S2, or (iii) flat space [216]. Case (iii) would give a black hole with T 3 horizon, which seems unlikely in view of the black-hole topology theorem discussed above (although this does not cover supersymmetric black holes, since they are necessarily extremal). An explicit form for supersymmetric solutions of this theory is known [101] and can be exploited, together with an additional assumption10, to show that the only black hole of type (i) is the BMPV black hole itself. The supersymmetric black ring of [74] belongs to class (ii). The remaining step required to complete the classification would be to prove that this is the only solution in this class. These results can be extended to minimal d = 5 supergravity coupled to vector multiplets [123].

These results show that much more is known about d = 5 supersymmetric black holes than about more general stationary d = 5 black holes. Note that no assumptions about the topology of the horizon, or the number of rotational symmetries are required to obtain these results; they are outputs, not inputs. One might interpret this as weak evidence that, for general d = 5 black holes, topologies other than 3 S and 1 2 S × S cannot occur and that the assumption of two rotational symmetries is reasonable. However, one should be cautious in generalizing from BPS to non-BPS black holes, since it is known that many properties of the former (e.g., stability) do not always generalize to the latter.

8.6 Algebraic classification

In d = 4 dimensions, spacetimes can be classified according to the algebraic type of the Weyl tensor. Associated with the Weyl tensor are four “principal null vectors” [244]. In general these are distinct, but two or more of them coincide in an algebraically special spacetime. For example, the Kerr-Newman spacetime is type D, which means that it has two pairs of identical principal null vectors.

Given that known d = 4 black-hole solutions are algebraically special, it is natural to investigate whether the same is true in d > 4 dimensions. Before doing this, it is necessary to classify possible algebraic types of the Weyl tensor in higher dimensions. This can be done using a spinorial approach for the special case of d = 5 [65]. The formalism for all dimensions d ≥ 4 has been developed in [50] (and reviewed in [51]). It is based on “aligned null directions”, which generalize the concept of principal null vectors in d = 4. A general d > 4 spacetime admits no aligned null directions. The Weyl tensor is said to be algebraically special if one or more aligned null directions exist.

The algebraic types of some higher-dimensional black-hole solutions have been determined. The Myers–Perry black hole belongs to the higher-dimensional generalization of the d = 4 type D class [66213Jump To The Next Citation Point215Jump To The Next Citation Point]. The black ring is also known to be algebraically special, although not as special as the Myers–Perry black hole [213215]. Further analysis of the Weyl tensor and principal null congruences in higher dimensions can be found in [214].

In d = 4 dimensions, interesting new solutions (e.g., the spinning C-metric) were discovered by solving the Einstein equations to determine all solutions of type D [159]. Perhaps the same strategy would be fruitful in higher dimensions. A particular class of spacetimes, namely Robinson-Trautman, admitting a hypersurface-orthogonal, nonshearing and expanding geodesic null congruence, has been studied in [209]. However, unlike in four dimensions, this class does not contain the analogue of the C-metric.

8.7 Laws of black-hole mechanics

The laws of black-hole mechanics are generally valid in any number of dimensions. The only novelty arises in the first law due to new possibilities afforded by novel black holes. A nontrivial extension is to include dipoles charges that are independent of the conserved charges. An explicit calculation in [79] shows that black rings with a dipole satisfy a form of the first law with an additional ‘work’ term φdq, where q is the dipole charge and φ its respective potential. The appearance of the dipole here was surprising, since the most general derivation of the first law seems to allow only conserved charges into it. However, [52] shows that a new surface term enters due to the impossibility of globally defining the dipole potential φ in such a way that it is simultaneously regular at the rotation axis and at the horizon. Then one finds

dM = κ--𝒜 + Ω dJ + ΦdQ + φdq, (115 ) 8π H H
where Q and Φ are the conserved charge and its potential, respectively (see also [6]).

The next extension is not truly specific to d > 4, but it refers to a situation for which there are no known four-dimensional examples: stationary multiple-black-hole solutions with nondegenerate horizons. As we have seen, there are plenty of these in d ≥ 5. In this case, the first law can be easily shown to take the form [73Jump To The Next Citation Point]

( (i) ) dM = ∑ κ---d𝒜 (i)+ Ω(i)dJ (i)+ Φ (i)dQ (i) . (116 ) 8π H j j i
Here the index i labels the different disconnected components of the event horizon and j their independent angular momenta. The Komar angular momentum Jj(i) and the charge Q (i) are computed as integrals on a surface that encloses a single component of the horizon, generated by the vector (i) ∂t + Ωj ∂ φj. The potential (i) Φ is the difference between the potential at infinity and the potential on the i-th component of the horizon; in general we cannot choose a globally defined gauge potential that simultaneously vanishes on all horizon components. Presumably the result can be extended to include dipoles but the possible subtleties have not been dealt with yet. A Smarr relation also exists that relates the total ADM mass to the sums of the different ‘heat’ and ‘work’ terms on each horizon component [73].

