Thermodynamic instability of doubly spinning black objects
aa r X i v : . [ h e p - t h ] A p r AEI-2010-039
Thermodynamic instability of doubly spinning blackob jects
Dumitru Astefanesei, Maria J. Rodriguez, and Stefan Theisen Max-Planck-Institut f¨ur Gravitationsphysik,Albert-Einstein-Institut, 14476 Golm, Germany
ABSTRACT
We investigate the thermodynamic stability of neutral black objects with (at least)two angular momenta. We use the quasilocal formalism to compute the grand canonicalpotential and show that the doubly spinning black ring is thermodynamically unstable. Weconsider the thermodynamic instabilities of ultra-spinning black objects and point out asubtle relation between the microcanonical and grand canonical ensembles. We also findthe location of the black string/membrane phases of doubly spinning black objects. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] ontents
The physics of event horizons in higher-dimensional General Relativity (GR) is an inter-esting area of research not just for its intrinsic relevance to string theory. An investigationof black hole solutions in higher dimensions is also important because it has revealed newfeatures: a richer rotation dynamics and the existence of regular extended black hole solu-tions.It is clear by now that some of the remarkable properties of four-dimensional black holesdo not hold in general. A notorious example of particular importance concerns their horizontopology. In four dimensions, the spherical S topology is the only allowed horizon topol-ogy for asymptotically flat (AF) stationary black holes. A related result is the ‘uniquenesstheorem’, which states that a stationary AF vacuum black hole in four dimensions is char-acterized by its mass and angular momentum and has no other independent characteristic(hair).The spectrum of stationary black objects is far richer in dimensions bigger than four(see [1] for a concise review). The restrictions on the topology of AF black holes requirethat spatial sections of the event horizon must be positive Yamabe type [2] and if spinning(stationary) they have to be axisymmetric [3]. The most obvious indication is the existenceof a neutral AF solution describing a spinning black ring in five dimensions [4, 5]. As itwas shown in [4], there are three solutions with the same asymptotic conserved charges (thesame mass and angular momentum). On top of the well known Myers-Perry (MP) black1ole [6] with an S horizon there are two different black rings with an S × S horizon( S × S D − in D dimensions). Therefore, unlike in four dimensions, the black objects inhigher dimensions are not completely determined by a few conserved asymptotic charges. But the space of solutions of Einstein’s equations in D dimensions also includes extendedblack holes. The black p-branes [8], dubbed black strings when p = 1 or otherwise mem-branes, are transverse asymptotically flat (only AF in D − p directions), evade the no-hairtheorems, and exhibit horizon topologies S D − − p × R p . Interestingly enough, due to thericher rotational dynamics, in certain regimes in higher dimensions the thermodynamicalproperties of the compact black holes resemble those of the extended black objects. In thispaper we aim to make progress towards a better understanding on these properties of blackobjects.We first study in detail the thermodynamic instabilities of doubly spinning black objects.We identify the ultra-spinning regimes (in parameter space) from their thermodynamicquantities. Our motivation stems from the observation made in [9] that, in dimensionsgreater than four, the thermodynamics of certain fast spinning MP black holes show aqualitative change in behaviour. That is, there is a transition towards a black membrane-like behaviour. This is due to the fact that, as one increases the angular momentum, thetemperature of these ultra-spinning black holes reaches a minimum and then starts to growas expected for the black membrane. Black rings also exhibit a change in the thermodynamic behaviour that resembles theultra-spinning regime of black holes when ultra-spinning along the S direction. In the‘thin ring approximation’, when the radius of S is much larger than the radius of S D − ,the singly spinning black ring can be approximated by a boosted black string. An inter-esting question we would like to address in this paper is how the ultra-spinning regime ofthe D = 5 black ring is affected by adding the second angular momentum, along the S .In other words, whether neutral doubly spinning black ring is thermodynamically stable inthe grand canonical ensemble and whether there is any connection with its ultra-spinningregime.Since there is no known background subtraction method for these black objects, we usea slightly modified version of the quasilocal formalism of Brown and York [12] to addressthese questions. Supplemented with counterterms [13], the quasilocal formalism becomesa very powerful tool to study the thermodynamics of black objects that are AF. Recently,several concrete five-dimensional examples were discussed in detail in [7, 14] and in [15] forthe unbalanced black ring.The relevant thermodynamic potential in the grand canonical ensemble is the Gibbspotential that is the Euclidean action divided by the periodicity of the Euclidean time [16].The Euclidean method was applied to the black ring thermodynamics in [17]. Since theblack ring does not have a real non-singular Euclidean section, the ‘quasi-Euclidean’ method[18] was adopted to analyze the black ring thermodynamics. In this approach, the horizon In [7] it was proposed that the necessary information to distinguish between black objects with differenthorizon topologies is encoded in the subleading terms in the boundary stress tensor. The numerical evidence of [10] suggests that the onset of the ultra-spinning regime for singly spinningMP black holes corresponds to a zero mode associated to the Gregory-Laflamme instability [11].
2s described by the ‘bolt’ in a complexified
Euclidean geometry rather than a real one. Itwas also pointed out in [17] that the neutral black ring with one angular momentum isunstable to angular fluctuations — a more detailed analysis can be found in [7, 19, 20].By employing this method to compute the Gibbs potential, we find the response func-tions directly in the grand canonical ensemble. We observe that the second angular mo-mentum changes the situation in the sense that, unlike the black ring with one angularmomentum, the doubly spinning ring is stable against perturbations in the angular velocityin some specific region of the parameter space. However, a careful analysis of all responsefunctions that characterize the system reveals that the doubly spinning black ring is ther-modynamically unstable in the grand canonical ensemble. That is, there is no region in theparameter space in which all response functions are positive definite.On general physical grounds it is expected that the microcanonical ensemble of asymp-totically flat black holes is dominated by diffuse radiation states rather than black holestates [21]. In other words, it is favorable for the black hole to decay away (the heat ca-pacity is negative — see Appendix B for a detailed discussion on local thermodynamicconditions) and so pure thermal radiation is a local equilibrium state. Indeed, it was shownin [22] that, for any vacuum black hole characterized by its mass and angular momenta, theHessian of the entropy always has a negative eigenvalue. Since the Hessian of the entropyis related to the inverse of the Hessian of the Gibbs potential [23], this implies at genericpoints in the moduli space, i.e. away from the hypersurfaces defined by a vanishing eigen-value, an instability in the grand canonical ensemble [22].Identifying which doubly spinning black hole solutions exhibit this ultra-spinning regimewill be our next objective. We will first find the threshold of the black membrane regime forMP black holes with at least two large angular momenta. To compare the ultra-spinningregimes of both, the black hole and black ring, we present a careful analysis of their ultra-spinning regime. For the black ring we observe a similar but slightly different ultra-spinningphase. That is, the temperature does not have a minimum but rather there is a ‘turningpoint’, which is responsible for the boosted black string behaviour. We also find that evenafter adding the second angular momentum the ring can have a membrane-like behavior.However, for large enough values of the second angular momentum the ‘membrane phase’disappears.The dynamical instabilities were related to the thermodynamic ones by (a conjectureof) Gubser and Mitra [24]: gravitational backgrounds with translationally invariant horizondevelop a tachyonic mode (negative non-conformal mode) whenever the specific heat of theblack brane geometry becomes negative. Our goal is not to study dynamical instabilities,which would require a perturbative analysis, but rather to generalize the arguments of [9]to black rings. However, to connect the work of [9] to [24], one has to understand the blackhole thermodynamics in the grand canonical ensemble (see, e.g., [25]). At first sight, it issurprising that an analysis in the microcanonical ensemble [9] provides information aboutthe membrane phase. We point out a subtle relation between the microcanonical and grand The complex geometry is obtained by the usual analytic continuation of time coordinate, τ = it . For a black ring with one angular momentum the ‘isothermal compressibility’ (moment of inertia) isalways negative [7].
