On explicit thermodynamic functions and extremal limits of Myers-Perry black holes
aa r X i v : . [ h e p - t h ] A p r On explicit thermodynamic functions and extremal limitsof Myers-Perry black holes
Jan E. ˚Aman and Narit Pidokrajt Department of PhysicsStockholm UniversitySE-106 91 StockholmSweden
Abstract
We study thermodynamic properties of Myers-Perry black holes by deriving explicit fun-damental relations from which we can obtain the temperature and specific heat in terms ofexplicit control parameters in arbitrary dimensions. Using the definition of extremal blackholes we establish the generalized Kerr bound in arbitrary dimension. We study thermody-namic geometries of the Myers-Perry black holes with equal angular momenta in arbitrarydimensions and draw thermodynamic cone diagrams which capture the extremal limits of theblack holes. Thermodynamic state space is represented geometrically as a wedge embedded inMinkowski space. The opening angle of such a wedge is uniquely determined by the numberof spacetime dimensions and the number of angular momenta. Our results can potentially beused to generalize thermodynamic instability analysis and other studies in which extremallimits of the Myers-Perry black holes are required.
Black hole solutions in more than four spacetime dimensions have been the subject of increasingattention in recent years. Of particular interest are the Myers-Perry (MP) black holes [1] whoseuncharged rotating version is a direct generalization of the Kerr black hole solution in GeneralRelativity. The MP solutions are significant mainly because of the richness of the solutionsthemselves. This is due to the possibility of rotation in N independent rotation planes with therotation group SO ( d −
1) having Cartan subgroup U (1) N with N ≡ (cid:22) d − (cid:23) , (1)where ⌊ ⌋ denotes the integer part. To each of these rotations there is an associated angularmomentum component J i . Dimensionality also plays a role when one consider the dynamics ofthe black hole solutions as pointed out by Emparan and Reall [2]. The other aspect of rotationthat changes qualitatively as we increase the number of dimensions is the competition betweenthe gravitational and centrifugal potentials. The radial fall-off of the Newtonian potential − GMr d − depends on the number of dimensions, whereas the centrifugal barrier J M r does not as rotationis confined to a plane. It is readily seen that the competition between the two quantities is [email protected] [email protected], [email protected] d = 4, d = 5, and d >
6. This shows that the dimensionality will have dramaticconsequences for the behavior of black holes. Vacuum black hole solutions in five dimensionsinclude the so-called black rings when we consider stationary solutions with two rotational Killingvectors [3]. The black ring solutions in five dimensions are exact solutions having the horizontopology S × S . The other feature that appears in higher dimensions is the presence of blackobjects with extended horizons, i.e. black strings and in general black p -branes which are unstableobjects [4]. We take note that these solutions are not asymptotically flat but they give us intuitionfor black holes in higher dimensions.For a comprehensive classification of black hole species in higher dimensions we refer thereader to a review by Rodriguez [5]. Recently an interesting study of particle injections to MPblack holes and black rings was done by Bouhmadi-Lopez et al. [6] finding that this particularway of destroying a black hole is not possible and that Cosmic Censorship is preserved.In this paper we study merely the uncharged MP black hole solutions in asymptotically flatspacetime with arbitrary rotation in each of the N ≡ (cid:4) d − (cid:5) independent rotation planes. Thesolutions have to be treated separately depending on whether the number of dimensions is oddor even. The black holes have ( d − / d is odd and ( d − / d is even.The multiple-spin Kerr black hole’s metric in Boyer-Lindquist coordinates for