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4 Myers–Perry Solutions

The generalization of the Schwarzschild solution to d > 4 is, as we have seen, a rather straightforward problem. However, in general relativity it is often very difficult to extend a solution from the static case to the stationary one (as exemplified by the Kerr solution). Impressively, in 1986 Myers and Perry (MP) managed to find exact solutions for black holes in any dimension d > 4, rotating in all possible independent rotation planes [200Jump To The Next Citation Point]. This feat was possible as the solutions belong in the Kerr-Schild class
ρ gμν = ημν + 2H (x )kμkν, (31 )
where k μ is a null vector with respect to both g μν and the Minkowski metric η μν. This entails a sort of linearization of the problem, which facilitates greatly the resolution of the equations. Of all known vacuum black holes in d > 4, only the Myers–Perry solutions seem to have this property.

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 d ≥ 6 is emphasized. The symmetries and stability of the MP solutions are also discussed.

4.1 Rotation in a single plane

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 form

μ ( ) Σ ds2 = − dt2 + ------ dt − asin2θ dφ 2 + --dr2 + Σd θ2 + (r2 + a2) sin2 θdφ2 rd− 5Σ Δ +r2 cos2θ dΩ2(d−4), (32 )
Σ = r2 + a2cos2 θ, Δ = r2 + a2 − --μ--. (33 ) rd−5
The physical mass and angular momentum are easily obtained by comparing the asymptotic field to Equations (14View Equation) and (19View Equation), and are given in terms of the parameters μ and a by
(d-−-2)Ωd−-2 --2--- M = 16πG μ , J = d − 2M a . (34 )
Hence, one can think of a as essentially the angular momentum per unit mass. We can choose a ≥ 0 without loss of generality.4

As in Tangherlini’s solution, this metric seems to follow from a rather straightforward extension of the Kerr solution, which is recovered when d = 4. The first line in Equation (32View Equation) looks indeed like the Kerr solution, with the 1∕r falloff replaced, in appropriate places, by d−3 1∕r. The second line contains the line element on a (d − 4)-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

Δ μ a2 -2 − 1 = − -d−-3 + -2 . (35 ) r r r
Roughly, the first term on the right-hand side (RHS) corresponds to the attractive gravitational potential and falls off in a dimension-dependent fashion. In contrast, the repulsive centrifugal barrier described by the second term does not depend on the total number of dimensions, since rotations always refer to motions in a plane.

Given the similarities between Equation (32View Equation) and the Kerr solution, it is clear that the outer event horizon lies at the largest (real) root r0 of g−r1r = 0, i.e., Δ (r) = 0. 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 r = r 0,

r20 + a2 − -μ---= 0 , (36 ) rd0−5
its area will be
𝒜H = rd0−4(r20 + a2)Ωd−2 . (37 )
For d = 4, a regular horizon is present for values of the spin parameter a up to the Kerr bound: a = μ ∕2 (or a = GM), which corresponds to an extremal black hole with a single degenerate horizon (with vanishing surface gravity). Solutions with a > GM correspond to naked singularities. In d = 5, the situation is apparently quite similar since the real root at ∘ ------- r0 = μ − a2 exists only up to the extremal limit μ = a2. However, this extremal solution has zero area, and in fact, has a naked ring singularity.

For d ≥ 6, Δ(r) is always positive at large values of r, but the term d−5 − μ ∕r makes it negative at small r (we are assuming positive mass). Therefore Δ always has a (single) positive real root independent of the value of a. Hence, regular black-hole solutions exist with arbitrarily large a. 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 a ≫ r0 shows that the black holes flatten along the plane of rotation [81Jump To The Next Citation Point]; the extent of the horizon along this plane is ∼ a, while, in directions transverse to this plane, its size is ∼ r0. In fact, a limit can be taken in which the ultraspinning black hole becomes a black membrane with horizon geometry ℝ2 × Sd−4. This turns out to have important consequences for black holes in d ≥ 6, 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

( ) TH = -1- --2r0-- + d-−-5- . (38 ) 4π r20 + a2 r0
( a ) ∘ d-−-3- -- = ------, (39 ) r0 mem d − 5
this temperature reaches a minimum. For a∕r 0 smaller than this value, quantities like T H and 𝒜 H decrease, in a manner similar to the Kerr solution. However, past this point they rapidly approach the black membrane results in which TH ∼ 1∕r0 and 2 d−4 𝒜H ∼ a r0, with 2 a characterizing the area of the membrane worldvolume.

