On the black hole species (by means of natural selection)
aa r X i v : . [ h e p - t h ] M a r October 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main AEI-2010-037
ON THE BLACK HOLE SPECIES(BY MEANS OF NATURAL SELECTION)
MARIA J. RODRIGUEZ,
Max-Planck-Institut f¨ur Gravitationsphysik,Albert-Einstein-Institut, 14476 Golm, Germany ∗ E-mail: [email protected]
Recently our understanding of black holes in D-spacetime dimensions, as solutions of theEinstein equation, has advanced greatly. Besides the well established spherical black holewe have now explicitly found other species of topologies of the event horizons. Whetherin asymptotically flat, AntideSitter or deSitter spaces, the different species are reallynon-unique when D ≥
5. An example of this are the black rings. Another issue in higherdimensions that is not fully understood is the struggle for existence of regular blackhole solutions. However, we managed to observe a selection rule for regular solutions ofthin black rings: they have to be balanced i.e. in vacuum, a neutral asymptotically flatblack ring incorporates a balance between the centrifugal repulsion and the tension. Theequilibrium condition seems to be equivalent to the condition to guarantee regularity onthe geometry of the black ring solution. We will review the tree of species of black holesand present new results on exotic black holes with charges.
Keywords : Black Holes, Higher Dimensions of Space-Time
1. Introduction
Black holes are the most elementary and intriguing objects of General Relativity(GR). The fact that the effects of the spacetime curvature are dramatic in theirpresence explains why it is relevant studying these systems.String/M-theory is currently the best candidate for a unified theory of all inter-actions and, in particular, is expected to describe quantum gravity. One of the mostsurprising outcomes of the theory was its prediction of the dimensionality of space-time. This, perhaps contrary to expectation, was required to be ten rather thanfour. As its low-energy limit, higher dimensional GR can be regarded as a powerfultool to gain insights into the more fundamental theory, as well as deserving fur-ther study in its own right. As it has been argued, higher dimensional black holescould form at very high energies and, actually, form at the Large Hadron Collider(LHC) at CERN. Bearing in mind the deep reasons to consider GR in dimensionshigher than four, we aim to analyze its most remarkable solutions, black holes, in ahigher-dimensional setting.The vast number of black holes a that exist in the Universe, usually lying at thecentre of galaxies, are exactly represented by the black hole solution found by RoyKerr, a neutral(electrically uncharged) b rotating Schwarzschild black hole in four a Note, incidentally, that the name black hole was coined by John Archibald Wheeler in 1967. If electric charges are allowed, the only possible four-dimensional black hole solutions are the ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main spacetime dimensions. This species is unique as well as the topology of the eventhorizon which can only be spherical, namely S and characterized only by its massand angular momentum (called charges in this context).These objects have a theoretical counterpart in higher spacetime dimensions.But, unlike in four, in higher dimensions there are other examples of black holesolutions with new exciting properties. In fact, the four-dimensional uniquenesstheorems break down for D > in five dimensions, there are also the blackring, the black saturn, the di-ring and the bicycling black ring. The listis as well enlarged by axisymmetric black holes known approximately such as thehigher dimensional black rings and its more general cousins the blackfolds. But before plunging into the different examples of black objects let us go backto the theory we will be interested in, namely GR in D dimensions. The centralfield equation of GR in vacuum is the commonly called Einstein equation R µν − g µν R + ( D − g µν = 0 (1)( µ, ν = 1 , , ..., D ), of a remarkable simplicity which nevertheless hides an extraordi-nary mathematical complexity. The geometry of spacetime is encoded in the metric g µν , which features explicitly and within the Ricci tensor R µν and scalar R , thatmeasure the curvature of spacetime. We will allow, in general, for a cosmologicalconstant Λ. Typically, Λ = 0 , ± ( D − L − , where L is the radius of the curved space.It follows from (1) that vacuum solutions are either Ricci-flat ( R µν = 0), if Λ = 0,or Einstein ( R µν = Λ g µν ), otherwise. Immediate solutions include D -dimensionalMinkowski space (if Λ = 0), D -dimensional de Sitter space, dS (if Λ > D -dimensional Anti-de Sitter space, AdS (if Λ < D directions or in asmaller number of directions (that we will call transverse). At the practical level,this will translate in the imposition of appropriate boundary conditions.The difficulty in solving Einstein’s equations (1) increases as the number D ofspacetime dimensions does too. Indeed, the larger number of degrees of freedom, ( D − D − −
