However, despite the paucity of exact solutions, there are strong indications that the variety of black
holes that populate general relativity in is vastly larger than in
. A first indication came
from the conjecture in [81
] of the existence of black holes with spherical horizon topology but with
axially-symmetric ‘ripples’ (or ‘pinches’). The plausible existence of black rings in any
was
argued in [144, 76
]. More recently, [80
] has constructed approximate solutions for black rings in
any
and then exploited the conjecture of [81
] to try to draw a phase diagram with
connections and mergers between the different expected phases. In the following we summarize these
results.
In the absence of exact techniques, [80] resorts to approximate constructions, in particular to the method of
matched asymptotic expansions previously used in the context of black holes localized in Kaluza–Klein circles
in [126, 115
, 156, 116]7.
The basic idea is to find two widely separated scales in the problem, call them
and
, with
, and then try to solve the equations in two limits; first, as a perturbative expansion for small
, and then in an expansion in
. The former solves the equations in the far region
in
which the boundary condition, e.g., asymptotic flatness, fixes the integration constants. The second
expansion is valid in the near region
. In order to fix the integration constants in this case, one
matches the two expansions in the overlap region
in which both approximations are valid.
The process can then be iterated to higher orders in the expansion; see [115] for an explanation of the
systematics involved.
In order to construct a black ring with horizon topology ,
we take the scales
to be the radii of the
and
,
respectively8.
To implement the above procedure, we take
, the horizon radius of the
of a straight
boosted black string, and
the large circle radius of a very thin circular string. Thus, in
effect, to first order in the expansion what one does is: (i) find the solution within the linearized
approximation, i.e., for small
, around a Minkowski background for an infinitely-thin circular string
with momentum along the circle; (ii) perturb a straight boosted black string so as to bend it
into an arc of very large radius
. Step (ii) not only requires matching to the previous
solution in order to provide boundary conditions for the homogeneous differential equations;
one also needs to check that the perturbations can be made compatible with regularity of the
horizon.
It is worth noting that the form of the solution thus found exhibits a considerable increase in complexity
when going from , where an exact solution is available, to
; simple linear functions of
in
change to hypergeometric functions in
. We take this as an indication that exact closed
analytical forms for these solutions may not exist in
.
We will not dwell here on the details of the perturbative construction of the solution (see [80] for this),
but instead we shall emphasize that adopting the view that a black object is approximated by a certain very
thin black brane curved into a given shape can easily yield nontrivial information about new kinds of black
holes. Eventually, of course, the assumption that the horizon remains regular after curving needs to be
checked.
Consider then a stationary black brane, possibly with some momentum along its worldvolume, with
horizon topology , with
. When viewed at distances much larger than the size
of the
, we can approximate the metric of the black-brane spacetime by the gravitational field
created by an ‘equivalent source’ with distributional energy tensor
, with nonzero
components only along directions tangent to the worldvolume, and where
corresponds to the
location of the brane. Now we want to put this same source on a curved, compact
-dimensional spatial
surface in a given background spacetime (e.g., Minkowski, but possibly (anti-)de Sitter or others, too).
In principle we can obtain the mass
and angular momenta
of the new object by
integrating
and
over the entire spatial section of the brane worldvolume. Moreover,
the total area
is similarly obtained by replacing the volume of
with the volume of
the new surface. Thus, it appears that we can easily obtain the relation
in this
manner.
There is, however, the problem that having changed the embedding geometry of the brane, it is not
guaranteed that the brane will remain stationary. Moreover, will be a function not only of
,
but will also depend explicitly on the geometrical parameters of the surface. However, we would expect that
in a situation of equilibrium, some of these geometrical parameters would be fixed dynamically by the
mechanical parameters
of the brane. For instance, take a boosted string and curve it into a
circular ring so that the linear velocity turns into angular rotation. If we fix the mass and the radius, then
the ring will not be in equilibrium for every value of the boost, i.e., of the angular momentum;
so, there must exist a fixed relation
. This is reflected in the fact that, in the
new situation, the stress-energy tensor is in general not conserved,
; additional
stresses would be required to keep the brane in place. An efficient way of imposing the brane
equations of motion is, in fact, to demand conservation of the stress-energy tensor. In the absence
of external forces, the classical equations of motion of the brane derived in this way are [28]
In general, Equation (110) constrains the allowed values of the parameters of a black brane that can be
put on a given surface. [80
] easily derives, for any
, that the radius
of thin rotating black rings
of given
and
is fixed to
Using the dimensionless area and spin variables (21), Equation (113
) allows one to compute the asymptotic
form of the curve
in the phase diagram at large
for black rings. However, when
is of order
one, the approximations in the matched asymptotic expansion break down, and the gravitational interaction
of the ring with itself becomes important. At present we have no analytical tools to deal with this
regime for generic solutions. In most cases, numerical analysis may be needed to obtain precise
information.
Nevertheless, [80] contains advanced heuristic arguments, which propose a completion of the curves that
is qualitatively consistent with all the information available at present. A basic ingredient is the
observation in [81
], discussed in Section 4.1, that in the ultraspinning regime in
, MP black
holes approach the geometry of a black membrane
spread out along the plane of
rotation.
We have already discussed how using this analogy, [81] argues that ultraspinning MP black holes should
exhibit a Gregory–Laflamme-type of instability. Since the threshold mode of the GL instability gives rise to
a new branch of static nonuniform black strings and branes [117, 120, 248], [81
] argues that it is natural to
conjecture the existence of new branches of axisymmetric ‘lumpy’ (or ‘pinched’) black holes, branching off
from the MP solutions along the stationary axisymmetric zero-mode perturbation of the GL-like
instability.
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[80] develops further this analogy, and draws a correspondence between the phases of black membranes
and the phases of higher-dimensional black holes, illustrated in Figure 11
. Although the analogy has several
limitations, it allows one to propose a phase diagram in
of the form depicted in Figure 12
, which
should be compared to the much simpler diagram in five dimensions, Figure 10
. Observe the presence of an
infinite number of black holes with spherical topology, connected via merger transitions to
MP black holes, black rings, and black Saturns. Of all multiple-black-hole configurations, the
diagram only includes those phases in which all components of the horizon have the same surface
gravity and angular velocity; presumably, these are the only ones that can merge to a phase with
connected horizon. Even within this class of solutions, the diagram is not expected to contain all
possible phases with a single angular momentum; blackfolds with other topologies must likely be
included as well. The extension to phases with several angular momenta also remains to be
done.
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Indirect evidence for the existence of black holes with pinched horizons is provided by the
results of [180], which finds ‘pinched plasma-ball’ solutions of fluid dynamics that are CFT
duals of pinched black holes in six-dimensional AdS space. The approximations involved in
the construction require that the horizon size of the dual black holes be larger than the AdS
curvature radius, and thus do not admit a limit to flat space. Nevertheless, their existence
provides an example, if indirect, that pinched horizons make an appearance in (and not in
).
The situation in is very similar to what we described for
in Section 5.4; most of what we
know is deduced by heuristic analogies and approximate methods. The following prototypic instabilities can
be easily identified:
We end this section emphasizing that, presumably, new concepts and tools are required for the
characterization of black holes in , let alone for their explicit construction.
The general problem of the dynamical linearized stability of MP black holes, in particular in the case
with a single rotation, becomes especially acute for the determination of possible black-hole phases in
. The arguments in favor of an ultraspinning instability seem difficult to evade, so a most pressing
problem is to locate the point (i.e., the value of
) at which this instability appears as a stationary mode,
and then perturb the solution along this mode to determine the direction in which the new branch of
solutions evolves.
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