The properties of the solutions are conveniently encoded using the dimensionless variables aH, j introduced in Equation (21View Equation). For the solutions (32View Equation) the curve a (j) H can be found in parametric form, in terms of the dimensionless ‘shape’ parameter r0- ν = a, as

π Ω ν5− d jd−3 = -------d−3 -d−3------2-, (40 ) (d − 3) 2 Ωd−2 1 + ν ( ) d−23 2 ad−3 = 8π d −-4- Ωd-−3 --ν----. (41 ) H d − 3 Ωd −2 1 + ν2
The static and ultraspinning limits correspond to ν → ∞ and ν → 0, respectively. These curves are represented for d = 5,6, and 10 in Figure 1View Image. The inflection point where 2 2 d aH ∕dj changes sign when d ≥ 6, occurs at the value (39View Equation).
View Image

Figure 1: Horizon area vs. angular momentum for Myers–Perry black holes with a single spin in d = 5 (black), d = 6 (dark gray), and d = 10 (light gray).

4.2 General solution

[200Jump To The Next Citation Point] also gives black-hole solutions with arbitrary rotation in each of the ⌊d−1⌋ N ≡ -2- independent rotation planes. The cases of odd and even d are slightly different. When d is odd, the solution is

2 2 2 2 2 2 2 μr2 2 2 ΠF 2 ds = − dt + (r + ai)(dμi + μid φi) + ---(dt − aiμidφi) + -------2dr . (42 ) ΠF Π − μr
Here and below i = 1,...,N and we assume summation over i. The mass parameter is μ, not to be confused with the direction cosines μi, which satisfy μ2 = 1 i. For even d, the general solution is
μr ΠF ds2 = − dt2 + r2dα2 + (r2 + a2i)(dμ2i + μ2idφ2i) + ----(dt − aiμ2id φi)2 +-------dr2 , (43 ) ΠF Π − μr
where now μ2i + α2 = 1.

For both cases we can write the functions F (r,μi) and Π (r ) as

2 2 ∏N F (r,μ ) = 1 − -aiμ-i-, Π(r) = (r2 + a2). (44 ) i r2 + a2i i i=1
The relation between μ and ai and the mass and angular momenta is the same as in Equation (34View Equation). The event horizon is again at the largest real root of grr, that is,
Π (r0) − μr2 = 0 (odd d) , Π (r0) − μr0 = 0 (even d). (45 ) 0
The horizon area is
( 2 ) 𝒜H = Ωd−-2μ d − 3 − --2ai-- , (46 ) 2κ r20 + a2i
and the surface gravity κ is
′ ′ κ = lim Π--−-2μr- (odd d) , κ = lim Π--−-μ- (even d). (47 ) r→r0 2μr2 r→r0 2 μr
Extremal solutions are obtained when κ = 0 at the event horizon.

4.2.1 Phase space

The determination of r 0 involves an equation of degree 2N, which in general is difficult, if not impossible, to solve algebraically. So the presence of horizons for generic parameters in Equation (42View Equation) and (43View Equation) 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 (21View Equation), we can fix the mass and define dimensionless quantities j i for each of the angular momenta. Up to a normalization constant, the rotation parameters ai at fixed mass are equivalent to the ji. We take (j1,...,jN ) 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 d = 4 Kerr black hole [200Jump To The Next Citation Point]. So we can expect that, as long as all spin parameters take values not too dissimilar, j1 ∼ j2 ∼ ⋅⋅⋅ ∼ jN, all spins must remain parametrically O (1), i.e., there is no ultraspinning regime in which all j ≫ 1 i.

Next, observe that for odd d, 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 ai vanish, a horizon will always exist, irrespective of how large the remaining spin parameters are. For even d, 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 j i in odd (even) dimensions.

Assume, then, an ultraspinning regime in which n rotation parameters are comparable among themselves, and much larger than the remaining N − n ones. A limit then exists to a black 2n-brane of limiting horizon topology Sd−2 → ℝ2n × Sd− 2(n+1). The limiting geometry is in fact the direct product of ℝ2n and a (d − 2n)-dimensional Myers–Perry black hole [81Jump To The Next Citation Point]. Thus, in an ultraspinning regime the allowed phase space of d-dimensional black holes can be inferred from that of (d − 2n )-dimensional black holes. Let us then begin from d = 5,6 and proceed to higher d.