1, carried by the unknown metric g µν to be solved for makes of (1)an increasingly involved system of coupled, nonlinear, partial differential equations.On the other hand, in a higher-dimensional spacetime there is more room than infour dimensions for solutions to be able to display richer features. For example,solutions can now rotate in up to N = [ D − ] independent rotation planes, thenumber of Casimir operators (independent angular momenta J i ) of the spacelike so-called Reissner-Nordstrom (non-rotating) and Kerr-Newman black holes (rotating). ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main rotation group SO ( D − S . Interestingly enough, this restriction drops if spacetime is allowed tohave a higher number of spacelike directions, in which case much richer possibilitiesdo indeed arise. We will present a catalogue according to the topology of their eventhorizon and point out a selection rule that might explain the struggle for existenceof regular black hole solutions.From here on we will be mainly concerned with stationary (time independent)black hole solutions of higher-dimensional GR. In the following section, 2, the pos-sible event horizon topologies of higher-dimensional single black holes are reviewed.The explicit metrics of the known examples are recorded in section 3 and the generalproperties of multi black holes are discussed in section 4. In section 5 we compilethe explicit known solutions of black holes in curved backgrounds as well as theproperties of more exotic cases such as the higher dimensional black ring. In thefinal sections we comment on the phase diagram, a selection rule for regular blackhole solutions and discuss some open problems in the subject. This review is alsointended as a brief guide to the higher-dimensional black hole solutions Conventions
We will refer to “extra” dimensions of spacetime when considering more than4 spacetime dimensions and we label them n while setting D = 4 + n where D isthe total number of spacetime dimensions. G is Newton’s constant in D dimensionsand the conventions used for unities are the speed of light, and the Planck andBoltzmann constants respectively c = ~ = k B = 1. In order to make comparisonsbetween the different asymptotically flat D -black holes we introduce dimensionlessquantities (factoring out the mass M ) for the spin j , the area a H , the angularvelocity ω H and the temperature t H as j n +1 = c j J n +1 GM n +2 , a n +1 H = c a A n +1 ( GM ) n +2 , (2) ω H = c ω Ω H ( GM ) n +1 , t H = c t ( GM ) n +1 T H , (3) where the numerical constants (defining Ω n the n -volume of a unit sphere) are c j = Ω n +1 n +5 ( n + 2) n +2 ( n + 1) n +12 , c a = Ω n +1 π ) n +1 ( n + 2) n +2 (cid:18) nn + 1 (cid:19) n +12 , (4a) c ω = √ n + 1 (cid:18) n + 216 Ω n +1 (cid:19) − n +1 , c t = 4 π r n + 1 n (cid:18) n + 22 Ω n +1 (cid:19) − n +1 . (4b) ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main We find convenient to introduce different dimensionless magnitudes for asymp-totically (A)dS black holes, denoted by sans-serif fonts, corresponding to quantitiesmeasured in units of the cosmological radius L or ‘cosmological mass’ scale L D − /G .For instance, for the S -radius, mass, angular momentum and horizon area of thering we define R = RL , M = GML D − , J = GJL D − , A H = A H L D − . (5) Equivalently, we might have set L = 1 = G , but the meaning of some formulas isclearer if we retain L .
2. Topological classification of black holes
Black objects in any dimension can be characterized and classified according tothe topology of their event horizon. On the one hand, the classification of neutral,asymptotically flat, static black holes (non-rotating solutions with null Killing vectorfields on the horizon) is simple and complete. The Schwarzschild-Tangherlini blackhole has been proved to be the only allowed static black hole in all dimensions D ≥
4, and the existence of static black holes with non spherical S D − topologiesis accordingly ruled out c .In contrast, stationary black holes (those with intrinsic rotation), can give riseto event horizons with more sophisticated topologies. The current status of the clas-sification of stationary black holes by horizon topologies is far from being complete,and most of the higher-dimensional black hole solutions allowed in principle remainunknown. Let us first review what the situation is for stationary, asymptoticallyflat black holes in all dimensions (see figure 1 for a quick summary). • D ≤
4. There are no asymptotically flat black holes below four dimensions: thelowest dimensional known black hole is the well known Kerr black hole in fourdimensions. As we discussed in section 3, this spherical ( S ) rotating blackhole is the only one in D = 4, and is uniquely characterized by its conservedcharges. • D = 5 ( n = 1). In five dimensions the situation is different since there are noequivalent general uniqueness theorems. Now the possible topologies of theevent horizon are not only S , but also S × S . The Myers-Perry black holesolution (the extension of the Kerr rotating black hole to higher dimensions)corresponds to the former, S , case. The latter possibility, S × S , is also realizedin the black ring of Emparan and Reall, with one angular momentum, and thatof Pomeransky and Sen’kov, with two angular momenta in orthogonal planes.These are, in fact, the only possibilities (aside from the Lens L ( p, q ) topologywhere no explicit solution is known), according to rigidity theorems. Allhigher-dimensional black holes, regardless of their asymptotics, have been arguedto be axisymmetric, that is, to display an axial U (1) symmetry. The existence c The proof can be generalized to Einstein-Maxwell-dilaton theories. ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main Summary of neutral stationary uni horizon black hole solutions in higher dimen-sions D ≥ D = 5, thefirst outer shell from the center, there are S and S × S topologies of the event horizonscorresponding to the Myers-Perry black hole and the black ring (BR) and helical BR. Forthe higher dimensional black rings and blackfolds the solutions have been found perturba-tively using the matched asymptotic expansions. These are characterized by two scales R and r corresponding to the radii of the spheres that conform its event horizon – a large S m i and a smaller s D − − m respectively. Note that for blackfolds p ≥ P i m i = m while ∀ i m i are odd. The perturbative analytical metrics of black rings (with S × s D − event horizons topologies) and p-tuboids (topologically T p × s D − − p ) have been found inthe thin approximation, when r << R . However, helical black rings and the more generalblackfolds which are products of odd spheres are only known to linear order. The samesummary is devised for black holes with asymptotical (A)dS boundary conditions. of black hole solutions with exactly one U (1) symmetry were conjectured in. The first evidence for such a solution was provided in and dubbed helical blackrings. • D = 6 ( n = 2). When going one dimension further up, to six dimensions, theterritory becomes vast and not many solutions are known. From cobordism the-ories, restrictions in the type of allowed topologies leave the open possibilitiesjust to the following: S , S × S and S × Σ g where Σ g is a genus g Riemannsurface (for example the S , with g = 0). For several years, the only knownblack hole solution in D = 6 was, again, the MP black hole with two rotationalsymmetries, with an S event horizon. The more extravagant topologies S × S were recently shown to be realized in the D = 6, the thin black rings andthe thin helical black rings (black rings in the weak gravity approximation, forwhich self gravitational effects are absent. See details in section 3). Also there isevidence of existence of black 2-tuboids with T × S horizons topologies, rep-resenting a particular type of blackfold in six dimensions. Explicit D = 6 blackhole metrics realizing the remaining possibility, S × S for the event horizon,are unknown. • D > n > ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main topologies. In fact, there are no analog rigidity theorems to restrict the topologiesof the black holes’ event horizons in dimensions greater than 6. However, we doknow of possible topologies: those realized by some explicitly known solutions.These include S D − (realized in the Myers-Perry solutions), S × S D − (realizedin the approximate solutions of thin black rings, see section 3), and finally, T p × S D − − p with p ≥ Q i S m i × S D − − m horizon topologies where2 ≤ m i ≤ n , where m i ∈ N odd and m = P i m i ≤ n . Collectively black holeswith horizons that are products of spheres and tori (particularly dubbed black p-tuboids) will be called blackfolds here. Note that even-ball blackfolds are claimedto describe the ultraspinning MP black holes.New evidence for the existence of exotic event horizon topologies (such as S × S )can be found in. Much less is known about black hole solutions which asymptotically approachglobal Anti-de Sitter space, AdS, at spatial infinity, the so called
AdS black holes .The reason is to be put down to the extra term that arises from the non-vanishingcosmological constant in the Einstein equation, which further complicates the prob-lem. In fact, only AdS black holes with spherical horizons S D − for n ≥ static Schwarzschild-AdS solution and the stationary
Kerr-AdS black holes.
Thissituation has now changed with the discovery of the thin AdS black rings in all di-mensions greater than four, the details of which are presented in section 5. Notethat by topological censorship a four dimensional AdS/dS black ring can be ruledout.
3. The asymptotically flat black hole solutions
This section includes the known metrics of the most general exact black objects withasymptotically flat boundary conditions. We review the asymptotically flat metricsof Myers-Perry black holes in all dimensions and of the doubly-spinning black ringin D = 5. Then we proceed to recall the conserved charges for the approximatehigher dimensional black rings, helical black rings and blackfolds. The comparisonamong them is performed in section 7. Black Hole
The Myers-Perry black hole, the higher-dimensional counterpart of Kerr’s blackhole, exhibits rotation in all possible N = [( D − /
2] planes. In D dimensions, itsline element is given by ds = − dt + dr + r dα ǫ + Π F Π − µr dr + X i ( r + a i )( dµ i + µ i dφ i )+ µr Π F X i ( dt + a i µ i dφ i ) (6)where i = 1 , . . . , N , ǫ = mod ( D − µ i are the direction cosines, φ i theazimuthal angles, µ and a i are free parameters. The coordinates are restricted as ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main P i µ i + α ǫ = 1, and F = 1 − a i µ i r + a i and Π = Q Ni =1 ( r + a i ). There exists an eventhorizon, with spherical topology S D − , situated at r the largest root ofΠ − µ r − ǫ = 0 . (7)The black hole is characterized by the mass parameter µ and the rotation parameter a i by which we can express the thermodynamics M = Ω D − πG ( D − µ, S = Ω D − G µ r , (8a) T H = 12 πr r N X i =1 r + a i −
