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.
Logically, the first step in the proof of the black-hole uniqueness theorem is Hawking’s black-hole
topology theorem [132
], which states that (a spatial cross-section of) the event horizon must be
topologically
. 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 [153, 95], black
holes.
The existence of black rings demonstrates that topologically nonspherical horizons are possible for
. But there are still restrictions on the topology of the event horizon. It has been shown [98, 96]
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
, the strongest restriction is for
, in which case
the topology must be a connected sum of spherical spaces (3-spheres with identifications) and
’s.
Another topological restriction arises from cobordism theory. Consider surrounding the black hole with
a large sphere. Let denote a spacelike hypersurface that runs from this sphere down to
the event horizon, which it intersects in some compact surface
. Then
is a cobordism
between
and a sphere. The existence of such a cobordism imposes topological restrictions on
. 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 . 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
, the two factors differ in that the
, being contractible, needs rotation to be stabilized,
whereas the
is already a minimal surface. In
we can envisage even more complicated
situations arising from the bending into different surfaces of the worldvolume directions of a
variety of black
-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.
In 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
-dimensions to
establish uniqueness of the
-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
-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 vacuum gravity [44, 46]. 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
Einstein–Maxwell theory [218, 220].
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.
In , 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 [138
]. More
precisely, it can be shown that a stationary, nonextremal, asymptotically-flat, rotating, black-hole
solution of
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
[230] and
[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 dimensions there are
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 static black holes). One could look for evidence that such solutions exist by considering
perturbations of known solutions [216
]. 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 [216
]. Alternatively, bifurcations could occur without breaking any rotational
symmetry. As we have discussed, [80
, 81
] have conjectured that such bifurcations will happen in
; see Figure 12
. Either case could lead to nonuniqueness within solutions of spherical
topology.
The formalism of [151] allows one to show that the only regular stationary perturbations of a
Schwarzschild black hole lead into the MP family of black holes [162
]. 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 ; 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
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 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 [80, 81] 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 are insufficient to specify the
solutions, an adequate physical parameterization of the phase space of higher-dimensional black holes is still
missing. [225, 241] 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 . We have seen that, very likely, an even larger nonuniqueness occurs in all
. 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
. 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.
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 .
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 ,
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
,
supergravity.
This success has been partially extended to minimal 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)
, or (iii) flat space [216]. Case (iii) would give a black hole
with
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
supergravity
coupled to vector multiplets [123].
These results show that much more is known about supersymmetric black holes than about
more general stationary
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
black holes, topologies other than
and
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.
In 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 black-hole solutions are algebraically special, it is natural to investigate
whether the same is true in
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
[65]. The formalism for all dimensions
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
. A general
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 type D
class [66, 213
, 215
]. The black ring is also known to be algebraically special, although not as special as the
Myers–Perry black hole [213, 215]. Further analysis of the Weyl tensor and principal null congruences in
higher dimensions can be found in [214].
In 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.
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 , where
is
the dipole charge and
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
in
such a way that it is simultaneously regular at the rotation axis and at the horizon. Then one finds
The next extension is not truly specific to , 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
. In this case, the first law can be easily shown to take the
form [73
]
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 and chemical potentials
,
,
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 [30, 155] for reviews.
There are also peculiarities with wave propagation that depend on the parity of the number of
dimensions [206, 24], 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 and
, to a fixed asymptotic value of
.
The spectrum of radiation from a multiple-black-hole configuration will contain several components with
parameters , 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.
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 gets compacted into a region whose circumference in every
direction is
[236]. A generalization of this conjecture to
using a hoop
of spatial dimension 1, in the form
, 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
in the form
, although black rings may require hoops with
nonspherical topology [145
, 8
]. 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
[145, 8]. See [257, 256, 224] 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 [182, 170], 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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