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5 Vacuum Solutions in Five Dimensions

In Section 4 we discussed the MP solutions, which can be regarded as higher-dimensional versions of the Kerr solution. However, in recent years it has been realized that higher dimensions allow for a much richer landscape of black-hole solutions that do not have four-dimensional counterparts. In particular, there has been great progress in our understanding of five-dimensional vacuum black holes, insofar as we consider stationary solutions with two rotational Killing vectors. The reason is that this sector of the theory is completely integrable, and solution-generating techniques are available. We begin by analyzing in Section 5.1 a qualitatively new class of solutions with connected horizons: black rings with one and two angular momenta. Then, in Section 5.2, we present a general study of stationary solutions with two rotational symmetries; actually, we can discuss the general case of d − 3 commuting U (1) spatial isometries. The simplest of these are the generalized Weyl solutions (Section 5.2.1). The general case is addressed in Section 5.2.2, and we discuss the characterization of solutions by their rod structures. The powerful solution-generating technique of Belinsky and Zakharov, based on inverse-scattering methods, is then introduced: the emphasis is on its practical application in generating old and new black-hole solutions. Section 5.3 discusses multiple-black-hole solutions (black Saturns, di-rings and bicycling black rings) obtained in this way. Work towards determining the stability properties of black rings and multiple black holes is reviewed in Section 5.4.

5.1 Black rings

5.1.1 One angular momentum

Five-dimensional black rings are black holes with horizon topology 1 2 S × S in asymptotically flat spacetime. The 1 S 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 S1 was presented in [83Jump To The Next Citation Point]. Its most convenient form was given in [79Jump To The Next Citation Point] as5

( ) 2 F (y) 1 + y 2 ds = − ----- dt − C R ------dψ F (x ) F(y) R2 [ G (y ) dy2 dx2 G (x) ] + --------F (x) − -----dψ2 − ----- + ----- + -----dφ2 , (50 ) (x − y)2 F (y ) G (y) G (x) F (x)
2 F (ξ) = 1 + λξ, G (ξ) = (1 − ξ )(1 + ν ξ), (51 )
∘ --------------- C = λ(λ − ν )1 +-λ. (52 ) 1 − λ
The dimensionless parameters λ and ν must lie in the range
0 < ν ≤ λ < 1 . (53 )
The coordinates vary in the ranges − ∞ ≤ y ≤ − 1 and − 1 ≤ x ≤ 1, with asymptotic infinity recovered as x → y → − 1. The axis of rotation around the ψ direction is at y = − 1, and the axis of rotation around φ is divided into two pieces: x = 1 is the disk bounded by the ring, and x = − 1 is its complement from the ring to infinity. The horizon lies at y = − 1∕ν. Outside it, at y = − 1∕λ, lies an ergosurface. A detailed analysis of this solution and its properties can be found in [84Jump To The Next Citation Point] and [76Jump To The Next Citation Point], so we shall only discuss it briefly.

In the form given above, the solution possesses three independent parameters: λ, ν, and R. Physically, this sounds like one too many: given a ring with mass M and angular momentum J, 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 (50View Equation): in general it has a conical defect on the plane of the ring, x = ±1. In order to avoid it, the angular variables must be identified with periodicity

∘ ------- √ ------ Δ ψ = Δφ = 4π --F-(−-1)-= 2 π--1-−-λ- (54 ) |G ′(− 1)| 1 − ν
and the two parameters λ, ν must satisfy
-2-ν--- λ = 1 + ν2 . (55 )
This eliminates one parameter, and leaves the expected two-parameter (ν,R ) family of solutions. The mechanical interpretation of this balance of forces for thin rings is discussed in [80Jump To The Next Citation Point]. The Myers–Perry solution with a single rotation is obtained as a limit of the general solution (50View Equation[79Jump To The Next Citation Point], but cannot be recovered if λ is eliminated through Equation (55View Equation).

The physical parameters of the solution (mass, angular momentum, area, angular velocity, surface gravity) in terms of ν and R can be found in [84Jump To The Next Citation Point]. It can be seen that while R provides a measure of the radius of the ring’s S1, the parameter ν can be interpreted as a ‘thickness’ parameter characterizing its shape, corresponding roughly to the ratio between the S2 and the S1 radii.

More precisely, one finds two branches of solutions, whose physical differences are seen most clearly in terms of the dimensionless variables j and aH introduced above. For a black ring in equilibrium, the phase curve aH (j) can be expressed in parametric form as

∘ --------- ∘ --------- (1-+-ν)3 aH = 2 ν(1 − ν), j = 8ν , (56 )
and is depicted in Figure 5View Image.
View Image

Figure 5: Curve aH (j) of horizon area vs. spin for five-dimensional black rings rotating along their 1 S (solid). The dashed curve corresponds to five-dimensional MP black holes (see Figure 1View Image). The solid curve for black rings has two branches that meet at a regular, nonextremal minimally-rotating black ring at ∘ ------ j = 27∕32: an upper branch of thin black rings, and a lower branch of fat black rings. Fat black rings always have a smaller area than MP black holes. Their curves meet at the same zero-area naked singularity at j = 1.