odd d is given by ds = − d ¯ t + ( r + a i )( dµ i + µ i d ¯ φ i ) + µr Π F ( d ¯ t + a i µ i d ¯ φ i ) + Π F Π − mr dr , (2)where d ¯ t = dt − mr Π − mr dr, (3) d ¯ φ i = dφ i + ΠΠ − mr a i r + a i dr, (4)with the constraint F = 1 − a i µ i r + a i , (5) µ i = 1 . (6)The function Π is defined as follows: Π = ( d − / Y i =1 ( r + a i ) . (7)The metric is slightly modified for even d . The event horizons in the Boyer-Lindquist coordinatesoccur where g rr = 1 /g rr vanishes. They are the largest roots ofΠ − mr = 0 even d (8)Π − mr = 0 odd d. (9)The areas of the event horizon are given by A = Ω ( d − r + Y i ( r + a i ) odd d, (10)2 = Ω ( d − Y i ( r + a i ) even d. (11)In d = 5 there can be only two angular momenta associated with the Kerr black hole, thus thearea of the event horizon reads A = 2 π r + ( r + a )( r + a ) . (12)The Bekenstein-Hawking entropy is given by S = k B A G , and we can choose k B = π and G = Ω ( d − π so that the Bekenstein-Hawing entropy for the MP black holes is simplified as S = 1 r + Y i ( r + a i ) odd d, (13) S = Y i ( r + a i ) even d. (14)Parts of this paper are devoted to a study of extremal limits of MP black holes. Extremalblack holes are those with vanishing temperature (because zero surface gravity is zero in theextremal limit). In a thermodynamic sense such the black holes do not radiate thermally. In this section we study and derive a fundamental relation of MP black holes with all the allowednumber of spins appearing in any number of dimensions. We utilize the manipulation employedin [7] in obtaining the fundamental relation. For the single spin case we begin by observing that S = r + m. (15)Inserting r + = Sm in Eqs. (8) and (9) we can solve for m in terms of S and a where m = 4 Md − , (16)and a = d − JM . (17)Next we can solve for M which is given by M = d − S d − d − (cid:18) J S (cid:19) / ( d − . (18)In the multiple-spin case we can follow the same procedure used in the single-spin case but wenow have a i = d − J i M , (19)and the desired mass formula M ( S, J , J , ..., J n ) is found to be M = d − S d − d − n Y i (cid:18) J i S (cid:19) nd − . (20)3ero spins can be ignored but the number n must obey n (cid:4) d − (cid:5) . Performing differentiation, T = ∂M∂S , we obtain the black hole’s temperature as T = d − n Y i (cid:18) J i S (cid:19) − n X k J k S n Y i = k (cid:18) J i S (cid:19) S d − n Y i (cid:18) J i S (cid:19) d − d − . (21)As an example we will compute the mass function and the black hole temperature in d = 7 withthree independent angular momenta M = 54 S (cid:18) J S (cid:19) (cid:18) J S (cid:19) (cid:18) J S (cid:19) . (22)The temperature in d = 7 is as follows: T = ∂M∂S = 2 + J S + J S + J S − J S J S J S S (cid:16) J S (cid:17) (cid:16) J S (cid:17) (cid:16) J S (cid:17) , (23)which becomes zero in the extremal limit.We study thermodynamic functions of the MP black holes and for simplicity concentrate ontwo special cases i.e. when (i) when a certain number of spins are zero and the remaining spinsare nonzero and equal (ii) when all spins are turned on and are equal in magnitude. We willmake attempts to investigate more general cases in forthcoming papers. J equal For the particular case when we have n nonzero equal spins J , and remaining spins are zero wecan write the mass function as M = d − S d − d − (cid:18) J S (cid:19) nd − (24)where n ⌊ d − ⌋ . The temperature becomes after factorization T = ∂M∂S = ( d − (cid:16) q n − d +3 d − JS (cid:17) (cid:16) − q n − d +3 d − JS (cid:17) S d − (cid:16) J S (cid:17) d − n − d − . (25)When T = 0 we have an extremal limit. If 2 n > d − SJ (cid:12)(cid:12)(cid:12) extr = 2 r n − d + 3 d − . (26)This limit exists for 2 n > d − M, S ) coordinates as S d − M d − (cid:12)(cid:12)(cid:12) extr = 2 d − n − (2 n − d + 3) n ( d − d − n n (27)4igure 1: Temperatures of the MP black holes (atfixed J ) in various dimensions when there is only oneangular momentum present. Note that in d = 5 the temperature vanishes at zero entropy, whereasin other dimensions the temperatures tend to infin-ity showing that there are no extremal limits for theblack hole solutions in d > when there is only oneangular momentum turned on. Figure 2:
