In this section, we review these solutions and their properties, beginning from black holes with a single rotation, and then extending them to arbitrary rotation. The existence of ultraspinning regimes in is emphasized. The symmetries and stability of the MP solutions are also discussed.
Let us begin with solutions that rotate in a single plane. These are not only simpler, but also exhibit more clearly the qualitatively new physics afforded by the additional dimensions.
The metric takes the form4
As in Tangherlini’s solution, this metric seems to follow from a rather straightforward extension of the Kerr solution, which is recovered when . The first line in Equation (32) looks indeed like the Kerr solution, with the falloff replaced, in appropriate places, by . The second line contains the line element on a ()-sphere, which accounts for the additional spatial dimensions. It might, therefore, seem that, again, the properties of these black holes should not differ much from their four-dimensional counterparts.
However, this is not the case. Heuristically, we can see the competition between gravitational attraction and centrifugal repulsion in the expression
Given the similarities between Equation (32) and the Kerr solution, it is clear that the outer event horizon lies at the largest (real) root of , i.e., . Thus, we expect that the features of the event horizons will be strongly dimension dependent, and this is indeed the case. If there is an event horizon at ,
For , is always positive at large values of , but the term makes it negative at small (we are assuming positive mass). Therefore always has a (single) positive real root independent of the value of . Hence, regular black-hole solutions exist with arbitrarily large . Solutions with large angular momentum per unit mass are referred to as “ultraspinning”.
An analysis of the shape of the horizon in the ultraspinning regime shows that the black holes flatten along the plane of rotation ; the extent of the horizon along this plane is , while, in directions transverse to this plane, its size is . In fact, a limit can be taken in which the ultraspinning black hole becomes a black membrane with horizon geometry . This turns out to have important consequences for black holes in , as we will discuss later. The transition between the regime in which the black hole behaves like a fairly compact, Kerr-like object, and the regime in which it is better characterized as a membrane, is most clearly seen by analyzing the black hole temperature
The properties of the solutions are conveniently encoded using the dimensionless variables , introduced in Equation (21). For the solutions (32) the curve can be found in parametric form, in terms of the dimensionless ‘shape’ parameter , as
 also gives black-hole solutions with arbitrary rotation in each of the independent rotation planes. The cases of odd and even are slightly different. When is odd, the solution is
For both cases we can write the functions and as
The determination of involves an equation of degree , which in general is difficult, if not impossible, to solve algebraically. So the presence of horizons for generic parameters in Equation (42) and (43) is difficult to ascertain. Nevertheless, a number of features, in particular the ultraspinning regimes that are important in the determination of the allowed parameter range, can be analyzed.
Following Equation (21), we can fix the mass and define dimensionless quantities for each of the angular momenta. Up to a normalization constant, the rotation parameters at fixed mass are equivalent to the . We take as the coordinates in the phase space of solutions. We aim to determine the region in this space that corresponds to actual black-hole solutions.
Consider first the case in which all spin parameters are nonzero. Then an upper extremality bound on a combination of the spins arises. If it is exceeded, naked singularities appear, as in the Kerr black hole . So we can expect that, as long as all spin parameters take values not too dissimilar, , all spins must remain parametrically , i.e., there is no ultraspinning regime in which all .
Next, observe that for odd , a sufficient (but not necessary) condition for the existence of a horizon is that any two of the spin parameters vanish, i.e., if two vanish, a horizon will always exist, irrespective of how large the remaining spin parameters are. For even , the existence of a horizon is guaranteed if any one of the spins vanishes. Thus, arbitrarily large (i.e., ultraspinning) values can be achieved for all but two (one) of the in odd (even) dimensions.
Assume, then, an ultraspinning regime in which rotation parameters are comparable among themselves, and much larger than the remaining ones. A limit then exists to a black -brane of limiting horizon topology . The limiting geometry is in fact the direct product of and a -dimensional Myers–Perry black hole . Thus, in an ultraspinning regime the allowed phase space of -dimensional black holes can be inferred from that of -dimensional black holes. Let us then begin from and proceed to higher .