This curve is easily seen to have a cusp at ν = 1∕2, which corresponds to a minimum value of ∘ ------ j = 27∕32 and a maximum aH = 1. Branching off from this cusp, the thin black-ring solutions (0 < ν < 1∕2) extend to j → ∞ as ν → 0, with asymptotic aH → 0. The fat black-ring branch (1∕2 ≤ ν < 1) has lower area and extends only to j → 1, ending at ν → 1 at the same zero-area singularity as the MP solution. This implies that in the range ∘ ------ 27 ∕32 ≤ j < 1 there exist three different solutions (thin and fat black rings and MP black holes) with the same value of j. The notion of black-hole uniqueness that was proven to hold in four dimensions does not extend to five dimensions.

[76Jump To The Next Citation Point] 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 [76Jump To The Next Citation Point] considers them for tests of stability. [143Jump To The Next Citation Point] is a more complete analysis of geodesics.

5.1.2 Two angular momenta

Rotation in the second independent plane corresponds to rotation of the S2 of the ring. In the limit of infinite S1 radius, a section along the length of the ring gives an S2 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 2 S is expected (actually, the motion of the ring along its 1 S yields a momentum that can be viewed as an electric Kaluza–Klein charge, so instead of a Kerr solution, the R → ∞ limit yields a rotating electric KK black hole).

Solutions with rotation only along the S2, but not on the S1, are fairly straightforward to construct and have been given in [192Jump To The Next Citation Point87]. 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 [212Jump To The Next Citation Point] (the techniques employed are reviewed in Section 5.2). They have furthermore managed to present it in a fairly compact form:

2 H-(y,x-) 2 F-(x,-y)- 2 J-(x,-y)- -F(y,x-) 2 ds = − H (x,y )(dt + Ω ) − H (y,x)dψ − 2 H (y,x)dψd φ + H (y, x)dφ 2 ( 2 2 ) + ---2k-H-(x,y)--- -dx-- − -dy-- . (57 ) (x − y)2(1 − ν)2 G (x) G (y)
Here we follow the notation introduced in [212Jump To The Next Citation Point], except that we have chosen mostly plus signature, and exchanged φ ↔ ψ to conform to the notation in Equation (50View Equation). The reader should be warned that, although the meanings of x and y are essentially the same in both solutions, the same letters are used in Equation (57View Equation) as in Equation (50View Equation) for different parameters and functions. In particular, the angles φ and ψ have been rescaled here to have canonical periodicity 2π.

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 [196Jump To The Next Citation Point]), but they simplify significantly when a balance of forces (i.e., cancellation of conical singularities) is imposed. In this case the one-form Ω characterizing the rotation is [212Jump To The Next Citation Point]

∘ ------------- 2kλ (1 + ν)2 − λ2 [ √ -- Ω = − ------------------- (1 − x2)y νdφ H (y,x) --1-+-y---( 2 ) ] + 1 − λ + ν 1 + λ − ν + x yν(1 − λ − ν) + 2νx (1 − y) d ψ , (58 )
and the functions G, H, J, and F become
G (x) = (1 − x2 )(1 + λx + νx2 ) , 2 2 2 2 2 2 2 2 2 H (x, y) = 1 + λ − ν + 2 λν(1 − x-)y + 2xλ (1 − y ν ) + x y ν (1 − λ − ν ), 2k2(1 − x2)(1 − y2)λ√ ν ( 2 2 2 2 ) J(x, y) = ------------------2----- 1 + λ − ν + 2(x + y)λν − xy ν(1 − λ − ν ) , (59 ) (x − y)(1 − ν) ------2k2-------[ 2 [( 2 2) 2 2 ] F(x, y) = (x − y)2(1 − ν)2 G (x)(1 − y ) (1 − ν) − λ (1 + ν) + yλ(1 − λ + 2ν − 3ν ) [ 2 2 2 2 ( 2 2) 3 2 2 3 +G (y) 2λ + xλ ((1 − ν ) + λ] ) + x (1 − ν ) − λ (1 + ν) + x λ(1 − λ − 3ν + 2ν ) 4 2 2 ] − x (1 − ν )ν(− 1 + λ + ν ) .
When λ = 0 we find flat spacetime. In order to recover the metric (50View Equation) one must take ν → 0, identify 2 2 2 R = 2k (1 + λ ) and rename λ → ν.

The parameters ν and λ are restricted to

√ -- 0 ≤ ν < 1 , 2 ν ≤ λ < 1 + ν (60 )
for the existence of regular black-hole horizons. The bound λ ≥ 2 √ν- is actually a Kerr-like bound on the rotation of the 2 S. To see this, consider the equation for vanishing G (y),
2 1 + λy + νy = 0 , (61 )
which determines the position of the horizon within the allowed range − ∞ < y < − 1. If we identify y → − k∕r, λ → 2m ∕k and 2 2 ν → a ∕k, this becomes the familiar 2 2 r − 2mr + a = 0 (this is not to say that m and a correspond to the physical mass and angular momentum parameters, although they are related to them). Requiring the roots of Equation (61View Equation) to be real yields the required bound. When it is saturated, λ = 2√ ν-, the horizon is degenerate, and when exceeded, it becomes a naked singularity. The parameter k sets a scale in the solution and gives (roughly) a measure of the ring radius. The extremal Myers–Perry solution is recovered as a limit of the extremal solutions in which ν → 1, λ → 2. However, in order to recover the general Myers–Perry solution as a limit, one needs to relax the equilibrium condition that has been imposed to obtain Equation (59View Equation), and use the more general form of these functions given in [196].

