Five-dimensional black rings are black holes with horizon topology in asymptotically
flat spacetime. The
describes a contractible circle, not stabilized by topology but
by the centrifugal force provided by rotation. An exact solution for a black ring with
rotation along this
was presented in [83
]. Its most convenient form was given in [79
]
as5
In the form given above, the solution possesses three independent parameters: ,
, and
.
Physically, this sounds like one too many: given a ring with mass
and angular momentum
, we
expect its radius to be dynamically fixed by the balance between the centrifugal and tensional forces. This
is also the case for the black ring (50
): in general it has a conical defect on the plane of the
ring,
. In order to avoid it, the angular variables must be identified with periodicity
The physical parameters of the solution (mass, angular momentum, area, angular velocity, surface
gravity) in terms of and
can be found in [84
]. It can be seen that while
provides a measure of
the radius of the ring’s
, the parameter
can be interpreted as a ‘thickness’ parameter
characterizing its shape, corresponding roughly to the ratio between the
and the
radii.
More precisely, one finds two branches of solutions, whose physical differences are seen most clearly in
terms of the dimensionless variables and
introduced above. For a black ring in equilibrium, the
phase curve
can be expressed in parametric form as
|
This curve is easily seen to have a cusp at , which corresponds to a minimum value of
and a maximum
. Branching off from this cusp, the thin black-ring solutions
(
) extend to
as
, with asymptotic
. The fat black-ring branch
(
) has lower area and extends only to
, ending at
at the same zero-area
singularity as the MP solution. This implies that in the range
there exist three different
solutions (thin and fat black rings and MP black holes) with the same value of
. The notion
of black-hole uniqueness that was proven to hold in four dimensions does not extend to five
dimensions.
[76] and [89] contain detailed analyses of the geometrical features of black-ring horizons. Some geodesics
of the black-ring metric have been studied with a view towards different applications: [205] studies them in
the context of the Penrose process, and [76
] considers them for tests of stability. [143
] is a more complete
analysis of geodesics.
Rotation in the second independent plane corresponds to rotation of the of the ring. In the limit of
infinite
radius, a section along the length of the ring gives an
that is essentially like that of a
four-dimensional black hole: setting it into rotation is thus similar to having a Kerr-like black hole. Thus,
an upper, extremal bound on the rotation of the
is expected (actually, the motion of
the ring along its
yields a momentum that can be viewed as an electric Kaluza–Klein
charge, so instead of a Kerr solution, the
limit yields a rotating electric KK black
hole).
Solutions with rotation only along the , but not on the
, are fairly straightforward to
construct and have been given in [192
, 87]. However, these black rings cannot support themselves
against the centripetal tension and thus possess conical singularities on the plane of the ring.
Constructing the exact solution for a black ring with both rotations is a much more complicated task,
which has been achieved by Pomeransky and Sen’kov in [212
] (the techniques employed are
reviewed in Section 5.2). They have furthermore managed to present it in a fairly compact form:
The metric functions take a very complicated form in the general case in which the black ring is not in
equilibrium (their explicit forms can be found in [196]), but they simplify significantly when a balance of
forces (i.e., cancellation of conical singularities) is imposed. In this case the one-form
characterizing the
rotation is [212
]
The parameters and
are restricted to
The physical parameters ,
,
, and
of the solution have been computed
in [212
]. An analysis of the physical properties of the solution, and in particular the phase space, has been
presented in [78
]. To plot the parameter region where black rings exist, we fix the mass and employ the
dimensionless angular-momentum variables
and
introduced in Equation (21
). The phase space of
doubly-spinning black rings is in Figure 6
for the region
(the rest of the plane is obtained by
iterating and exchanging
). It is bounded by three curves (besides the axis, which is not a boundary
in the full phase plane):
|
For doubly-spinning black rings the angular momentum along the is always bounded above by the
one in the
as
Figure 7 shows the phase space covered by all five-dimensional black holes with a single horizon. Two
kinds of black rings (thin and fat) and one MP black hole, the three of them with the same values of
, exist in small spandrels near the corners of the MP phase-space square. It is curious that,
once black rings are included, the available phase space for five-dimensional black holes resembles more
closely that of six-dimensional MP black holes, Figure 2
(b).
|
[78] contains sectional plots of the surface
for black rings at constant
, for
, from which it is possible to obtain an idea of the shape of the surface. In the complete
range of
and
the phase space of five-dimensional black holes (with connected event
horizons) consists of the ‘dome’ of MP black holes (Figure 4
replicated on all four quadrants), with
‘romanesque vaults’ of black rings protruding from its corners, and additional substructure in
the region of nonuniqueness – our knowledge of architecture is insufficient to describe it in
words.