In this section we apply the counterterm method to doubly spinning five-dimensional vac-uum solutions of Einstein gravity. We explicitly show how to compute the boundary stresstensor and the conserved charges for Myers-Perry black hole, doubly spinning black ring,and doubly spinning black branes.
To begin our considerations on thermodynamics of doubly spinning black objects in fivedimensions, we recall the description of quasilocal formalism [12] supplemented with coun-terterms.To define the conserved charges we use the divergence-free boundary stress tensor pro-posed in [17]: τ ij ≡ p | h | δIδh ij = 18 πG (cid:16) K ij − h ij K − Ψ( R ij − R h ij ) − h ij (cid:3) Ψ + Ψ ; ij (cid:17) (2.1)where Ψ = √ / √ R , h ij is the induced boundary metric, and R ij is its Ricci scalar. Arigorous justification and more details about this proposal can be found in [26, 27, 28].Here, I is the renormalized action that includes counterterms, I = 116 πG Z M R √− g d x + ǫ πG Z ∂M K − r R ! p | h | d x (2.2) K is the extrinsic curvature of ∂M and ǫ = +1( −
1) if ∂M is timelike (spacelike).The boundary metric can be written locally in ADM-like form h ij dx i dx j = − N dt + σ ab ( dy a + N a dt )( dy b + N b dt ) (2.3)4here N and N a are the lapse function and the shift vector respectively and { y a } arethe intrinsic coordinates on a (closed) hypersurface Σ. If the boundary geometry has anisometry generated by a Killing vector ξ i , a conserved charge Q ξ = I Σ d y √ σn i τ ij ξ j (2.4)can be associated with the hypersurface Σ (with normal n i ). The Einstein equations in higher dimensions have spinning black hole solutions [6]. In fivedimensions, the Myers-Perry black hole in Boyer-Lindquist type coordinates is ds BH = − dt + Σ (cid:18) r ∆ dr + dθ (cid:19) + ( r + a ) sin θ dφ + ( r + b ) cos θ dψ + m Σ (cid:0) dt − a sin θ dφ − b cos θ dψ (cid:1) (2.5)where Σ = r + a cos θ + b sin θ, ∆ = ( r + a )( r + b ) − m r (2.6)and m is a parameter related to the physical mass of the black hole, while the parameters a and b are associated with its two independent angular momenta. This metric dependsonly on two coordinates, 0 < r < ∞ and 0 ≤ θ ≤ π/
2, and it is independent of time, −∞ < t < ∞ , and the azimuthal angles, 0 < φ, ψ < π .Since r is playing the role of a radial coordinate in this coordinate system, the eventhorizon is also the null surface determined by the equation g rr = 0 . So, the event horizonof the black hole can be computed by using (A.5), which implies ∆ = 0. The largest rootof this equation gives the radius of the black hole’s outer event horizon r h = 12 (cid:16) m − a − b + p ( m − a − b ) − a b (cid:17) (2.7)Notice that the horizon exists if and only if a + b + 2 | a b | ≤ m (2.8)so that the condition m = a + b + 2 | a b | or, equivalently, r h = | a b | defines the extremalhorizon of a five dimensional black hole (when one angular momentum vanishes, the horizonarea goes to zero in the extremal limit). Otherwise, the metric describes a naked singularity.In the asymptotic limit, r → ∞ , the metric (2.5) approaches Minkowski space ds = − dt + dr + r ( dθ + sin θ dφ + cos θ dψ ) (2.9)5e use the expression of black hole metric in Boyer-Lindquist coordinates to compute theboundary stress tensor and we obtain the following non-vanishing components: τ tt = 18 πG (cid:18) − m r −
53 ( a − b ) cos 2 θr + O (1 /r ) (cid:19) ,τ tφ = 18 πG (cid:18) − a m sin θr + O (1 /r ) (cid:19) ,τ tψ = 18 πG (cid:18) − b m cos θr + O (1 /r ) (cid:19) ,τ θθ = 18 πG (cid:18) (cid:0) a − b (cid:1) cos 2 θr + O (1 /r ) (cid:19) , (2.10) τ φφ = 18 πG (cid:18) (cid:0) a − b (cid:1) ( − θ ) sin θr + O (1 /r ) (cid:19) ,τ ψψ = 18 πG (cid:18) (cid:0) a − b (cid:1) (1 + 2 cos 2 θ ) cos θr + O (1 /r ) (cid:19) ,τ φψ = 18 πG (cid:18) − a b m cos θ sin θr + O (1 /r ) (cid:19) This stress tensor is covariantly conserved with respect to the boundary metric (2.9). Wealso notice that, for equal angular momenta, the diagonal ‘angular’ components of the stresstensor vanish — this is intuitively expected due to the enhanced symmetry.Using the definition (2.4), it is straightforwardly to obtain the conserved charges asso-ciated with the surface Σ as M = I Σ d y √ σn i τ ij ξ jt , J φ = I Σ d y √ σn i τ ij ξ jφ , J ψ = I Σ d y √ σn i τ ij ξ jψ where the normalized Killing vectors associated with the mass and angular momenta are ξ t = ∂ t , ξ φ = ∂ φ , and ξ ψ = ∂ ψ respectively. We find M = 3 π m G , J φ = π m a G , J ψ = π m b G (2.11)which is in perfect agreement with the ADM calculation.