The physical parameters M, J ψ = J1, J φ ≡ J2, and 𝒜H of the solution have been computed in [212Jump To The Next Citation Point]. An analysis of the physical properties of the solution, and in particular the phase space, has been presented in [78Jump To The Next Citation Point]. To plot the parameter region where black rings exist, we fix the mass and employ the dimensionless angular-momentum variables j 1 and j 2 introduced in Equation (21View Equation). The phase space of doubly-spinning black rings is in Figure 6View Image for the region j1 > j2 ≥ 0 (the rest of the plane is obtained by iterating and exchanging ±j1,2). It is bounded by three curves (besides the axis, which is not a boundary in the full phase plane):

View Image

Figure 6: Phase space of doubly-spinning black rings (j1 ≡ jψ, j2 ≡ jφ), restricted to the representative region j1 > j2 ≥ 0. The dashed line j1 + j2 = 1 corresponds to extremal MP black holes (see Figure 2View Image(a)). The (upper) thin black curve corresponds to regular extremal black rings with degenerate horizons at maximal S2 spin j 2, for given S1 rotation j 1. It ends on the extremal MP curve at (3∕4,1∕4). The (lower) thick black curve corresponds to regular nonextremal black rings with minimal spin j1 along 1 S for given j2 on 2 S. It ends on the extremal MP curve at (4∕5,1∕5). Black rings exist in the gray-shaded parameter regions bounded by the black curves, the segment j1 ∈ [3∕4,4∕5] of the extremal MP dashed line, and the j1 axis with j ≥ ∘27--∕32 1. In the light-gray region there exist only thin black rings. In the dark-gray spandrel between the dashed MP line, the thick black curve, and the axis ∘ ------ 27∕32 < j1 < 1, there exist thin and fat black rings and MP black holes: there is discrete three-fold nonuniqueness.
  1. Extremal black rings, with maximal j2 for given j1, along the curve
    √-- 1 + 4 ν + ν ν1∕4 j1 = --1∕4----√----, j2 = -----√----, 0 ≤ ν ≤ 1 (62) 4ν (1 + ν ) 2(1 + ν )
    (thin solid curve in Figure 6View Image). This curve extends between j1 = 3∕4, j2 = 1∕4 (as ν → 1) to j1 → ∞, j2 → 0 (as ν → 0). (See [217] for more discussion of extremal rings.)
  2. Nonextremal minimally spinning black rings, with minimal j1 for given j2, along the curve
    ( 2 ∘ --------------- )3∕2 3(1 + ν ) + (1 + ν) (9 + ν )(1 + 9ν ) − 26ν j1 = ---------∘-(-∘------------------------)(-----------∘-----------------)-, 8(1 − ν)2 (9 + ν)(1 + 9 ν) − 1 − ν 5(1 + ν) − (9 + ν)(1 + 9 ν) ∘ -(--√-------------(-√----------------(---√--------------))) ν-3+--(9+ν)(1+9ν)+ν-18--(√9+ν)(1+9ν)−-103+ν-3ν+--(9+ν)(1+9ν)−-103--- (9+ ν)(1+9ν)−1− ν j2 = --------------------------√----------------------------------- (63) 2 2(1 − ν)2
    (thick solid curve in Figure 6View Image). This curve extends between j = 4 ∕5 1, j = 1∕5 2 (as ν → 1) and ∘ ------ j1 = 27∕32, j2 = 0 (as ν → 0).
  3. Limiting extremal MP black holes, with j1 + j2 = 1 within the range j1 ∈ [3∕4,4 ∕5] (dashed line in Figure 6View Image). There is a discontinuous increase in the area when the black rings reach the extremal MP line.

For doubly-spinning black rings the angular momentum along the S2 is always bounded above by the one in the S1 as

|j | < 1-|j |. (64 ) 2 3 1
This is saturated at the endpoint of the extremal black-ring curve j = 3 ∕4 1, j = 1∕4 2.

Figure 7View Image 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 (M, J1, J2), 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 2View Image(b).

View Image

Figure 7: The phase space (j1,j2) covered by doubly-spinning MP black holes and black rings, obtained by replicating Figure 6View Image taking ±j1 ↔ ±j2. The square |j1| + |j2| ≤ 1 corresponds to MP black holes (see Figure 2View Image(a)). The light-gray zones contain thin black rings only, and the medium-gray zone contains MP black holes only. At each point in the dark-gray spandrels near the corners of the square there exist one thin and one fat black ring, and one MP black hole.

[78Jump To The Next Citation Point] contains sectional plots of the surface aH(j1,j2) for black rings at constant j2, for j1 > j2 ≥ 0, from which it is possible to obtain an idea of the shape of the surface. In the complete range of j1 and j2 the phase space of five-dimensional black holes (with connected event horizons) consists of the ‘dome’ of MP black holes (Figure 4View Image 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 [78Jump To The Next Citation Point]. It is curious to notice that even if the two angular momenta can never be equal, the two angular velocities Ω1 and Ω2 have equal values, for a given mass, when

∘ -----√---------- λ = 2ν − ν (1 + ν ), (65 )
which lies in the allowed range (60View Equation). We can easily understand why this is possible: Ω1 becomes arbitrarily small for thin rings, even if j1 is large, so it can be made equal to any given Ω2.

