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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. Subject: phase transitions in BH Physics Black Hole Phases:

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On the Black Hole/Black Ring Transition

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  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 twoasymptotically flatrotating 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: [“Poincare method” of stability] [Katz ‘79] [Kaburaki ‘94] Stability of the Lorentzian solution ↔ Microcanonical ensemble (fixed M, fluctuations in Temperature) Off-Equilibrium Equilibrium series S(M) (Natural extension: Legendre Transform) Near the equilibrium series:

  8. However, changes of stability only occur at a “turning point”: Typical plot of β(M): This method predicts stability of Schwarzschild and Kerr BHs 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

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

  10. 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

  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 Hole Small BR Large 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

  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: Change in stability (Small BR becomes unstable against axisymmetric perturbations) 2nd-order phase transition at extremality

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