11 Myers-Perry black holes
Robert C. MyersIn this chapter, we will continue the exploration of black holes in higher di-mensions with an examination of asymptotically flat black holes with spher-ical horizons, i.e., in d spacetime dimensions, the topology of the horizonand of spatial infinity is an S d − . In particular, we will focus on a familyof vacuum solutions describing spinning black holes, known as Myers-Perry(MP) metrics. In many respects, these solutions admit the same remarkableproperties as the standard Kerr black hole in four dimensions. However,studying these solutions also begins to provide some insight into the newand unusual features of event horizons in higher dimensions.These metrics were discovered in 1985 as a part of my thesis work as aPh.D. student at Princeton [1]. My supervisor, Malcolm Perry, and I hadbeen lead to study black holes in higher dimensions, in part, by the renewedexcitement in superstring theory which had so dramatically emerged in theprevious year. We anticipated that examining black holes in d > While I will not have space todiscuss these extensions, the interested reader may find a description of thegeneralized solutions in ref. [2].
Before considering spinning black holes, we should mention that the Schwarzschildsolution is easily generalized to d ≥ ds = − (cid:16) − µr d − (cid:17) dt + (cid:16) − µr d − (cid:17) − dr + r d Ω d − (1.1) There is more than one such parameter in higher dimensions. a r X i v : . [ g r- q c ] N ov Myers-Perry black holes where d Ω d − denotes the line element on the unit ( d –2)-sphere. While thisvacuum solution of the d -dimensional Einstein equations was first found byTangherlini in the early 1960’s [3], it is still traditionally referred to as aSchwarzschild black hole. In part, this nomenclature probably arose becausefor any value of d >
4, the features of this spacetime (1.1) are essentiallyunchanged from its four-dimensional predecessor.In particular, the constant µ emerges as an integration constant in solv-ing the Einstein equations. In Appendix A, we derive expressions for themass and angular momentum in a d -dimensional spacetime by examiningthe asymptotic structure of the metric. There one finds that µ fixes themass of the black hole (1.1) — see eq. (1.65) — with M = ( d −
2) Ω d − πG µ (1.2)where Ω d − is the area of a unit ( d –2)-sphere, i.e., Ω d − = 2 π d − Γ (cid:0) d − (cid:1) . (1.3)As long as µ >
0, the surface r d − = µ is an event horizon. It is a straight-forward exercise to generalize the discussion presented in Chapter 1 in con-structing good coordinates across this surface and finding the maximal an-alytic extension of the geometry. The corresponding Penrose diagram thentakes precisely the same form as given in Figure 1.1 of Chapter 1 where eachpoint now represents a ( d –2)-sphere. Notably, there is a future (past) cur-vature singularity at r = 0 in region II (III), where R µνρσ R µνρσ ∝ µ /r d − as r →
0. Of course, if µ < r =0 and the corresponding Penrose diagram matches that given in Figure1.4 of Chapter 1.Another simple exercise is to extend Birkhoff’s theorem to higher dimen-sions. That is, one can solve Einstein’s vacuum equations in any d ≥ i.e., the solution has an SO ( d −
1) isometry, but withoutassuming that the spacetime is static. The Schwarzschild-Tangherlini metric(1.1) remains the most general solution and so any spherically symmetricsolution of R µν = 0 must also be static. It is also possible to prove a unique-ness theorem indicating that this metric (1.1) is the only solution of thevacuum Einstein equations in higher dimensions if one assumes that the ge-ometry is asymptotically flat and static [4]. Hence all such static solutionsare spherically symmetric and completely determined by their mass M . Of course, the past and future horizons should now be labeled as r d − = µ . .2 Spinning Black Holes The generalization of the four-dimensional Reissner-Nordstr¨om metricto solutions describing static charged black holes in higher dimensions isalso straightforward. Again, the features of these solutions of the Einstein-Maxwell equations in d >
Before writing the metric for a spinning black hole, it is useful to first orientthe discussion by writing the metric for flat space in higher dimensions. Tobegin, consider the case d = 2 n + 1 (with n ≥ ds = − dt + n (cid:88) i =1 (cid:0) dx i + dy i (cid:1) = − dt + dr + r n (cid:88) i =1 (cid:0) dµ i + µ i dφ i (cid:1) . (1.4)In the first line, we have paired all of the spatial coordinates as Cartesiancoordinates ( x i , y i ) in n orthogonal planes. In the second line, we have in-troduced polar coordinates which can be expressed with: x i = r µ i cos φ i , y i = r µ i sin φ i . (1.5)Implicitly, we are defining r = (cid:80) ni =1 (cid:0) x i + y i (cid:1) and so the direction cosines µ i are constrained to satisfy n (cid:88) i =1 µ i = 1 . (1.6)Hence not all of the dµ i in the flat space metric (1.4) are independentand one of these terms can be eliminated using this constraint. However,we have left this replacement implicit for the sake of keeping the metricsimple. For completeness, we note that the range of each of the coordinatesis: t ∈ ( −∞ , ∞ ), r ∈ [0 , ∞ ), µ i ∈ [0 ,
1] and φ i ∈ [0 , π ], where the latterare periodically identified φ i = φ i + 2 π . We will adopt polar coordinates Myers-Perry black holes analogous to those in eq. (1.4) to present the MP metrics for d = 2 n + 1below. In particular then, the black hole geometry will approach the flatspace metric (1.4) asymptotically.For an even number of dimensions, i.e., d = 2 n + 2 (with n ≥ z = r α with α ∈ [ − , . (1.7)Hence the flat space metric becomes ds = − dt + dr + r n (cid:88) i =1 (cid:0) dµ i + µ i dφ i (cid:1) + r dα . (1.8)while the constraint on the direction cosines becomes n (cid:88) i =1 µ i + α = 1 . (1.9)Eq. (1.8) exhibits the polar coordinates which we adopt below for the MPmetric with d = 2 n + 2.One outstanding feature of the polar coordinates in eqs. (1.4) and (1.8)is that there are n commuting Killing vectors in the angular directions φ i .The corresponding rotations in each of the orthogonal planes (1.5) matchthe n generators of the Cartan subalgebra of the rotation groups SO (2 n )or SO (2 n + 1) for odd and even d , respectively. This feature highlightsthe fact that in higher dimensions we must think of angular momentumas an antisymmetric two-tensor J µν , e.g., see eq. (1.62). In considering ageneral rotating body, we may simplify this angular momentum tensor bygoing to the center-of-mass frame, which eliminates the components with atime index. Then a suitable rotation of the spatial coordinates brings theremaining spatial components J ij into the standard form J ij = J − J J − J . (1.10)Here each of the J i denote the angular momentum associated with motionsin the corresponding plane. Note that for even d , the last row and column ofthe above matrix vanishes. Therefore a general angular momentum tensoris characterized by n = (cid:98) ( d − / (cid:99) independent parameters J i . Hence thegeneral spinning black hole metrics, which are considered below, will bespecified by n + 1 parameters: the mass M and the n commuting angular .2 Spinning Black Holes momenta J y i x i . In four dimensions, these parameters would completely fixthe black hole solution but, as we will see in section 1.2.8 and in subsequentchapters, these parameters alone will not fix a unique black hole metric inhigher dimensions. As can be anticipated from eqs. (1.4) and (1.8), the form of the metricsdiffers slightly for odd and even dimensions. Hence let us begin with themetric describing a spinning black hole in an even number of spacetimedimensions, i.e., d = 2 n + 2 with d ≥ ds = − dt + µr Π F (cid:32) dt + n (cid:88) i =1 a i µ i dφ i (cid:33) + Π F Π − µr dr + n (cid:88) i =1 ( r + a i ) (cid:0) dµ i + µ i dφ i (cid:1) + r dα (1.11)where F = 1 − n (cid:88) i =1 a i µ i r + a i (1.12)Π = n (cid:89) i =1 ( r + a i ) . (1.13)With n = 1, we have d = 4 and the above metric reduces to the well knownKerr solution, discussed in Chapter 1. For d = 2 n +1 with d ≥
5, the metricbecomes ds = − dt + µr Π F (cid:32) dt + n (cid:88) i =1 a i µ i dφ i (cid:33) + Π F Π − µr dr + n (cid:88) i =1 ( r + a i ) (cid:0) dµ i + µ i dφ i (cid:1) (1.14)with F and Π again given by eqs. (1.12) and (1.13). Examining the asymp-totic structure of these metrics — see eq. (1.65) — one finds that the n +1free parameters, µ and a i , determine the mass and angular momentum ofthe black hole with M = ( d −
2) Ω d − πG µ (1.15) To make the connection more explicit, we would set a = a , µ = sin θ and α = cos θ . Myers-Perry black holes J y i x i = Ω d − πG µ a i = 2 d − M a i where Ω d − is the area of an S d − given in eq. (1.3). Setting all of the spinparameters a i = 0, both eqs. (1.11) and (1.14) reduce to the d -dimensionalSchwarzschild metric (1.1). Now also setting µ = 0 yields the flat spacemetric in eqs. (1.4) and (1.8), respectively.With general spin parameters a i , both metrics have n +1 commutingKilling symmetries, corresponding to shifts in t and φ i . These symmetriesare enhanced when some of the spin parameters coincide. In particular,with a i = a for i = 1 , · · · , m , the corresponding rotational symmetry isenhanced from U (1) m to U ( m ), where the latter acts on the complex co-ordinates z i = µ i e iφ i in the associated subspace. A particularly interestingcase is d = 2 n + 1 with all n spin parameters equal. Then with the U ( n )symmetry, the solution reduces to cohomogeneity-one, i.e., it depends on asingle (radial) coordinate. Of course, if k of the spin parameters vanish, an SO (2 k ) symmetry emerges in the corresponding subspace. When d is even,this enhanced rotational symmetry extends to SO (2 k + 1) by including the z direction.Of course, as with the Kerr metric, these geometries are only stationary,rather than static, reflecting the rotation of the corresponding black holes. Inparticular, the metric components g tφ i are nonvanishing when a i (cid:54) = 0 and as aresult, one finds frame dragging in these higher dimensional spacetimes, justas was described in Chapter 1 for four dimensions. We might also note thateqs. (1.11) and (1.14) also contain nonvanishing g φ i φ k . Further, implicitlythere are also nonvanishing g µ i µ k (as well as g µ i α with even d ), which wouldappear explicitly if one of the direction cosines were eliminated with eq. (1.6)or (1.9). Various components of the metrics, (1.11) and (1.14), will diverge if eitherΠ