The phase space is fairly easy to determine in ; see Figure 2. In Equation (45) admits a real root for
In the phase space of regular black-hole solutions is again bounded by a curve of extremal black holes. In terms of the dimensionless parameter , the extremal curve is
In , with three angular momenta , it is more complicated to obtain the explicit form of the surface of extremal solutions that bind the phase space of MP black holes, but it is still possible to sketch it; see Figure 3(a). There are ultraspinning regimes in which one of the angular momenta becomes much larger than the other two. In this limit the phase space of solutions at, say, large , becomes asymptotically of the form , i.e., of the same form as the five-dimensional phase space (48), only rescaled by a factor (which vanishes as ).
A similar ‘reduction’ to a phase space in two fewer dimensions along ultraspinning directions appears in the phase space of MP black holes; see Figure 3(b); a section at constant large becomes asymptotically of the same shape as the six-dimensional diagram (49), rescaled by a -dependent factor.
These examples illustrate how we can infer the qualitative form of the phase space in dimension if we know it in , e.g., in , with four angular momenta, the sections of the phase space at large approach the shapes in Figure 3 (a) and (b), respectively.
If we manage to determine the regime of parameters where regular black holes exist, we can express other (dimensionless) physical magnitudes as functions of the phase-space variables . Figure 4 is a plot of the area function in , showing only the quadrant ; the complete surface allowing is a tent-like dome. In the shape of the area surface is a little more complicated to draw, but it can be visualized by combining the information from the plots we have presented in this section. In general, the ‘ultraspinning reduction’ to dimensions also yields information about the area and other properties of the black holes.
Let us now discuss briefly the global structure of these solutions, following . The global topology of the solutions outside the event horizon is essentially the same as for the Kerr solution. However, there are cases in which there can be only one nondegenerate horizon: even with at least one spin vanishing; odd with at least two spins vanishing; odd with one and There is also the possibility, for odd and all nonvanishing spin parameters, of solutions with event horizons with negative . However, they contain naked closed causal curves.
The MP solutions have singularities where for even , for odd . For even and all spin parameters nonvanishing, the solution has a curvature singularity where , which is the boundary of a -ball at , thus generalizing the ring singularity of the Kerr solution; as in the latter, the solution can be extended to negative . If one of the , then itself is singular. For odd and all , there is no curvature singularity at any . The extension to contains singularities, though. If one spin parameter vanishes, say , then there is a curvature singularity at the edge of a -ball at , ; however, in this case, the ball itself is the locus of a conical singularity. If more than one spin parameter vanishes then is singular. The causal nature of these singularities varies according to the number of horizons that the solution possesses; see  for further details.
The Myers–Perry solutions are manifestly invariant under time translations, as well as under the rotations generated by the Killing vector fields . These symmetries form a isometry group. In general, this is the full isometry group (up to discrete factors). However, the solutions exhibit symmetry enhancement for special values of the angular momentum. For example, the solution rotating in a single plane (32) has a manifest symmetry. If angular momenta are equal and nonvanishing then the associated with the corresponding 2-planes is enhanced to a non-Abelian symmetry. This reflects the freedom to rotate these 2-planes into each other. If angular momenta vanish then the symmetry enhancement is from to an orthogonal group or for odd or even respectively . Enhancement of symmetry is reflected in the metric depending on fewer coordinates. For example, in the most extreme case of equal angular momenta in dimensions, the solution has isometry group and is cohomogeneity-1, i.e., it depends on a single (radial) coordinate [111, 112].