A black ring is a five-dimensional black hole with an event horizon of topology S × S and the metric was presented in [4] — the solution of Emparan and Reall has one angularmomentum. In five dimensions, a more general solution for a black ring with two angularmomenta was presented by Pomeransky and Sen’kov [5]. Some properties of the solutionincluding the structure of the phases in the microcanonical ensemble are discussed in [29].A study of its geodesics has been performed in [30] and a careful investigation of globalproperties appeared recently in [31]. We provide here a brief account of the doubly spinningblack ring solution and compute the boundary stress tensor and the conserved charges.We will use the solution in the form presented in [5]. The metric depends just on thecoordinates x and y defined within the following intervals − ≤ x ≤ −∞ < y < − x is like an angular coordinate — this observation will be useful when we willdefine new coordinates that make asymptotic flatness clear.The metric has a coordinate singularity where g yy diverges. The event horizon of thedoubly spinning black ring is located at the smallest absolute value of 1 + λ y + ν y = 0,namely y h = − λ + √ λ − ν ν (2.12)For a regular black ring solution, the parameters ν and λ are constrained to satisfy [5]:0 ≤ ν < , √ ν ≤ λ < ν (2.13)In the limit ν → J φ ) is recovered ( J ψ is theangular momentum on S ). The limit λ → √ ν was carefully studied in [29] and shown tocorrespond to regular extremal black rings.We use a coordinate transformation similar to the one in [29]: x = − k α cos θr , y = − − k α sin θr , α = r ν − λ − ν (2.14)In these coordinates ∂ t , ∂ φ , and ∂ ψ are Killing vectors and the asymptotic metric is thesame as (2.9).The boundary stress tensor in these new coordinates is τ tt = 18 πG (cid:18) − k λ (1 + ν − λ ) 1 r − k F [ ν, λ ]3(1 + ν − λ )(1 − ν ) cos 2 θr + O (1 /r ) (cid:19) ,τ tφ = 18 πG k λ (1 + λ − ν + λν + ν ) p (1 + ν ) − λ (1 + ν − λ ) (1 − ν ) sin θr + O (1 /r ) ! ,τ tψ = 18 πG k λ q ν (cid:2) (1 + ν ) − λ (cid:3) (1 + ν − λ )(1 − ν ) cos θr + O (1 /r ) ,τ θθ = 18 πG (cid:18) k F [ ν, λ ]3(1 + ν − λ )(1 − ν ) cos 2 θr + O (1 /r ) (cid:19) , (2.15) τ φφ = 18 πG (cid:18) k ( − F [ ν, λ ] + F [ ν, λ ] cos 2 θ )3(1 + ν − λ )(1 − ν ) sin θr + O (1 /r ) (cid:19) ,τ ψψ = 18 πG (cid:18) k ( F [ ν, λ ] + F [ ν, λ ] cos 2 θ )3(1 + ν − λ )(1 − ν ) cos θr − O (1 /r ) (cid:19) ,τ φψ = 18 πG (cid:18) k λ √ ν (4 λν − ( λ + (1 − ν ) )(1 + ν ))(1 + ν − λ )(1 − ν ) cos θ sin θr + O (1 /r ) (cid:19) where F [ ν, λ ] = 1 − ν − ν + 5 ν + λ (3 + 7 ν ) + λ (1 − ν − ν ) ,F [ ν, λ ] = 1 − ν − ν + 11 ν + λ (3 + 13 ν ) + 4 λ (1 − ν − ν ) ,F [ ν, λ ] = 5 − ν − ν + 7 ν + λ (15 + 17 ν ) − λ (1 + 13 ν + 2 ν ) ,F [ ν, λ ] = 7 − ν − ν + 29 ν + λ (21 + 43 ν ) + λ (4 − ν − ν )7s in the case of doubly spinning black hole, this stress tensor is covariantly conservedwith respect to the boundary metric (2.9). However, since for the doubly spinning blackring the angular momenta can not be equal, namely3 J ψ ≤ J φ , (2.16)there is no similar symmetry enhancement as in the black hole case in the angular part.By plugging the expressions of the boundary stress-energy components (2.15) in (2.4)we find the following expressions for the conserved charges: M = 3 π k G λ ν − λ , J ψ = 4 π k G λ q ν (cid:2) (1 + ν ) − λ (cid:3) (1 + ν − λ )(1 − ν ) , (2.17) J φ = 2 π k G λ (1 + λ − ν + ν λ + ν ) p (1 + ν ) − λ (1 + ν − λ ) (1 − ν ) (2.18)As expected, the charges computed by using the quasilocal formalism recover correctlythe ADM results [5].In principle, one can obtain a black hole and a black ring with the same conservedcharges. However, an asymptotic observer can not distinguish between a black hole and ablack ring just by computing the conserved asymptotic charges. We would like to emphasizethat it is expected that the subleading terms of the quasilocal stress tensor encode theinformation necessary to distinguish between black objects with different horizon topologies. Here we would like to apply the quasilocal formalism to doubly spinning black p-branes,dubbed also black membranes (BM) or black strings if p = 1. The black membrane metricwe are interested in is obtained by adding flat directions to a 5-dimensional black hole withtwo angular momenta. Therefore, the metric is ds BM = ds BH + D − X i =1 dx i (2.19)where ds BH is the black hole metric defined in (2.5).Since the number of dimensions and the topology are changed, one expects changes withrespect to the former discussion. For example, the form of the counterterm leading to afinite actions may be different when the number of dimensions is increased. However, inthis particular case, what is important is the ‘seed’ 5-dimensional solution to which we addthe flat directions. Thus, the form of the counterterm does not change but the stress tensorwill have new components.A similar computation as for the doubly spinning black hole reveals that the stress tensorof the BM is the one in (2.10) supplemented with the components in the new directions: τ x i x i = 18 πG (cid:18) − m r −
53 ( a − b ) cos 2 θr + O (1 /r ) (cid:19) (2.20)This result resembles the tension (per unit length) of the black string.8 Thermodynamic instability of the black ring
In this section, we discuss the thermodynamics of a doubly spinning ring in the grandcanonical ensemble.So far, we have computed the conserved charges of neutral spinning black objects withtwo angular momenta by using the quasilocal formalism. However, the quasilocal formalismis a very powerful tool for understanding the thermodynamics in more detail. In particular,one can compute the action and, therefore, the thermodynamic potential.In what follows, we present a detailed analysis of thermodynamic stability of the doublyspinning black ring — an analysis of the thermodynamic stability of Myers-Perry black holewith two angular momenta can be found in [20].Let us start by computing the angular velocities and the temperature for this solution.From (A.6) we obtain the following expressions for the angular velocities:Ω ψ = λ (1 + ν ) − (1 − ν ) √ λ − ν λ √ νk r ν − λ ν + λ , Ω φ = 12 k r ν − λ ν + λ (3.1)The area of the event horizon and the temperature (A.7) are A H = 32 π k λ (1 + λ + ν )(1 − ν ) ( y − h − y h ) , T = √ λ − ν (1 − ν )( y − h − y h )8 πk λ (1 + λ + ν ) (3.2)Note that y h = − λ + √ λ − ν ν is the biggest root of (A.5) which corresponds to the outer eventhorizon — at this point, it might be useful to emphasize again that −∞ < y < − Z ( β ) = Z d [ g, φ ] e − I [ g,φ ] (3.3)where φ is a collective notation for the matter fields, d [ g, φ ] is the measure, and I [ g, φ ] isthe Euclidean classical action. The gravitational partition function is defined by a sum overall smooth geometries (including black holes) that are periodic with period β = T − in thesame class of boundary conditions e.g., AF spacetimes.For our purpose it is enough to consider the saddle point approximation. The grandcanonical partition function is then Z = T re − β ( H − Ω a J a ) ≃ e − I cl (here we are interested inblack objects with two angular momenta), where I cl is the classical action. The saddlepoint is usually referred to as a gravitational instanton . The thermodynamic (effective) potential associated to grand canonical ensemble is G [ T, Ω a ] ≡ I cl β = M − T S − Ω a J a (3.4) It should be understood as a low energy effective theory rather than a proper theory of quantum gravity. A quantum field can be treated as a small perturbation about the gravitational instanton. The nextorder contribution, which gives the one loop correction, includes also the thermal radiation outside theblack hole.