F/r γ = 0 or Π − µ r γ = 0, where γ = 2 and 1 for d odd and even, re-spectively. The former indicates a true curvature singularity while the lattercorresponds to an event horizon. To consider the former in more detail, onemust examine a list of separate cases, i.e., odd or even d and different num-bers of vanishing spin parameters. In most cases, one finds that Π F/r γ = 0at r = 0 and this entire surface is singular. There are three exceptional caseswhich we consider in more detail below: a) even d and all a i (cid:54) = 0, b) odd d and only one a i = 0, and c) odd d and all a i (cid:54) = 0. We should add that all ofour comments with regards to curvature singularities can be confirmed by .2 Spinning Black Holes directly examining the behaviour of the curvatures. For example, we exam-ine the particular case of the d = 5 MP metric in detail in Appendix B andour results there explicitly match those discussed in (b) and (c) below. a) even d and all a i (cid:54) = 0 : This case would include the Kerr metric with d = 4 and the results are similar to those found there, as described inChapter 1. First it is useful here to use the constraint (1.9) to re-expresseq. (1.12) as F = α + r n (cid:88) i =1 µ i r + a i for even d . (1.16)From this expression, we can see that in order for Π F/r to vanish we musthave both r = 0 and α = 0. Further intuition comes from noting thatit is most appropriate to think of the surfaces of constant r as describingellipsoids of the form z r + n (cid:88) i =1 x i + y i r + a i = 1 . (1.17)For example, if we set µ = 0 in the black hole metric (1.11), the resultingmetric describes flat space foliated by these surfaces. Hence as we approach r = 0, these ( d –2)-dimensional ellipsoids collapse to a ( d –2)-dimensionalball in the hyperplane z = 0. Now the direction cosine α = z/r acts as aradial coordinate in this ball with α = 1 corresponding to the origin and α = 0 being the boundary of the ball where the curvature diverges. Hencein higher even dimensions, the ring-like singularity of the Kerr metric iselevated to a singularity on a ( d –3)-sphere. The ( d –2)-ball at r = 0 acts as atwo-sided aperture. Passing through the aperture to negative values of r , weenter a new asymptotically flat space with negative mass (and no horizons).Further, as noted in Chapter 1 for the Kerr metric, this region also containsclosed time-like curves. Passing through the aperture a second time in thesame direction, we reach a space isometric to the original r > b) odd d and only one a i = 0 : For simplicity, let us denote the vanishingspin parameter as a . We begin again by rewriting eq. (1.12), this time usingthe constraint (1.6) F = µ + r n (cid:88) i =2 µ i r + a i for odd d and a = 0 . (1.18)Hence in this case, for Π F/r to vanish, we require both r = 0 and µ = 0— note that Π contributes a factor of r here. In this case, the appropriate Myers-Perry black holes geometric intuition comes from regarding constant r surfaces as ellipsoids ofthe form x + y r + n (cid:88) i =2 x i + y i r + a i = 1 . (1.19)Hence as we approach r = 0, these ( d –2)-dimensional ellipsoids collapse to aball in the hyperplane x = 0 = y . As above, µ acts as a radial coordinatein this ball with µ = 0 corresponding to the boundary of the ball wherethe curvature diverges. However, a key difference from the previous case isthat here as r →
0, the ellipsoids (1.19) become very narrow and collapse toa point in the ( x , y )-plane at r = 0. Hence the ball at r = 0 extends onlyin d –3 dimensions. A careful examination of the geometry shows that thereis also a conical singularity in the ( x , y )-plane for any µ (cid:54) = 0. Hencethe entire r = 0 surface is in fact singular here, although with a mildersingularity than in the generic cases. c) odd d and all a i (cid:54) = 0 : If we apply the constraint (1.6), eq. (1.12) becomes F = r n (cid:88) i =1 µ i r + a i for odd d . (1.20)In this case, we observe that Π approaches a finite constant at r = 0 andeq. (1.6) does not allow all of the µ i can vanish simultaneously. Hence,Π F/r remains finite at r = 0 and so there is no curvature singularity here.However, the metric (1.11) remains problematic at this location since onefinds that g rr ∝ r as r →
0. However, this is only a coordinate singularitywhich is avoided by choosing a new radial coordinate ρ = r . Now in passingto negative values of ρ , the function Π F/r ( ρ ) eventually vanishes and acurvature singularity arises at ρ = − a s , where a s is the absolute value ofthe spin parameter(s) with the smallest magnitude. If more than one spinparameter has the value ± a s , the entire surface ρ = − a s is singular. If onlyone spin parameter, say a , has the value ± a s , the singularity at ρ = − a s only appears at µ = 0. In this case, if a s (cid:48) is the absolute value of thenext smallest spin parameter, the geometry extends smoothly to values of − a s (cid:48) ≤ ρ ≤ − a s in certain directions. However, the curvature singularityextends throughout this range of ρ since F can vanish for certain angulardirections. Hence ultimately all trajectories moving towards smaller valuesof ρ end on a singularity in this region. Of course, this statement assumes that the mass parameter µ is nonvanishing. .2 Spinning Black Holes In considering the event horizons for these metrics, we must again considerseparately the cases where the spacetime dimension is even or odd. Let usstart with d = 2 n + 2, which includes the Kerr metric for d = 4. The eventhorizons arise where g rr vanishes and so from eq. (1.11), we requireΠ − µr = 0 . (1.21)Thus the horizons correspond to the roots of a polynomial, which is order d − r . Unfortunately, apart from d = 4 or 6, there will be no generalanalytic solutions (in terms of radical expressions) for the position of thehorizon. Hence a complete set of necessary and sufficient conditions for theexistence of a horizon is unavailable for higher d . However, we can still makesome general observations.First of all if it exists the horizon must have the topology of S d − since itis a surface of constant r . Further to avoid a naked singularity, we requirethe mass ( i.e., µ ) to be positive. The latter can be deduced with two obser-vations: first, the singularity appears at r = 0 and second, the function Π iseverywhere positive (or zero) — recall the definition in eq. (1.13). Hence foreq. (1.21) to have a root at positive r , we must have µ >
0. With a closerexamination of the polynomial in eq. (1.21), we see that, in fact, it is largeand positive for large | r | and has a single minimum. Hence we conclude thatthere are only three possible scenarios: two, one or zero horizons. Hence inthis regard, the higher dimensional metrics (1.11) are the same as the fa-miliar Kerr metric in d = 4. However, an interesting difference arises if one(or more) of the spin parameters vanishes. Recall that P i is monotonicallyincreasing and grows as r n at large r . However, in this case, Π vanishesat r = 0 and grows as r m for small r , with m vanishing spin parameters.Hence the right-hand side of eq. (1.21) is negative for small r while it stillbecomes large and positive for large r . Hence there must always be one non-degenerate root at positive r , corresponding to a single horizon. This resultholds irrespective of how large the remaining spin parameters are and hencethe event horizon appears even when the angular momentum grows arbi-trarily large, as long as there is no rotation in at least on of the orthogonalplanes. These solutions with very large angular momenta have been dubbed‘ultra-spinning’ black holes in [6]. As we will see in section 1.2.8, the latterhave further interesting consequences.For d = 2 n + 1, the location of the horizon in eq. (1.14) is determined byΠ − µr = 0 . (1.22) Myers-Perry black holes
It is more useful to present this expression using the new radial coordinate ρ = r introduced in the previous discussion of singularities. In terms of ρ ,eq. (1.22) becomes n (cid:89) i =1 ( ρ + a i ) − µρ = 0 . (1.23)Hence we are looking for the roots of a polynomial of order n and so analyticsolutions only exist for n = 2, 3 and 4, i.e., d=5, 7, 9 — these are givenin Appendix B for d = 5. Of course, the horizon has the topology of S d − since it is a surface of constant ρ . Finding a root with ρ > µ . In fact, a positive root requires µ > (cid:88) i (cid:89) j (cid:54) = i a j , (1.24)which ensures that the coefficient of the linear term is negative in eq. (1.23).This constraint is necessary but not sufficient for the absence of a nakedsingularity. Provided that µ is sufficiently large, we will again only find oneor two horizons with positive ρ , just as in the case of even d . Note thatfor odd d , a single vanishing spin parameter is insufficient to guaranteethe existence of a horizon, since the constraint (1.24) remains nontrivial.However if two or more of the spin parameters vanish, eq. (1.23) has onepositive root, as well as a root at ρ = 0. Further in this particular case, wecan have regular ultra-spinning solutions where the event horizon appearseven when the remaining spin parameters become arbitrarily large.Recall that the singularity structure distinguished the case of odd d andall a i (cid:54) = 0. In particular, in this case, the surface ρ = 0 is nonsingular and thegeometry extends to negative values of ρ . To avoid naked singularities here,we only need that the outermost horizon, i.e., the largest root of eq. (1.23),appears for ρ > − a s where the singularity appears. Now with positive µ ,the only possibility is that the horizon appears at positive ρ provided µ issufficiently large, as described above. On the other hand, we have Π( ρ = − a s ) = 0 and hence for any negative µ , a root appears in eq. (1.23) in therange − a s < ρ <
