Microscopic entropy of black holes :

a two-dimensional approachM. Cadoni, Capri 2004

AbstractTwo-dimensional gravity models allow in many situations to compute the microscopic entropy of black holes and black branes in higher dimensions. We present recent results achieved using this approach. The relevance of these results for the AdS/CFT correspondence (or more in general for the holographic principle) is also discussed.

IntroductionThe problemThe problem: : Microscopic derivation of the Microscopic derivation of the

Bekenstein-Hawking area law S=A/4 Bekenstein-Hawking area law S=A/4 for black for black holes (BH) and black branes (BBholes (BH) and black branes (BB))

Solution of the problem relevant also for related topicsSolution of the problem relevant also for related topics

Holographic principle ( Fundamental or Holographic principle ( Fundamental or Emergent?)Emergent?)

Information loss for black holesInformation loss for black holes

• Two different strategies1. Detailed Knowledge of the fundamental dynamics

(e.g string theory) controlling the BH microscopic degrees of freedom is necessary

2. Area law should have some explanation also at the level of the low-energy effective theory describing the BH

• Usually 1. uses non-perturbative solutions of string theory and some duality (e.g AdS/CFT) to map the BH into a weak-coupled system• Nice example: Strominger and Vafa calculation of

the entropy of the 5D extremal RN black hole. • Bad features: Rely on SUSY Works only for BPS states (extremal BH and BB)

It is not easy to implement strategy 2.It is not easy to implement strategy 2. Because of no-hair theorems it seems impossible to reproduce the Because of no-hair theorems it seems impossible to reproduce the

huge degeneracy of BH states using low-energy gravity theoryhuge degeneracy of BH states using low-energy gravity theory We can circumvent this difficulty using an effective 2D description of the We can circumvent this difficulty using an effective 2D description of the

BH or BBBH or BB For spherical symmetric BH and BB thermodynamics seems to be For spherical symmetric BH and BB thermodynamics seems to be

determined completely by the 2D (r,t) sections of the spacetime. Only determined completely by the 2D (r,t) sections of the spacetime. Only spherical perturbations should enter in the entropy computationspherical perturbations should enter in the entropy computation

2D gravity is a topological theory ( as 3D gravity)2D gravity is a topological theory ( as 3D gravity) Pure gauge degrees in the bulk become physical on the boundaryPure gauge degrees in the bulk become physical on the boundary We can calculate the entropy of the 2D BH first finding a duality (e.g ADS/CFT) We can calculate the entropy of the 2D BH first finding a duality (e.g ADS/CFT)

between bulk gravity and a boundary theory then counting states in the 1D between bulk gravity and a boundary theory then counting states in the 1D boundary theoryboundary theory

SummarySummary1.1. Microscopic entropy of 2D AdS BHMicroscopic entropy of 2D AdS BH2.2. Microscopic entropy of non-dilatonic Microscopic entropy of non-dilatonic

Black branesBlack branesa)a) Thermodynamics of non-dilatonic Thermodynamics of non-dilatonic

BBBBb)b) Dimensional reductionDimensional reductionc)c) Statistical entropy of the BBStatistical entropy of the BB

3.3. ConclusionsConclusions

1. Microscopic entropy of 2D AdS BH

The simplest 2D gravity model allowing for AdS solutions isThe simplest 2D gravity model allowing for AdS solutions is

€

A = 12

d2∫ x −gφ R − 2λ2( )

• The BH solutions are (M= BH mass)

€

ds2 = − λ2r2 − 2Mλ

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt 2 + λ2r2 − 2M

λ ⎛ ⎝ ⎜

⎞ ⎠ ⎟−1

dr2,

φ = φ0 λr( )

• Thermodynamical entropy of the BH is

€

S = 4π Mφ0

2λ• To compute the microscopic entropy of the 2D BH we use a method that works for 3D ( Brown-Hennaux, Strominger) and 2D ( M.C. and S. Mignemi) BH:

BH entropy is computed by considering the deformation algebra generated on the 1D boundary of AdS by the action of the 2D bulk diffeos

This defines a CFT1 living on the boundary of AdS2

Can be considered as a AdS2/CFT1 correspondence

In more detail1. Choose suitable boundary conditions for the metric (r=)

2. Identify the group of asymptotic symmetries (ASG) preserving the r= behaviour of the metric

3. Show that the ASG is generated by a Virasoro algebra with (possible) central extension

€

Lk ,Ll[ ] = k − l( )Lk +l + c12

k 3δk +l

4. Compute the central charge c using a canonical realisation of the ASG (Regge, Teitelboim):

€

H χ( ) = dr χ tH t + χ rH r( )∫ + J χ( ),

J χ( ),J ω( )[ ]PB= J χ ,ω[ ][ ] + c χ ,ω( )

H is the Hamiltonian, are the Killing vector associated with the ASG and J a surface term (charge) 5. Use the Cardy formula

€

S = 2π cL0

6

To compute the entropy of the CFT1

• The orthogonality problem (the bundary of AdS2 is a point!) can be solved introducing time-integrated charges

