XVI European Workshop on Strings Theory Madrid– 14 June 2010 arXiv: [hep-th] arXiv: [hep-th] Dumitru Astefanesei, MJR, Stefan Theisen Dumitru Astefanesei, Robert B. Mann, MJR, Cristian Stelea Maria J. Rodriguez Thermodynamic instabilityThermodynamic instabilityof doubly spinning black objectsdoubly spinning black objects & [hep-th]

1 Motivation Black holes are the most elementary and fascinating objects in General Relativity In their presence the effects of the space-time curvature are dramatic In string theory, mathematics and recent cutting edge experiments black objects are also relevant. The study of the properties of higher dimensional black holes is essential to understand the dynamics of space-time

2 Asymptotically flat On the BH species (by means of natural selection) Vacuum Einstein´s equation Boundary conditions Equilibrium Stationary – no time dependence Regular solutions on and outside the event horizon R µ ν =0 [*] [*] We start from a five dimensional continuum which is x 1,x 2,x 3,x 4,x 0. In it there exists a Riemannian metric with a line element ds 2 = g μν dx μ dx ν μ, ν = 1,2,…,D

3 On the BH species (by means of natural selection) S5S5 S D-2 S4S4 S3S3 S2S2 The boundary stress tensor satisfies a local conservation law Summary of neutral D-dim BHs classified by its horizon topologies

4 In D>4 many black holes have been found: Motivation Galloway+SchoenTopology: Rigidity :Hollands et al Stationary & axisymmetric In D=4 stationary black holes are spherical and unique The main feature of high-D is the richer rotation dynamics Study thermodynamical properties of spinning BH in D-dim to learn how these solutions connect. Our goal: NEW FEATURE

5 One&Two angular momenta +Vacuum + Asymtotically flat Phase diagram of black objects

6 What do we know about black objects? In D=4 dimensions In D=5 dimensions -Kerr black hole In D>5 dimensions BH w/ one J in D-dim -the Myers-Perry black hole aHaH j It seems that there is an infinite number of BHs. aHaH j2j2 1 aHaH j 1 -black ring - thin black ring and black saturn

j12j12 7 The generalization of the black hole solution with ANY # of angular momenta is the Myers-Perry (MP) solution. singular The gray curve is the phase of zero temperature BH’s Representative phase of MP-BHs with one of the two angular momenta fixed BH w/ two J in D-dim D=5 The dashed lines show MP for fixed values of =0.1,0.3,0.5 right to left j2j2

8 The generalization of the black ring with TWO angular momenta is known. BR w/ two J in D-dim D=5 The dark gray curve is the phase of zero temperature BR’s Representative phase of the doubly spinning BR with the S 2 angular momenta fixed The dashed lines show BR for fixed values of (right towards left) The black dashed curve is the phase of zero temperature MP BH’s The angular momenta are bounded The fat ring branch disappears for

9 What do we know about these black objects? In D=5 dimensions In D>5 dimensions -Myers-Perry Black Hole (BH) -Black Ring (BR) -Myers-Perry Black Hole -Black Ring (BR) BH w/ two J in D-dim -Helical BH -Black Saturn -Bicycling BR j 2 is fixed Not shown here -Helical BH -Black Saturn -Bicycling BR -Blackfolds Not shown here aHaH j1j1 j 2 is fixed aHaH j1j1 1

10 Why are we interested doubly spinning solutions? Black Holes with T=0 are interesting because they can teach us about the microscopic origin of their physical properties Black Holes with T=0 are interesting because they can teach us about the microscopic origin of their physical properties SUSY Asymptotically flat Non SUSY T=0 BH w/ two J in D-dim Doubly spinning black rings, in contrast to the singly spinning black rings, can be extremal.