0. Below, we will see that these negative mass solutionsare even more pathological since they contain causality violating regionsextending beyond the horizon. To close this discussion, we recall that whenonly one spin parameter has the minimal value, the geometry extends furtherto the range − a s (cid:48) < ρ < − a s . In this case, for small positive µ , one finds Implicitly we are assuming ρ > As in the previous section, to discuss this case, we adopt the notation that a s and a s (cid:48) are themagnitudes of the smallest and second smallest spin parameters, respectively. .2 Spinning Black Holes two roots or one degenerate root in this new range. However, these surfacesintersect the singular surface and so the latter is not entirely concealed bythese horizons. Further if horizons occur in the range − a s (cid:48) < ρ < − a s , onemay show no other horizons appear for positive ρ . Therefore these spacetimescontain naked singularities. Turning now to ergosurfaces, we must determine the surfaces where g tt van-ishes. From the metrics in eqs. (1.11) and (1.14), the latter correspond tothe roots of F Π − µr = 0 , even dF Π − µr = 0 , odd d (1.25)for r >
0. These surfaces still have the topology of S d − but, of course, thefactor F introduces a more complicated directional dependence than appearsfor the horizons. As above, while there is no analytic solution for theseequations, one is still able to deduce the general properties of the surfaces.In particular, one such surface always appears outside of the outer horizonand another may appear inside the inner horizon, if the latter exists. As canbe seen from eq. (1.25), the ergosurface will touch the horizons where F = 1.If m spin parameters vanish when d is even, then the latter corresponds tothe (2 m )-dimensional sphere described by 1 = α + (cid:80) mk =1 µ k , where thesum runs over the m indices for which a k = 0. Hence if no spin parametersvanish, the two surfaces only touch at the two points on the horizon where α = ±
1, as found for the four-dimensional Kerr metric. Similarly if m spinparameters vanish when d is odd, the ergosurface and horizon will touchalong the S m − described by 1 = (cid:80) mk =1 µ k . In particular, the two surfaceswill not coincide anywhere if all of the spin parameters are nonvanishing inthe case of odd d . Further in this case, one finds that with positive µ , therewill be an ergosurface outside of the outer horizon but no such surface insidethe inner horizon. On the other hand, if µ is negative, no ergosurfaces existat all.As described in Chapter 1, the outer ergosurface marks the boundarywithin which particles cannot remain at rest with respect to infinity. Further,the spinning black holes in higher dimensions can be mined with Penroseprocesses, just as in four dimensions. Another analogy with d = 4 arisesin the scattering waves propagating in these geometries, which producessuperradiance for the MP solutions as in the Kerr metric.We close this section by turning to the question of causality violation. For Myers-Perry black holes many of the black holes under consideration, we need only consider r > a i (cid:54) = 0. First for even d , r can be extended to negativevalues in the second asymptotic region. In this region, the metric components g φ i φ i can become negative leading to closed time-like loops, as occurs in theKerr metric. For odd d and all a i (cid:54) = 0, the geometry extends beyond r = 0to negative values of ρ = r . In this case for a each angle φ i , eq. (1.14) gives g φ i φ i = ( ρ + a i ) (cid:18) µa i Π (cid:19) (1.26)in the plane µ i = 1. The above expression will become negative if the secondfactor has a zero, i.e., for radii inside that where Π + µa i = 0. Now recallthat with µ <
0, the horizon arises at the root of eq. (1.23) which lies be-tween ρ = − a s and 0. Hence the more important observation is that for anyangle φ i for which the corresponding spin parameter satisfies a i > a s , theabove metric component will be negative for some values of ρ outside of thehorizon (since Π is a monotonically increasing function). That is, the neg-ative mass solutions typically contain causality violating regions extendingbeyond the horizon — the only exception would be the case when all of thespin parameters are precisely equal. For completeness, we also note that inthis case with µ > a i taking the value ± a s , there is the pos-sibility that eq. (1.26) may vanish for a i = a s in the range − a s (cid:48) < ρ < − a s . In examining the maximal analytic extension of the solutions (1.11) and(1.14), one can use the usual techniques developed to study four-dimensionalblack holes and the results are essentially the same as for d = 4. In particular,one finds two separate extensions of the spacetime at each horizon, i.e., an infalling coordinate patch which extends the geometry across the futurehorizon and an outgoing patch which smoothly traverses the past horizon. Inthe following, our discussion will focus on the case of even d and the extensionof eq. (1.11). However, with the obvious changes, the same discussion iseasily adapted to the case of odd d , as we briefly examine near the end ofthis section.Towards the construction of the maximal analytic extension of these space-time geometries, it is straightforward to construct Eddington-like coordi- .2 Spinning Black Holes nates dt = dt ± ∓ µr Π − µr dr , (1.27) dφ i = dφ ± ,i ± ΠΠ − µr a i drr + a i . With these new coordinates, the metric (1.11) becomes ds = − dt ± + dr + n (cid:88) i =1 ( r + a i ) (cid:0) dµ i + µ i dφ ± ,i (cid:1) + r dα ± n (cid:88) i =1 a i µ i dφ ± ,i dr + µr Π F (cid:32) dt ± ± dr + n (cid:88) i =1 a i µ i dφ ± ,i (cid:33) . (1.28)Hence the metric is well-behaved in either coordinate system at the horizons, i.e., Π − µr = 0. Of course, various metric components are still singular atΠ F/r = 0 since the latter corresponds to a true curvature singularity. Ascan be seen from eq. (1.28), each of these coordinate systems are adaptedto a particular family of radial geodesics following the null vectors k µ ± ∂∂x µ = ∂∂t ± ∓ ∂∂r . (1.29)That is, the ‘+’ and ‘–’ coordinates are well-behaved along infalling andoutgoing geodesics, respectively, which cross the horizons. Hence t + remainsfinite on the future horizon, where r → r H and t → + ∞ , while t − remainsfinite on the past horizon, where r → r H and t → −∞ .The above Eddington-like coordinates (1.27) indicate that the structureof the horizons is essentially the same as that found in four dimensions. Inparticular, let us consider the case where eq. (1.21) has two distinct roots atpositive r — recall this requires that all of the spin parameters are nonvan-ishing. Hence we have an outer event horizon at r = r H and an inner Cauchyhorizon at r = r C ( < r H ). The corresponding Penrose diagram is shown infigure 1.1. A typical Eddington coordinate patch covers three regions in thisdiagram: the asymptotically flat exterior region where r > r H ; the centralregion between the inner and outer horizons where r C < r < r H ; and theinner region where r < r C which contains a time-like “ring” singularity andwhich can be extended to an asymptotically flat region (with r < { t + , φ + ,i } , theneach of these three regions can be separately extended by transforming tothe outgoing coordinates, { t − , φ − ,i } . Hence the maximally extended space-time becomes a tower in which the basic geometry illustrated in figure 1.1 isrepeated an infinite number of times. We might note that, as illustrated in Myers-Perry black holes the figure, the horizons at r = r H and r C have the characteristic ‘X’ struc-ture of a bifurcate Killing horizon. Here the various branches of the horizonare connected at the bifurcation surface at the center of the X, which corre-sponds to a fixed point of the associated Killing vector. Strictly speaking todemonstrate that the regions of the various overlapping Eddington patchesare in fact smoothly connected at the bifurcation surface, one should findKruskal-like coordinates, which are simultaneously well-behaved across boththe future and past horizons (as well as the bifurcation surface). While thisis certainly possible, the construction of these coordinates is a more involvedexercise and we refer the interested reader to [1] for a discussion of this point. Figure 1.1 Penrose diagram for spinning black hole with two horizons foreven d . The shaded regions indicate a single coordinate patch covered byinfalling Eddington coordinates. As noted above the inner horizon at r = r C is a Cauchy horizon, represent-ing the boundary for the unique evolution of initial data on some space-like .2 Spinning Black Holes surface stretched across the Einstein-Rosen bridge joining two asymptot-ically flat regions. Now we expect that these Cauchy horizons should beunstable since the same simple arguments, which indicate such a surface isunstable in the four-dimensional Kerr metric, can be applied equally wellhere in higher