It is also interesting to study other properties of black rings, such as temperature and horizon angular
velocities, expressions for which can be found in [78]. It is curious to notice that even if the two angular
momenta can never be equal, the two angular velocities
and
have equal values, for a given mass,
when
On the other hand, the temperature of the black ring – which for thin rings with a single spin is bounded below and diverges as the ring becomes infinitely long and thin (at fixed mass) – decreases to zero when the second spin is taken to the extremal limit, so there exist ‘cold’ thin black rings.
Some consequences of these features to properties of multiple-ring solutions will be discussed in Section 5.3.
A sector of five-dimensional vacuum general relativity in which a complete classification of
black-hole solutions may soon be achieved is the class of stationary solutions with two angular
Killing vectors. Integration of the three Killing directions yields a two-dimensional nonlinear
sigma model that is completely integrable. Solutions can be characterized in terms of their rod
structure along multiple directions, introduced in [82] and extended in [127
]. It has been proven
that these data (whose relation to physical parameters is unfortunately not quite direct), in
addition to the total mass and angular momenta, uniquely characterize asymptotically-flat
solutions [139
].
Since most of the analysis is applicable to any number of dimensions, we will keep arbitrary,
although only in
can the solutions be globally asymptotically flat. So, henceforth, we assume that
the spacetime admits
commuting, non-null, Killing vectors
,
(we
assume, although this is not necessary, that the zero-th vector is asymptotically timelike and all other
vectors are asymptotically spacelike). Then it is possible to prove that, under natural suitable conditions,
the two-dimensional spaces orthogonal to all three Killing vectors are integrable [82
]. In this case the
metrics admit the form
Equations (68) and (69
) are the equations for the principal chiral field model, a nonlinear sigma model
with group
, which is a completely integrable system. In the present case, it is also subject
to the constraint (67
). This introduces additional features, some of which will be discussed
below.
In order to understand the structure of the solutions of this system, it is convenient to first analyze a
simple particular case [82].
Consider the simplest situation in which the Killing vectors are mutually orthogonal. In this case the solutions admit the diagonal form6
Equations (68 At a rod source for , the orbits of the corresponding Killing vector vanish: if it is an angular Killing
vector
, then the corresponding one-cycles shrink to zero size, and the periodicity of
must be
chosen appropriately in order to avoid conical singularities; if it is instead the timelike Killing
, then it
becomes null there. In both cases, a necessary (but not sufficient) condition for regularity at the rod is that
the linear density be
The rod structures for the four and five-dimensional Schwarzschild and Tangherlini solutions are
depicted in Figure 8. The Rindler space of uniformly-accelerated observers is recovered as the horizon rod
becomes semi-infinite,
. [82
] gives a number of ‘rules of thumb’ for interpreting rod
diagrams.
In the general case where the Killing vectors are not orthogonal to each other, the simple construction in terms of solutions of the linear Laplace equation does not apply. Nevertheless, the equations can still be completely integrated, and the characterization of solutions in terms of rod structure can be generalized.
More precisely, the condition (67) implies that the matrix
must have at least one zero
eigenvalue. It can be shown that regularity of the solution (analogous to the requirement of density
in
the orthogonal case) requires that only one eigenvalue is zero at any given interval on the axis [127
]. Each
such interval is called a rod, and, for each rod
, we assign a direction vector
such that
The rod is referred to as timelike or spacelike according to the character of the rescaled norm
at the rod
,
. For a timelike rod normalized such that the
generator of asymptotic time translations enters with coefficient equal to one, the rest of the
coefficients correspond to the angular velocities of the horizon (if this satisfies all other regularity
requirements).