11 + ǫ ! , (8b) J i = Ω D − πG a i µ, Ω i = a i r + a i . (8c)The event horizons of black holes are not at all rigid. On the contrary, they have beenobserved to be very elastic. For large enough angular momenta the behaviour ofsome black holes changes to that of extended black branes (black rings also exhibit asimilar behavior for large spins and act like black strings – see the following sectionand for details.) Qualitatively, as the spin becomes large, the event horizon spreadsout in the plane of rotation and becomes a higher dimensional ‘pancake’ approachingthe geometry of a black brane. Our focus will be on the particular case in whichthe black hole has one large angular momenta and all others are zero. However, adetailed analysis of the more general situations in which black holes present blackmembrane phases can be found in. An important simplification occurs in the ultra-spinning regime of J → ∞ withfixed M , which corresponds to a → ∞ . Then (7) becomes µ → a r n − (9)leading to simple expressions for the eqs. (8b) in terms of r and a , which in thisregime play roles analogous to those of r and R for the black ring. Specifically, a is a measure of the size of the horizon along the rotation plane and r a measure ofthe size transverse to this plane. In fact, in this limit M → ( n + 2)Ω n +2 πG a r n − , S → Ω n +2 πG a r n , T H → n − πr (10)take the same form as the expressions characterizing a black membrane extendedalong an area ∼ a with horizon radius r . This identification d lies at the core ofthe ideas in. The reader may rightly wonder what happens to J → Ω n +2 πG a r n − , Ω H → a , (11) d The entropy corresponds precisely to a membrane of planar area Ω n +2 Ω n a . This value also givesthe precise membrane mass once the dimension dependence of the mass normalization is takeninto account. ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main under this identification. Both turn out to disappear, since the black membranelimit is approached in the region near the axis of rotation of the horizon and so inthe limit the membrane is static. Observe that the value of (9), (10) and (11), arevalid up to O ( r /a ) corrections.The transition to this membrane-like regime is signaled by a qualitative changein the thermodynamics of the MP black holes. At (cid:18) ar (cid:19) mem = r D − D − , (12) the temperature reaches a minimum and (cid:0) ∂ S/∂J (cid:1) M changes sign. This pointshould not be considered as a sign for an instability or a new branch but rather atransition to an infinitesimally nearby solution along the same family of solutions.The numerical evidence of supports this connection with the zero-mode perturba-tion of the solution. For a/r smaller than this value, the thermodynamic quantitiesof the MP black holes such as T and S behave similarly to those of the Kerr solutionand so we should not expect any membrane-like behaviour. However, past this pointthey rapidly approach the membrane results and develop a Gregory-Laflamme typeof instability. Black Rings and helical black ring
Black Rings, whose horizon exhibits an S × S topology, were first found by Em-paran and Reall (see for a review). Following this development, but now using the inverse scattering method , Pomeransky and Sen’kov managed to build whatis usually called the doubly spinning black ring that had long been anticipated. It isbalanced by angular momentum in the plane of the ring, with angular momentumalso in the orthogonal plane corresponding to rotation of the two-sphere. The lattersolution can be written as ds = − H ( y, x ) H ( x, y ) ( dt + Ω) + F ( x, y ) H ( y, x ) dφ + 2 J ( x, y ) H ( y, x ) dφdψ − F ( y, x ) H ( y, x ) dψ − k H ( x, y )( x − y ) (1 − ν ) (cid:18) dx G ( x ) − dy G ( y ) (cid:19) (13)Here, k , ν , λ are parameters, k = ν (1 − λ − ν ), k = λ (1 − λ − ν + 2 ν ), theone-form Ω is defined as Ω = − kλ √ (1+ ν ) − λ H ( y,x ) ((1 − x ) y √ νdψ + (1+ y )(1 − λ + ν ) (1 + λ − ν + x yν (1 − λ − ν ) + 2 νx (1 − y )) dφ ) and the functions G , H , J , F as G ( x ) = (1 − x )(1 + λx + νx ) ,H ( x, y ) = 1 + λ − ν + 2 λν (1 − x ) y + 2 xλ (1 − y ν ) + x y k ,J ( x, y ) = 2 k (1 − x )(1 − y ) λ √ ν ( x − y )(1 − ν ) (1 + λ − ν + 2( x + y ) λν − xy k ,F ( x, y ) = 2 k ( x − y ) (1 − ν ) ( G ( x )(1 − y ) (cid:0)(cid:0) (1 − ν ) − λ (cid:1) (1 + ν )+ yλ (1 − λ + 2 ν − ν ) (cid:1) + G ( y )(2 λ + xλ ((1 − ν ) + λ )+ x (cid:0) (1 − ν ) − λ (cid:1) (1 + ν ) + x k + x (1 − ν ) k )) . ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main The solution e is parameterized by a scale k and two dimensionless parameters λ and ν which are required to satisfy 0 ≤ ν < √ ν ≤ λ < ν . The metric hasa coordinate singularity at the roots where g yy diverges. The roots of the equation1 + λ y + ν y = 0 determine the locations of the inner and outer horizons; the eventhorizon is located at y h = − λ + √ λ − ν ν . (14) The properties of these black rings, such as the phase diagrams and limits, were firstanalyzed in. A summary of the results is presented here. There are three limitingcases of the parameters in this solution: ν → λ → ν / and ν → , λ →
2. Onone hand it was found that the latter two limiting cases correspond to extremal so-lutions. The extremal black ring with λ = 2 ν / is regular and has zero temperatureas expected. Physically it corresponds to the S rotating maximally, i.e. saturat-ing the Kerr bound. There exists zero temperature black rings for any S angularmomentum j ψ > /
