13 0895 tellechea el

On the Black Hole/Black Ring Transition: 

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]

Introduction: 

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?

BH Phase Transitions : 

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…]

Black Rings in d=5: 

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!

Dynamical vs. Thermodynamical Stability: 

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)

In this talk:: 

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

Stability of Non-Extensive Systems: 

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:

However, changes of stability only occur at a “turning point”: Typical plot  of β(M):: 

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

Black Hole/Black Ring System: 

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

Stability: 

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

Critical Exponents: 

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”?

Thermodynamic Geometry: 

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

Thermodynamic Curvature for the BH/BR System: 

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

Summary: 

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)