XVI European Workshop on Strings Theory Madrid– 14 June 2010

1. Thermodynamic instability of doubly spinning black objects Maria J. Rodriguez XVI European Workshop on Strings Theory Madrid– 14 June 2010 Dumitru Astefanesei, Robert B. Mann, MJR, Cristian Stelea arXiv:0909.3852 [hep-th] & 1003.2421[hep-th] Dumitru Astefanesei, MJR, Stefan Theisen arXiv:0909.0008 [hep-th]

2. Motivation Black holes are the most elementary and fascinating objects in General Relativity The study of the properties of higher dimensional black holes is essential to understand the dynamics of space-time 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.

3. On the BH species (by means of natural selection) [*] Rµν=0 Vacuum Einstein´s equation μ,ν = 1,2,…,D Asymptotically flat Boundary conditions Equilibrium Stationary – no time dependence Regular solutions on and outside the event horizon [*] We start from a five dimensional continuum which is x1,x2,x3,x4,x0. In it there exists a Riemannian metric with a line element ds2= gμνdxμdxν

4. On the BH species (by means of natural selection) Summary of neutral D-dim BHs classified by its horizon topologies Black Holes Blackfolds Black Ring & Helical BR SD-2 S5 πiSmixsD-2-m S4 S3 S3x s2 S1x S2 S1x s3 S2 S1x S1xs2 S1x s4 T3xs2 T2xs3 S1x SD-3 Tpx SD-2-p The boundary stress tensor satisfies a local conservation law

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

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

7. BH w/ one J in D-dim What do we know about black objects? In D=4 dimensions -Kerr black hole In D=5 dimensions -the Myers-Perry black hole -black ring aH In D>5 dimensions aH aH - the Myers-Perry black hole - thin black ring and black saturn It seems that there is an infinite number of BHs. j2 j 1 1 j

8. BH w/ two J in D-dim The generalization of the black hole solution with ANY # of angular momenta is the Myers-Perry (MP) solution. D=5 singular j12 j2 The dashed lines show MP for fixed values of =0.1,0.3,0.5 right to left The gray curve is the phase of zero temperature BH’s Representative phase of MP-BHs with one of the two angular momenta fixed

9. BR w/ two J in D-dim D=5 The generalization of the black ring with TWO angular momenta is known. The angular momenta are bounded The fat ring branch disappears for The dashed lines show BR for fixed values of (right towards left) The dark gray curve is the phase of zero temperature BR’s The black dashed curve is the phase of zero temperature MP BH’s Representative phase of the doubly spinning BR with the S2 angular momenta fixed

10. BH w/ two J in D-dim What do we know about these black objects? In D=5 dimensions -Myers-Perry Black Hole (BH) -Black Ring (BR) Not shown here -Helical BH -Black Saturn -Bicycling BR j2 is fixed aH In D>5 dimensions aH -Myers-Perry Black Hole -Black Ring (BR) j2 is fixed -Helical BH -Black Saturn -Bicycling BR -Blackfolds Not shown here j1 1 j1

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

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

13. Ultra-spinning black objects where Parameters in the solution Balance condition R • + • + R ∞ Thin Black ring R • - • - SD-2 S1 x SD-3

14. 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 Thin Black ring Black string ≡ R ∞ ro ro R ro Having a better understanding of the properties of BO may be useful to construct new solutions Emparan + Harmark + Niarchos + Obers + MJR 0708.2181 [hep-th]

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

16. Instabilities from thermodynamics Q1: If black objects are thermally unstable in the grand-canonical ensemble for jth does this imply that there they are classically unstable? But to investigate this and where the threshold of the classical instability is one has to perform a linearized analysis of the perturbations. 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. 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? Q2b: Which is the threshold of the membrane phase, jm? Q2c: Is there any relation between zeros of eigenvalues of Hess(G) and jm?

17. Thermodynamics of black objects

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

19. Quasilocal thermodynamics 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 r =const. Brown + York gr-qc/9209012 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. Compute directly the Gibbs-Duhem relation by integrating the action supported with counterterms. Mann + Marolf

20. Instabilities from Thermodynamics

21. 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?

22. Black hole thermal stability Singly rotating Myers-Perry black hole Heat capacity Compressibility 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. For doubly spinning MP-BH the response functions are positive for different (complementary) regions of the parameter space implying its instability. Monterio + Perry + Santos 0903.3256[gr-qc]

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

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

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

26. Critical points & turning points

27. Instabilities from thermodynamics The instabilities and the threshold of the membrane phase of the singly spinning MP BH are D=5 D=10 D=6 D=5 D=10 D=6 0 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. Numerical evidence supports this connection with the zero-mode perturbation of the solution. Note that the relation between ensembles is not in general valid.

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

29. Critical points: MP BH where for the ultra spinning MP BH while the angular velocity reaches its maximum value. The transition to a membrane-like phase of the rapidly spinning black holes is established from the study of the thermodynamics of the system. 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.

30. Critical points: MP BH 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? These points seem to be related to the classical instabilities. We´ve checked that at least one of the eigenvalues of the Hess[G] is zero. And also checked that the Ruppeiner curvature pinpoints the zero of the determinant of the Gibbs potential’s hessian The Ruppeiner metric measures the complexity of the underlying statistical mechanical model is the so called Ruppeiner metric A curvature singularity is a signal of critical behavior.

31. Turning points : BR λ=0.5 At the cusp in s vs j The Ruppeiner curvature diverges. We´ve checked that at least one of the eigenvalues of the Hess[G] is zero there. 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

32. Turning points : BR III III I I λ [ν] At the cusp in s vs j Indicate where the transition tothe thermodynamicalblack membrane phase. No eigenvalue of the Hess[G] is zero there. Particular BR solutions with jψ > 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.

33. Summary and outlook 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 BLACK HOLES It would be interesting to investigate numerically whether these correspond to the zero-mode perturbations. 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.

34. Gracias.

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

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

37. Jargon and reminder On the number of angular momenta Maximum # angular momenta: j1 j2 . . . j[(D-1)/2] On the number of horizons One compact horizons: uni horizon black hole solutions Disconnected compact horizons: multi horizon black hole solutions On how we compare solutions Compare by drawing diagrams i.e. a phase diagram To compare solutions we need to fix a common scale Classical GR aH  We'll fix the mass M equivalently and factor it out to get dimensionless quantities … aH j2 ω j2

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

39. Schwarze LÖcher BLACK HOLES

40. Ultra spinning multi BH