dimensions. However, it must be said that this issue has notbeen studied in the same detail as in four dimensions and so an accurate de-scription of the resulting singularity remains lacking for higher dimensions.Above, we considered the spinning black holes (1.11) (with all of the a i (cid:54) = 0) in the regime where there were two distinct horizons. Now if themass of this solution is fixed and some of the spin parameters are increased,eventually the two horizons will coalesce producing an extremal black hole.In this case, the individual Eddington coordinate patches cover the exteriorregion and the inner region, and connecting these patches results in themaximal extension illustrated in figure 1.2(a). In this case, the near-horizonanalysis of [7] can also be extended to higher dimensions to find that thethroat region of the extremal black hole corresponds to an analog of thegeometry AdS × S n [8]. If any of the spin parameters are further increasedthen the horizon disappears and one is left with a naked singularity, asshown in figure 1.2(b). Hence the extended black hole geometries describedhere and above provide a direct analogue in higher dimensions of the four-dimensional story for the Kerr solution, described in Chapter 1.Another possibility, which we have not yet considered for even d , is whenone or more of the spin parameters vanish. In this case, there is a singlehorizon but that it corresponds to a simple zero in eq. (1.21). There will bea second root but it occurs at the singularity at r = 0. One finds that thissingular surface is space-like and so the Penrose diagram is similar to thatof the Schwarzschild solution. In particular, there is no infinite tower of con-nected regions here but rather the singularities form space-like boundariesfor the future and past interior regions. Here an analogy might be drawnwith the d = 4 Kerr metric in the limit that a → r C → a i become. Hence, as noted above, with one a i = 0 (and d is even), we can construct ultra-spinning black holes carrying an arbitraryamount of angular momentum.The above discussion was restricted to even d but there are no essentialdifferences for the case of odd d . Of course as mentioned in section 1.2.2,with all of the spin parameters nonvanishing, the surface r = 0 is nonsingularand the geometry extends to negative values of ρ = r . Further one finds Myers-Perry black holes
Figure 1.2 Further Penrose diagrams for even d : a) an extremal spinningblack hole with single degenerate horizon, b) an over-rotating solution with-out horizon, and c) a spinning black hole with one or more a i = 0. As before,the shaded regions indicate a single coordinate patch covered by infallingEddington coordinates. a time-like singularity in the latter domain but there is no connection to asecond asymptotically flat region. Another difference is that the cases wherethe Penrose diagram takes a Schwarzschild form includes either two or more a i =0 and µ > a i =0 and µ > (cid:80) i (cid:81) j (cid:54) = i a j . The same structure alsoarises when all a i (cid:54) = 0 and µ < g µν = η µν + h ( k ± ) µ ( k ± ) ν (1.30)where h = µr/ Π F . Of course, a further coordinate transformation wouldbe required to introduce Cartesian coordinates so that the flat space line-element takes the conventional form. Here I might note that one of theremarkable features of the four-dimensional Kerr metric is that it can bewritten in this particular form [9]. Ultimately, it was the fact that the MPmetrics can also be written in the Kerr-Schild form that allowed us to deriveeqs. (1.11) and (1.14).It is also interesting to examine the null vectors (1.29) in the original .2 Spinning Black Holes coordinate system given in eq. (1.11): k µ ± ∂∂x µ = ΠΠ − µr (cid:32) ∂∂t − n (cid:88) i =1 ω i ∂∂φ i (cid:33) ∓ ∂∂r (1.31)where ω i = a i r + a i . From these expressions, we see that upon approachingthe horizon, k µ ± ∂∂x µ ∝ ∂∂t − n (cid:88) i =1 Ω i ∂∂φ i , (1.32)with Ω i = a i r H + a i . That is, k µ − becomes the generator of the future horizon at r = r H in the infalling Eddington coordinate patch described by { t + , φ + ,i } .Similarly with infalling Eddington coordinates, k µ + matches the generator ofthe past horizon at r = r H . A final comment is that these two vector fieldsgiven in eq. (1.29) or (1.31) correspond to the principal null vectors thatappear in the algebraic classification, discussed in Chapter 9. In the four-dimensional Kerr metric, particle motion is easily studied be-cause the geodesics are completely soluble by quadratures. That is, thereare four constants of motion, which allow us to write the complete solutionfor geodesic motion in terms of a set of indefinite integrals. At first sight,this is a rather remarkable property since the Killing symmetries and thefixed norm of the four-velocity only provide three such constants. The fourthconstant is more subtle and relies on the existence of a Killing-Yano tensorin this particular background [10] – see below. The existence of this tensoris also responsible for the separability of the wave equation for spin-0, -1/2,-1 and -2 fields in this background. Recent work uncovered a rich structureof analogous relationships in higher dimensions, e.g., [11, 12, 13, 14]. Inparticular, the required hidden symmetries were found for the Myers-Perrymetrics [11], from which one can infer the integrability of geodesic motionin these backgrounds [12].Central to this discussion is the existence of a rank-two closed conformalKilling-Yano tensor (CCKY) h µν which is a two-form satisfying ∇ ( µ h ν ) ρ = 1 d − (cid:0) g µν ∇ σ h σρ − ∇ σ h σ ( µ g ν ) ρ (cid:1) . (1.33)As this two-form is closed, it also satisfies dh = 0 and so at least locallythere exists a one-form potential b such that h = db . In the case of the MP Myers-Perry black holes metrics, (1.11) and (1.14), the CCKY tensor can be explicitly written as h = n (cid:88) i =1 a i µ i dµ i ∧ (cid:0) a i dt + ( r + a i ) dφ i (cid:1) + r dr ∧ (cid:32) dt + n (cid:88) i =1 a i µ i dφ i (cid:33) . (1.34)Following the standard construction in four dimensions, one constructs asecond-rank Killing tensor [10] K ( µν ) = − h µσ h νσ + 12 g µν h ρσ h ρσ (1.35)which then satisfies the identity ∇ ( µ K νρ ) = 0 . (1.36)It follows then that along a geodesic described by the d -velocity u µ , thefollowing is a constant of the motion: K µν u µ u ν . In higher dimensions, thelatter is only the first in a series of new conserved quantities. We will notdescribe the construction here but one finds the following tower of second-rank Killing tensors [11, 12] K ( (cid:96) ) µν = (2 (cid:96) )!(2 (cid:96) (cid:96) !) (cid:16) δ µν h [ µ ν · · · h µ (cid:96) ν (cid:96) ] h [ µ ν · · · h µ (cid:96) ν (cid:96) ] − (cid:96) h µ [ ν · · · h µ (cid:96) ν (cid:96) ] h ν [ ν · · · h µ (cid:96) ν (cid:96) ] (cid:17) . (1.37)Note that comparing this expression to eq. (1.35), we see K (1) µν = K µν . Nowusing eq. (1.33) for the CCKY tensor, it follows that all of these tensorssatisfy the identity (1.36) and hence each provides a constant of the motionalong a geodesic: c (cid:96) = K ( (cid:96) ) µν u µ u ν .From the above expression (1.37), it appears that this construction ex-tends to (cid:96) = 1 , · · · , n + 1 for d = 2 n + 2. However, one finds that for (cid:96) = n + 1 that the right-hand side vanishes as an identity. On the otherhand, one naturally extends this series to (cid:96) = 0 with K (0) µν = δ µν , in whichcase c = K (0) µν u µ u ν = g µν u µ u ν is simply the norm of the d -velocity. Hencethe Killing tensors then provide n + 1 constants of motion. An essential fea-ture of this construction is that these constants are in fact all independent.The latter statement is related to the fact that the CCKY tensor contains n + 1 independent ‘eigenvalues’ for even d , when it is put in the standardform analogous to eq. (1.10). Of course, the Killing symmetries (time trans-lations and the n rotations in each φ i ) provide a further n + 1 constants of .2 Spinning Black Holes the motion. Hence in total, there are d = 2 n + 2 constants which allow usto solve for the geodesics in quadratures.For d = 2 n + 1, there is a similar counting of the constants of motion. Inthis case, the Killing tensors provide n + 1 independent constants c (cid:96) with (cid:96) = 0 , , · · · , n . Further the Killing symmetries provide n + 1 independentconstants. At this point, it may seem that we have too many integrationconstants but, in the case of odd d , it turns out that c n is reducible. That is, c ( n ) = ( ξ ν g µν u µ ) where ξ ν is a Killing vector [12]. This result is related todescribing eq. (1.37) as the contraction of a CCKY tensor of rank d − (cid:96) (dualto the wedge product of (cid:96) h ’s). Hence for (cid:96) = n , the latter is a one-form forwhich the analog of eq. (1.33) reduces to Killing’s equation. Hence this tensoris in fact simply a linear combination of the Killing vectors. Consequently,the total number of independent constants is precisely d = 2 n + 1 and thegeodesic motion is again completely integrable [12].We comment that it has also been shown that the Killing(-Yano) tensorsalso lead to the separability of the Klein-Gordon and Dirac equations, as wellas the Hamilton-Jacobi equations in these backgrounds, e.g., [14]. While wedo not have room to describe these results in detail here, a key element inthis analysis is to construct ‘symmetry operators’ which commute with theappropriate wave operator. For example, in the case of the Klein-Gordonequation [13], we can start with simple operators constructed for each of theKilling coordinates, i.e., i ∂ t and i ∂ φ i , each of which commute with ∇ − m .Various components of the separated solution of ( ∇ − m ) ψ = 0 can thenbe identified as