For a spacelike rod, the following two regularity requirements are important. First, given a rod-direction vector
with norm the length Second, the presence of time components on a spacelike rod creates causal pathologies. Consider a
vector that is timelike in some region of spacetime, and whose norm does not vanish at a
given spacelike rod
. If the direction vector
associated with this rod is such that
Further analysis of Equations (68), (70
) and their source terms can be found in [130].
Since Equations (88) are linear, we can construct new solutions by ‘dressing’ a ‘seed’ solution
. The
seed defines matrices
and
through Equations (69
). Equations (88
) can then be solved to
determine
. Then, we ‘dress’ this solution using a matrix
to find a new solution of the
form
The solution for the matrices , where
labels the solitons, can be constructed by first
introducing a set of
-dimensional vectors
using the seed as
All the information about the solitons that are added to the solution is contained in the soliton positions
and the soliton-orientation vectors
. This is all we need to determine the dressing
matrix in Equation (91
), and then the new metric
,
There is, however, one problem that turns out to be particularly vexing in : the new metric
in Equation (96
) does not satisfy, in general, the constraint (67
); the introduction of the
solitons gives
a determinant for the new metric
A possible way out of this problem is to restrict oneself to transformations that act only on a
block of the seed
[165]. In this case it is possible to apply the above renormalization to only this part
of the metric – effectively, the same as in four dimensions – and thus obtain a solution with the correct,
physical rod densities. However, it is clear that, if we start from diagonal seeds, this method
cannot deal with solutions with off-diagonal terms in more than one
block, e.g., with a
single rotation. It cannot be applied to obtain solutions with rotation in several planes. Still, [7]
applies this method to obtain solutions with arbitrary number of rods, with rotation in a single
plane.
Fortunately, a clever and very practical way out of this problem has been proposed by
Pomeransky [211], that can deal with the general case in any number of dimensions. The key idea is the
observation that Equation (97
) is independent of the ‘realignment’ vectors
. One may
then start from a solution with physical rod densities, ‘remove’ a number of solitons from it
(i.e., add solitons or antisolitons with negative densities
), and then re-add these same
solitons, but now with different vectors
, so the rods affected by these solitons acquire,
in general, new directions. If the original seed solution satisfied the determinant constraint
(67
), then by construction so will the metric obtained after re-adding the solitons (including, in
particular, the sign). And more importantly, if the densities of the initial rods are all
and
negative densities do not appear in the end result, the final metric will only contain regular
densities.
In the simplest form of this method, one starts from a diagonal, hence static, solution and
then removes some solitons or antisolitons with ‘trivial’ vectors aligned with one of the Killing basis vectors
, i.e.,
(recall that in the diagonal case this alignment of rods is indeed possible).
Removing a soliton or antisoliton
at
aligned with the direction
amounts to changing
If the vectors for the re-added solitons mix the time and spatial Killing directions, then this
procedure may yield a stationary (rotating) version of the initial static solution. The method requires
the determination of the function
that solves Equation (88
) for the seed. This
is straightforward to determine for diagonal seeds (see the examples below), so for these the
method is completely algebraic. Although even a two-soliton transformation of a multiple-rod
metric can easily result in long expressions for the metric coefficients, the method can be readily
implemented in a computer program for symbolic manipulation. The procedure can also be applied,
although it becomes quite more complicated, to nondiagonal seeds. In this case, the function
for the nondiagonal seed is most simply determined if this solution itself is constructed
starting from a diagonal seed. The doubly-spinning black ring of [212
] was obtained in this
manner.
The simplest case, which demonstrates one of the basic tools for adding rotation in more complicated
cases, is the Kerr solution – in fact, one generates the Kerr–Taub-NUT solution, and then sets the nut
charge to zero. Begin from the Schwarzschild solution, generated, e.g., using the techniques available
for Weyl solutions [82], and whose rod structure is depicted in Figure 8
. The seed metric is
So, following the method above, add an antisoliton at and a soliton at
, with
respective constant vectors
and
. For this step, we need to construct the
matrix
in Equation (94
), which in turn requires the matrix
The Myers–Perry black hole with two angular momenta is obtained in a very similar way [211] starting
from the five-dimensional Schwarzschild–Tangherlini solution, whose rod structure is shown in Figure 8
. We
immediately see that
The black ring with rotation along the requires a more complicated seed, but, on the other hand, it
requires only a one-soliton transformation [77
] (the first systematic derivations of this solution used a
two-soliton transformation [148
, 240]). The seed is described in Figure 9
. The static black ring of [82
]
(which necessarily contains a conical singularity) is recovered for
. However, one needs to
introduce a ‘phantom’ soliton point at
and a negative density rod, in order to eventually obtain the
rotating black ring. Thus, we see that the initial solution need not satisfy any regularity requirements. To
obtain the rotating black ring, we remove an antisoliton at
with direction
, and re-add it
with vector
. At the end of the process one must adjust the parameters,
including
and the rod positions, to remove a possible singularity at the phantom point
.