4. This is quite similar to the case of supersymmetric blackrings.
Remarkably, it was shown that the entropy of this non-supersymmetricextremal black ring can be reproduced from a microscopic calculation.
Thelimit ν → , λ → a , a and µ / = a + a . In this collapse limit the area is discontinuous,just like in the similar collapse limits of supersymmetric black rings. So this areendpoints where the ring has collapsed to the zero temperature Myers-Perry blackhole. On the other hand, when ν →
0, the solution is the balanced black ring withrotation only in the plane. In this case, note that since the balance condition hasalready been imposed the unbalanced black ring with angular momentum only onthe S cannot be obtained from the Pomeransky-Sen’kov solution. The moregeneral unbalanced doubly spinning black ring metric contains this limit.Another qualitative feature is the disappearance of the “fat ring branch” as j φ ≥ /
5, becomes large. Diagonally opposite 2-spheres of the ring carry j φ angularmomentum which creates an attractive spin-spin interaction. This is what causesthe diminishing and the disappearance of the fat ring branch as j φ increases.The analysis of, in concordance with, suggested that the thin black ringbranch solutions are stable to radial perturbations and the fat rings unstable. Ex-trapolating these results, doubly spinning rings with large enough S angular mo-mentum, j φ ≥ /
5, may be expected to be radially stable.The physical parameters of the doubly spinning black ring can be written M = 3 π k G λ ν − λ , S = 8 π k λ (1 + ν + λ ) G (1 − ν ) ( y − h − y h ) , (15a) e We have analytically verified that the solution presented in indeed satisfies the Einstein vacuumequations, R µν = 0. Note that this form of the metric is also in except that we interchange φ and ψ , so that φ is the azimuthal angle of the S and ψ parameterizes the circle of the ring ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main T H = ( y − h − y h )(1 − ν ) √ λ − ν π k λ (1 + ν + λ ) , (15b) J φ = 4 π k G λ q ν (cid:0) (1 + ν ) − λ (cid:1) (1 + ν − λ )(1 − ν ) , Ω φ = λ (1 + ν ) − (1 − ν ) √ λ − ν k λ √ ν r ν − λ ν + λ (15c) J ψ = 2 π k G λ (1 + λ − ν + ν λ + ν ) p (1 + ν ) − λ (1 + ν − λ ) (1 − ν ) , Ω ψ = 12 k r ν − λ ν + λ . (15d) Examining the ranges of the angular momenta one finds that the angular momentacan never be equal, and the ratio J φ /J ψ ≤ / Black ring in all dimensions
Heuristically, a black ring can be defined by taking a black string (see the followingsection), bending and wrapping it into a circle, S , and spinning it in order tobalance its self-gravitational attraction .The method employed in the construction of the higher dimensional black ringswas the matched asymptotic expansion. The general idea was to match thelinearized gravity solution for a thin black ring away from the horizon to a near-horizon solution for a bent boosted black string. An important result of this exerciseis that the perturbed event horizon remains regular.For the convenience of the reader we collect here the entire thermodynamics : M = Ω n +1 G R r n ( n + 2) , S = π Ω n +1 G R r n +10 r n + 1 n , (16a) T H = n π r nn + 1 1 r , (16b) J = Ω n +1 G R r n √ n + 1 , Ω H = 1 √ n + 1 1 R . (16c)These results are valid up to O ( r /R ) corrections. Helical black ring in all dimensions
Due to its elasticity, the thin black ring can be bent and balanced in an helicoidalshape(a spring ring) as was shown in. The horizon being S × S D − , it preservesonly two commuting Killing vector fields in agreement with the rigidity theoremsof. The physical parameters characterizing the helical black ring are M = Ω n +1 G ( n + 2) r n qX n a R a , S = π Ω n +1 G r n +10 qX n a R a r n + 1 n , (17a) T H = n π r nn + 1 1 r , (17b) ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main J a = ± Ω n +1 G √ n + 1 r n n a R a , Ω a = 1 √ n + 1 n a pP n a R a , (17c)where at least two strands n i > n j > Blackfolds
The horizons topology of blackfolds, p-tuboids or in its more general form as prod-ucts of odd-spheres, are Q i S m i × S D − − m horizon topologies where 2 ≤ m i ≤ n ,where m i ∈ N odd and m = P i m i ≤ n . For completeness we present its physicalparameters M = R m Ω m Ω n +1 πG r n ( n + m + 1) (18a) S = R m Ω m Ω n +1 G r n +10 r n + mn , T = n π r nn + m r , (18b) J i = 1 k + 1 R m +1 Ω m Ω n +1 πG r n p m ( n + m ) , Ω i = r mn + m R . (18c)where R is the radius of the m -sphere. A detailed analysis can be found in.