9n the Euclidean section, the topology near the horizon is modified and one has todeal with manifolds with conical singularities. It was shown in [32, 33] that the conicaldefect has a contribution to the curvature and, consequently, the path integral is rescaledby e S . However, this can be intuitively interpreted as a consequence of a trace over themacroscopically indistinguishable microstates.Let us now compute the action for the doubly spinning black ring. Since the Ricci scalarvanishes on-shell, the only contribution to the action is coming from the surface terms. Toevaluate these terms, it is convenient to use the ( r, θ ) coordinate system instead of the ( x, y )coordinates — the reason is that the normal to the boundary has just one non-vanishingcomponent. We findlim r →∞ p | h | r R − K ! = 2 k ( λ (1 − ν ) − F [ ν, λ ] cos 2 θ ) sin 2 θ (1 + ν − λ )(1 − ν ) + O (1 /r ) (3.5)where F [ ν, λ ] = 1 + 3 λ + 6 ν + 5 ν − λ − λν . The expression for the total action is I cl = β πk G λ (1 + ν − λ ) (3.6)and satisfies (3.4), which is the quantum statistical relation for the doubly spinning blackring. This can also be regarded as a non-trivial check that the entropy S of this solutionis, indeed, one quarter of the event horizon area A H .We have checked that the usual thermodynamic relations S = − (cid:18) ∂G∂T (cid:19) Ω a , J a = − (cid:18) ∂G∂ Ω a (cid:19) T, Ω b (3.7)are satisfied and so the Gibbs potential G [ T, Ω φ , Ω ψ ] is indeed the Legendre transform ofthe energy M [ S, J φ , J ψ ] with respect to S , J φ , and J ψ .We want to also point out that, in the light of the new developments in understandingthe balance condition for gravity solutions [15], the form of quantum statistical relationhints to the fact that this solution is balanced. Indeed, our results are in perfect agreementwith the recent detailed analysis of the global properties of the doubly spinning black ring[31].Now, we are ready to discuss the thermodynamic stability in the grand canonical ensem-ble — in Appendix B we summarize the thermal stability conditions and present explicitexpressions for some response functions we are interested in. We analyze in detail theresponse functions that signal the (in)stability of the black ring against fluctuations.We consider first the specific heat at constant angular velocities C Ω ≡ T (cid:18) ∂S∂T (cid:19) Ω φ , Ω ψ (3.8) The origin in the Euclidean spacetime translates to the horizon surface in the Lorentzian spacetime.The Euclidean section can be understood as an effective description where the microstates can not bedistinguished. . . . . . . λ νλ ν Figure 1:
Scatter plots in parameter phase space ( ν, λ ) for the doubly spinning black ring. Theplot on the left shows the regions (10,000 points) where the heat capacities are negative, C Ω < gray ) and C J < black ). The regions where the compressibility ǫ φφ ( gray ) and the det[ ǫ ] ( black )are negative cover the entire parameter space (plot on the right) implying the local thermalinstability of the doubly spinning black ring. The region in the parameter space is bounded:0 ≤ ν < λ by the functions 1 + ν and 2 √ ν , shown as the dashed and solid lines respectively. The analytic form of this quantity is too complicated to be written down here. Instead, weshow on the left hand side in Fig. 1 a scatter region in the parameter space of the doublyspinning black ring where this heat capacity C Ω is negative ( gray – 10,000 points). Notealso that the parameters in the solution (2.13) are constrained and represented as a dashedline for λ = 1 + ν and solid line for the extremal black ring with λ = 2 √ ν .In a similar way, we explore the region where the specific heat at constant angularmomentum C J ≡ T (cid:18) ∂S∂T (cid:19) J φ ,J ψ (3.9)is negative. The region (in black) where C J < C Ω and C J , can be positive simultaneously. However, this condition is notsufficient to draw the conclusion of thermodynamic stability: one should also investigatethe matrix of ‘isothermal moment of inertia’.These response functions are defined as ǫ ab ≡ (cid:18) ∂J a ∂ Ω b (cid:19) T, Ω a = b (3.10)We observe in Fig. 1 that the spectrum of the matrix of isothermal moment of inertia,spec[ ǫ ab ], is nowhere positive definite in the parameter space.Since there is no overlap region in parameter space in which all the response functionsof interest are positive definite, we conclude that the doubly spinning black ring is unstablein the grand canonical ensemble. 11 Instabilities from thermodynamics
Many known stationary black holes in higher dimensions present a black string or, moregeneral, black brane phase — we will refer to it as the ‘membrane phase’. That is, as theangular momenta are sufficiently increased (the ultra-spinning regime), the behaviour ofsome black holes and black rings changes to that of extended black branes and strings.In the next subsection , we deal with the ultra-spinning black holes. From the study ofthe Gibbs potential’s Hessian, we show the existence and find the locus of the transitionpoints to the membrane phase. We also argue that there is a subtle relation between themicrocanonical and grand canonical ensembles that may be at the basis of some of theresults for ultra-spinning black holes discussed recently in [10].The analysis can be extended to (doubly) spinning black rings . These results arepresented in Section 4.2.To compare different examples we will make use of the dimensionless expressions forthe temperature t , the spin j , and the angular velocity ω defined by t D − = c t GM T D − ω D − = c ω GM Ω D − , j D − = c j J D − GM D − (4.1)where the numerical constants are c t = 2( D −
2) (4 π ) D − Ω D − (cid:18) D − D − (cid:19) D − , c ω = 16( D −
2) ( D − D − Ω D − , c j = Ω D − D +1 ( D − D − ( D − D − . (4.2) Due to the qualitative changing behaviour of black holes as the dimensions are increased, theauthors of [9] have argued that the ultra-spinning black holes — those in D ≥ a r h = D − D − blows-up exactly at the value (4.3) signaling a thermalinstability of the system. The doubly spinning solutions we consider are with the spins in orthogonal planes. Other black holesolutions, where the two spins are parallely oriented, were also studied [34]. The Ruppeiner curvature is the scalar curvature of the Hessian matrix of the entropy.
12 qualitative understanding of this fact is related to the observation that, as the spinbecomes large, the event horizon spreads out in the plane of rotation: it becomes a higherdimensional ‘pancake’ approaching the geometry of a black brane.The existence of the ultra-spinning limit resembling black branes has a remarkableconsequence. Black branes were shown to be classically unstable [11] so that the ultra-spinning black holes would inherit the Gregory-Laflamme instability. The threshold of theclassical instabilities and the connection to the thermal instability as conjectured by [24](see, also, [25]) requires a linearized analysis of the perturbations of the black hole solutions.However, the transition to a membrane-like phase of the rapidly spinning black holescan be established from the study of the thermodynamics of the system. The existence andlocation of the threshold of this regime is signaled by the minimum of the temperature andthe maximum angular velocity as functions of the angular momentum.It was observed in [10] that, for ultra-spinning black holes, this is in tight correspondencewith a vanishing eigenvalue of the Hessian of the Gibbs potential. A complete thermody-namic analysis, though, should be based on the full Hessian of the thermodynamic potentialrather than only a study of the determinant. We will see in the next subsection that themembrane phase of a doubly spinning ring is not signaled by a zero-eigenvalue of the Gibbspotential’s Hessian.For ultra-spinning black holes, there is a direct relation between (some response functionsin) the microcanonical and grand canonical ensembles. To see that, let us compare theexpressions of two particular response functions in these two ensembles: (cid:18) ∂ S∂J (cid:19) M = − T (cid:18) ∂ Ω ∂J (cid:19) M + Ω T (cid:18) ∂T∂J (cid:19) M and (cid:18) ∂ G∂ Ω (cid:19) T = − (cid:18) ∂J∂ Ω (cid:19) T (4.4)We have checked that in the particular case of the singly spinning black hole, indeed,these two response functions are inverse proportional at the particular point where thetemperature has a minimum. Therefore, an inflexion point in the microcanonical ensemblecorresponds to a divergence of the corresponding response function in the grand canonicalensemble. This may well be an explanation for the results obtained in [10]. Moreover, thispoint should not be considered as a sign for an instability or a new branch but a transitionto an infinitesimally nearby solution along the same family of solutions. The numericalevidence of [10] supports this connection with the zero-mode perturbation of the solution.We now examine the situation for a more general family of ultra-spinning Myers-Perryblack holes with multiple spin parameters, a i , where i = 1 , , ..., N and N = [( D − / µ and the horizon radius r h (thelargest root of) µ = 1 r ǫh N Y i =1 ( r h + a i ) , (4.5) A spinodal is defined as a line separating the regions of stability and instability of a homogeneoussystem. It is important to emphasize that all spinodals are zero-determinant lines, but in general not allzero-determinant lines are spinodal. Different ensembles correspond to different physical conditions and so, in more general cases, one doesnot expect such a relation.