11 One&Two angular momenta +Vacuum + Asymtotically flat Ultra-spinning black objects

12 Ultra-spinning black objects R Balance condition Parameters in the solution where S 1 x S D-3 R Thin Black ring R ∞ S D

13 Thin Blackring/fold Recently the matched asymptotic expansion has been applied to solve Einstein´s equations to find thin Black ring/folds in D>4 dimensions The basic idea Black ringBlack string ≡ R ∞ R ro Emparan + Harmark + Niarchos + Obers + MJR [hep-th] Having a better understanding of the properties of BO may be useful to construct new solutions Thin

14 Black Holes and black rings in ultra-spinning regime will inherit the instabilities. In certain regimes black holes and black rings behave like black strings and black p-branes. Ultra-spinning black objects Black strings and branes exhibit Gregory-Laflamme instability Gubser + Wiseman Branch of static lumpy black strings A black hole solution which is thermally unstable in the grand- canonical ensemble will develop a classical instability. Gubser + Mitra Emparan + Myers

15 Q1: If black objects are thermally unstable in the grand-canonical ensemble for j th does this imply that there they are classically unstable? Instabilities from thermodynamics But to investigate this and where the threshold of the classical instability is one has to perform a linearized analysis of the perturbations. Q2c: Is there any relation between zeros of eigenvalues of Hess(G) and j m ? Q2: What information can we get from the study of the thermodynamical instabilities? We can establish a membrane phase signaled by the change in its thermodynamical behavior which could imply the classical instability. Q2b: Which is the threshold of the membrane phase, j m ? We can study the zeros of the Hess(G) which seem to be linked to the classical instabilities Q2a: How to establish the membrane phase?

16 Thermodynamics of black objects

17 Which ensemble is the most suitable for this analysis? Entropy – microcanonical ensemble Thermal ensembles Gibbs potential – grand canonical ensemble Enthalpy Helmholtz free energy – canonical ensemble

18 Due to the equivalence principle, there is no local definition of the energy in gravitational theories Basic idea of the quasilocal energy: enclose a region of space-time with some surface and compute the energy with respect to that surface – in fact all thermodynamical quantities can be computed in this way For asymptotical flat space-time, it is possible to extend the quasilocal surface to spatial infinity provided one incorporates appropriate boundary (counterterms) in the action to remove divergences from the integration over the infinite volume of space-time. r =const. Brown + York gr-qc/ Mann + Marolf Quasilocal thermodynamics Compute directly the Gibbs-Duhem relation by integrating the action supported with counterterms.

19 Instabilities from Thermodynamics

20 Thermal stability In analogy with the definitions for thermal expansion in the liquid-gas system, the specific heat at a constant angular velocity, the isothermal compressibility, and the coefficient of thermal expansion can be defined The conditions for thermal stability in the grand-canonical ensemble or What do we know about the thermal stability of black objects?

21 Black hole thermal stability Monterio + Perry + Santos [gr-qc] The response functions are positive for different values of the parameters implying there is no region in parameters space where both are simultaneously positive. The black holes is thermally unstable, both in the canonical and grand- canonical ensembles. Compressibility Heat capacity Singly rotating Myers-Perry black hole For doubly spinning MP-BH the response functions are positive for different (complementary) regions of the parameter space implying its instability.

22 Black ring thermal stability The black ring is thermally unstable, both in the canonical and grand- canonical ensembles. The C Ω →0 as T→0 which is expected and can be drawn from Nernst theorem. Heat capacity Compressibility Singly spinning black ring

23 We investigated the stability of the doubly spinning black ring The doubly spinning black hole and the singly spinning black ring are thermally unstable in the grand- canonical ensembles. A second rotation could help to stabilize the solution Doubly spinning black ring What about the thermal stability of the doubly spinning black ring?

24 Doubly spinning black ring The grand canonical potential for doubly spinning black ring (using the quasi local formalism) The Hessian should be negatively defined The doubly spinning black ring is local thermally unstable. where

25 Critical points & turning points

26 These points should not be considered as a sign for an instability or a new branch but a transition to an infinitesimally nearby solution along the same family of solutions. Instabilities from thermodynamics The instabilities and the threshold of the membrane phase of the singly spinning MP BH are 0 D=5 D=10 D=6 D=5 D=10 D=6 Numerical evidence supports this connection with the zero-mode perturbation of the solution. Note that the relation between ensembles is not in general valid.