€

J χ( ) = λ2π

dtJ χ( )0

2π / λ∫

Going through the various steps of the calculations one finds perfect agreement between the entropy of the CFT1 and the thermodynamical entropy of the AdS BH

• Origin of the non-vanishing central charge: breaking of the conformal symmetry of the ASG due to a non constant scalar field :

€

φ=φ0 λr( )

• The method can be used to calculate the entropy of 4D (or higher-D) BHs, which admit AdS2 as effective description

Example : The BH (zero branes) solutions of the model

€

A = 116π

d4∫ x −g R − 2 ∂Φ( )2 − 1

4e−αΦF 2 ⎡

⎣ ⎢ ⎤ ⎦ ⎥,α =1/ 3

in the near-horizon regime

• One can also identify the boundary conformal theory (M.C., P. Carta, D. Klemm, S. Mignemi): DFF conformal mechanics with external source :

€

A = dt 12

dqdt ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

− g2q2 + 1

4λ2γq2

⎡

⎣ ⎢

⎤

⎦ ⎥∫

2. Microscopic entropy of non-dilatonic Branes

• The 2D approach can be used to calculate the entropy of non-dilatonic p-branes of SUGRA theories (M. C. , Class. Quant. Grav. 21(2004)251, M.C. and N. Serra, hep-th/0406153)

€

A = 12kD

dD x −g∫ R − 12

∂Φ( )2 − 1

2n!Fn

2eaΦ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

is the dilaton, Fn is the RR field strength of a n-1-form potential, n=p+2

• We consider only non dilatonic branes

1. M-branes (2-brane and 5-brane in D=11) (no dilaton)

2. Self dual dyonic branes ( 1-brane in D=6 and 3-brane in D=10) (constant dilaton)

• Non dilatonic branes play a crucial role in the AdSp+2/CFTp+1 correspondence

In the extremal limit the near-horizon geometry of the p-brane becomes AdSp+2 SD-p-2

€

ds2 = λ p2 u2 −dt 2 + dx idx i

i=1

p∑ ⎛ ⎝ ⎜ ⎞

⎠ ⎟+ du2

λ p2 u2 + Rp

2dΩD− p−22

Maldacena conjecture

Duality between type II string theory on AdSp+2 SD-p-2 and CFTp+1

• Most of the progress about the string/CFT duality has come from comparing the two theories at zero temperature. For the gravity side this means extremal branes

• Finite temperature effects are important both for testing the duality and for discussing the thermodynamics of the brane from a microscopic point of view

They can be discussed considering the near-extremal brane

• Excitations near extremality break conformal invariance of the AdSp+2 background the brane acquires finite temperature

• We are interested in the near-horizon, near-extremal regime: r0, u,E=fixed (E= energy above extremality)

€

ds2 = λ p2 u2 −dt 2 1− u0

u ⎛ ⎝ ⎜

⎞ ⎠ ⎟p +1 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ dx idx i

i=1

p∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ du2

λ p2 u2 1− u0

u ⎛ ⎝ ⎜

⎞ ⎠ ⎟p +1 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

+ Rp2dΩD− p−2

2

• Using Bekenstein-Hawking area law we get the entropy as a function of the brane volume V and temperature T

€

Sbrane = apVT p

€

a1 = πN 2,a2 = 27 / 2π 2

27N 3 / 2,a3 = π 2

2N 2,a5 = 27

36 π 3N 3

N= number of coincident p-branes (related with the RR charge)

• In the spirit of the AdS/CFT correspondence Sbraneshould be matched by the entropy of the dual field theory at finite temperature

• Klebanov et al tried to do this using a gas of weak interacting brane excitation. The scaling with V and T was the right one but the coefficients ap turned out to depend on n , the number of fields of the model

€

a1 = π2

n,a2 = 78π

ζ 3( )n,a3 = π 2

12n,a5 = π 3

40n

• For the 3- and 1-brane the AdS/CFT duality allows an identification of n

For p=3 the dual CFT is N=4, U(N) SYM. This allows the identification n=8N2. We have still a mismatch of a 3/4 factor, which is only qualitatively understood gauge theory computation performed at weak ‘t Hooft coupling, gravity description holds at strong coupling

For p=1 brane becomes the BTZ BH times a 3-sphere, the dual theory is a 2D CFT, whose entropy has been calculated by Strominger. We get n=2N 2, matching exactly Sbrane with SCFT

• For the two M-branes the situation is more involved

In this case AdS/CFT is poorly understood It is required n ~ N3/2 and n ~ N3 (for p=2,5 respectively), hard to achieve with a field theory

Despite some progress using D-brane anti-D-brane systems (Danielssons et al) remains a puzzling point

Let us use the 2D approach

Dimensional reduction• As a first step we performe the dimensional reduction D2 in the D-dimensional gravity action. We use the ansatz