8.8 Hawking radiation and black-hole thermodynamics

The extension of Hawking’s original calculation to most of the black holes that we have discussed in this review presents several difficulties, but we regard this as mostly a technical issue. In our opinion, there is no physically reasonable objection to the expectation that Hawking radiation is essentially unmodified in higher dimensions; a black hole emits radiation that, up to grey-body corrections, has a Planckian spectrum of temperature T = κ∕2 π and chemical potentials Ω, Φ, etc.

Some of the technical difficulties relate to the problem of wave propagation in the black-hole background; this can only be dealt with analytically for Myers–Perry black holes, since only in this case have the variables been separated (and then only for scalars and vectors in the general rotating case). There is in fact a considerable body of literature on the problem of radiation from MP black holes, largely motivated by their possible detection in scenarios with large extra dimensions. As mentioned in Section 2, this is outside the scope of this paper and we refer to [30155] for reviews.

There are also peculiarities with wave propagation that depend on the parity of the number of dimensions [20624], but these are unlikely to modify in any essential way the Planckian spectrum of radiation. This is, in fact, confirmed by detailed microscopic derivations of Hawking radiation in five dimensions based on string theory [188]. Other approaches to Hawking radiation that do not require one to analyze wave propagation have been applied to black rings [193], confirming the expected results. An early result was the analysis of vacuum polarization in higher-dimensional black holes in [92]. More recently, [204] claims that the evolution of a five-dimensional rotating black hole emitting scalar Hawking radiation leads, for arbitrary initial values of the two rotation parameters a and b, to a fixed asymptotic value of 2 2 a∕M = b∕M = const ⁄= 0.

The spectrum of radiation from a multiple-black-hole configuration will contain several components with parameters (T(i),Ω (i),Φ(i),...), so it will not really be a thermal distribution unless all the black holes have equal intensive parameters. This is, of course, the conventional condition for thermal equilibrium.

The Euclidean formulation of black-hole thermodynamics remains largely the same as in four dimensions. For rotating solutions, it is more convenient not to continue analytically the angular velocities and instead to work with complex sections that have real actions. In fact, black rings do not admit nonsingular real Euclidean sections [76]. Multiple-black-hole configurations with disconnected components of the horizon with different surface gravities, angular velocities, and electric potentials clearly do not admit regular Euclidean sections. Still, the Euclidean periodicity of the horizon generator formally yields the horizon temperature in the usual fashion.

8.9 Apparent and isolated horizons and critical phenomena

A number of other interesting studies of higher-dimensional black holes have been made. The properties of higher-dimensional apparent horizons have been analyzed in [223], which provides simple criteria to determine them. The isoperimetric inequalities and the hoop conjecture, concerning bounds on the sizes of apparent horizons through the mass they enclose, involve new features in higher dimensions. For instance, the four-dimensional hoop conjecture posits that a necessary and sufficient condition for the formation of a black hole is that a mass M gets compacted into a region whose circumference in every direction is 𝒞 ≤ 4πGM [236]. A generalization of this conjecture to d > 4 using a hoop of spatial dimension 1, in the form 1∕(d− 3) 𝒞 ≤ #(GM ), is unfeasible; the existence of black objects whose horizons have arbitrary extent in some directions (e.g., black strings, black rings, and ultraspinning black holes) shows that this condition is not necessary. It seems possible, however, that plausible, necessary, and sufficient conditions exist using the area of hoops of spatial codimension d − 3 in the form 𝒞d−3 ≤ #GM, although black rings may require hoops with nonspherical topology [145Jump To The Next Citation Point8Jump To The Next Citation Point]. There is also some evidence that the isoperimetric inequalities, which bound the spatial area of the apparent horizon by the mass that it encloses [208], may be extended in the form d−2 𝒜 ≤ # (GM )d−3 [1458]. See [257256224] for further work on these subjects.

The study of possible topologies for black-hole event horizons may be helped by the study of possible apparent horizons in initial data sets. [222] shows that it is possible to construct time-symmetric initial-data sets for black holes with apparent-horizon topology with the form of a product of spheres. Time symmetry, however, implies that these apparent horizons cannot correspond to rotating black holes and it is likely that their evolution in time leads to collapse into a spherical horizon.

The formalism of isolated horizons and the laws of black-hole mechanics for them, have been extended to higher dimensions in [182170], and then to five-dimensional Einstein–Maxwell theory with the Chern–Simons term [183] and anti-de Sitter asymptotics [5].

Finally, the critical phenomena in the collapse of a massless scalar at the onset of black-hole formation, discovered by Choptuik [41], have been studied in [227].


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