The doubly-spinning ring has resisted all attempts at deriving it directly from a diagonal, static seed.
Instead, [212] obtained it in a two-step process. Rotation of the of the ring is similar to
the rotation of the Kerr solution. In fact, the black-ring solutions with rotation only along
the
can be obtained by applying to the static black ring the same kind of two-soliton
transformations that yielded Kerr from a Schwarzschild seed (101
) [239]. Hence, if we begin from the
black ring rotating along the
and perform similar soliton and antisoliton transformations
at the endpoints of the horizon rod, we can expect to find a doubly-spinning ring. The main
technical difficulty is in constructing the function
for the single-spin black-ring
solution (50
). However, if we construct this solution via a one-soliton transformation as we have just
explained, this function is directly obtained from Equation (90
). In this manner, solution (57
) was
derived.
This method has also been applied to construct solutions with disconnected components of the horizon, which we shall discuss next. The previous examples provide several ‘rules of thumb’ for constructing such solutions. However, there is no precise recipe for the most efficient way of generating the sought solution. Quite often, unexpected pathologies show up, of both local and global type, so a careful analysis of the solutions generated through this method is always necessary.
Finally, observe that there are certain arbitrary choices in this method; it is possible to
choose different solitons and antisolitons, with different orientations, and still get essentially the
same final physical solution. Also, the intermediate rescaling, and the form for , admit
different choices. All this may lead to different-looking forms of the final solution, some of them
possibly simpler than others. Occasionally, spurious singularities may be introduced through bad
choices.
[114] develops a different algebraic method to obtain stationary axisymmetric solutions in five
dimensions from a given seed. An subgroup of the “hidden”
symmetry of solutions
with at least one spatial Killing vector (the presence of a second one is assumed later) is identified that
preserves the asymptotic boundary conditions, and whose action on a static solution generates a
one-parameter family of stationary solutions with angular momentum, e.g., one can obtain the Myers–Perry
solution from a Schwarzschild–Tangherlini seed. It is conjectured that all vacuum stationary
axisymmetric solutions can be obtained by repeated application of these transformations on static
seeds.
In it is believed that there are no stationary multiple-black-hole solutions of vacuum gravity.
However, such solutions do exist in
. ‘Black Saturn’ solutions, in which a central MP-type of black
hole is surrounded by a concentric rotating black ring, have been constructed in [77
]. They exhibit a
number of interesting features, such as rotational dragging of one black object by the other, as well as both
co- and counter-rotation. For instance, we may start from a static seed and act with the kind of one-soliton
BZ transformation that turns on the rotation of the black ring. This gives angular momentum
(measured by a Komar integral on the horizon) to the black ring but not to the central black hole.
However, the horizon rod of this black hole is reoriented and acquires a nonzero angular velocity:
the black hole is dragged along by the black-ring rotation. It is also possible (this needs an
additional one-soliton transformation that turns on the rotation of the MP black hole) to have a
central black hole with a static horizon that nevertheless has nonvanishing angular momentum;
the ‘proper’ inner rotation of the black hole is cancelled at its horizon by the black-ring drag
force.
The explicit solutions are rather complicated, but an intuitive discussion of their properties is presented
in [73]. The existence of black Saturns is hardly surprising; since black rings can have arbitrarily large
radius, it is clear that we can put a small black hole at the center of a very long black ring, and the
interaction between the two objects will be negligible. In fact, since a black ring can be made arbitrarily
thin and light for any fixed value of its angular momentum, for any nonzero values of the total mass and
angular momentum, we can obtain a configuration with larger total area than any MP black hole or black
ring; put almost all the mass in a central, almost-static black hole, and the angular momentum in a very
thin and long black ring. Such configurations can be argued to attain the maximal area (i.e., entropy) for
given values of
and
. Observe also that for fixed total
we can vary, say, the
mass and spin of the black ring, while adjusting the mass and spin of the central black hole
to add up to the total
and
. These configurations, then, exhibit doubly-continuous
nonuniqueness.