€

dsD2 = ds2

2 + φ2 / p dx i

i=1

p

∑ dx i + Rp2dΩD− p−2

2

The scalar field parametrizes the volume W of the brane embedded in the p+2-dim spacetime, W= V • The 2D, dimensionally reduced gravity model, is

€

A2D = k d2∫ x −gφ R + p −1p

⎛ ⎝ ⎜

⎞ ⎠ ⎟∂φ( )

2

φ2 + Λ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

kk and and are function of the brane parameters are function of the brane parameters N,VN,V and of the and of the radius of AdS radius of AdS RRpp

The model has 2D BH solutions which are asymptotically AdS The model has 2D BH solutions which are asymptotically AdS

€

ds2 = − b2r2 − A2 br( )1− p

( )dt 2 + b2r2 − A2 br( )1− p

( )−1

dr2,

φ = φ0 br( )p

• The mass and entropy of the BH are

€

MBH = p2

φ0A2b,SBH = 2πφoA2 p / p +1( ),b = Λ/ p p +1( )( )

• The 2D gravity model gives an effective description of the spherical excitations of the near-extremal, near-horizon brane solution. After identification of the integration constant A:

2D sections of the brane solutions match the 2D BH solution

The thermodynamical parameters of the BH reproduce exactly those of the near-extremal brane with TBH=Tbrane, SBH=Sbrane, MBH= E ( energy of excitations above extremality)

Statistical entropy• Let us now compute the entropy of the 2D BH (or equivalently the entropy of the boundary CFT) using the canonical realization of the ASG

• In principle we just need to repeat the calculations of the pure AdS BH for the 2D model under consideration

• Problem: we get divergent charges need a renormalization procedure

• To separate a finite from a divergent part in the charges we change the radial coordinate

€

br( )p−1 → br( )

p−1 + βA2 p−1( )

p +1

is an arbitrary dimensionless renormalisation parameter the final result (entropy) will depend on this parameter

• We can now go through the various steps of the calculations : 1. Define appropriate boundary conditions for the metric 2. Identify the ASG 3. Show that it is generate by a Virasoro algebra 4. Compute the associated charges J

• The charges J are divergent, we can eliminate its divergent part subtracting the contribution of the AdS background (A=0)

€

ds2 = − b2r2( )dt 2 + b2r2( )−1

dr2,

φ = φ0 br( )p

• We can now define renormalized charges

€

JR = J − Jbg

Using JR in the canonical realization of the ASG we find the central charge

€

c12

= φ0βpA2 p−1( )

p +1

2p +2

• The central charge depends on the arbitrary dimensionless parameter . We expect to be a rational number ( c is a rational function of the conformal weights of the boundary fields)

• We choose for the renormalization parameter the value

€

=2p +2

p2

• Using Cardy formula we find the entropy of boundary thermal CFT

€

S = 2πφ0A2 pp +1

• Expressing 0 and A in terms of the brane parameters T, V, N, we reproduce exactly (including the right factor ap) the brane thermodynamical entropy S= apVTp

• Notice: even though we leave the parameter unfixed, we can reproduce exact scaling of Sbrane on V, T, N S will be determined up to a dimensionless numerical factor

Conclusions The main question raised at the beginning was: The main question raised at the beginning was: does does

the AdS/CFT correspondence survive finite temperature the AdS/CFT correspondence survive finite temperature effects ( breaking of the conformal symmetry)?effects ( breaking of the conformal symmetry)?

Answering to this question may be crucial also for Answering to this question may be crucial also for related problems (related problems (e.g. use of the correspondence for e.g. use of the correspondence for describing the non-perturbative phase of QCDdescribing the non-perturbative phase of QCD))

Although the results for the entropy Although the results for the entropy for the 1- and the for the 1- and the 3-brane (AdS3-brane (AdS33/CFT/CFT2 2 and AdSand AdS55/CFT4/CFT4)) give some hope give some hope that the answer may be yes, the results for the 2- and that the answer may be yes, the results for the 2- and 5 brane indicate that this in general may not be the case5 brane indicate that this in general may not be the case

• There is also a general scaling argument that points against a yes answer:

If the dual theory is ( at least in some regime) a weak coupled field theory, the energy E(S,V) must have an extensive (or sub-extensive if it has a Casimir part) behaviour under SS, VV

This scaling behaviour can be achieved only if we interpret V as the intrinsic volume of the brane, which it is OK for the extremal brane but unnatural for the BB (V should be taken as the volume of the brane embedded in AdS)

• On the other hand the results our 2D approach points in a completely different direction

Finite temperature effects in higher dimensional AdS/CFT dualities can be described by an effective AdS2/CFT1 duality endowed with a scalar field that breaks the conformal symmetry and produces a non-vanishing central charge in the conformal algebra

• Moreover, the thermodynamics of the 2D models has a nice scaling behaviour and can be classified in terms of t violation of Eulero identity (extensivity) E=TS, through E=kTS

• From this point of view the holographic principle looses any fundamental character - becomes emergent- due to a fundamental feature of the gravitational interaction: its non-extensivity