27 Indicates where the transition to the black membrane phase. More general black holes with N spins ultra-spin iff Critical points: MP BH jmjm Black holes with one spin 0 where Are there other ultra spinning MP black holes? And for and

28 The existence and location of the threshold of this regime is signaled by the minimum of the temperature and the maximum angular velocity as functions of the angular momentum. The transition to a membrane-like phase of the rapidly spinning black holes is established from the study of the thermodynamics of the system. where for the ultra spinning MP BH Critical points: MP BH while the angular velocity reaches its maximum value.

29 But let´s take a closer look to the Hessian, which has to be negatively defined, Do the zeros of the eigenvalues of this Hessian have any physical interpretation? We´ve checked that at least one of the eigenvalues of the Hess[G] is zero. Critical points: MP BH And also checked that the Ruppeiner curvature pinpoints the zero of the determinant of the Gibbs potential’s hessian These points seem to be related to the classical instabilities. is the so called Ruppeiner metric The Ruppeiner metric measures the complexity of the underlying statistical mechanical model A curvature singularity is a signal of critical behavior.

30 Turning points : BR We´ve checked that at least one of the eigenvalues of the Hess[G] is zero there. λ=0.5 At the cusp in s vs j In this case the temperature does not have a minimum, but there exists a turning point and plays a similar role as the minimum of the temperature for the BH In this case the temperature does not have a minimum, but there exists a turning point and plays a similar role as the minimum of the temperature for the BH The Ruppeiner curvature diverges.

31 Indicate where the transition to the thermodynamical black membrane phase. λ [ν] At the cusp in s vs j No eigenvalue of the Hess[G] is zero there. Turning points : BR Particular BR solutions with j ψ > 1/5 fall into the same category as other black holes with no membrane phase as the four dimensional Kerr black hole and the five dimensional Myers-Perry black hole. I III I

32 Summary and outlook It would be interesting to investigate numerically whether these correspond to the zero- mode perturbations. We showed,in parameter space, that doubly spinning black rings are thermally unstable Found the thresholds of the transition to the black membrane phase of black holes and black rings with at least two spins. Identified particular cases of doubly spinning BR with no membrane phase Study the ultraspinning behavior of multi black holes, such as the bicycling black ring and saturn, which can be relevant in finding new higher dimensional multi black hole solutions.

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34 In five dimensions stationary implies axisymmetric To calculate the physical quantities we employ the complex instanton method The ADM decomposition of the full spacetime We can write (B) in the (A) form The Wick transformation changes the intensive variables but not the extensive ones (B) (A) Lapse function Shift function Angular velocity Temperature Quasilocal thermodynamics

35 Black String =0 The stress energy tensor is conserved for any value of the parameters Observe that =0 Corresponds to the thin black ring limit Boundary stress energy tensor for black strings

36 To compare solutions we need to fix a common scale Classical GR  We'll fix the mass M equivalently and factor it out to get dimensionless quantities aHaH aHaH j2j2 j2j2 ω ω … … Disconnected compact horizons: multi horizon black hole solutions One compact horizons: uni horizon black hole solutions On the number of angular momenta On the number of horizons Maximum # angular momenta: On how we compare solutions Compare by drawing diagrams i.e. a phase diagram aHaH j2j2 j1j1 j2j2 Jargon and reminder j [(D-1)/2]...

37 Outline Introduction Thermodynamics Instabilities from Thermodyn. Summary and outlook Motivation - Ultra-spinning BH - BH solutions in D-dim - Membrane phase - Critical points & turning points - Thermal ensembles - Thermodynamic stability - Grand-canonical ensemble

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39 Ultra spinning multi BH