eigenfunctions of these operators, e.g., e iωt and e imφ i . Nowthe Killing tensors provide an additional set of symmetry operators: ˆ K ( (cid:96) ) = ∇ µ ( K ( (cid:96) ) µν ∇ ν ), which also satisfy [ ∇ − m , ˆ K ( (cid:96) ) ]. Again, various separatedcomponents of the desired solutions can then be written as eigenfunctionsof these new operators. It remains an open question as to whether a similarset of symmetry operators can be constructed for the field equations of aMaxwell field or linearized gravitons and whether separability extends tothese equations. We might note that some progress in analyzing linearizedmetric perturbations has been made for the particular case of odd d and all a i equal [15]. As already commented in chapter 1, the basic framework of black hole ther-modynamics extends from four to higher dimensions in a straightforwardway. We might add that implicitly this relies on the fact that our discus-sion of higher dimensional black holes is restricted to solutions of Einstein’s Myers-Perry black holes equations. There have also been interesting extensions of black hole thermo-dynamics to include both higher curvature actions and higher dimensions[16]. In any event, we will keep our comments here brief — see also commentsin the following section.The zeroth law, namely, that the surface gravity or temperature ( i.e., T = κ/ π ) is constant across any stationary event horizon, is essential ifthe corresponding black holes are to behave like a thermal bath. This resultis easily established if the horizon is a bifurcate Killing horizon, which iscertainly the case here, following the discussion of section 1.2.5. As notedthere, the horizon generator is given by χ µ ∂ µ = ∂ t − n (cid:88) i =1 Ω i ∂ φ i . (1.38)Recall that Ω i = a i r H + a i . Hence using χ σ ∇ σ χ µ = κχ µ to evaluate the surfacegravity, one finds κ = ∂ r Π − µ µr (cid:12)(cid:12)(cid:12) r = r H for even d , ∂ r Π − µr µr (cid:12)(cid:12)(cid:12) r = r H for odd d . (1.39)While these are somewhat formal expressions, they clearly illustrate that κ is constant across the entire horizon.Of course, the first law takes precisely the same form as in four dimensions: δM = κ πG δ A + n (cid:88) i =1 Ω i δJ i (1.40)which leads to the interpretation of the area of (a cross-section of) thehorizon A as the entropy of the black hole with the celebrated formula: S = A / G . (Of course, in an d -dimensional spacetime, this area A actuallyhas the dimensions of length to the power d − d − d − M = n (cid:88) i =1 Ω i J i + κ πG A . (1.41)Following [17], the irreducible mass of the black hole may be identifiedfrom the first law. This is the mass associated with the area of the horizon, i.e., one integrates the area term in eq. (1.40), M ir = 18 πG (cid:90) A κ ( A (cid:48) , J i = 0) d A (cid:48) .2 Spinning Black Holes = d − πG Ω / ( d − d − A d − d − = d − d − κ A πG (1.42)Hence M − M ir is the mass or energy connected to the rotation of the blackhole and we expect that it may be removed through Penrose processes.In four dimensions, this can be explicitly verified because the geodesics inthe Kerr metric are completely soluble by quadratures. Given the recentdevelopments described in section (1.2.6), it would be interesting to extendthis analysis to higher dimensions.To close this section, we note that the second law ( i.e., δ A ≥
0) is alsoeasily extended to higher dimensions, following the discussion in Chapter1. One proof of the latter relies on the matter falling across the horizonsatisfying the null energy condition and also on cosmic censorship [18]. Whilethe former still seems a reasonable assumption in higher dimensions, thelatter may appear more dubious given the recent results discussed in Chapter3. However, the second law may also be proved by using the null energycondition and by demanding that the null generators of the horizon arecomplete [18]. In fact, the latter is consistent with our current understandingof the final state of the Gregory-Laflamme instability and hence it seems thatthe second law remains to have a firm foundation in higher dimensions.
While there is strong evidence for the stability of Kerr black holes in fourdimensions, in fact, the opposite is true for spinning black holes in higherdimensions. That is, we believe that in higher dimensions, various instabili-ties arise for MP black holes when the angular momentum becomes large. Infact, it has been argued that these instabilities are related to the appearanceof a rich fauna of new black holes in higher dimensions [19, 20].A precise understanding of instabilities would require an analysis of thelinearized perturbations of the MP metrics, (1.11) and (1.14). While thisis possible in four dimensions, as noted in section 1.2.6, limited progresshas been made in higher dimensions. However, insight into the situationin higher dimensions comes from making connections with the Gregory-Laflamme instability of black branes — see Chapter 2. As described below,this approach led to the conjecture that ultra-spinning black holes should beunstable for d ≥ Myers-Perry black holes relativity in higher dimensions imposes a dynamical ‘Kerr bound’ on thespin of the form J d − (cid:46) GM d − in d dimensions.To illustrate this point, let us consider the spinning black hole solutionswith a single nonvanishing spin parameter. With this restriction, for eitherodd or even d , the metric reduces to ds = − dt + µr d − ρ (cid:0) dt + a sin θ dϕ (cid:1) + Σ∆ dr (1.43)+Σ dθ + ( r + a ) sin θ dϕ + r cos θ d Ω d − , where Σ = r + a cos θ and ∆ = r + a − µr d − . (1.44)Here we have set a = a and µ = sin θ (as well as a i> = 0). Now the eventhorizon is determined as the largest root r H of ∆( r ) = 0. That is, r H + a − µr d − H = 0 . (1.45)In examining this equation, it is not hard to see that for d = 4 or 5, there is anextremal limit ( i.e., an upper bound on a ) beyond which no horizon exists.However, as our discussion in section 1.2.3 indicated, the more interestingcase is d ≥
6. For the latter, we may note that the term r makes the left-hand side of eq. (1.45) large and positive as r → ∞ . On the other hand,the term − µ/r d − makes ∆( r ) negative for small r and hence there mustbe a (single) positive root independent of the value of a . That is, we havethe possibility of ultra-spinning solutions, for which a regular event horizonremains even when the angular momentum (per unit mass) grows arbitrarilylarge.Let us examine the geometry of the horizon of eq. (1.43) in this ultra-spinning regime. In the limit of very large a and fixed µ , the solution ofeq. (1.45) is approximately given by r H (cid:39) (cid:16) µa (cid:17) / ( d − (cid:28) a . (1.46)Hence we observe that r H is shrinking as a grows (and µ is kept fixed).However, r H is simply some coordinate expression and one must instead ex-amine the horizon in a covariant way to uncover the true geometry. Variousapproaches may be taken here, all with the same simple result. If we charac-terize the size of the horizon along and orthogonal to the plane of rotationas (cid:96) (cid:107) and (cid:96) ⊥ , respectively, then (cid:96) (cid:107) ∼ a and (cid:96) ⊥ ∼ r H . (1.47) .2 Spinning Black Holes That is, the horizon of these rapidly rotating black holes spreads out inthe plane of rotation while contracting in the transverse directions, taking a‘pancake’ shape in this plane. Considering the area of the horizon, we find A = Ω d − r d − H ( r H + a ) (cid:39) Ω d − r d − H a (cid:39) Ω d − (cid:18) µ d − a (cid:19) / ( d − . (1.48)Note that the area decreases as a grows because the contraction in thetransverse directions overcomes the spreading in the plane of rotation. Weemphasize that this result (1.48) only applies for d ≥
6. The horizon areaalso decreases with increasing a in d = 4 or 5, but it is only for larger d thatwe can consider the ultra-spinning regime with a → ∞ , in which case thearea shrinks to zero size.Hence from the perspective of an observer near the axis of rotation andnear the horizon ( i.e., near θ ∼ r ∼ r H ), the horizon geometry appearssimilar to that of a black membrane, i.e., it has roughly the geometry R × S d − . However, as we saw in Chapter 2, Gregory and Laflamme foundthat a black membrane would be classically unstable when the size in thebrane directions is larger than that of the transverse sphere [24]. Hence itis natural to expect that the ultra-spinning MP solutions are unstable inthe limit a → ∞ but also that the instability actually sets in at some finitevalue of a [6].The transition between the horizon behaving similar to the Kerr blackhole and behaving like a black membrane is easily seen using black holethermodynamics. One simple quantity to consider is the black hole temper-ature of the metric (1.43). Beginning from zero spin, T decreases as a grows,just like in the familiar case of the Kerr black hole. In d = 4 and d = 5 thetemperature continues to decrease reaching zero at extremality, however,in d ≥ T reaches a minimum andthen starts growing again, as expected for a black membrane. The minimum,where this behavior changes, can be determined exactly [6] a r H (cid:12)(cid:12)(cid:12)(cid:12) crit = d − d − a d − µ (cid:12)(cid:12)(cid:12)(cid:12) crit = d − d − (cid:18) d − d − (cid:19) d − . (1.49)Following [21], we can use this critical ratio (1.49) to define the boundaryof the ultra-spinning regime. That is, ultra-spinning solutions are definedto be those for which the ratio a d − /µ exceeds the critical value given ineq. (1.49). Explicitly evaluating eq. (1.49) for the latter ratio, we finds some This statement can be made mathematically precise in the limit a → ∞ [6]. Myers-Perry black holes of these critical values to be a d − µ (cid:12)(cid:12)(cid:12)(cid:12) crit = 1 . , . , . , .