13y which we can express the thermodynamics M = Ω D − πG D ( D − µ, J i = Ω D − πG D a i µ, Ω i = a i r h + a i , A H = Ω D − µ r h , T = 12 πr h r h N X i =1 r h + a i − ǫ ! , (4.6)where ǫ = mod D . A sufficient, but not necessary, condition for the existence of ultra-spinning black holes was given in [9]. In even(odd) dimensions at least one(two) of thespins should be much smaller than the rest. The ultra-spinning regime is obtained in thelimit 0 ≤ a , a , ..., a k << a k +1 , ..., a N → ∞ (4.7)where N − ≥ k ≥ ǫ . The generic limiting black brane metric whether static, with allfinite angular momenta a , .., a k vanishing, or spinning, with some a , .., a k non-vanishing,is the product S D − N − k +1) × R N − k ) .Our focus will be on the case in which the black hole has at least two large spins andwe set the remaining angular momenta to zero. When the angular momenta are equal, J k +1 = ... = J N = J , the Ruppeiner curvature scalar blows up at a r h = D − k − − ǫ . (4.8)According with the arguments in [23], this signals a thermodynamic instability. However,the expected new phase should correspond to the black membrane phase of ultra-spinningblack holes and not to a new branch of solutions.This is further supported by examining the eigenvalues of the Hessian of the Gibbspotential. Indeed, we find that the divergences of the Ruppeiner curvature pinpoint thezero of the determinant of the Gibbs potential’s Hessian.Also, by studying the temperature T = ( D − (cid:16) n ( D −
3) 4 J S (cid:17) S D − (cid:0) J S (cid:1) D − n D − , n = 2 k − − ǫ (4.9)we find that the temperature has a minimum at exactly (4.8), while the angular velocityΩ reaches its maximum value. Therefore for these more general ultra-spinning black holes,similarly to the singly spinning situation discussed in [9], once the minimum is reachedthe temperature increases and the angular velocity decreases signaling a transition to amembrane phase. This conduct is shown in figure 2 ( I points on the solid thin line) for theparticular case of D = 7, k = 2 so j = j ≡ j φ and j = 0.Another case of interest is the ultra-spinning black holes that resemble spinning blackbranes, when some of the slower spins are non-zero a , ..., a k = 0. It is not our goal to makea detailed analysis of this case here. Nevertheless, in all the cases where the non-vanishingspins are set equal, we find divergences of the Ruppeiner scalar curvature which could helpto detect the threshold of their membrane phase. Note that a r h = J S and in the particular case when k = k max our results agree with those of [35]. φ t I IIIII IVV j φ ω φ I II I IIIII
Figure 2:
Plots of the temperature (left hand side) and angular velocity (right hand side) asfunctions of the angular momentum j φ , for a fixed mass, for different black objects. These includethe singly spinning Myers-Perry black holes in five dimensions of space-time ( black dashed line ) andits seven dimensional cousin ( solid thin line ). The singly ( solid thick ) and doubly spinning blackring ( light gray ) for different values of angular momenta (right towards left) j ψ = 0 . , . S , are also shown here. For this same values of the second angular momenta the five-dimensionaldoubly spinning black holes are represented by the dashed light gray lines. Other solutions, e.g. the black ring with one angular momentum, also exhibit an ultra-spinning behavior. The black ring, which is characterized by the radii r and R of thespheres S D − and S , respectively, becomes thin in this limit (when r << R ). Since the final expressions for the response functions are very complicated for the doublyspinning ring (see Appendix B), we prefer to present the ‘conjugacy diagram’ of the angularvelocity versus the angular momentum and the plot of the temperature as a function of theangular momentum, both for a fixed mass (see Fig. 2).For the singly spinning black ring, an analysis of the temperature as a function of theangular momentum was presented in [38]. In this case (solid thick line in the plot on theleft hand side of Fig. 2), the temperature does not have a minimum, but there exists aturning point. In our analysis, the turning point II for the black ring plays a similarrole as the minimum of the temperature for the black hole. That is, it signals a changein the thermodynamic behaviour of the black ring. In fact, it is the starting point of theultra-spinning regime where the black ring can be approximated by a boosted black string.Using the Poincar´e ‘turning point’ method, this special point was carefully studied in[38]. In particular, they found a divergence of the Ruppeiner curvature. In the conjugacydiagram (on the right) there is also a turning point II at the same minimum value of theangular momentum j φ .The question is then if there still is a relation between the microcanonical and grand This thin regime was essential to find perturbatively the higher dimensional black rings cousins. More-over, a generalization of this construction to black branes led to the construction of blackfolds [36, 37]. We would like to point out that in (Sherk-Schwarz-)Anti-de Sitter, there is also a turning point [39]. II while the second eigenvalue never changes its sign. Therefore, we conclude that the turningpoint is the onset of the ultra-spinning black string phase.A far more richer structure is found for the doubly spinning black ring. The angularmomentum on S is bounded as j ψ ∈ [0 , /
4] and for a specific j ψ the black ring can alwaysbe extremal (in the limit λ → √ ν as shown in [29]). But besides extremality, accordingto how large j ψ is, the behaviour of the doubly spinning black ring changes. There aretwo distinctive regions in the microcanonical ensemble. On one hand, for 0 ≤ j ψ < / / ≤ j ψ ≤ /
4, the fat black ring branch disappearsand so there are no cusps. As we will discuss in what follows, this will become relevant tounderstand the physics and regimes of the doubly spinning black ring.To explore how the physics of the black rings at fixed mass is modified as we turn onthe angular momenta along the S , j ψ , we will study the temperature and angular velocityas functions of the S angular momentum, j φ , for different fixed values of j ψ . Doubly spinning black rings with ≤ j ψ < / j ψ , there also are turning points withtangents of infinite slope signaling the onset of the black membrane phase that coincidewith the cusps (in parameter space), namely at λ = − (1 / ν − (9 + ν ) / (1 + 9 ν ) / ).Figure 2 shows this change in behavior explicitly. On the left (light gray curve) in the t vs. j φ diagram, we observe the turning point III that corresponds to the minimum value of j φ angular momentum. In the conjugacy diagram ω vs. j φ (also the light gray curve), thecorresponding point ( III ) is a turning point.But an interesting difference with the single spin black ring occurs for a large enough S angular momentum. The temperature of the black ring in its membrane phase increaseswhile the area of the event horizon decreases up to a point where the spin-spin interactionis large enough making a turn to abruptly become extremal with zero temperature. This isthe maximum critical value labeled IV in figure 2. Therefore, the black membrane phaseexists between points III and IV .Finally, as for the singly spinning black ring, we computed the eigenvalues of the Hess[G]and found that none of them are zero the relevant turning points. Doubly spinning black rings with / ≤ j ψ ≤ / S angular momentum show noturning points and therefore have no membrane phase. This mimics its behavior in thephase diagram of microcanonical ensemble where the cusp and fat branch of these doublyspinning black ring disappear [29]. The lack of a fat black ring branch seems to coincidewith the lack of a black membrane phase. Therefore such solutions would never be capturedwith long distance effective approaches [36, 37].16ote that for certain fixed j ψ ∈ [1 / , /
4] the temperature grows, reaching a maximumat V and rapidly decreasing to zero to become extremal. It would be interesting to explorethe physical meaning of these points which we observe to correspond to an inflection point ∂ S/∂ J = 0 for fixed mass.In summary, we have found examples for which the zero eigenvalues of the Hessian ofthe Gibbs potential can also be turning points (with tangents of infinite slope) and notjust critical points as for the Myers-Perry black holes. Other less symmetric solutions,such as the doubly spinning black ring, do not show this connection between the zerosof the Hess[G] and the onset of the black membrane phase. Moreover, we showed thatcertain doubly spinning black ring, those with 1 / ≤ j ψ ≤ /
4, have no membrane phase.Therefore, these particular solutions fall into the same category as other black holes with nomembrane phase as the four dimensional Kerr black hole and the five dimensional Myers-Perry black hole.