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.

5.2 Stationary axisymmetric solutions with d − 3 rotational symmetries

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 [82Jump To The Next Citation Point] and extended in [127Jump To The Next Citation Point]. 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 [139Jump To The Next Citation Point].

Since most of the analysis is applicable to any number of dimensions, we will keep d arbitrary, although only in d = 4,5 can the solutions be globally asymptotically flat. So, henceforth, we assume that the spacetime admits d − 2 commuting, non-null, Killing vectors a ξ(a) = ∂ ∕∂x, a = 0,...,d − 3 (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 [82Jump To The Next Citation Point]. In this case the metrics admit the form

2 a b 2ν(r,z) ( 2 2) ds = gab(r,z) dx dx + e dr + dz . (66 )
Without loss of generality we can choose coordinates so that
2 det gab = − r . (67 )
For this class of geometries, the Einstein equations divide into two groups, one for the matrix g,
∂rU + ∂zV = 0 , (68 )
U = r(∂rg)g−1 , V = r(∂zg)g− 1, (69 )
and a second group of equations for ν,
∂r ν = − 1--+ -1-Tr (U 2 − V 2), ∂zν = 1--Tr(U V ). (70 ) 2r 8r 4r
The equations for ν satisfy the integrability condition ∂r∂zν = ∂z∂rν as a consequence of Equation (68View Equation). Therefore, once g (r,z) ab is determined, the function ν(r,z) is determined by a line integral, up to an integration constant that can be absorbed by rescaling the coordinates.

Equations (68View Equation) and (69View Equation) are the equations for the principal chiral field model, a nonlinear sigma model with group GL (d − 2,ℝ ), which is a completely integrable system. In the present case, it is also subject to the constraint (67View Equation). 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 [82Jump To The Next Citation Point].

5.2.1 Weyl solutions

Consider the simplest situation in which the Killing vectors are mutually orthogonal. In this case the solutions admit the diagonal form6

d∑−3 ds2 = − e2U0dt2 + e2Ua(dxa)2 + e2ν(dr2 + dz2) . (71 ) a=1
Equations (68View Equation) require that Ua (r,z), a = 0, ...,d − 3, be axisymmetric solutions of the Laplace equation
( ) ∂2 + 1-∂r + ∂2 Ua = 0 , (72 ) r r z
in the auxiliary three-dimensional flat space
2 2 2 2 2 ds = dz + dr + r dφ , (73 )
while equations (70View Equation) for ν(r,z) become
1 r d∑−3[ 2] ∂rν = − ---+ -- (∂rUa ) − (∂zUa ) , (74 ) 2r 2 a=0 d∑−3 ∂zν = r ∂rUa∂zUa . (75 ) i=0
The constraint (67View Equation) implies that only d − 3 of the Ua are independent, since they must satisfy
d∑−3 Ua = logr . (76 ) a=0
Thus we see that solutions are fully determined once the boundary conditions for the Ui are specified at infinity and at the z-axis. Note that log r is the solution that corresponds to an infinite rod of zero thickness and linear mass density 1∕2 along the axis r = 0. The solutions are in fact characterized by the ‘rod’ sources of U i along the axis, which have to add up to an infinite rod (76View Equation). The potential for a semi-infinite rod along [ak, +∞ ) with linear density ϱ is
U = ϱ log μk , (77 )
∘ -2-----------2 μk = r + (z − ak) − (z − ak ). (78 )
If the rod instead extends along (− ∞, ak] then
U = ϱ log ¯μk , (79 )
-------------- 2 ¯μ = − ∘ r2 + (z − a )2 − (z − a ) = −-r-. (80 ) k k k μk
Given the linearity of equations (72View Equation), one can immediately construct the potential for a finite rod of density ϱ along [ak−1,ak] as
( μ ) U = ϱ log -k−1- . (81 ) μk
The functions Ui for any choice of rods are sums of these. Integration of Equation (74View Equation) is then a straightforward, if tedious, matter; see appendix G in [82Jump To The Next Citation Point]. [165Jump To The Next Citation Point] has applied the inverse scattering method (to be reviewed below) to provide explicit diagonal solutions with an arbitrary number of rods.

At a rod source for Ui, the orbits of the corresponding Killing vector vanish: if it is an angular Killing vector ∂ φi, then the corresponding one-cycles shrink to zero size, and the periodicity of φ i must be chosen appropriately in order to avoid conical singularities; if it is instead the timelike Killing ∂t, 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

1- ϱ = 2 . (82 )
Otherwise when r → 0 at the rod, the curvature diverges. If all rods are of density 1 ∕2, then given the constraint (76View Equation), at any given point on the axis there will be one cycle of zero (or null) length with all others having finite size. This phenomenon of some cycles smoothly shrinking to zero with other cycles blowing up to finite size as one moves along the axis (essentially discovered by Weyl in 1917 [247]), is referred to in string-theoretical contexts as ‘bubbling’. When it is ∂ t that becomes null, the rod corresponds to a horizon.