4. Multi black holes
All the examples of higher-dimensional black holes that we have discussed so farpresent a single event horizon and can, accordingly, be referred to as uni black holes .However, unlike its four-dimensional counterpart, higher-dimensional GR also ad-mits black-hole solutions with several, disconnected horizons: the so called multiblack holes . Examples of multi back holes include a five-dimensional black saturn , acombination of a black ring with a Myers-Perry black hole at its centre, di-ring , a coplanar configuration of two concentric rings or the bicycling black ring , con-sisting of two five-dimensional black rings rotating in orthogonal planes. Due tothe lengthy expressions for the metrics of this multi black holes we will not includethem here.Black holes, and black rings in particular, have been usually found by means ofeducated guesses. In some cases, however, a systematic procedure to generate blackhole solutions can be used. This is the inverse scattering method. The under-lying idea behind the method is to make use of the complete integrability of thesystem of non-linear equations that follow from Einstein’s equations for solutionswith sufficient symmetry. Remarkably, among the solutions with the required degreeof symmetry are the rotating black holes in various dimensions. And, perhaps moresurprisingly, the technique can also be used even to generate multi black holes so-lutions. Note that solutions in curved backgrounds and
D > ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main The metric of a D -dimensional stationary vacuum space with D − Furthermore, thetwo-planes orthogonal to the Killing vector fields are integrable. This means that onecan always introduce a coordinate system that is independent of the corresponding D − ds = G ab ( ρ, z ) dx a dx b + e ν ( ρ,z ) ( dρ + dz ) (19)where the conformal factor e ν ( ρ,z ) is a function of ρ, z and G ab ( ρ, z ) is an inducedmetric in a D − G ab ) = − ρ . The Einstein equations for this kind of metricsdecouple and the system is completely integrable. Several strategies were developedto deal with the problem of the appearance of singularities. For singly spinning blackholes a uniform rescaling or renormalization was introduced. And, in or-der to generate healthy solutions with rotations along any number of planes a moregeneral method was proposed in and applied to generate many multi black holes.A remarkable feature of this type of solutions is that they can be characterized by Black Saturn Black Di-RingBicycling Black Ring
Fig. 2.
Rod structure of all multi horizon black holes (besides combinations between them) in fivedimensions. In the most general solutions the ( upper ) horizon rods have mixed directions (1 , Ω ( i ) φ , Ω ( i ) ψ ), i = 1 , lower ) rodsare oriented purely along φ and ψ . their rod structure , as defined in generalizing. It involves the specification of therods and its directions to characterize a solution. A graphical representation of therod structure that determines each solution uniquely f can be found (see Fig. 2).
5. The black hole solutions in curved backgrounds
This section is devoted to the (A)dS black holes solutions. (A)dS black holes in all dimensions
The stationary black hole solution with dS or AdS asymptotics was found in fourdimensions and thirty years later in five dimensions. The extension of this f The static solutions are characterized uniquely by the rod diagrams. However, a unique charac-terization of five-dimensional stationary solutions is more subtle.