In this paper we have analyzed in detail the thermodynamic stability of neutral doublyspinning black objects. We have analytically computed the response functions and pre-sented strong evidence showing that the doubly spinning black ring is thermodynamicallyunstable. That is, there is no region in the parameter space in which all the response func-tions are positive definite.We have provided an explanation of why the microcanonical and grand canonical en-sembles for ultra-spinning black holes are related in a very specific way. An inflexion pointin the microcanonical ensemble corresponds to a divergence of the corresponding responsefunction in the grand canonical ensemble. We will comment on the validity of this argumentand on the significance of these results in the last part of this section.The onset of a membrane phase of different doubly spinning black objects was identified.We have found that the onset of the black membrane phase for all black holes that we havestudied (except for the less symmetric doubly spinning black ring) is characterized by atleast one zero eigenvalue of the Hessian of the Gibbs potential. A tight relation with theclassical perturbations, where a transition to an infinitesimally nearby solution of the samefamily branch happens, is expected in the cases where the connection between membraneand zero-eigenvalue of Hess[G] exists. The numerical evidence of [10, 22] supports thisconnection precisely with the zero-mode perturbation of the solution.We now discuss the thermodynamics of doubly spinning black rings in the grand canon-ical ensemble. An analysis of this solution in the microcanonical ensemble was presentedin [29]. In general, for black holes, the entropy is used to obtain the phase diagrams inthe microcanonical ensemble while the mass is kept fixed. However, in general relativity itmakes more sense to use the total energy instead of the entropy. The reason is that thiswould require appropriate boundary conditions.It is important to emphasize that it is not known how the background subtraction17 -
25 000 -
50 000025 00050 00075 000 Ε ΦΦ √ .
15 2 √ . √ .
42 1 .
39 1 . Figure 3:
Plot of the response function ǫ φφ as a function of λ . The gray curve corresponds to thecompressibility of the singly spinning black ring, namely ν = 0. As the angular momentum alongthe S is increased ( dashed line left towards right) the isothermal moment of inertia for differentvalues of ν = 0 . , . , . , . , .
53 changes and becomes positive for values of ν > . λ > . gray region ). method can be applied to black rings, because it is not clear how to choose the backgroundsolution. We therefore used the counterterm method to compute the action and the grandthermodynamic potential (which is a Legendre transform of the energy). Since the expres-sions of the response functions are too complicated for analytical treatment (see AppendixB), we have plotted the regions in the parameter space where they are positive.Let us know compare some of our results with other well understood examples — atthis point we are interested just in the thermodynamic instabilities, not in the relation withdynamical ones. The fact that the Schwarzschild black hole has a negative heat capacitymeans that the thermodynamic ensemble is dominated by diffuse radiation states ratherthan black holes states but is classically dynamically stable at the linearized level. Whenadding angular momentum the situation changes and the heat capacity can be positivefor a large enough angular momentum. However, this condition is not enough to concludethat the system is thermodynamically stable. The stability also implies that when angularmomentum is added to the system the angular velocity goes up.For a black ring with one angular momentum the heat capacity can be positive, butthe momentum of inertia is always negative [7]. Therefore, the singly spinning black ring isthermodynamically unstable. As in the case of one angular momentum, the heat capacityof a doubly spinning black ring can be positive in some region of the parameter space.However, there is a key difference when the second angular momentum is turned on. Thatis, the component of the momentum of inertia associated to S of the black ring can becomepositive — this is explicitly shown in Figure 3.Since there are two angular momenta one should also investigate the effect of coupled‘angular’ inhomogeneities. A careful study of the determinant of the momentum of inertiamatrix shows that there is no region in the parameter space with the desired properties andso the doubly spinning black ring is also thermodynamically unstable.18 j ωg A B A BC
Figure 4:
The microcanonical phase diagram - entropy, s, as a function of the angular momentum,j - for fixed mass (on the left ) of the singly spinning Myers-Perry black hole in D = 5 , , black , gray and lightgray lines respectively). The grand canonical phasediagram (on the right ) - Gibbs potential, g, as a function of the angular velocity, ω - for fixedtemperature of the black hole. The points A, B correspond to a/r h = √ , q for six and tendimensions, and in general for D > a/r h = p ( D − / ( D − a/r h = 1 / √ We would like now to discuss in more detail some of the results for ultra-spinning blackholes presented in Section 4. In Fig. 4, we show the points (A,B) in the grand canonicalensemble that correspond to inflexion points (A,B) in the microcanonical ensemble. Thiscan be quantitatively understood by comparing the particular response functions in eq.(4.4) at a very special point in the parameter space, namely where the temperature has aminimum.We have explained that this argument applies to this particular case but not in general.A counterexample is the 5-dimensional black hole with one angular momentum. In thiscase there is no relation between ensembles in the sense that there is no special point inthe microcanonical ensemble which corresponds to the inflexion point ( C ) in the grandcanonical ensemble. Moreover we have checked and there are no points where an eigenvalueof the Hessian of the Gibbs potential vanishes. Therefore, it should not be considered as asign for a membrane phase — most probably, it is similar with the Schwarschild black holeexample for which the thermodynamical instability is not related to a dynamical one.One can also consider the ultra-spinning black holes with some of the finite angularmomenta non-zero. In odd(even) spacetime dimensions, the metric of an ultra-spinningblack hole with all but two(one) of the spins finite and non-zero will reduce to that of aspinning black brane. As we have shown in Section 2, the counterterm method can alsobe applied to spinning black branes and the results are similar to the ones for the ‘seed’spinning black hole solution. We have computed the renormalized action to find the Gibbspotential and we expect similar thermal instabilities as for the corresponding black holes.As we already emphasized, though, in all these examples we expect that the thermodynamicinstabilities do not signal a dynamical instability or a new branch, but rather a transition19o an infinitesimally nearby solution along the same family of solutions [10].It is remarkable that our study of thermodynamic instabilities provides to some extentinformation about the zero-mode of the ‘Gregory-Laflamme instability’ and it may wellbe the starting point for a more detailed study of dynamical instabilities. The extensionof the dynamical stability studies to spinning black branes has not been yet developed.The analytic theory of perturbations is much more involved. However, we expect that thespinning black branes suffer from similar instabilities as the static ones. We hope that theobservations made in this paper will be useful in future investigations of the perturbationsof higher-dimensional spinning black rings, black holes and spinning black branes. Acknowledgments
We would like to thank Ido Adam for interesting conversations. DA and MJR would alsolike to thank Robb Mann and Cristian Stelea for collaboration on related projects andvaluable discussions.