The rod structures for the four and five-dimensional Schwarzschild and Tangherlini solutions are depicted in Figure 8View Image. The Rindler space of uniformly-accelerated observers is recovered as the horizon rod becomes semi-infinite, a2 → ∞. [82Jump To The Next Citation Point] gives a number of ‘rules of thumb’ for interpreting rod diagrams.

View Image

Figure 8: Rod structures for the (a) 4D Schwarzschild and (b) 5D Tangherlini black holes. From top to bottom, the lines represent the sources for the time, φ and ψ (in 5D) potentials Ut, U φ, U ψ.

5.2.2 General axisymmetric class

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. Rod structure and regularity
Let us begin by extending the characterization of rods [127Jump To The Next Citation Point]. In general, a rod is an interval along the z-axis, where the action of a Killing vector v has fixed points (for a spacelike vector) or it becomes null (for a timelike vector). In both cases |v|2 → 0 at the rod. In general, v will be a linear combination v = va ξ(a) in a given Killing basis ξ(a), a = 0,...,d − 3; usually, this basis is chosen so that it becomes a coordinate basis of orthogonal vectors at asymptotic infinity. In the orthogonal case of the previous section 5.2.1 one could assign a basis vector (and only one, for rod density 1 ∕2) ξ(a) to each rod, but this is not possible, in general, if the vectors are not orthogonal. For instance, at a rod corresponding to a rotating horizon, the Killing vector that vanishes is typically of the form ∂t + Ωa∂ φa.

More precisely, the condition (67View Equation) implies that the matrix g(r = 0,z ) must have at least one zero eigenvalue. It can be shown that regularity of the solution (analogous to the requirement of density 1∕2 in the orthogonal case) requires that only one eigenvalue is zero at any given interval on the axis [127Jump To The Next Citation Point]. Each such interval is called a rod, and, for each rod z ∈ [a, b], we assign a direction vector v such that

g(0,z)v = 0 for z ∈ [a,b]. (83 )
At isolated points on the axis the kernel of g(r = 0,z) is spanned by two vectors instead of only one: this will happen at the points where two intervals with different eigenvectors meet. The rod structure of the solution consists of the specification of intervals a1 < a2 < ⋅⋅⋅ < an, plus the n + 1 directions v(k) ∈ ker(g(0,z)) for each rod [ak−1,ak], k = 1,...,n + 1 (with a = − ∞ 0, a = + ∞ n+1[127]. The vector v is defined up to an arbitrary normalization constant.

The rod is referred to as timelike or spacelike according to the character of the rescaled norm gab(r,z)vavb∕r2 at the rod r → 0, z ∈ [ak−1,ak]. 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

∂-- v = ∂ψ (84 )
with norm
2 i j |v| = gij(r,z)v v (85 )
the length L = |v|Δ ψ of its circular orbits at constant r, z, vanishes near the rod like ∼ r. Since the proper radius of these circles approaches eν(0,z)dr, a conical singularity will be present at the rod unless these orbits are periodically identified with period
( ) −1 Δ ψ = 2πe ν(0,z) ∂|v|(0,z) . (86 ) ∂r
When several rods are present, it may be impossible to satisfy simultaneously all the periodicity conditions. The physical interpretation is that the forces among objects in the configuration cannot be balanced and as a result, conical singularities appear. If the geometry admits a Wick rotation to Euclidean time, then Equation (86View Equation) gives the temperature TH = 2π ∕Δ ψ of the horizon associated with the rod.

Second, the presence of time components on a spacelike rod creates causal pathologies. Consider a vector ζ that is timelike in some region of spacetime, and whose norm does not vanish at a given spacelike rod [a,b]. If the direction vector v associated with this rod is such that

v ⋅ ζ ⁄= 0 at r = 0, z ∈ [a,b], (87 )
then making the orbits of v periodic introduces closed timelike curves; periodicity Δ ψ along orbits of v requires that the orbits of ζ be identified as well with periods equal to an integer fraction of (v ⋅ ζ)Δ ψ. Then, closed timelike curves will appear wherever ζ is timelike. These are usually regarded as pathological if they occur outside the horizon, so such time components on spacelike rods must be avoided.

Further analysis of Equations (68View Equation), (70View Equation) and their source terms can be found in [130]. The method of Belinsky and Zakharov
As shown by Belinsky and Zakharov (BZ), the system of nonlinear equations (67View Equation) and (68View Equation) is completely integrable [1110Jump To The Next Citation Point]. It admits a Lax pair of linear equations (spectral equations) whose compatibility conditions coincide with the original nonlinear system. This allows one to generate an infinite number of solutions, starting from known ones, following a purely algebraic procedure. Since, as we have seen, we can very easily generate diagonal solutions, this method allows us to construct a vast class of axisymmetric solutions. It seems likely, but to our knowledge has not been proven, that all axisymmetric solutions can be generated in this way. The spectral versions of Equations (67View Equation) and (68View Equation) are

D1 Ψ = rV-−--λU- Ψ , D2 Ψ = rU-+--λV- Ψ , (88 ) λ2 + r2 λ2 + r2
where λ is the (in general complex) spectral parameter, independent of r and z, and D1 and D2 are two commuting differential operators,
2λ2 2λr D1 = ∂z − ------- ∂ λ, D2 = ∂r + ------- ∂λ . (89 ) λ2 + r2 λ2 + r2
The function Ψ(λ, r,z ) is a (d − 2) × (d − 2) matrix such that Ψ (0,r,z) = g(r,z), where g is a solution of Equation (68View Equation). Compatibility of Equations (88View Equation) then implies Equations (68View Equation) and (69View Equation).