Temperatures of the MP black holes (atfixed J ) in various dimensions when all the angularmomenta are present and they are equal in magni-tude. The extremal limits can be read off at T = 0 .As the number of dimension increases the extremallimits tend to occur at lower entropies (but note e.g. the irregularities of how T approaches zero for T = 0 between d = 6 , and d = 8 , . or in ( M, J ) coordinates J d − M d − (cid:12)(cid:12)(cid:12) extr = 2 d − n − (2 n − d + 3) n − d +32 ( d − d − ( d − d − n n . (28)For the case 2 n = d − d ) we have S ext = 0 and J d − M d − (cid:12)(cid:12)(cid:12) extr = 2 d − ( d − d − . (29)For even d with all n = d − SJ (cid:12)(cid:12)(cid:12) extr = 2 √ d − , (30)and J d − M d − (cid:12)(cid:12)(cid:12) extr = 2 d − ( d − d − ( d − d − . (31)For odd d with all n = d − spins we have SJ (cid:12)(cid:12)(cid:12) extr = 2 √ √ d − , (32)5 n = 1 n = 2 n = 3 n = 4 n = 54 SJ > SJ > SJ >
26 - SJ > √ SJ > SJ > √
28 - - SJ > √ SJ > SJ > √
10 - - - SJ > √
11 - - - SJ > SJ > Extremal limits in ( S, J ) coordinates for MP black holes in various spectrum of dimen-sions, d , for different angular momenta, n . A dash means that no extremal limit exists. and J d − M d − (cid:12)(cid:12)(cid:12) extr = 2 d ( d − d − ( d − d − ( d − d − . (33)The Schwarzschild limit is when J = 0 and this sets a physical bound to be S d − M d − d − ( d − d − . (34)We list the SJ (cid:12)(cid:12) extr for d
11 in Table 1. We present extremal limits in various coordinates for d up to 11 in Appendix B. It is straightforward to compute the specific heat for MP black holes of dimension d with n equalspins is C = T ∂ M∂S = ( d − J S )(1 − n − d +3 d − J S ) S (2 n − d +3)(2 n +1) d − ( J S ) + dn − d +3) d − J S − J S = dn − d + 3 − p n ( d n + 4 dn − d − n + 20 d − dn + d − n − n − , (36)and except for 2 n = d − J S = d − n − d + 3 . (37)6 .5 1 1.5 2 J-40-202040C Figure 3:
A plot of C as a function of J for d = 5 , n = 1 with S = 1 . It goes to infinity at J = √ . Aplot of C as a function of J for d = 5 , n = 2 with S = 1 . Figure 4:
A plot of C as a function of J for d = 5 , n = 2 with S = 1 . The specific heat in this casechanges sign at J = and goes to infinity at J = q √ − . J equal For the particular case when all allowed (cid:4) d − (cid:5) spins J are equal we still have M = d − S d − d − (cid:18) J S (cid:19) nd − , (38)where n = (cid:4) d − (cid:5) . For even d we have n = d − T = ( d − (cid:16) J √ d − S (cid:17) (cid:16) − J √ d − S (cid:17) S d − (cid:16) J S (cid:17) , (39)while for odd d we have n = d − and temperature T = ( d − (cid:16) √ J √ d − S (cid:17) (cid:16) − √ J √ d − S (cid:17) S d − (cid:16) J S (cid:17) d − d − . (40) In this section we study thermodynamic geometry (also known as Ruppeiner geometry [8]) ofthe MP black hole families. Black hole thermodynamic geometry has been studied over the past7ecade, see e.g. [9] and references therein for a review. The geometrical patterns are given bythe curvature of the Ruppeiner metric defined as the Hessian of the entropy on the state spaceof the thermodynamic system g Rij = − ∂ i ∂ j S ( M, N a ) , (41)where M denotes mass (internal energy) and N a are other mechanically conserved parameterssuch as charge and spin. The indices i, j run over these parameters. The minus sign ensures thatthe metric has positive signature when the entropy function is concave. This metric is conformalto the so-called Weinhold metric (defined as the Hessian of energy function) via g Wij = T g