A Temperature and angular velocities
Consider a general stationary 5-dimensional metric that corresponds to a black object withtwo angular momenta : ds = g tt ( ~x ) dt + 2 g tφ ( ~x ) dtdφ + 2 g tψ ( ~x ) dtdψ + g φφ ( ~x ) dφ + 2 g φψ ( ~x ) dφdψ + g ψψ ( ~x ) dψ + g αβ ( ~x ) dx α dx β (A.1) ∂ t , ∂ φ , and ∂ ψ are Killing vectors. Rewrite the metric in the ADM form ds = − N dt + γ ij ( dx i + N i dt )( dx j + N j dt ) (A.2)with lapse function N = − g tt + g φφ ( N φ ) + g ψψ ( N ψ ) + 2 g φψ N φ N ψ (A.3)and shift vector N φ = g tψ g φψ − g ψψ g tφ g φψ − g φφ g ψψ , N ψ = g tφ g φψ − g φφ g tψ g φψ − g φφ g ψψ (A.4)The event horizon is obtained for N = 0 (A.5)In other words, it is a Killing horizon of ∂ t + Ω φ ∂ φ + Ω ψ ∂ ψ , where Ω φ and Ω ψ are theangular velocities defined as the shift vectors at the horizon:Ω φ = − N φ (cid:12)(cid:12) H , Ω ψ = − N ψ (cid:12)(cid:12) H (A.6) A similar analysis for one angular momentum can be found in [7, 40]. it, r ), to obtain the periodicity of the Euclidean time. In this way, oneobtains the following expression for the temperature of the black hole: T = ( N ) ′ π p g rr N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (A.7)We have used these definitions to compute the corresponding physical quantities of thedoubly spinning ring. B Conditions for thermodynamic stability
In this appendix, we present the conditions for the thermodynamic stability and we alsogive some useful explicit expressions for the response functions used in Section 4 — wefollow closely [41].For simplicity, let us start with a black hole with one angular momentum. We areinterested in the thermodynamic potentials: the energy and its Legendre transforms.The basic extremum principle of thermodynamics (for the entropy S ) implies both that dS = 0 and that d S <
0. The second condition determines the stability of predicted equi-librium states. The stability criterion in energy representation requires that an equilibriumstate at fixed S and J is a state of minimum energy, namely a minimum of E [ S, J ]. Thelocal stability conditions ensure that inhomogeneities of either S and J separately (cid:18) ∂ E∂S (cid:19) J = (cid:18) ∂T∂S (cid:19) J ≥ , (cid:18) ∂ E∂J (cid:19) S = (cid:18) ∂ Ω ∂J (cid:19) S ≥ also that a coupled inhomogeneity of S and J togetherdet(Hess( E )) = ∂ E∂S ∂ E∂J − (cid:18) ∂ E∂S∂J (cid:19) ≥ convex functions of their extensive variables and concave functions of their intensive vari-ables (see, e.g., [41]).For a grand canonical ensemble defined at fixed temperature T and angular velocities Ω a (intensive variables) the associated potential, the Gibbs free energy, satisfies the followingrelations: G [ T, Ω] = E − T S − Ω J , dG = − SdT − J d
Ω (B.3)21n this case, the local stability conditions following from the convexity of the Gibbs functionyield (cid:18) ∂ G∂T (cid:19) Ω = − (cid:18) ∂S∂T (cid:19) Ω ≤ , (cid:18) ∂ G∂ Ω (cid:19) T = − (cid:18) ∂J∂ Ω (cid:19) T ≤ G )) = ∂ G∂T ∂ G∂ Ω − (cid:18) ∂ G∂T ∂ Ω (cid:19) ≥ C Ω , C J ) and the ‘isothermal moment of inertia’ or ’com-pressibility’ ǫ ≡ ( ∂J/∂ Ω) T should be positive definite.A generalization for two angular momenta is straightforward (see, also, [20]). TheHessian is a 3 × G ) = ( − (cid:18) C Ω T − α b α a ǫ ab (cid:19) where the matrix components are α a = (cid:0) ∂J a ∂T (cid:1) Ω , ǫ ab = (cid:16) ∂J a ∂ Ω b (cid:17) T , and the indices cover theangular directions a, b = φ, ψ .Considering the relationship between the specific heats C Ω = C J + T ( ǫ − ) ab α a α b itcan be shown that a thermodynamically stable system is characterized by positive heatcapacities C Ω > C J > ǫ ab ] > φφ )-component of the isothermal moment of inertia tensor is ǫ φφ = − k πλ (1 + λ + ν ) G ( ν − (1 − λ + ν ) F ( λ, ν ) √ λ − ν ( λ + λ ν − ν ) + λ (1 + ν ) − λ (4 ν + ν − F ( λ, ν ) = λ (1 + ν ) + λ (1 + ν ) (1 + √ λ − ν + ν )+ λ (1 + ν )[4 + √ λ − ν + ν (2(1 + √ λ − ν ) + ν ( −
24 + √ λ − ν + 2 ν ))] − √ λ − ν + ν [1 + ν ( −
14 + ν (18 + ( −
14 + ν ) ν ))]+2 λ (1 + ν )[2 + √ λ − ν + ν ( −
25 + √ λ − ν + ν (13 − √ λ − ν + ν ( − √ λ − ν + ν )))] + 2 λ (1 + ν )[2 + √ λ − ν + ν ( − − √ λ − ν + ν (32 + 14 √ λ − ν + ν (16 − √ λ − ν + ( − √ λ − ν ) ν )))] − λ [ − ν (10 + 4 √ λ − ν + ν ( − − √ λ − ν + ν ( − − √ λ − ν )+ ν ( −
23 + 4 √ λ − ν + ( − ν ) ν ))))]The determinant is ǫ = 4 k π λ ( λ − √ λ − ν )( λ + √ λ − ν ) √ ν (1 + λ + ν ) ( G ) ( − ν ) [ − λ + (1 + ν ) ] / [ ν ( − λ + (1 + ν ) )] / G ( λ, ν ) Z ( λ, ν ) (B.6)where Z ( λ, ν ) = 8 √ λ − νν − λ (1 + ν ) − λ √ λ − ν (1 + ν ) + 2 λ ( − ν + ν ) G ( λ, ν ) = λ + 7 λ (1 + ν ) − λ (1 + ν )[1 + 3 √ λ − ν + (2 − √ λ − ν ) ν + ν ]+ λ [5 − √ λ − ν + (26 + 6 √ λ − ν ) ν + 5 ν ] − ν (1 + 11 ν + 11 ν + ν ) − λ [(8 − √ λ − ν ) ν + 9(16 + 3 √ λ − ν ) ν + (8 + 3 √ λ − ν ) ν − √ λ − ν ]22 eferences [1] M. J. Rodriguez, “On the black hole species (by means of natural selection),”arXiv:1003.2411 [hep-th].[2] G. J. Galloway and R. Schoen, “A generalization of Hawking’s black hole topol-ogy theorem to higher dimensions,” Commun. Math. Phys. , 571 (2006)[arXiv:gr-qc/0509107].[3] S. Hollands, A. Ishibashi and R. M. Wald, “A Higher Dimensional Stationary Ro-tating Black Hole Must be Axisymmetric,” Commun. Math. Phys. , 699 (2007)[arXiv:gr-qc/0605106].[4] R. Emparan and H. S. Reall, “A rotating black ring in five dimensions,” Phys. Rev.Lett. , 101101 (2002) [arXiv:hep-th/0110260].