Since Equations (88View Equation) are linear, we can construct new solutions by ‘dressing’ a ‘seed’ solution g0. The seed defines matrices U0 and V0 through Equations (69View Equation). Equations (88View Equation) can then be solved to determine Ψ0. Then, we ‘dress’ this solution using a matrix χ = χ (λ,r,z) to find a new solution of the form

Ψ = χΨ0 . (90 )
Introducing Equation (90View Equation) into Equation (88View Equation) we find a system of equations for χ. The simplest and most interesting solutions are the solitonic ones, for which the matrix χ(λ,r,z) can be written in terms of simple poles
n χ = 1 + ∑ --Rk---. (91 ) λ − &tidle;μk k=1
The residue matrices Rk and the ‘pole-position’ functions μ&tidle;k depend only on r and z. For a dressing function of this form, it is fairly straightforward to determine the functions &tidle;μk and the matrices R k. The pole positions are
{ μ for a soliton, &tidle;μk = k (92 ) ¯μk for an antisoliton,
where μk, ¯μk were introduced in Equations (78View Equation) and (80View Equation). In principle the parameters ak may be complex and must appear in conjugate pairs, if the metric is to be real. However, in all the examples that we consider, the ak are real; complex poles appear to lead to naked singularities, for instance, in the Kerr solution they occur when the extremality bound on the angular momentum is violated.

The solution for the matrices Rk, where k labels the solitons, can be constructed by first introducing a set of d − 2-dimensional vectors m (k) using the seed as

(k) (k)[ −1 ] m a = m 0b Ψ 0 (&tidle;μk,r,z )ba . (93 )
The constant vectors m (0k) introduced here are the crucial new data determining the rod orientations in the new solution. Now defining the symmetric matrix
(k) (l) Γ = m-a--(g0)abm-b--, (94 ) kl r2 + μ&tidle;k &tidle;μl
the Rk are
∑n (Γ − 1) m (cl)(g ) (Rk )ab = m (ka) -----lk------0cb . (95 ) l=1 &tidle;μl

All the information about the solitons that are added to the solution is contained in the soliton positions ak and the soliton-orientation vectors m (0k). This is all we need to determine the dressing matrix in Equation (91View Equation), and then the new metric g(r,z) = Ψ (0, r,z ) = χ (0,r,z)Ψ0(0,r,z ),

∑n (k) − 1 (l) g = (g ) − (g0)ac-m-c-(Γ---)kl-m-d-(g0)db-. (96 ) ab 0 ab μ&tidle;k &tidle;μl k,l=1

There is, however, one problem that turns out to be particularly vexing in d > 4: the new metric g in Equation (96View Equation) does not satisfy, in general, the constraint (67View Equation); the introduction of the n solitons gives a determinant for the new metric

( ) n 2n ∏n − 2 detg = (− 1) r &tidle;μk detg0 . (97 ) k=1
A determinant of a new physical solution g(phys) must satisfy constraint (67View Equation), but g in Equation (97View Equation) does not. This problem can be expediently resolved by simply multiplying the metric obtained in Equation (96View Equation) by an overall factor
( 2 ) d1−2 g(phys) = ± --r---- g (98 ) ±detg
(we may ignore here the choice of ± signs). Now, observe that the problem that Equation (98View Equation) solves is that of making the rod densities at each point along the axis to add up to a total density of 1∕2. However, recall that regularity required that individual rod densities, and not just their sums, be exactly equal to 1∕2. This is a problem for Equation (98View Equation) whenever d > 4; since Equation (97View Equation) contains only solitons and antisolitons with regular densities ±1 ∕2 (with − 1∕2 allowed only at intermediate steps), the fractional power in Equation (98View Equation) introduces rods with fractional densities 1∕ (d − 2 ), which in d > 4 will always result in curvature singularities at the rod.

A possible way out of this problem is to restrict oneself to transformations that act only on a 2 × 2 block of the seed g 0 [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 2 × 2 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 [211Jump To The Next Citation Point], that can deal with the general case in any number of dimensions. The key idea is the observation that Equation (97View Equation) is independent of the ‘realignment’ vectors (k) m 0b. 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 − 1∕2), and then re-add these same solitons, but now with different vectors m(k) 0b, so the rods affected by these solitons acquire, in general, new directions. If the original seed solution satisfied the determinant constraint (67View Equation), 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 ±1 ∕2 and negative densities do not appear in the end result, the final metric will only contain regular 1∕2 densities.

In the simplest form of this method, one starts from a diagonal, hence static, solution 2ν0 (g0,e ) and then removes some solitons or antisolitons with ‘trivial’ vectors aligned with one of the Killing basis vectors m (k)= ξ 0 (a), i.e., m (k)= δ 0b ab (recall that in the diagonal case this alignment of rods is indeed possible). Removing a soliton or antisoliton μ&tidle;k at z = ak aligned with the direction a amounts to changing

2 (g′)aa = − &tidle;μ-k(g0)aa , (99 ) 0 r2
while leaving unchanged all other metric components (g′)bb = (g0)bb 0 with b ⁄= a. We now take this new metric ′ g0 as the seed to which, following the BZ method, we re-add the same solitons and antisolitons, but with more general vectors (k) ∑ m 0 = bCbξ(b). Note that there is always the freedom to re-scale each of these vectors by a constant. Finally, the two-dimensional conformal factor e2ν for the new solution is obtained from the seed as
2ν 2ν0-detΓ- e = e det Γ 0 , (100 )
where the matrices Γ 0 and Γ are obtained from Equation (94View Equation) using g0 and g respectively.