The static black hole solution with AdS asymptotics had been found preiously by Kottler. ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main solution, known as the Kerr-de Sitter and Anti-de Sitter metric, to higher dimensionswas carried out by Gibbons, Lu, Page and Pope and, in dimension D and Boyer-Lindquist coordinates, is given by ds = − W (1 − λ r ) dτ + 2 MU " W dτ − N X i =1 a i µ i dϕ i Ξ i + U dr V − M + r dα ǫ + N X i =1 r + a i Ξ i [ dµ i + µ i ( dϕ i + λa i dτ ) ] + λW (1 − λr ) " N + ǫ X i =1 ( r + a i )Ξ i µ i dµ i (20)where i = 1 , . . . , N = [( D − /
2] and ǫ = mod ( D − D = 2 N + 1 + ǫ .There are N the azimuthal angles ϕ i and ( N + ǫ ) direction cosines α, µ i obeyingthe constrain P Ni µ i + ǫ α = 1. Also the mass parameter M and the rotationalparameters a i are free. Finally, λ = Λ / ( D − U , V , W and Ξ i are defined by W ≡ N + ǫ X i =1 µ i Ξ i , U ≡ r ǫ N + ǫ X i =1 µ i r + a i N Y j =1 ( r + a j ) (21) V ≡ r ǫ − (1 − λ r ) N Y i =1 ( r + a i ) , Ξ i ≡ λ a i . (22)In the limit of vanishing cosmological constant, Λ → Their mass, angular momenta, area and surface gravity, as computedin, are M = m Ω D − π Q j Ξ j N X i =1 i − − ǫ ! , J i = m Ω D − a i π Ξ i Q j Ξ j , (23) A = Ω D − r − ǫ + Y i r + a i Ξ i , κ = r + (cid:18) r L (cid:19) X i r + a i + ǫ r ! − r + , (24)where N = (cid:2) D − (cid:3) is the maximal number of independent angular momenta, a i arethe N angular velocity parameters, m is the mass parameter, ǫ = ( D − i = 1 − a i /L . This solution displays a spherical S D − event horizon situated at r = r + , the highest real solution of V − M = 0. Note that this equation is the sameas the equation for the horizon of the asymptotically flat Myers-Perry black hole in D + 2 dimensions, where the additional rotation is a N +1 = L and mass parameteris µ = 2 L m . Hence, the root structure and the horizons of the Kerr-AdS D blackhole can be inferred from the MP D +2 solution; in particular, for odd D , an horizonalways exists provided that any two of the spin parameters vanish, while for even D , its existence is guaranteed if any one of the spins vanishes. Therefore, underthis assumption, an ultraspinning limit can be achieved for all but two (one) of the a i → L in odd (even) dimensions. ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main These black holes comply with the “BPS bound” M L ≥ N X i =1 | J i | (25)This bound can only be saturated in the ultra-spinning regime, in which oneor more spin parameters tend to L , but never when all the angular momenta arenon-zero. Indeed, suppose n spin parameters approach the ultraspinning limit. Tokeep the mass finite, we need to scale the parameters asΞ α =1 ...n = ξ α ν , m = µν n +1 , (26)where ν → ξ , . . . , ξ n and µ constant.As we observed previously, this limit is allowed provided any one (two) of the a i vanish in even (odd) dimensions. Then the root r + tends to zero, while the massand angular momenta reach the values (with α, β = 1 . . . n running on the spinparameters that tend to L and I = n + 1 , . . . , N denoting the others) M = µ Ω D − π Π α ξ α Π I Ξ I X α ξ α , J α = µ Ω D − πξ α Π β ξ β Π I Ξ I , J I = 0 , (27)and saturate the BPS bound (25). However, these black holes are not extremal,since the surface gravity diverges like κ → (2 k + ǫ − / r + , where k is the numberof vanishing spin parameters. The area vanishes in the limit, decreasing to zero like A H ∝ M k + ǫ − k + ǫ − (cid:18) − JM (cid:19) k + n + ǫ − k + ǫ − (1 + O ( M − J )) . (28)The limiting black holes are pancaked out along the planes of rotation (the geometrydescribes a black membrane with horizon topology R n × S D − n +1) ) and so, it isreasonable to presume that they will develop a Gregory-Laflamme type of instability. (A)dS black rings in all dimensions It is natural to ask whether black rings exist in higher dimensions. Their existence(or absence) in Anti-de Sitter space is of special interest for the possible implicationsin the context of the AdS/CFT duality. However, in spite of attempts since earlyon, an exact solution describing an (Anti-)de Sitter black ring remains elusive.Nevertheless, there appears no obvious physical reason why these solutionsshould not exist. Putting a black ring in Anti-de Sitter space should have the ef-fect of increasing the gravitational centripetal pull on it, but, at least within someparameter ranges, this can be plausibly balanced by spinning the black ring faster.On the other hand, if we put the ring in de Sitter space, the cosmological expan-sion should act against the tension, and so the required rotation should be smallerand possibly reach zero. Thin black rings have been constructed via approximatemethods in every dimension D ≥ The physical quantities are M = Ω n +1 G Lr n ( n + 2) R (cid:0) R (cid:1) / , S = π Ω n +1 G Lr n +10 R s n + 1 + ( n + 2) R n , ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main T H = n / p R πr q n + 1 + ( n + 2) R , (29a) J = r n L G Ω n +1 R (cid:2)(cid:0) n + 2) R (cid:1) (cid:0) n + 1 + ( n + 2) R (cid:1)(cid:3) / , (29b)Ω H = 1 L s (1 + R )(1 + ( n + 2) R ) R ( n + 1 + ( n + 2) R ) . (29c)in principle valid up to corrections of order r / min ( R, L ).