Rij where T is thermodynamic temperature of the system of interest. In ordinary thermodynamicsit has been argued that the curvature scalar of the Ruppeiner metric measures the complexity ofthe underlying interactions of the system, i.e. the metric is flat for the ideal gas whereas it hascurvature singularities for the van der Waals gas . However the story is different in black holethermodynamics in that results have been obtained but there has not been a consensus on howto interpret uncovered geometrical patterns of black hole thermodynamics. In [10] it is arguedthat local thermodynamic instability of black holes is encoded in the Ruppeiner metric, and thatthis method is consistent with the Poincar´e method of stability analysis. Other works in thisdirection can be found in [11–31].Apart from being a tool to analyze black hole’s stability, it is expected that geometricalpatterns (whether the metric is flat or nonflat) will play a role in the context of quantum gravity.Below we study both Ruppeiner and Weinhold geometries of the MP black holes in arbitrarydimensions. Instability of MP black holes have been investigated e.g. in [32–34]. Recently in [35]Astefanessei, Rodriguez and Theisen argue that the singularity of the Ruppeiner metric couldhelp detect the threshold of the membrane phase of MP black holes. In particular they study theRuppeiner curvature of doubly spinning MP black holes in arbitrary dimensions. Their resultswill be discussed in comparison with ours in this section.We use an algebraic computation package CLASSI [36] in computing all the metrics andassociated curvature scalars. The Weinhold metric in original coordinates is ds W = λ W (cid:16) [ − n + 1)( d − n − J + 8( dn − d + 3) J S − ( d − S ] dS +[64 n ( d − n − J S − n ( d − J S ] dSdJ +[ − n ( d − n − J S + 8 n ( d − S ] dJ (cid:17) (42)where λ W = 14( d − S + 4 J ) d − n − d − S d +2 n − d − . (43)For n = 1 this reduces to (37)-(38) in [7]. With the coordinate transformations u = JS (44) In this case the singularities are associated with phase transitions = r d − d − S d − d − (1 + 4 u ) n d − , (45)it becomes diagonal and reads ds W = − dτ + 2 n (cid:0) d − − ( d − − n )4 u (cid:1) ( d − u ) τ du . (46)This is a flat metric. It can be brought to Rindler coordinates ds W = − dτ + τ dσ (47)by an additional coordinate transformation σ = Z u s n [ d − n − d + 3)4 u ]( d − u ) du. (48)Finally we can transform the metric into Minkowski coordinates ds W = − dt + dx using t = τ cosh σ, x = τ sinh σ . (49)The transformation from u to σ is best studied in three subcases depending on 2 n is greaterthan, equal or less than d −
3. The flat metrics can be embedded as a state space wedge inthe Minkowskian-like diagram, which we call thermodynamic cone . The black hole temperaturevanishes on the edge of the wedge, whilst the entropy vanishes on the thermodynamic cone.Hence this diagram can be used to decide which black hole families possess genuine extremallimits, i.e. the black hole families without extremal limits will have no state space wedges in thethermodynamic cone, see Fig. 5. The case n > d − Here the transformation is σ = √ n − d + 3 √ √ d − √ n − d + 3 u √ d − √ d − n − √ d − √ d − n − u p d − n − d + 3) u . (50)As here u extr = √ d − √ n − d +3 we have σ extr = √ n − d + 3 √ √ d − √ d − n − √ d − √ √ d − n − u √ d − n = d − n = d − This resembles relativity’s light cone structure in the sense that the cone displays the causality of the thermo-dynamic state space. x n = d − [ , ] [ , ] [9,4] T = 0 S = 0 t x n = d − [ , ] [ , ] [8,3] t x n = d − [ , ] [ , ] [9,3] Figure 5:
We present the black hole solutions in three series: Series 1 is when n = d − , Series2 when n = d − , and Series 3 when n = d − . For each series we have three cases as shownin the figures. Numbers in the brackets refer to [ d, n ] with d the spacetime dimension, and n thenumber of nonzero spins. The case n = d − This applies to dimension 5 with 1 spin, dimension 7 with 2 spins etc. We use the followingtransformation σ = √ d − √ √ d − u (52) u = 12 tan √ √ d − √ d − σ ! (53)Here we obtain u extr = ∞ and we have σ extr = π √ d − √ √ d − n σ extr | xt | arcsinh1 ≈ . π √ ≈ . q arcsinh1 ≈ . (arcsinh1 + π √ ) ≈ . π √ ≈ . q (arcsinh1 + arctan √ ) ≈ . (arcsinh1 + 2 arctan √ ≈ . √ π ≈ . √ (arcsinh1 + π √ ) ≈ . (arcsinh1 + q π ) ≈ . π ≈ . √ (arcsinh1 + √ q ) ≈ . (arcsinh1 + 2 √ ≈ . n > d −