The phase space is fairly easy to determine in d = 5,6; see Figure 2View Image. In d = 5 Equation (45View Equation) admits a real root for

|j1| + |j2| ≤ 1, (48 )
which is a square, with extremal solutions at the boundaries, where the inequality is saturated. These extremal solutions have regular horizons if, and only if, both angular momenta are nonvanishing. There are no ultraspinning regimes: our arguments above relate this fact to the nonexistence of three-dimensional vacuum black holes.

In d = 6 the phase space of regular black-hole solutions is again bounded by a curve of extremal black holes. In terms of the dimensionless parameter 1∕3 ν = r0∕μ, the extremal curve is

( ) ∘ ----------√---------- ( ) ∘ ----------√---------- -π--- 1∕3 1-−-4ν3-±---1 −-16ν3- --π-- 1∕3 1 −-4ν3 ∓---1 −-16ν3- |j1| = √ -- 4ν , |j2| = √ -- 4ν , (49 ) 2 3 2 3
with 0 ≤ ν ≤ 2−4∕3. As ν → 0 we get into the ultraspinning regimes, in which one of the spins diverges while the other vanishes, according to the general behavior discussed above. In this regime, at, say, constant large j 1, the solutions approach a Kerr black membrane and thus the available phase space is of the form j2 ≤ f(j1), i.e., a rescaled version of the Kerr bound j ≤ 1∕4. Functions such as aH can be recovered from the four-dimensional solutions.
View Image

Figure 2: Phase space of (a) five-dimensional and (b) six-dimensional MP rotating black holes: black holes exist for parameters within the shaded regions. The boundaries of the phase space correspond to extremal black holes with regular horizons, except at the corners of the square in five dimensions, where they become naked singularities. The six-dimensional phase space extends along the axes to arbitrarily large values of each of the two angular momenta (ultraspinning regimes).

In d = 7, with three angular momenta j1,j2,j3, 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 3View Image(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 j3, becomes asymptotically of the form |j1| + |j2| ≤ f(j3), i.e., of the same form as the five-dimensional phase space (48View Equation), only rescaled by a factor f(j3) (which vanishes as j3 → ∞).

A similar ‘reduction’ to a phase space in two fewer dimensions along ultraspinning directions appears in the phase space of d = 8 MP black holes; see Figure 3View Image(b); a section at constant large j 3 becomes asymptotically of the same shape as the six-dimensional diagram (49View Equation), rescaled by a j3-dependent factor.

View Image

Figure 3: Phase space of (a) seven-dimensional, and (b) eight-dimensional MP rotating black holes (in a representative quadrant ji ≥ 0). The surfaces for extremal black holes are represented: black holes exist in the region bounded by these surfaces. (a) d = 7: the hyperbolas at which the surface intersects the planes ji = 0 (which are √ -- jkjl = 1∕ 6, i.e., √ -- akal = μ and r0 = 0) correspond to naked singularities with zero area; otherwise, the extremal solutions are nonsingular. The three prongs extend to infinity; these are the ultraspinning regimes in which one spin is much larger than the other two. The prong along ji becomes asymptotically of the form |jk| + |jl| ≤ f(ji), i.e., the same shape as the five-dimensional diagram in Figure 2View Image(a). (b) d = 8: ultraspinning regimes exist in which two spins are much larger than the third one. The sections at large constant ji asymptotically approach the same shape as the six-dimensional phase space Figure 2View Image(b).

These examples illustrate how we can infer the qualitative form of the phase space in dimension d if we know it in d − 2, e.g., in d = 9,10, with four angular momenta, the sections of the phase space at large j4 approach the shapes in Figure 3View Image (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 ja. Figure 4View Image is a plot of the area function aH (j1,j2) in d = 5, showing only the quadrant j1,j2 ≥ 0; the complete surface allowing j1,j2 < 0 is a tent-like dome. In d = 6 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 d − 2n dimensions also yields information about the area and other properties of the black holes.

View Image

Figure 4: Horizon area a (j ,j ) H 1 2 of five-dimensional MP black holes. We only display a representative quadrant j1,j2 ≥ 0 of the full phase space of Figure 2View Image(a), the rest of the surface being obtained by reflection along the planes j1 = 0 and j2 = 0.

4.2.2 Global structure

Let us now discuss briefly the global structure of these solutions, following [200Jump To The Next Citation Point]. 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 d with at least one spin vanishing; odd d with at least two spins vanishing; odd d with one a = 0 i and ∑ 2 μ > iΠj⁄=ia j There is also the possibility, for odd d and all nonvanishing spin parameters, of solutions with event horizons with negative μ. However, they contain naked closed causal curves.