35 for d = 6 , , , , respectively . (1.50)We note that these critical values seem to be only weakly dependent on d . Further, these results would seem to indicate that the membrane-likebehaviour, and hence the instability, arises for relatively small values of thespin parameter a .A further connection to black hole thermodynamics appears because itis expected that the classical Gregory-Laflamme instabilities should be con-nected to thermodynamic instabilities of the corresponding black branes[25]. More precisely, it was conjectured that the appearance of a negative‘specific heat’ for the black brane is connected to the appearance of thisclassical instability. Applying this reasoning in the present context wouldsuggest that the rotating black hole should become unstable at some pointafter ∂ S/∂J > i.e., after the point of inflection marked ‘x’ in fig-ure 1.3. Given the expression for the area (1.48), one finds that this pointcorresponds precisely to that identified above from the behaviour of the tem-perature. That is, the critical point ‘x’ where ∂ S/∂J = 0 is given preciselyby eq. (1.49). Figure 1.3 Phase diagram of entropy vs. angular momentum, at fixed mass,for MP black holes spinning in a single plane for d ≥
6. The point ‘x’indicates where ∂ S/∂J = 0. The subsequent points (a,b,c, . . . ) correspondto the threshold of axisymmetric instabilities which introduce increasingnumbers of ripples in the horizon. It is further conjectured that a new classof black holes with rippled horizons branches off from each of these points[19]. While resolving these issues analytically remains intractable at present,there has been remarkable progress coming from numerical investigations .2 Spinning Black Holes in recent years [21]. If one considers the instability just at threshold, i.e., precisely at the critical value of a , then the corresponding frequency is pre-cisely zero and the unstable mode becomes a time-independent zero-mode.In [21], with a particular ansatz for such zero-modes, the authors were ableto numerically locate the corresponding critical values of a for the singlyspinning MP black holes (1.43) in d = 6 to 11. In fact, they found such amode precisely where ∂ S/∂J = 0. However, the interpretation of this sta-tionary mode is more subtle. Rather than corresponding to an instability,this perturbation simply corresponds to shifting the solution to a nearbyMP black hole with a slightly larger spin. However, a small distance furtherinto the ultra-spinning regime, they were also found a new zero-mode which‘pinches’ the horizon at the axis of rotation, as illustrated for the point ‘a’ infigure 1.3. It is believed that this zero-mode does correspond to the onset ofa true instability for higher values of the angular momentum J . Further, thiswas only the first of a hierarchy of zero-modes which introduced an increas-ing number of pinches or ripples in the event horizon along the θ direction.While these numerical searches only identified the stationary modes (by de-sign), this provides strong evidence for a hierarchy of Gregory-Laflammeinstabilities in the ultra-spinning regime.These zero-modes also provide evidence for a new class of stationary rotat-ing black holes with spherical horizons but with a rippled profile in the polarangle θ . The existence of these solutions was also conjectured in [6, 19]. Ac-cording to the phase diagram suggested in [19], there would be a new branchof solutions beginning at the point ‘a’ and in moving along this branch, thepinch in the horizon at the axis of rotation would grow larger and larger.The conjecture is that this branch connects to yet another phase where thepinch produces a puncture in the horizon and the new phase would consistof spinning black rings, analogous to those discussed in Chapter 6 exceptthe horizon topology would be S × S d − . Similarly, it is conjectured thatthe branch starting from the point ‘b’ would connect the spinning MP blackholes to higher dimensional versions of the ‘black saturn’ found in [26] forfive dimensions. Hence the new spinning black holes with rippled sphericalhorizons appear only to be a precursor to a rich fauna of new solutions withcomplex horizon topologies in higher dimensionsImplicitly, the latter analysis was only considering modes which respectall of the rotational symmetries present in the original metric (1.43), i.e., U (1) × SO ( d − Myers-Perry black holes [23], full numerical simulations were carried out to describe evolution ofrapidly spinning MP black holes in higher dimensions — again with a singlenonvanishing spin parameter as in eq. (1.43). In all of the cases studied, itwas found that the solutions were unstable against non-axisymmetric per-turbations, with an initial profile proportional to sin(2 φ ). The critical valuewhere this ‘bar-mode’ instability set in was found to be: a d − µ (cid:12)(cid:12)(cid:12)(cid:12) bar = 0 . , . , . , .
27 for d = 5 , , , , respectively . (1.51)We should note that these values are considerably smaller than those iden-tified above, in eq. (1.50). Notably, these numerical simulations were able tofind an instability of the d = 5 MP black hole, where the previous discus-sion was unable to identify any instabilities. Further, following the nonlinearevolution of the unstable perturbation, the simulations [23] found that thedeformed black holes spontaneously emit gravitational waves causing themto spin down and settle again to a stable MP black hole with a spin pa-rameter smaller than the critical value in eq. (1.51). An open question is todetermine when such ‘bar-mode’ instabilities arise for MP black holes rotat-ing in more than one plane. As an aside, let us note here that in d = 5 withboth spin parameters equal, it was shown analytically that no instabilitiesappear whatsoever [27].In the preceding discussion, we have only considered MP black holes ro-tating in a single plane. However, this was only done to simplify the pre-sentation and because this case was the focus of the numerical studies in[21, 23]. As discussed in section 1.2.3, ultra-spinning black hole solutionscan also arise with several of nonvanishing spin parameters growing large,as long as one (or two) of the spin parameters vanish in even (or odd) d .It is natural to expect that the ultra-spinning regime also extends to theregime where several a i grow large while the remainder stay small. Guidedby this intuition, it is straightforward to extend the original discussion of theGregory-Laflamme-like instabilities to the case where several spin parame-ters, say m , grow without bound while the remainder stay finite (or vanish)[6]. The limiting metric describes a (rotating) black 2 m -brane, where thehorizon topology is R m × S d − − m . However, a Gregory-Laflamme-like in-stability is again expected to appear for these branes when the characteristicsize in the planes with large spins is somewhat larger than the characteristicsize in the transverse directions. In general, with many independent spins,the thermodynamic analysis mentioned above extends studying of the Hes-sian ∂ S/∂J i ∂J j for negative eigenvalues [21]. This expression provides amore refined definition of ultra-spinning black holes. In particular, following .2 Spinning Black Holes the discussion with a single nonvanishing J i , we define the boundary of theultra-spinning regime as the boundary where this Hessian first acquires azero eigenvalue.Further insights into ultra-spinning instabilities have been found for oneother example [20, 27, 28], namely, odd d = 2 n + 1 with all of the n spinparameters equal. As noted, in section 1.2.1, the rotational symmetry ofthese geometries is enhanced to U ( n ) and it can be shown that the metricinvolves a fibration over the complex projective space CP n [15]. Further themetric perturbations of these spacetimes can be decomposed as harmonics onthis CP n and their analysis reduces to the study of an ordinary differentialequation for the radial profile. Of course, in these metrics with all a i (cid:54) = 0,there is an extremal limit and so it is not immediately obvious that onecan reach an ultra-spinning regime or that any instabilities should appear.In fact, analysis of the above Hessian reveals an ultra-spinning regime forany odd d ≥
7. Ref. [20] explicitly identified unstable modes for d = 9 andsupplementary work [28] later found unstable modes appeared very close tothe extremal limit for d = 7 ,
9, 11 and 13. Ref. [27] was able to show thatno instabilities arise for d = 5. Hence these results suggest that instabilitieswill arise in these cohomogeneity-one black hole spacetimes for any odd d ≥
7. Recently these instabilities of the cohomogeneity-one black holeswere connected to those of the singly spinning black holes with the numericalwork of ref. [22]. They showed that the ultra-spinning instabilities in thesetwo sectors are continuously connected by examining perturbations of MPblack holes with all but one of the spin parameters being equal. While theirexplicit calculations were made for d = 7, similar results are expected forhigher odd d as well.To close, we observe that the construction of the threshold zero-modes in d = 9 suggest that there should be a new family of spinning black hole so-lutions characterized by 70 independent parameters [20]!! Generically, thesesolutions would have only two Killing symmetries, i.e., time translations andone U (1) rotation symmetry. Hence here again, the ultra-spinning instabili-ties open the window on a exciting panorama of new black hole solutions inhigher dimensions. Acknowledgements
Research at Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Min-istry of Research & Innovation. The author also acknowledges support froman NSERC Discovery grant and funding from the Canadian Institute for Myers-Perry black holes
Advanced Research. I would also like to thank the Aspen Center for Physicsfor hospitality while preparing this paper. I would also like to thank RobertoEmparan, Gary Horowitz, David Kubiznak and Jorge Santos for their com-ments on this manuscript.