In addition to isometries, the Kerr solution possesses a “hidden” symmetry associated with the existence of a second-rank Killing tensor, i.e., a symmetric tensor obeying . This gives rise to an extra constant of motion along geodesics, rendering the geodesic equation integrable. It turns out that the general Myers–Perry solution also possesses hidden symmetries [174, 91] (this was first realized for the special case of [93, 94]). In fact, it has sufficiently many hidden symmetries to render the geodesic equation integrable [207, 173]. In addition, the Klein–Gordon equation governing a free massive scalar field is separable in the Myers–Perry background . These developments have been reviewed in .
The classical linearized stability of these black holes remains largely an open problem. As just mentioned, it is possible to separate variables in the equation governing scalar-field perturbations [147, 26, 195]. However, little progress has been made with the study of linearized gravitational perturbations. For Kerr, the study of gravitational perturbations is analytically tractable because of a seemingly-miraculous decoupling of the components of the equation governing such perturbations, allowing it to be reduced to a single scalar equation [234, 235]. An analogous decoupling has not been achieved for Myers–Perry black holes, except in a particular case that we discuss below.
Nevertheless, it has been possible to infer the appearance of an instability in the ultraspinning regime of black holes in . We have seen that in this regime, when rotation parameters become much larger than the mass parameter and the rest of the , the geometry of the black-hole horizon flattens out along the fast-rotation planes and approaches a black -brane. As discussed in Section 3.4, black -branes are unstable against developing ripples along their spatial worldvolume directions. Therefore, in the limit of infinite rotation, the MP black holes evolve into unstable configurations. It is then natural to conjecture that the instability already sets in at finite values of the rotation parameters. In fact, the rotation may not need to be too large in order for the instability to appear. The GL instability of a neutral black brane horizon appears when the size of the horizon along the brane directions is larger than the size of the . We have seen that the sizes of the horizon along directions parallel and transverse to the rotation plane are and , respectively. This brane-like behavior of MP black holes begins when , which suggests that the instability will appear shortly after crossing thresholds like (39). This idea is supported by the study of the possible fragmentation of the rotating MP black hole: the total horizon area can increase by splitting into smaller black holes whenever . The analysis of  indicates that the instability should be triggered by gravitational perturbations. It is, therefore, not surprising that scalar-field perturbations appear to remain stable even in the ultraspinning regime [26, 195].
This instability has also played a central role in proposals for connecting MP black holes to new black-hole phases in . We discuss this in Section 6.
The one case in which progress has been made with the analytical study of linearized gravitational perturbations is the case of odd dimensionality, , with equal angular momenta [177, 197]. As discussed above, this Myers–Perry solution is cohomogeneity-1, which implies that the equations governing perturbations of this background are just ODEs. There are two different approaches to this problem, one for  and one for (i.e., ) .
For , the spatial geometry of the horizon is described by a homogeneous metric on , with isometry group. Since , one can define a basis of -invariant 1-forms and expand the components of the metric perturbation using this basis . The equations governing gravitational perturbations will then reduce to a set of coupled scalar ODEs. These equations have not yet been derived for the Myers–Perry solution (however, this method has been applied to study perturbations of a static Kaluza–Klein black hole with symmetry ).
For , gravitational perturbations can be classified into scalar, vector and tensor types according to how they transform with respect to the isometry group. The different types of perturbation decouple from each other. Tensor perturbations are governed by a single ODE that is almost identical to that governing a massless scalar field. Numerical studies of this ODE give no sign of any instability . Vector and scalar type perturbations appear to give coupled ODEs; the analysis of these has not yet been completed.
It seems likely that other MP solutions with enhanced symmetry will also lead to more tractable equations for gravitational perturbations. For example, it would be interesting to consider the cases of equal angular momenta in even dimensions (which resemble the Kerr solution in many physical properties), and MP solutions with a single nonzero angular momentum (whose geometry (32) contains a four-dimensional factor, at a constant angle in the , mathematically similar to the Kerr metric; in fact this four-dimensional geometry is type D). The latter case would allow one to test whether the ultraspinning instability is present.
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