[5] A. A. Pomeransky and R. A. Sen’kov, “Black ring with two angular momenta,”arXiv:hep-th/0612005.[6] R. C. Myers and M. J. Perry, “Black Holes In Higher Dimensional Space-Times,”Annals Phys. , 304 (1986).[7] D. Astefanesei, R. B. Mann, M. J. Rodriguez and C. Stelea, “Quasilocal formalismand thermodynamics of asymptotically flat black objects,” arXiv:0909.3852 [hep-th].[8] G. T. Horowitz and A. Strominger, “Black strings and P-branes,” Nucl. Phys. B ,197 (1991).[9] R. Emparan and R. C. Myers, “Instability of ultra-spinning black holes,” JHEP ,025 (2003) [arXiv:hep-th/0308056].[10] O. J. C. Dias, P. Figueras, R. Monteiro, J. E. Santos and R. Emparan, “Instabilityand new phases of higher-dimensional rotating black holes,” Phys. Rev. D , 111701(2009) [arXiv:0907.2248 [hep-th]].[11] R. Gregory and R. Laflamme, “Black strings and p-branes are unstable,” Phys. Rev.Lett. , 2837 (1993) [arXiv:hep-th/9301052].[12] J. D. Brown and J. W. . York, “Quasilocal energy and conserved charges derived fromthe gravitational action,” Phys. Rev. D , 1407 (1993) [arXiv:gr-qc/9209012].[13] S. R. Lau, “Lightcone reference for total gravitational energy,” Phys. Rev. D , 104034(1999) [arXiv:gr-qc/9903038];R. B. Mann, “Misner string entropy,” Phys. Rev. D , 104047 (1999)[arXiv:hep-th/9903229];P. Kraus, F. Larsen and R. Siebelink, “The gravitational action in asymptotically AdSand flat spacetimes,” Nucl. Phys. B , 259 (1999) [arXiv:hep-th/9906127].2314] B. Kleihaus, J. Kunz and E. Radu, “New nonuniform black string solutions,” JHEP , 016 (2006) [arXiv:hep-th/0603119].[15] D. Astefanesei, M. J. Rodriguez and S. Theisen, “Quasilocal equilibrium condition forblack ring,” JHEP , 040 (2009) [arXiv:0909.0008 [hep-th]].[16] G. W. Gibbons and S. W. Hawking, “Action Integrals And Partition Functions InQuantum Gravity,” Phys. Rev. D , 2752 (1977).[17] D. Astefanesei and E. Radu, “Quasilocal formalism and black ring thermodynamics,”Phys. Rev. D , 044014 (2006) [arXiv:hep-th/0509144].[18] J. D. Brown, E. A. Martinez and J. W. York, “Complex Kerr-Newman geometry andblack hole thermodynamics,” Phys. Rev. Lett. , 2281 (1991).[19] H. Elvang, R. Emparan and A. Virmani, “Dynamics and stability of black rings,”JHEP , 074 (2006) [arXiv:hep-th/0608076].[20] R. Monteiro, M. J. Perry and J. E. Santos, “Thermodynamic instability of rotatingblack holes,” Phys. Rev. D , 024041 (2009) [arXiv:0903.3256 [gr-qc]].[21] D. N. Page, “Hawking radiation and black hole thermodynamics,” New J. Phys. ,203 (2005) [arXiv:hep-th/0409024].[22] O. J. C. Dis, P. Figueras, R. Monteiro, H. S. Reall and J. E. Santos, “An instabilityof higher-dimensional rotating black holes,” JHEP , 076 (2010) [arXiv:1001.4527[hep-th]].[23] G. Ruppeiner, “Riemannian geometry in thermodynamic fluctuation theory,” Rev.Mod. Phys. , 605 (1995) [Erratum-ibid. , 313 (1996)].[24] S. S. Gubser and I. Mitra, “The evolution of unstable black holes in anti-de Sitterspace,” JHEP , 018 (2001) [arXiv:hep-th/0011127].[25] H. S. Reall, “Classical and thermodynamic stability of black branes,” Phys. Rev. D (2001) 044005 [arXiv:hep-th/0104071].[26] R. B. Mann and D. Marolf, “Holographic renormalization of asymptotically flat space-times,” Class. Quant. Grav. , 2927 (2006) [arXiv:hep-th/0511096].[27] R. B. Mann, D. Marolf and A. Virmani, “Covariant counterterms and conservedcharges in asymptotically flat spacetimes,” Class. Quant. Grav. , 6357 (2006)[arXiv:gr-qc/0607041].[28] D. Astefanesei, R. B. Mann and C. Stelea, “Note on counterterms in asymptoticallyflat spacetimes,” Phys. Rev. D , 024007 (2007) [arXiv:hep-th/0608037].[29] H. Elvang and M. J. Rodriguez, “Bicycling Black Rings,” JHEP , 045 (2008)[arXiv:0712.2425 [hep-th]]. 2430] M. Durkee, “Geodesics and Symmetries of Doubly-Spinning Black Rings,” Class.Quant. Grav. , 085016 (2009) [arXiv:0812.0235 [gr-qc]].[31] P. T. Chrusciel, J. Cortier and A. P. Gomez-Lobo, “On the global structure of thePomeransky-Senkov black holes,” arXiv:0911.0802 [gr-qc].[32] M. Banados, C. Teitelboim and J. Zanelli, “Black hole entropy and the dimen-sional continuation of the Gauss-Bonnet theorem,” Phys. Rev. Lett. , 957 (1994)[arXiv:gr-qc/9309026].[33] D. V. Fursaev and S. N. Solodukhin, “On The Description Of The RiemannianGeometry In The Presence Of Conical Defects,” Phys. Rev. D , 2133 (1995)[arXiv:hep-th/9501127].[34] J. Evslin and C. Krishnan, “Metastable Black Saturns,” JHEP , 003 (2008)[arXiv:0804.4575 [hep-th]].[35] J. E. Aman and N. Pidokrajt, “Geometry of higher-dimensional black hole thermody-namics,” Phys. Rev. D , 024017 (2006) [arXiv:hep-th/0510139].[36] R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodriguez, “The PhaseStructure of Higher-Dimensional Black Rings and Black Holes,” JHEP , 110(2007) [arXiv:0708.2181 [hep-th]].[37] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “Blackfolds,” Phys. Rev.Lett. , 191301 (2009) [arXiv:0902.0427 [hep-th]].[38] G. Arcioni and E. Lozano-Tellechea, “Stability and critical phenomena of black holesand black rings,” Phys. Rev. D , 104021 (2005) [arXiv:hep-th/0412118].[39] S. Lahiri and S. Minwalla, “Plasmarings as dual black rings,” JHEP , 001 (2008)[arXiv:0705.3404 [hep-th]].[40] D. Astefanesei and H. Yavartanoo, “Stationary black holes and attractor mechanism,”Nucl. Phys. B794