Appendix A: Mass and Angular Momentum
This appendix will consider the definition of the mass and angular momen-tum of an isolated gravitating system in d dimensions. Our approach is tosimply generalize the standard asymptotic analysis of four-dimensional so-lutions of Einstein’s equations [29] to higher dimensions. In particular, themass and angular momentum of any isolated gravitating system ( e.g., a blackhole) may be defined by comparison with a system which is both weaklygravitating and non-relativistic. The result then provides the d -dimensionalgeneralization of the ADM mass and angular momentum [29].So let us begin with the d -dimensional Einstein equations R µν − g µν R = 8 πG T µν , (1.52)where we have included the stress-energy tensor for some matter fields, as itwill be useful in the following discussion. Now we wish to consider solutionsof these equations when the gravitating system is both weakly gravitatingand non-relativistic. First, with a weakly gravitating system, the metric iseverywhere only slightly perturbed from its flat space form: g µν = η µν + h µν , (1.53)with | h µν | (cid:28)
1. Next, if the the system is non-relativistic, any time deriva-tives of fields will be much smaller than their spatial derivatives. Of course,this also implies that components of the stress energy tensor may be ordered | T | (cid:29) | T i | (cid:29) | T ij | . (1.54)These inequalities indicate that the dominant source of the gravitationalfield is the energy density while the momentum density provides the nextmost important source.The solutions are most conveniently examined in the harmonic gauge ∂ µ (cid:18) h µν − η µν h αα (cid:19) = 0 . (1.55) In the following, Greek indices run over all values µ, ν = 0 , , . . . d −
1, while Latin indicesonly run over spatial values i, j = 1 , , . . . d − .2 Spinning Black Holes With this choice, to leading order, the Einstein equations (1.52) can bewritten as ∇ h µν = − πG (cid:18) T µν − d − η µν T αα (cid:19) = − πG (cid:101) T µν (1.56)where ∇ is the ordinary Laplacian in flat d -dimensional space, i.e., wehave dropped the time derivatives of the metric perturbation. Note that T αα ≈ − T for non-relativistic sources. Eq. (1.56) is now readily solvedwith h µν ( x i ) = 16 πG ( N − d − (cid:90) (cid:101) T µν ( y i ) | (cid:126)x − (cid:126)y | d − d d − y (1.57)where the integral extends only over the ( d −
1) spatial directions. Recallthat Ω d − denotes the area of a unit ( d –2)-sphere, as given in eq. (1.3). Nowevaluating eq. (1.57) in the asymptotic region far from any sources, we have r = | (cid:126)x | (cid:29) | (cid:126)y | and so we may expand the result as h µν ( x i ) = 16 πG ( d − d − r d − (cid:90) (cid:101) T µν d d − y + 16 πG Ω d − x k r d − (cid:90) y k (cid:101) T µν d d − y + · · · (1.58)To simplify our results, we consider the system in its rest frame, whichimplies (cid:90) T i d d − x = 0 , (1.59)and we choose the origin to sit at the center of mass, which fixes (cid:90) x k T d d − x = 0 , . (1.60)Now the total mass and angular momentum are defined as M = (cid:90) T d d − x , (1.61) J µν = (cid:90) ( x µ T ν − x ν T µ ) d d − x . (1.62)One further simplification comes from the conservation of stress-energy,which reduces to ∂ k T kµ = 0 in the present case of interest and from whichwe can infer (cid:90) x (cid:96) T kµ d d − x = − (cid:90) x k T (cid:96) µ d d − x . (1.63) Myers-Perry black holes
Now this result, along with eqs. (1.59) and (1.60), allows us to simplify theangular momentum to J k = 0 J kl = 2 (cid:90) x k T (cid:96) d d − x . (1.64)Applying these results to the expansion in eq. (1.58), we find that toleading order far from the system h ≈ πG ( d − d − Mr d − ,h ij ≈ πG ( d − d − d − Mr d − δ ij , (1.65) h i ≈ − πG Ω d − x k r d − J ki . While these results were derived for a system which is both weakly gravitat-ing and non-relativistic, the asymptotic behaviour of the metric will be thesame for any isolated gravitating system. In particular then, we use theseexpressions to identify the mass and angular momentum of the black holesolutions discussed in the main text.
Appendix B: A Case Study of d=5
For d = 5 dimensions, we can write the metric (1.11) as ds = − dt + µ Σ (cid:0) dt + a sin θ dφ + b cos θ dφ (cid:1) + r ΣΠ − µr dr +Σ dθ + ( r + a ) sin θ dφ + ( r + b ) cos θ dφ (1.66)where Σ = r + a cos θ + b sin θ , (1.67)Π = ( r + a ) ( r + b ) . (1.68)Comparing our notation here to that in the main text, we have set a = a , a = b , µ = sin θ and µ = cos θ . Singularities:
Now with some computer assistance, one can easily calculatethe Kretchman invariant R µνρσ R µνρσ = 24 µ Σ (4 r − r − Σ) . (1.69) .2 Spinning Black Holes At r = 0, this expression yields R µνρσ R µνρσ | r =0 = 72 µ ( a cos θ + b sin θ ) . (1.70)Hence if b = 0 above, we see there is a divergence as θ → π/
2, as describedin case (b) in section 1.2.2. Further with b = 0, if we examine ( r, φ ) part ofthe metric near r = 0 but away from θ = π/
2, we find ds (cid:39) cos θ − µa (cid:16) dr + (cid:16) − µa (cid:17) r dφ (cid:17) + · · · . (1.71)Hence we see that there is an angular deficit of ∆ φ = 2 πµ/a on this axis.On the other hand with both a and b nonvanishing, curvature invariantin eq. (1.70) remains finite. In this case, we introduce the radial coordinate ρ = r and assuming 0 < a ≤ b , we find R µνρσ R µνρσ | ρ = − a = 24 µ (4 a + 3( b − a ) sin θ )(4 a + ( b − a ) sin θ )( b − a ) sin θ . (1.72)Hence in accord with the discussion of case (c) in section 1.2.2, the surface ρ = − a is entirely singular if b = a . However, if b (cid:54) = a , the singularityin eq. (1.72) only appears at θ = 0. Thus in this case, we can extend thegeometry into the region − b ≤ ρ ≤ − a . However, one finds that for anyvalue of ρ in this domain, there are singularities atsin θ = | ρ | − a b − a (1.73)where Σ = 0. Horizons:
With d = 5, eq. (1.22) for the horizon becomes a quadraticequation in r and the roots are given by the relatively simple expressions2 r H = µ − a − b + (cid:112) ( µ − a − b ) − a b , (1.74)2 r C = µ − a − b − (cid:112) ( µ − a − b ) − a b . Therefore the existence of a horizon requires µ ≥ a + b + 2 | a b | M ≥ π G ( J + J + 2 | J J | ) . (1.75)The definitions of the mass and angular momentum given by eq. (1.15) havebeen inserted to yield the second equation and we have defined J ≡ J y x and J ≡ J y x . Hence there are no ultra-spinning black holes in d = 5. Myers-Perry black holes
Rather, if the angular momentum exceeds the above condition (1.75), thesolution contains a naked ‘ring’ singularity without any event horizon.
