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Entropy for Black Holes & Black Rings

Wen-Yu Wen @ NTU

Ref: Isono,Tai,Wen [0812.4440]Isono, Wen [work in progress]

Motivation• Black holes entropy is obtained from the Bekenstein-

Hawking area law, macroscopically. • Some black holes can be realized in String theory and

therefore microstates counting is available in the underlying CFT. For 5D supersymmetric black holes, we have at least two descriptions from AdS/CFT and Kerr/CFT.

outline• Black holes in four dimensions• Entropy from black hole thermodynamics• (susy-) Black holes (rings) in five dimensions• String theory realization of 5D black holes (rings)• Microstates counting from AdS/CFT• Microstates counting from Kerr/CFT• Outlook

Black holes in D<5• Uniqueness Theorem:

Black holes solutions are completely determined by conserved charges, such as mass (M), em charge (Q), and angular momentum/spin (J).

• Given a Killing form dual to some Killing vector preserving isometry, one can apply the Komar integral (at the asymptotic infinity) to compute the conserved charges.

A simple example

• 4D Schwarzschild black holes [Schwarzschild,1916]

• The only charge here is the mass M and the associated Killing vector is time translation

))(sin()21()21( 22222122 ddrdrrGMdt

rGMds

MddrrGM

Gd

G

dtrGM

r

,

22 )sin()2(

81*

81

)21(

More black holes in D=4

Kerr-NewmanReissner-Nordström

Q≠0

Kerr[GRS 1915+105]

SchwarzschildQ=0

J≠0J=0

Event horizon of Black holes

• The boundary of a region from which null rays cannot escape to future null infinity.

• A null hypersurface where a null Killing vector field can be defined.

• It refers to the outer horizon if there exist both outer and inner (Cauchy) horizons.

• Extremal limit corresponds to degeneracy of two horizons.

Some examples…

• SS BH:

• Kerr BH:

horizonat 0 ,)0,0,0,1(2

021

aa

H

tt

GMrrGMg

JGM

MGJMMr

MGJrMGJMrrgrr

2

22

222

221

:limit Extremal

)/(

0cos)/(

)/(2

Black Holes in String Theory

• For D>4, horizon topology is richer than sphere.• Non-uniqueness: solutions are not uniquely

determined by conserved charges.• There exist small black holes which have

nonzero entropy but vanishing horizon area.• Microstates counting is available for some black

holes in String theory.

5D supersymmetric black holes

• Theorem [Reall,06] : any susy solution (D=5,N=1) with compact horizon has a horizon geometry that is locally isometric to-T3

-S1 x S2 Black Ring [Emparan-Reall,01;Elvang,03]

-S3 BMPV Black Hole [Breckenridge-Myers-Peet-Vafa,96]

BMPV black hole

S3

Horizon at ρ→ 0

Stringy realizationfrom IIB theory:

t, R4

T4

S1

z

Black Ring

S1xS2

-1≦x ≦1, -∞<y≦-1

Asymptotic infinity at x, y → -1Event horizon at y → -∞

Stringy realizationfrom M-theory

dipoles

T6

t, x, y, ψ, φ

T6

Non-uniqueness

• A susy black ring is completely specified by 7 parameters: 3 Qi,3 qi,R with 1 constraint, but only carries 3 gauge charges Qi and 2 spins.

• The m5 branes wrap a common circular direction ψ, which is contractible, therefore 3 dipole charges are not conserved.

Near-horizon geometry

• The near-horizon geometry of 5D super black hole (ring) is a direct product of the near-horizon geometry of the extremal BTZ black hole and a compact S2. The unique rotating black hole in 3D

• The entropy is given by the Bekenstein-Hawking area law:

• To reproduce the Bekenstein-Hawking entropy by microstates counting, we need a quantum theory of gravity, such as String theory.

• To be precise, we may expect the existence of some 2D CFT(s) dual to the near-horizon geometry (~AdS3).

Cardy formula• We start with a 2D CFT with central charge c and

Virasoro algebra

• The partition function on T2 of modulus τreads

62)(log

)6

2exp()(

cS

c

Eigenvalue of L0

[Cardy,86]

Entropy from AdS/CFT

• We have learnt that CFT living on (t, φ) dual to BTZ has central charge [Brown-Henneaux,86]

• The energy, or oscillator level, is given by Komarintegral evaluated at the horizon [Emparan-Mateos,05],

where the Killing form dual to is

3

Δ

62

cS

Kerr/CFT correspondence

• A novel CFT on the (r, φ) dual to Kerr black hole such that entropy can be reproduced by Cardy formula: [Guica-Hartman-Song-Strominger,08]

• The central charge c is obtained via the Dirac brackets:

Virasoro generators

• The Frolov-Thorne temperature is defined by

• The near-horizon geometry for Kerr black hole is

1

rr

FT JST

+…

)1(),2( URSL

• The CFT in Kerr/CFT is chiral, however the CFT in AdS/CFT is non-chiral.

• The near-horizon geometry of susy black holes (rings) can also be brought into near-horizon geometry of Kerr black holes. [Isono-Tai-Wen,08]

• They may agree simply by accident, or susy black holes (rings) serves as a good working example to study connection between both formalisms.

Comments

Thank You