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6 Vacuum Solutions in More Than Five Dimensions

With no available techniques to construct asymptotically-flat exact solutions beyond those found by Myers and Perry, the situation in d ≥ 6 is much less developed than in d = 5. Recall that the symmetry requirements for the class of solutions (66View Equation) are incompatible with the asymptotic symmetries of globally flat space in d > 5; metrics of type (66View Equation) necessarily involve directions with nonflat asymptotics (e.g., infinitely-extended black strings and branes) and/or asymptotic compact Kaluza–Klein circles. In order to preserve asymptotic flatness one would instead ask for rotational symmetry around ‘axes’ that are hyperplanes of a codimension larger than two, but the integrability of the resulting equations [3435] remains essentially an unsolved problem.

However, despite the paucity of exact solutions, there are strong indications that the variety of black holes that populate general relativity in d ≥ 6 is vastly larger than in d = 4,5. A first indication came from the conjecture in [81Jump To The Next Citation Point] 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 d ≥ 5 was argued in [14476Jump To The Next Citation Point]. More recently, [80Jump To The Next Citation Point] has constructed approximate solutions for black rings in any d ≥ 5 and then exploited the conjecture of [81Jump To The Next Citation Point] to try to draw a phase diagram with connections and mergers between the different expected phases. In the following we summarize these results.

6.1 Approximate solutions from curved thin branes

In the absence of exact techniques, [80Jump To The Next Citation Point] 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 [126115Jump To The Next Citation Point156116]7. The basic idea is to find two widely separated scales in the problem, call them R1 and R2, with R2 ≪ R1, and then try to solve the equations in two limits; first, as a perturbative expansion for small R2, and then in an expansion in 1 ∕R1. The former solves the equations in the far region r ≫ R2 in which the boundary condition, e.g., asymptotic flatness, fixes the integration constants. The second expansion is valid in the near region r ≪ R1. In order to fix the integration constants in this case, one matches the two expansions in the overlap region R2 ≪ r ≪ R1 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 S1 × Sd −3, we take the scales R ,R 1 2 to be the radii of the S1 and Sd−3, respectively8. To implement the above procedure, we take R2 = r0, the horizon radius of the d−3 S of a straight boosted black string, and R1 = R 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 r0∕r, 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 1∕R. 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 d = 5, where an exact solution is available, to d > 5; simple linear functions of r in d = 5 change to hypergeometric functions in d > 5. We take this as an indication that exact closed analytical forms for these solutions may not exist in d > 5.

We will not dwell here on the details of the perturbative construction of the solution (see [80Jump To The Next Citation Point] 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 p+1 q ℝ × S, with q = d − p − 2. When viewed at distances much larger than the size r0 of the Sq, we can approximate the metric of the black-brane spacetime by the gravitational field created by an ‘equivalent source’ with distributional energy tensor Tνμ ∝ rq−0 1δ(q+1)(r), with nonzero components only along directions tangent to the worldvolume, and where r = 0 corresponds to the location of the brane. Now we want to put this same source on a curved, compact p-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 M and angular momenta Ji of the new object by integrating Ttt and T it over the entire spatial section of the brane worldvolume. Moreover, the total area 𝒜 H is similarly obtained by replacing the volume of ℝp with the volume of the new surface. Thus, it appears that we can easily obtain the relation 𝒜H (M, Ji) 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, 𝒜H will be a function not only of (M, Ji), 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 (M, Ji) 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 R = f(M, J ). This is reflected in the fact that, in the new situation, the stress-energy tensor is in general not conserved, ∇μT μν ⁄= 0; 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]

K ρμνT μν = 0, (110 )
where ρ K μν is the second fundamental tensor, characterizing the extrinsic curvature of the embedding surface spanned by the brane worldvolume. For a string on a circle of radius R in flat space, parameterized by a coordinate z ∼ z + 2πR, this equation becomes
Tzz-= 0 . (111 ) R
In d = 5, this can be seen to correspond to the condition of the absence of conical singularities in solution (50View Equation), in the limit of a very thin black ring [71Jump To The Next Citation Point]. [80Jump To The Next Citation Point] shows that this condition is also required in d ≥ 6 in order to avoid curvature singularities on the plane of the ring.

In general, Equation (110View Equation) constrains the allowed values of the parameters of a black brane that can be put on a given surface. [80Jump To The Next Citation Point] easily derives, for any d ≥ 5, that the radius R of thin rotating black rings of given M and J is fixed to

-d-−-2--J- R = √ d − 3M ; (112 )
so large R corresponds to large spin for fixed mass. The horizon area of these thin black rings goes like
𝒜 (M, J) ∝ J− d1−4 M dd−−24 . (113 ) H
This is to be compared to the value for ultraspinning MP black holes in d ≥ 6 (cf. Equations (40View Equation), (41View Equation) as ν → 0),
− d2−5 dd−−25 𝒜H (M, J) ∝ J M . (114 )
This shows that in the ultraspinning regime the rotating black ring has larger area than the MP black hole.

6.2 Phase diagram

Using the dimensionless area and spin variables (21View Equation), Equation (113View Equation) allows one to compute the asymptotic form of the curve aH(j) in the phase diagram at large j for black rings. However, when j 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, [80Jump To The Next Citation Point] 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 [81Jump To The Next Citation Point], discussed in Section 4.1, that in the ultraspinning regime in d ≥ 6, MP black holes approach the geometry of a black membrane 2 d−4 ≈ ℝ × S spread out along the plane of rotation.

We have already discussed how using this analogy, [81Jump To The Next Citation Point] 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 [117120248], [81Jump To The Next Citation Point] 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.

View Image

Figure 11: Correspondence between phases of black membranes wrapped on a two-torus of side L (left) and quickly-rotating MP black holes with rotation parameter a ∼ L ≥ r0 (right: must be rotated along a vertical axis): (i) uniform black membrane and MP black hole; (ii) nonuniform black membrane and pinched black hole; (iii) pinched-off membrane and black hole; (iv) localized black string and black ring (reproduced from [80Jump To The Next Citation Point]).

[80Jump To The Next Citation Point] 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 11View Image. Although the analogy has several limitations, it allows one to propose a phase diagram in d ≥ 6 of the form depicted in Figure 12View Image, which should be compared to the much simpler diagram in five dimensions, Figure 10View Image. 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.

View Image

Figure 12: Proposal of [80Jump To The Next Citation Point] for the phase curves aH(j) of thermal equilibrium phases in d ≥ 6. The solid lines and figures have significant arguments in their favor, while the dashed lines and figures might not exist and admit conceivable, but more complicated, alternatives. Some features have been drawn arbitrarily; at any given bifurcation, and in any dimension, smooth connections are possible instead of swallowtails with cusps; also, the bifurcation into two black-Saturn phases may happen before, after, or right at the merger with the pinched black hole. Mergers to di-rings or multiple-ring configurations that extend to asymptotically large j seem unlikely. If thermal equilibrium is not imposed, the whole semi-infinite strip 0 < aH < aH(j = 0), 0 ≤ j < ∞ is covered, and multiple rings are possible.

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 d = 6 (and not in d = 5).

6.3 Stability

The situation in d ≥ 6 is very similar to what we described for d = 5 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 d ≥ 6, 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 d ≥ 6. 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 j) 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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