We can similarly consider multiple-ring solutions. Di-rings, with two concentric black rings rotating on
the same plane, were first constructed in [150]; [85] re-derived them using the BZ approach. Each new
ring adds two parameters to the continuous degeneracy of solutions with given total and
.
Note that the surface gravities (i.e., temperatures) and angular velocities of disconnected components of
the horizon are in general different. Equality of these ‘intensive parameters’ is a necessary condition for
thermal equilibrium – and presumably also for mergers in phase space to solutions with connected horizon
components; see Section 6.2. So these multiple-black-hole configurations cannot, in general,
exist in thermal equilibrium (this is besides the problems of constructing a Hartle–Hawking
state when ergoregions are present [157]). The curves for solutions, where all disconnected
components of the horizon have the same surface gravities and angular velocities, are presented in
Figure 10 (see [73
]). All continuous degeneracies are removed, and black Saturns are always
subdominant in total horizon area. It is expected that no multiple-ring solutions exist in this
class.
|
It is also possible to have two black rings lying and rotating on orthogonal, independent planes. Such
bicycling black rings have been constructed using the BZ method [152, 78
], and provide a way of obtaining
configurations with arbitrarily large values of both angular momenta for fixed mass – which cannot be
achieved simultaneously for both spins, either by the MP black holes or by doubly-spinning black
rings. The solutions in [152, 78] are obtained by applying to each of the two rings the kind of
transformations that generate the singly-spinning black ring. Thus each black ring possesses angular
momentum only on its plane, along the
, but not in the orthogonal plane, on the
–
nevertheless, they drag each other so that the two horizon angular velocities are both nonzero on each
of the two horizons. The solutions contain four free parameters, corresponding to, e.g., the
masses of each of the rings and their two angular momenta. It is clear that a more general,
six-parameter solution must exist in which each black ring has both angular momenta turned
on.
It can be argued, extending the arguments in [73], that multiple-black-hole solutions allow one to cover
the entire
phase plane of five-dimensional solutions. It would be interesting to determine for which
parameter values these multiple black holes have the same surface gravity and angular velocities on all
disconnected components of the horizon. With this constraint, multiple-black-hole solutions still allow one
to cover a larger region of the
plane than already covered by solutions with a connected
horizon;see Figure 7
. For instance, it can be argued that some bicycling black rings (within the
six-parameter family mentioned above for which the four angular velocities of the entire system can be
varied independently) should satisfy these ‘thermal equilibrium’ conditions; as we have seen, a
doubly-spinning black ring can have
. Thus, if we consider two identical doubly-spinning thin
black rings, one on each of the two planes, then we can make them have
-angular velocity equal to the
-angular velocity of the other ring in the orthogonal plane. These solutions then lie along the lines
, reaching arbitrarily large
, which is not covered by the single-black-hole phases in
Figure 7
. Clearly, there will also exist configurations extending continuously away from this
line.
Black Saturns with a single black ring that satisfy ‘thermal equilibrium’ conditions should also exist. In
fact, the possibility of varying the temperature of the ring by tuning the rotation in the might allow
one to cover portions of the
plane beyond Figure 7
. If so, this would be unlike the situation with a
single rotation, where thermal-equilibrium Saturns lie within the range of
of black rings,
Figure 10
.
The Weyl ansatz of Section 5.2.1 enables one to easily generate solutions in with multiple black
holes of horizon topology
, which are asymptotically flat [82, 231]. However, all these solutions possess
conical singularities reflecting the attraction between the different black holes. It seems unlikely that the
extension to include off-diagonal metric components (rotation and twists) could eliminate these singularities
and yield balanced solutions.
The linearized perturbations of the black-ring metric (50) have not yielded to analytical study. The
apparent absence of a Killing tensor prevents the separation of variables even for massless scalar-field
perturbations. In addition, the problem of decoupling the equations to find a master equation for linearized
gravitational perturbations, already present for the Myers–Perry solutions, is, if anything, exacerbated for
black rings.