Ergosurfaces:
The equation for the ergosurface reduces to Σ − µ = 0 or r E ( θ ) = µ − a cos θ − b sin θ . (1.76)When both a and b are nonvanishing, it is not hard to show that r E > r H , i.e., the ergosurface nowhere touches the horizon. Cohomogeneity-One:
It is also interesting to observe the simplificationsthat arise when b = a . First note that in this case, we haveΣ = r + a , and Π = ( r + a ) . (1.77)Further then, we see that the angular components in the second line ofeq. (1.66) now combine to give ( r + a ) d Ω , i.e., the round metric on athree-sphere. Hence this portion of the metric is symmetric under SO (4) (cid:39) SU (2) × SU (2). However, this symmetry does not survive for the full metricbecause there are other angular contributions in the first line of eq. (1.66).However these terms can be written in terms of the potential A = i (¯ z dz + ¯ z dz ) = sin θ dφ + cos θ dφ , (1.78)where z = sin θ e iφ and z = cos θ e iφ . Writing A in terms of these complexcoordinates makes clear that the surviving symmetry is U (1) × SU (2) = U (2), as discussed in section 1.2.1. The metric (1.66) with b = a is calledcohomogeneity-one because after imposing this U (2) symmetry, the metriccomponents are entirely functions of the single (radial) coordinate r .This enhanced symmetry also leads to a simplicity in other aspects ofthe geometry. For example, the Kretchman invariant (1.69) is now only afunction of r , R µνρσ R µνρσ = 24 µ ( r + a ) ( r − a ) (3 r − a ) (cid:39) r → µ a . (1.79)Hence the singularity at ρ = − a in eq. (1.72) simplifies to R µνρσ R µνρσ | ρ = − a + ε = 384 µ a ε , (1.80)where we are assuming that ε (cid:28) a . We might also note that for these blackholes, the location of ergosurface (1.76) reduces to r E = µ − a and so thelatter is now also independent of θ . Given this simple result, it is also astraightforward exercise to write r E − r H = µ (cid:16) − (cid:112) − a /µ (cid:17) > , (1.81) .2 Spinning Black Holes confirming that the ergosurface does not touch the horizon at any point inthese cohomogeneity-one black hole spacetimes. eferences [1] R. C. Myers and M. J. Perry, “Black Holes in Higher Dimensional Space-Times,” Annals Phys. (1986) 304.[2] W. Chen, H. Lu and C. N. Pope, “General Kerr-NUT-AdS metrics in all di-mensions,” Class. Quant. Grav. (2006) 5323 [arXiv:hep-th/0604125].[3] F.R. Tangherlini, “Schwarzschild Field in n Dimensions and the Dimensionalityof Space Problem,” Nuovo Cim. (1963) 636.[4] S. Hwang, “A Rigidity Theorem for Ricci Flat Metrics,” Geometriae Dedicata (1998) 5;G. W. Gibbons, D. Ida and T. Shiromizu, “Uniqueness and nonuniqueness ofstatic vacuum black holes in higher dimensions,” Prog. Theor. Phys. Suppl. (2003) 284 [arXiv:gr-qc/0203004].[5] R. C. Myers, “Higher Dimensional Black Holes in Compactified Space-times,”Phys. Rev. D (1987) 455.[6] R. Emparan and R. C. Myers, “Instability of ultra-spinning black holes,” JHEP (2003) 025 [arXiv:hep-th/0308056].[7] J. M. Bardeen and G. T. Horowitz, “The Extreme Kerr throat geometry: AVacuum analog of AdS × S ,” Phys. Rev. D (1999) 104030 [arXiv:hep-th/9905099].[8] H. K. Kunduri, J. Lucietti and H. S. Reall, “Near-horizon symmetries of ex-tremal black holes,” Class. Quant. Grav. (2007) 4169 [arXiv:0705.4214 [hep-th]];P. Figueras, H. K. Kunduri, J. Lucietti and M. Rangamani, “Extremal vac-uum black holes in higher dimensions,” Phys. Rev. D (2008) 044042[arXiv:0803.2998 [hep-th]].[9] R. P. Kerr, “Gravitational Field of a Spinning Mass as an Example of Alge-braically Special Metrics,” Phys. Rev. Lett. (1963) 237.[10] M. Walker and R. Penrose, Comm. Math. Phys. (1970) 265;R. Penrose, Ann. N. Y. Acad. Sci. (1973) 125;R. Floyd, “The dynamics of Kerr fields,” PhD Thesis, London (1973).[11] V. P. Frolov and D. Kubiznak, “Hidden Symmetries of Higher Dimensional Ro-tating Black Holes,” Phys. Rev. Lett. (2007) 011101 [arXiv:gr-qc/0605058].[12] D. N. Page, D. Kubiznak, M. Vasudevan and P. Krtous, “Complete integra-bility of geodesic motion in general Kerr-NUT-AdS spacetimes,” Phys. Rev.Lett. (2007) 061102 [arXiv:hep-th/0611083]; eferences (2007) 004 [arXiv:hep-th/0612029].[13] A. Sergyeyev and P. Krtous, “Complete Set of Commuting Symmetry Op-erators for Klein-Gordon Equation in Generalized Higher-Dimensional Kerr-NUT-(A)dS Spacetimes,” Phys. Rev. D (2008) 044033 [arXiv:0711.4623[hep-th]].[14] V. P. Frolov, P. Krtous and D. Kubiznak, “Separability of Hamilton-Jacobiand Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes,” JHEP (2007) 005 [arXiv:hep-th/0611245];T. Oota and Y. Yasui, “Separability of Dirac equation in higher di-mensional Kerr-NUT-de Sitter spacetime,” Phys. Lett. B (2008) 688[arXiv:0711.0078 [hep-th]];M. Cariglia, P. Krtous and D. Kubiznak, “Dirac Equation in Kerr-NUT-(A)dS Spacetimes: Intrinsic Characterization of Separability in All Dimen-sions,” arXiv:1104.4123 [hep-th].[15] H. K. Kunduri, J. Lucietti and H. S. Reall, “Gravitational perturbations ofhigher dimensional rotating black holes: Tensor perturbations,” Phys. Rev. D (2006) 084021 [arXiv:hep-th/0606076].[16] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D (1993) 3427 [arXiv:gr-qc/9307038];V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposalfor dynamical black hole entropy,” Phys. Rev. D (1994) 846 [arXiv:gr-qc/9403028];T. Jacobson, G. Kang and R. C. Myers, “On Black Hole Entropy,” Phys. Rev.D (1994) 6587 [arXiv:gr-qc/9312023].[17] L. Smarr, Phys. Rev. Lett. (1973) 71.[18] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, 1973).[19] R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodriguez,“The Phase Structure of Higher-Dimensional Black Rings and Black Holes,”JHEP (2007) 110 [arXiv:0708.2181 [hep-th]].[20] O. J. C. Dias, P. Figueras, R. Monteiro, H. S. Reall and J. E. Santos, “Aninstability of higher-dimensional rotating black holes,” JHEP (2010) 076[arXiv:1001.4527 [hep-th]].[21] O. J. C. Dias, P. Figueras, R. Monteiro, J. E. Santos and R. Emparan, “Insta-bility and new phases of higher-dimensional rotating black holes,” Phys. Rev.D (2009) 111701 [arXiv:0907.2248 [hep-th]];O. J. C. Dias, P. Figueras, R. Monteiro and J. E. Santos, “Ultraspinning insta-bility of rotating black holes,” Phys. Rev. D (2010) 104025 [arXiv:1006.1904[hep-th]].[22] O. J. C. Dias, R. Monteiro and J. E. Santos, “Ultraspinning instability: themissing link,” JHEP (2011) 139 [arXiv:1106.4554 [hep-th]].[23] M. Shibata and H. Yoshino, “Bar-mode instability of rapidly spinning blackhole in higher dimensions: Numerical simulation in general relativity,” Phys.Rev. D (2010) 104035 [arXiv:1004.4970 [gr-qc]];M. Shibata and H. Yoshino, “Nonaxisymmetric instability of rapidly rotatingblack hole in five dimensions,” Phys. Rev. D (2010) 021501 [arXiv:0912.3606[gr-qc]].6 References [24] R. Gregory and R. Laflamme, “Black Strings And P-Branes Are Unstable,”Phys. Rev. Lett. (1993) 2837 [arXiv:hep-th/9301052]; “The Instability ofcharged black strings and p-branes,” Nucl. Phys. B (1994) 399 [arXiv:hep-th/9404071].[25] S. S. Gubser and I. Mitra, “Instability of charged black holes in Anti-de Sitterspace,” arXiv:hep-th/0009126;H.S. Reall, “Classical and thermodynamic stability of black branes,” Phys.Rev. D (2001) 044005 [arXiv:hep-th/0104071].[26] H. Elvang and P. Figueras, “Black Saturn,” JHEP (2007) 050 [arXiv:hep-th/0701035].[27] K. Murata and J. Soda, “Stability of Five-dimensional Myers-Perry BlackHoles with Equal Angular Momenta,” Prog. Theor. Phys. (2008) 561[arXiv:0803.1371 [hep-th]].[28] J. E. Santos, unpublished;M. Durkee and H. S. Reall, “Perturbations of near-horizon geometries andinstabilities of Myers-Perry black holes,” Phys. Rev. D (2011) 104044[arXiv:1012.4805 [hep-th]].[29] R. Arnowitt, S. Deser and C. Misner, “Canonical Variables for General Rela-tivity,” Phys. Rev. (1960) 1595;R. Arnowitt, S. Deser and C. Misner, “Energy and the Criteria for Radiationin General Relativity,” Phys. Rev. (1960) 1100;R. Arnowitt, S. Deser and C. Misner, “Coordinate Invariance and Energy Ex-pressions in General Relativity,” Phys. Rev.122