If the vectors m (0ka) 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 Ψ0 (λ, r,z) that solves Equation (88View Equation) 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 Ψ0 (λ,r,z) 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 [212Jump To The Next Citation Point] was obtained in this manner. BZ construction of the Kerr, Myers–Perry and black-ring solutions
We sketch here the method to obtain all known black-hole solutions with connected horizons in four and five dimensions. These solutions illustrate how to add rotation to S2, S3 and S1 × S2 components of multiple-black-hole horizons.

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 [82Jump To The Next Citation Point], and whose rod structure is depicted in Figure 8View Image. The seed metric is

{ } g0 = diag − μ1,r2μ2- , (101 ) μ2 μ1
with a1 < a2. Then, remove an antisoliton at z = a1, with vector (1) m 0 = (1,0), and a soliton at z = a2 with the same vector. Following rule (99View Equation) we obtain the matrix
{ } ′ μ2- 2μ2- g0 = diag − μ1,r μ1 . (102 )
It is now convenient (but not necessary) to re-scale the metric by a factor μ1∕μ2. This yields the new metric
{ } &tidle;g′0 = diag − 1,r2 . (103 )
This is nothing but flat space; what we have done here is undo the generation of the Schwarzschild metric out of Minkowski space. But now the idea is to retrace our previous steps, after re-adding the solitons with new vectors m (01,2).

So, following the method above, add an antisoliton at z = a1 and a soliton at z = a2, with respective constant vectors (1) m 0 = (1,A1 ) and (2) m 0 = (1,A2 ). For this step, we need to construct the matrix Γ in Equation (94View Equation), which in turn requires the matrix

&tidle;Ψ0 (λ,r,z) = diag{ − 1, r2 − 2z λ − λ2} (104 )
that solves the spectral Equations (88View Equation) for &tidle;g′0. Equations (93View Equation), (94View Equation), and (96View Equation) then give a new metric &tidle;g. But we still have to undo the rescaling we did to get &tidle;g0′ from g′0. That is,
g = μ2&tidle;g . (105 ) μ1
By construction, g is correctly normalized, i.e., it satisfies Equation (67View Equation). Finally, the function 2ν e is obtained using Equation (100View Equation), which is straightforward since we obtained Γ when the solitons were re-added, and Γ 0 = Γ (A1 = A2 = 0). The new solution contains two more parameters than the Schwarzschild seed: these correspond to the rotation and nut parameters. The latter can be set to zero once the parameters are correctly identified. For details about how the Boyer-Lindquist form of the Kerr solution is recovered, see [10].

The Myers–Perry black hole with two angular momenta is obtained in a very similar way [211Jump To The Next Citation Point] starting from the five-dimensional Schwarzschild–Tangherlini solution, whose rod structure is shown in Figure 8View Image. We immediately see that

{ } μ1- r2- g0 = diag − μ2 ,μ2,μ1 . (106 )
Now remove an antisoliton at z = a1 and a soliton at z = a2, both with vectors (1,0,0), to find
{ 2} g′ = diag − μ2-,μ2, r- . (107 ) 0 μ1 μ1
An overall rescaling by μ1 ∕μ2 simplifies the metric to
{ 2 } g&tidle;′0 = diag − 1, μ1, r = diag {− 1,μ1,− ¯μ2} . (108 ) μ2
This is the metric that we dress with solitons applying the BZ method. Observe that it does not satisfy Equation (67View Equation) and so it is not a physical metric, but this is not a problem. The associated &tidle; Ψ0 is
&tidle;Ψ (λ, r,z) = diag{− 1,μ − λ, −μ¯ + λ } . (109 ) 0 1 2
Now add the antisoliton at z = a1 and the soliton at z = a2, with vectors (1) m 0 = (1,0,B1 ) and m (2)= (1,A2, 0) 0 (more general vectors give singular solutions). A final rescaling of the metric by μ2∕ μ1 yields a physically-normalized solution. The two new parameters that we have added are associated with the two angular momenta. The MP solution with a single spin can be obtained through a one-soliton transformation, which is not possible for Kerr. See [211] for the transformation to the coordinates used in Section 4.2.

The black ring with rotation along the S1 requires a more complicated seed, but, on the other hand, it requires only a one-soliton transformation [77Jump To The Next Citation Point] (the first systematic derivations of this solution used a two-soliton transformation [148Jump To The Next Citation Point240]). The seed is described in Figure 9View Image. The static black ring of [82Jump To The Next Citation Point] (which necessarily contains a conical singularity) is recovered for a1 = a2. However, one needs to introduce a ‘phantom’ soliton point at a1 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 z = a1 with direction (1,0,0), and re-add it with vector (1) m 0 = (1,0,B1 ). At the end of the process one must adjust the parameters, including B1 and the rod positions, to remove a possible singularity at the phantom point a1.