Studies of the classical stability of black rings have, therefore, been mostly heuristic. Already the original
paper [83] pointed out that very thin black rings locally look like boosted black strings (this was made
precise in [71]), which were expected to suffer from GL-type instabilities. The instability of boosted black
strings was indeed confirmed in [144
]. Thus, thin black rings are expected to be unstable to the formation
of ripples along their
direction. This issue was examined in further detail in [76
], which found that
thin black rings seem to be able to accommodate unstable GL modes down to values
.
Thus, it is conceivable that a large fraction of black rings in the thin branch, and possibly
all of this branch, suffer from this instability. The ripples rotate with the black ring and then
should emit gravitational radiation. However, the timescale for this emission is much longer
than the timescale of the fastest GL mode, so the pinchdown created by this instability will
dominate the evolution, at least initially. The final fate of this instability of black rings depends on
the endpoint of the GL instability, but it is conceivable, and compatible with an increment of
the total area, that the black-ring fragments separate into smaller black holes that then fly
away.
Another kind of instability was discussed in [76]. By considering off-shell deformations of the black ring
(namely, allowing for conical singularities), it is possible to compute an effective potential for radial
deformations of the black ring. Fat black rings sit at maxima of this potential, while thin black rings sit at
minima. Thus, fat black rings are expected to be unstable to variations of their radius, and presumably
collapse to form MP black holes. The analysis in [76
] is in fact consistent with a previous,
more abstract analysis of local stability in [2]. This is based on the ‘turning-point’ method of
Poincaré, which studies equilibrium curves for phases near bifurcation points. For the case of
black rings, one focuses on the cusp, where the two branches meet. One then assumes that
these curves correspond to extrema of some potential, e.g., an entropy, that can be defined
everywhere on the plane
. The cusp then corresponds to an inflection point of this
potential at which a branch of maxima and a branch of minima meet. By continuity, the branch
with the higher entropy will be the most stable branch, and the one with lower entropy will be
unstable. Thus, for black rings an unstable mode is added when going from the upper (thin) to the
lower (fat) branch. This is precisely as found in [76
] from the mechanical potential for radial
deformations.
Thus, a large fraction of all single-spin neutral black rings are expected to be classically
unstable, and it remains an open problem whether a window of stability exists for thin black rings
with . The stability, however, can improve greatly with the addition of charges and
dipoles.
Doubly-spinning black rings are expected to suffer from similar instabilities. Insofar as a fat ring branch
that meets at a cusp with a thin ring branch exists, the fat rings are expected to be unstable. Very
thin rings are also expected to be unstable to GL-perturbations that form ripples. The angular
momentum on the may be redistributed nonuniformly along the ring, with the larger blobs
concentrating more spin. In addition, although it has been suggested that super-radiant ergoregion
instabilities associated to rotation of the
might exist [68], a proper account of the asymptotic
behavior of super-radiant modes needs to be made before concluding that the instability is actually
present.
Much of what we can say about the classical stability of black Saturns and multiple rings follows from what we have said above for each of its components, e.g., if their rings are thin enough, they are expected to be GL-unstable. We know essentially nothing about what happens when the gravitational interaction among the black objects involved is strong. For instance, we do not know if the GL instability is still present when a thin black ring lassoes at very close range a much larger, central, MP black hole.
Massive geodesics on the plane of a black ring (see [143]) show that a particle at the center of the
is unstable to migrating away towards the black ring. This suggests that a black Saturn with a small black
hole at the center of a larger black ring should be unstable. One possibility for a different instability of black
Saturns appears from the analysis of counter-rotating configurations in [77]. For large enough
counter-rotation, the Komar-mass of the central black hole vanishes and then becomes negative.
By itself, this does not imply any pathology, as long as the total ADM mass is positive and
the horizon remains regular, which it does. However, it suggests that the counter-rotation in
this regime becomes so extreme that the black hole might tend to be expelled off the plane of
rotation.
Clearly, the classical stability of all, old and new, rotating black-hole solutions of five-dimensional general relativity remains largely an open problem, where much work remains to be done.
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