3. Note that as the number of angular momenta increases the wedgetends to fill up the thermodynamic cone.
The case n < d − Here there are no extremal limits as T cannot become 0. Integration gives σ as σ = − √ d − n − √ √ d − √ d − n − u √ d − √ d − n − √ d − √ d − n − u p d − − d − n − u (55)This formula will however not hold up to u → ∞ .We present additional special cases in Appendix A. Given the conformal transformation, ds R = T ds W , the Ruppeiner metric is easily obtained inthe same coordinates ( S , J ) and differs from the Weinhold metric in Eq. (42) with λ W replacedby λ R = 1( d − d − S (1 + J S ) d − d − (1 + n − d +3 d − J S ) . (56)The Ruppeiner metric is not flat and its curvature is given by R R = − S n − d +3 d − J S (1 + n − d +3 d − J S )(1 − n − d +3 d − J S ) . (57)11f 2 n > d − JS = √ d − √ n − d + 3 , (58)whereas if 2 n < d − JS = √ d − √ d − n − n = d − R R = − S without a divergence. In this paper it is shown that MP black holes of dimension d with n equal nonzero spins and2 n > d − of 2 n − d + 3 is 0, 1 or 2. Forblack holes with 2 n < d − i.e. infinitely large spin at least one of the possible ⌊ d − ⌋ spins must be exactly 0. This follows from the general temperature formula (21) whichalways allows T = 0 unless some J i are zero, see also example of d = 7 with three independentspins in Eq. (23). We have derived explicit thermodynamic functions of Myers-Perry black holes namely the massformula in any dimension for the MP black holes with an arbitrary number of angular momen-tum. We have also derived the Bekenstein-Hawking temperature and the specific heat of theMP black hole in general. It is readily seen that thermodynamic metrics vary with the num-ber of dimensions, d and the number of angular momenta, n . We establish extremal limits ofblack holes for the MP black holes with arbitrary n in any dimension, in other words we areable to establish the generalized Kerr bound of multiply spinning Kerr black holes in higherdimensions. We study thermodynamic geometries of the Myers-Perry black holes with arbitraryangular momenta in various dimensions and present the outcomes by drawing thermodynamiccone diagrams which capture the extremal limits of the black holes. Thermodynamic state spacecan be geometrically represented as a wedge embedded in Minkowski space. The opening angle ofsuch the wedge is uniquely determined by the number of spacetime dimensions and the numberof angular momenta. We believe that these results will be useful for the purpose of studyinghigher dimensional black holes.We conjecture that the membrane phase ultraspinning MP black holes is reached at theminimum temperature in the case 2 n < d − where n denotes the number of angular momenta and d the number of dimensions. cknowledgments Narit Pidokrajt acknowledges the KoF group, Fysikum, Stockholms Universitet for the kind ofhospitality. We thank Ingemar Bengtsson for his enlightening and many useful comments. NPwould like to thank Roberto Emparan for enlightening discussions on MP black holes with equalspins while he was a visitor in Barcelona.
Appendix A
We extend our discussion on the opening angles of the Weinhold metrics here with two subcases:
The subcase n = d − u to σ is σ = 12 (cid:18) arcsinh 2 u √ d − √ d − √ d − u √ d − u (cid:19) (60)Here we have u extr = √ d − which gives σ extr = 12 (cid:18) arcsinh1 + √ d − √ d − √ (cid:19) (61) The subcase n = d − σ = √ d − √ √ d − arcsinh 2 √ u √ d − √ d − √ √ d − u √ d − u ! (62)In this case u extr = √ d − √ so we obtain σ extr = √ d − √ √ d − (cid:18) arcsinh1 + √ d − √ √ d − (cid:19) (63) Appendix B
In the table below we present extremal limits in various coordinates.13 n Extr limits Extr limits Extr limits4 1 SJ > JM SM
45 1 SJ > J M S M SJ > J M S M SJ > √ J M S M
17 2 SJ > J M S M SJ > √ J M S M SJ > √ J M S M SJ > J M S M SJ > √ J M S M
10 4 SJ > √ J M S M
11 4 SJ > J M S M