The MP solutions have singularities where μr∕ΠF → ∞ for even d, μr2∕ΠF → ∞ for odd d. For even d and all spin parameters nonvanishing, the solution has a curvature singularity where F = 0, which is the boundary of a (d − 2)-ball at r = 0, thus generalizing the ring singularity of the Kerr solution; as in the latter, the solution can be extended to negative r. If one of the ai = 0, then r = 0 itself is singular. For odd d and all ai ⁄= 0, there is no curvature singularity at any r2 ≥ 0. The extension to r2 < 0 contains singularities, though. If one spin parameter vanishes, say a1 = 0, then there is a curvature singularity at the edge of a (d − 3 )-ball at r = 0, μ1 = 0; however, in this case, the ball itself is the locus of a conical singularity. If more than one spin parameter vanishes then r = 0 is singular. The causal nature of these singularities varies according to the number of horizons that the solution possesses; see [200] for further details.

4.3 Symmetries

The Myers–Perry solutions are manifestly invariant under time translations, as well as under the rotations generated by the N Killing vector fields ∂∕∂ φi. These symmetries form a N ℝ × U (1) 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 (32View Equation) has a manifest ℝ × U (1 ) × SO (d − 3) symmetry. If n angular momenta are equal and nonvanishing then the U (1)n associated with the corresponding 2-planes is enhanced to a non-Abelian U(n ) symmetry. This reflects the freedom to rotate these 2-planes into each other. If n angular momenta vanish then the symmetry enhancement is from n U (1) to an orthogonal group SO (2n) or SO (2n + 1) for d odd or even respectively [243]. Enhancement of symmetry is reflected in the metric depending on fewer coordinates. For example, in the most extreme case of N equal angular momenta in 2N + 1 dimensions, the solution has isometry group ℝ × U (N ) and is cohomogeneity-1, i.e., it depends on a single (radial) coordinate [111Jump To The Next Citation Point112Jump To The Next Citation Point].

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 K μν obeying K (μν;ρ) = 0 [245]. 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 [17491] (this was first realized for the special case of d = 5 [9394]). In fact, it has sufficiently many hidden symmetries to render the geodesic equation integrable [207Jump To The Next Citation Point173Jump To The Next Citation Point]. In addition, the Klein–Gordon equation governing a free massive scalar field is separable in the Myers–Perry background [90Jump To The Next Citation Point]. These developments have been reviewed in [88].

4.4 Stability

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 [14726Jump To The Next Citation Point195Jump To The Next Citation Point]. 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 [234235]. 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 d ≥ 6 [81Jump To The Next Citation Point]. We have seen that in this regime, when n rotation parameters ai become much larger than the mass parameter μ and the rest of the ai, the geometry of the black-hole horizon flattens out along the fast-rotation planes and approaches a black 2n-brane. As discussed in Section 3.4, black p-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 p q ℝ × S appears when the size L of the horizon along the brane directions is larger than the size r0 of the q S. We have seen that the sizes of the horizon along directions parallel and transverse to the rotation plane are ∼ ai and ∼ r0, respectively. This brane-like behavior of MP black holes begins when ai >∼ r0, which suggests that the instability will appear shortly after crossing thresholds like (39View Equation). 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 > ai∼ r0 [81Jump To The Next Citation Point]. The analysis of [81Jump To The Next Citation Point] 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 [26195].

This instability has also played a central role in proposals for connecting MP black holes to new black-hole phases in d ≥ 6. 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, d = 2N + 1, with equal angular momenta [177Jump To The Next Citation Point197Jump To The Next Citation Point]. 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 N > 1 [177Jump To The Next Citation Point] and one for N = 1 (i.e., d = 5[197Jump To The Next Citation Point].

For d = 5, the spatial geometry of the horizon is described by a homogeneous metric on 3 S, with SU (2 ) × U (1) isometry group. Since S3 ∼ SU (2), one can define a basis of SU (2)-invariant 1-forms and expand the components of the metric perturbation using this basis [197]. 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 SU (2) × U (1 ) symmetry [158]).

For N > 1, gravitational perturbations can be classified into scalar, vector and tensor types according to how they transform with respect to the U (N ) 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 [177Jump To The Next Citation Point]. 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 (32View Equation) contains a four-dimensional factor, at a constant angle in the Sd −4, 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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