View Image

Figure 9: Rod structures for (a) the seed used to generate (b) the rotating black ring. The seed metric is diagonal, and the dotted rod has negative density − 1∕2. In the final solution the parameters can be adjusted so that the metric at z = a1 on the axis is completely smooth. The (upper) horizon rod in the final solution has mixed direction (1,0,Ω ψ), while the other rods are aligned purely along the φ or ψ directions.

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 S2 of the ring is similar to the rotation of the Kerr solution. In fact, the black-ring solutions with rotation only along the S2 can be obtained by applying to the static black ring the same kind of two-soliton transformations that yielded Kerr from a Schwarzschild seed (101View Equation[239]. Hence, if we begin from the black ring rotating along the S1 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 Ψ0(λ,r,z ) for the single-spin black-ring solution (50View Equation). However, if we construct this solution via a one-soliton transformation as we have just explained, this function is directly obtained from Equation (90View Equation). In this manner, solution (57View Equation) 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 Ψ&tidle;0, 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. Other methods
In four dimensions there exists another technique, akin to the Bäcklund transformation that adds solitons to a seed solution, to integrate the equations for the stationary axisymmetric class of vacuum solutions [20313729]. [192Jump To The Next Citation Point149] have extended this to higher dimensions. Unfortunately, though this method can produce simpler expressions than the BZ technique, it cannot deal with more than two off-diagonal terms, and hence, no more than a single angular momentum. [192] applied this method to derive a black ring with rotation in the S2 (but not along the S1). The same authors used this technique to provide the first systematic derivation (i.e., through explicit integration of Einstein’s equations, instead of guesswork) of the black ring with rotation along 1 S [148]. The connection between the Bäcklund transformation method and the BZ technique has been studied in [238], where it is argued that all the solutions obtained by a two-soliton Bäcklund transformation on an arbitrary diagonal seed are contained among those that the BZ method generates from the same seed. This is not too surprising in view of similar, and more general, results in four dimensions [5354]. It may be interesting to investigate the application of related but more efficient, axisymmetric solution-generating methods [226189] to higher dimensions.

[114] develops a different algebraic method to obtain stationary axisymmetric solutions in five dimensions from a given seed. An SO (2,1) subgroup of the “hidden” SL (3,ℝ ) 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.

5.3 Multiple-black-hole solutions

In d = 4 it is believed that there are no stationary multiple-black-hole solutions of vacuum gravity. However, such solutions do exist in d = 5. ‘Black Saturn’ solutions, in which a central MP-type of black hole is surrounded by a concentric rotating black ring, have been constructed in [77Jump To The Next Citation Point]. 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 [73Jump To The Next Citation Point]. 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 M and J. Observe also that for fixed total (M, J) 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 M and J. 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 M and J.

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 10View Image (see [73Jump To The Next Citation Point]). 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.

View Image

Figure 10: Curves aH (j) for phases of five-dimensional black holes with a single angular momentum: MP black hole (black), black ring (dark gray), black Saturn (light gray). We only include those black Saturns, where the central black hole and the black ring have equal surface gravities and angular velocities. The three curves meet tangentially at a naked singularity at j = 1, aH = 0. The cusp of the black-ring curve occurs at ∘ ------ j = 27∕32 ≈ 0.9186, aH = 1. The cusp of the black-Saturn curve is at j ≈ 0.9245, with area aH ≈ 0.81.

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 [152Jump To The Next Citation Point78Jump To The Next Citation Point], 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 [15278] 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 S1, but not in the orthogonal plane, on the S2 – 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 [73Jump To The Next Citation Point], that multiple-black-hole solutions allow one to cover the entire (j1,j2) 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 (j1,j2) plane than already covered by solutions with a connected horizon;see Figure 7View Image. 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 Ω1 = Ω2. Thus, if we consider two identical doubly-spinning thin black rings, one on each of the two planes, then we can make them have S1 -angular velocity equal to the S2-angular velocity of the other ring in the orthogonal plane. These solutions then lie along the lines |j1| = |j2|, reaching arbitrarily large |j1,2|, which is not covered by the single-black-hole phases in Figure 7View Image. 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 S2 might allow one to cover portions of the (j1,j2) plane beyond Figure 7View Image. If so, this would be unlike the situation with a single rotation, where thermal-equilibrium Saturns lie within the range of j of black rings, Figure 10View Image.

The Weyl ansatz of Section 5.2.1 enables one to easily generate solutions in d = 5 with multiple black holes of horizon topology 3 S, which are asymptotically flat [82231]. 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.

5.4 Stability

The linearized perturbations of the black-ring metric (50View Equation) 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 [71Jump To The Next Citation Point]), which were expected to suffer from GL-type instabilities. The instability of boosted black strings was indeed confirmed in [144Jump To The Next Citation Point]. Thus, thin black rings are expected to be unstable to the formation of ripples along their 1 S direction. This issue was examined in further detail in [76Jump To The Next Citation Point], which found that thin black rings seem to be able to accommodate unstable GL modes down to values j ∼ O(1). 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 [76Jump To The Next Citation Point]. 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 [76Jump To The Next Citation Point] 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 (j,aH ). 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 [76Jump To The Next Citation Point] 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 j ∼ O (1). 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 S2 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 S2 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 1 S 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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