6. Transverse asymptotically flat black holes
We proceed now to recall the metrics for the simplest black strings ( p = 1) and black p -branes, characterized by transverse asymptotically flat boundary conditions, andextended horizons with topologies S D − − p × R p .Black p -branes and black strings (1-branes) are D -dimensional solutions thatarise from the combination of a D − p dimensional Schwarzschild-Tangherlini metricwith a flat, Euclidean metric on the remaining R p . These extended black holes aretransverse asymptotically flat (namely, asymptotically flat in only D − p directions),evade the no-hair theorems, and exhibit horizon topologies S D − − p × R p . Theirmetric, in D dimensions, is of the form ds = − V dt + 1 V dr + r d Ω D − p − + dx i dx i . (30)where V = 1 − ( r + r ) D − p − and i = 1 , , . . . , p . The event horizon is situated at r = r + . The x i directions correspond to the flat part R p ; alternatively, they can beperiodically identified, x i = x i + 2 π R i , yielding the so-called localized black objectsin Kaluza-Klein circles (see e.g. the review ).
7. The phase diagram
The understanding of the black hole phases in five dimensions has advanced greatlyin recent years. As we have seen, besides the well known uni horizon black holes,namely the Myers-Perry black hole and the black ring, there also exist multi blackhole solutions. In fact, in five dimensions it is possible that essentially all uni ormulti horizons black holes with two axial Killing vectors have been found by now(up to iterations between them). All these findings are represented in Fig. 3 thatinclude the doubly spinning black objects. In contrast, the situation in six or moredimensions is much more obscure. Only the MP black hole is explicitly knownand the black rings, helical black rings and blackfolds only perturbatively. Forblack holes with asymptotically (A)ds boundary conditions exact black holes and thin black rings. These phases, and the proposal of, are shown in Fig. 4. Themost interesting analysis comes from comparing the different black hole solutionsin higher dimensions to elucidate and learn which properties change when tuningthe number of dimensions. We present the phase diagrams in the micro-canonicalensemble (area vs. angular momenta with fixed mass) in this section. ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main Phase diagram of all known uni horizon black holes in five space times dimensions. The doublyspinnign black objects have its second angular momentum equaly fixed j = 0 . Fig. 4.
Proposal for the completion of phase curves in D ≥
6. The plot on the left are the patternsfor asymptotically flat black holes with a single spin proposed in. In AdS, the plot on the right, thepattern is compressed to the range J ≤ M at small M . We stress that the details of the connections ( e.g., first order vs. second order transitions) remain unknown and are arbitrarily drawn.
8. The selection rule
Black objects in certain regimes have a black membrane phase and behave accord-ingly. One could then use the inverse logic and build new black holes, by bend-ing these horizons to form compact objects with appropriate boundary conditions,from a black string/brane. This idea was widely employed in for generating thin black rings, helical black rings and blackfolds. In the process of construct-ing the higher dimensional black ring, it was found that the absence of nakedsingularities required a zero-tension condition that corresponds to balancing thestring/brane tension against the centrifugal repulsion. In other words, General Rel-ativity encodes(selects) in the equations of motion of black holes the regularityconditions on the geometry.This condition is in tight correspondence with the conservation of the stressenergy tensor. The quasilocal formalism gives the appropriate definition forthe stress energy tensor in higher dimensions that, in absence of matter, satisfiesa local conservation law D a τ ab = 0 (31)where the covariant derivative is with respect to the boundary metric h ab . Thecondition (31) is then satisfied in the absence of conical singularities. The con-servation and explicit expressions of the stress tensor can be found in. This extraingredient is the balance(zero tension) condition encoding the selection rule of GRfor regular black hole configurations. ctober 5, 2018 21:46 WSPC - Proceedings Trim Size: 9.75in x 6.5in main As an example we find the balance(zero tension) condition for a D-dimensional blackring with dipole charges, as solution of Einstein-Maxwell-dilaton theory with thedilaton coupling a = 4 /N − n/ ( n + 2). At high spin its geometry will be that of astraight black string with boost and charge (parametrized by α and γ respectively).Therefore, (31) determines the specific value for the boost parameter required forthis agreement between the two geomteries. A straightforward computation fixessinh α = (1 /n ) + N sinh γ (32)The charges of the thin D-dipole black ring are the ones of the charged boostedblack string with a fixed boost value (32). In D = 5 these agree with.
9. Outlook
We were able to provide a catalogue for current known species of D -black holes. Inspite of all this headway, the complete list of all possible topologies that the eventhorizon of a higher dimensional black hole can display, for each of the three relevantasymptotic behaviors (Minkowski, AdS and dS), is still unknown. Only few explicitmetrics of higher dimensional black holes are known and so, it would be worthcompleting the task and find the more exotic species. It would be also interesting toinvestigate the stability and to further explore the selection rule for regular blackhole solutions in GR. Acknowledgements
MJR wants to thank D. Astefanesei, M. Caldarelli, H. Elvang, R. Emparan, T.Harmark, J. Kunz, R. B. Mann, V. Niarchos, N. Obers, E. Radu, S. Theisen andO. Varela for the valuable discussions.
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