1. On the Black Hole/Black Ring Transition Ernesto Lozano-Tellechea Weizmann Institute of Science Israel ICHEP-04 Beijing Based on colaboration with: Giovanni Arcioni (Hebrew University) [to appear]

2. Introduction Subject: phase transitions in BH Physics Black Hole Phases: –In 4d: BH uniqueness –In d>4: different phases ( ↔ horizon topolgy) (BHs, black stings, branes…) Phase transitions between them?

3. BH Phase Transitions Noncompact dimensions: Gregory-Laflamme Black String  Black Hole Compact Dimensions: Small BH ↔ Compact Black String Relevance: –Gravitation: Cosmic Censorship, singularities,… [Kol, Harmark, Obers, Sorkin, Weiseman, … ] –Field Theory (AdS/CFT): Confinement/Deconfinement,… [Aharony, Gubser, Minwalla, Witten…]

4. Black Rings in d=5 In d=5 pure Gravity, in addition to 5d Kerr BH, there are two asymptotically flat rotating BHs with horizon topology [Emparan & Reall, 2002] “Black Rings” Is this describing different phases of the same system? BLACK HOLE NON UNIQUENESS!

5. Dynamical vs. Thermodynamical Stability True question: dynamical stabilility Can it be derived from thermodynamics? [Davies ‘77] [Gubser & Mitra 2000] In ordinary (extensive) systems: STABILITY ↔ NOT APPLICABLE TO BHs !! (prime example: Schwarzschild BH) (  positivity of the Hessian of S)

6. In this talk: We will try to address the issues of Stability Study of critical points in the Black Hole/Black Ring system using appropriate tools for the study of non-extensive thermodynamics

7. Stability of Non-Extensive Systems Let us use only the entropy-max principle: Equilibrium series S(M) Off-Equilibrium (Natural extension: Legendre Transform) [“Poincare method” of stability] [Katz ‘79] [Kaburaki ‘94] Stability of the Lorentzian solution ↔ Microcanonical ensemble (fixed M, fluctuations in Temperature) Near the equilibrium series:

8. However, changes of stability only occur at a “turning point”: Typical plot of β(M): A: change in sign of only along the equilibrium series. B: true change in stability (along the axis of fluctuations).  stability analysis based on sign[Hessian(S)] only valid around a turning point This method predicts stability of Schwarzschild and Kerr BHs

9. Black Hole/Black Ring System Behaviour of Large BRSmall BRBlack Hole Change in sign along eq. series only Divergent specific heat but NO CHANGE IN STABILITY (in the microcanonical) Fluctuations Diverge ?

10. Stability We will see that At x=x min  change in stability (Small BR is unstable against axisymmetric perturbations – const J) At x=1  2nd-order phase transition BH/Small BR

11. Critical Exponents One can define the appropriate susceptibilities And order parameter Obey scaling relations of the “first kind” both at: ↔ BH/SBR ↔ SBR/LBR What about the correlation length and scalings of the “second kind”?

12. Thermodynamic Geometry Proposal: [Ruppeiner ‘79] Suitable for nonextensive thermodynamics Allows to compute ξ and check scalings of the 2nd kind

13. Thermodynamic Curvature for the BH/BR System Black HoleLarge BRSmall BR  At x = 1 (BH/SBR): Scaling relations are obeyed assuming d=2  At x = x min (SBR/LBR): Incompatible with scaling relations (for any effective d) OK with the geometry in the extremal limit

14. Summary We have used: –“turning point method”  stability –thermodynamic geometry  critical points Both seem appropriate for the study of nonextensive systems Applied to the 5d Black Hole/Black Ring System: 2nd-order phase transition at extremality Change in stability (Small BR becomes unstable against axisymmetric perturbations)