On the Black Hole/Black Ring On the Black Hole/Black Ring TransitionTransition

Ernesto Lozano-TellecheaWeizmann 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:– In 4d: BH uniqueness– In d>4: different phases ( ↔ horizon topolgy)

(BHs, black stings, branes…)

Phase transitions between them?

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…]

nnnn SSSS

Black Rings in d=5In d=5 pure Gravity, in addition to 5d Kerr BH, there aretwo asymptotically flat rotating BHs with horizon topology

[Emparan & Reall, 2002] “Black Rings”

Is this describing different phases of the same system?

12 SS

23// MJx

91903227

bound"Kerr " 1

min ./

x

x

BLACK HOLE NON UNIQUENESS!

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)

02

2

M

S

)()()( MSMMSMMS 2

( positivity of the Hessian of S)

S(M)S(M)

M

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

• Let us use only the entropy-max principle:

Equilibrium series

S(M)

Off-Equilibrium

),(ˆ MS

02

2

M

S0

2

2

S

(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: 1

2

21

2

2

M

S

M

S

ˆ

However, changes of stability only occur at a “turning point”:

Typical plot of β(M):

1/T

M

A

B

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

2

2

M

S

This method predicts stability of Schwarzschild

and Kerr BHs

Black Hole/Black Ring System

Behaviour of 2

2

M

S

Large BRSmall BRBlack Hole

23// MJx

Change in sign along eq. series onlyDivergent specific heat but

NO CHANGE IN STABILITY(in the microcanonical)

FluctuationsDiverge?

Stability

T

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

• One can define the appropriate susceptibilities

• And order parameter

2

2

2

2

J

SM

S

J

M

/T)( 2 phase1 phase /1

JM

) to(analog or 1 critmin TTxxx )()(

Obey scaling relations of the “first kind”

both at: ↔ BH/SBR

↔ SBR/LBRmin

1

xx

x

1 2,2

, ,

,)(

´´

What about the correlation length and scalings of the

“second kind”?

Thermodynamic Geometry

• Proposal:

[Ruppeiner ‘79]

Suitable for nonextensive thermodynamics

point critical aat

length)n correlatio( 1

2

dji

m

ij RXX

XSg

)(

Allows to compute ξand check scalings of the 2nd kind

Thermodynamic Curvature for the BH/BR System

Black Hole Large BRSmall 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 geometryin the extremal limit

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 transitionat extremality

Change in stability

(Small BR becomes unstable against

axisymmetric perturbations)