Rong-Gen Cai ( 蔡荣根）

Institute of Theoretical PhysicsChinese Academy of Sciences

Holography and Black Hole Physics

®= ¯®= ¯

Contents:

1. Black Hole Mechanics and Black Hole Thermodynamics

2. Bekenstein Bound and D-Bound

3. Holography in AdS Space and dS Space

4. Friedmann equations and first law of thermodynamics

1. Black Hole mechanics and Black Hole Thermodynamics

18

2R g R GT

Einstein’s Equations (1915):

{Geometry matter (energy-momentum)}

Black Holes:

horizonSchwarzschild Black Hole: Mass M

More general:

Kerr-Newmann Black Holes

M, J, Q

No Hair Theorem

2 2 1 2 2 22 22(1 ) (1 )GM GM

r rds dt dr r d

The 0th law k =const.

The 1st law d M=k dA/8πG + Ω dJ +Φd Q

The 2nd law d A > 0

The 3rd law K -> 0

Four Laws pf Black Hole mechanics:

K: surface gravity (J.M. Bardeen,B. Carter, S. Hawking, CMP,1973)

The 0th law T=Const. on the horizon

The 1st law d M= T d S + J d Ω+Φ d Q

The 2nd law d (SBH +Smatter)>=0

The 3rd law T->0

Four Laws of Black Hole Thermodynamics:

Key Points: T = k/2π S= A/4G(S. Hawking, 1974, J. Bekenstein, 1973)

2. What we learn from black hole thermodynamics:

Holography

/ 2T

2/ 4 / 4 pS A G A l

Hawking Temperature:

Bekenstein-Hawking Entropy:

(1) Black hole entropy:

(2) Bekenstein Bound and Holographic Bound:

R

V, A

Bekenstein Bound (Bekenstein 1981):

2bS ER

Energy: E

Holography Bound:

2/ 4 pS A l

(‘t Hooft,1993, L. Susskind, 1994)

To be consistent with the second Law of thermodynamics

Holographic Bound:

Bekenstein Bound and Geroch Process.

E, R

2bS ER

(R.G. Cai and Y.S. Myung, PLB 559 (2003)60)

Consider a spherically symmetric black hole

Its horizon and Hawking temperature

First law of black hole thermodynamics

The red shift factor near the horizon is given by

Therefore near the horizon the proper distance R has the relation to the coordinate distance x

The absorbed energy is given by

and the increased entropy of the black hole

(3) de Sitter Space and D-bound:

(Willem de Sitter,1872-1934)

112 ( )R R g g g g

0C

Definition:

2 2 2 2 2 20 1 2 3 4z z z z z l 3R S

2 2 2 2 2 20 1 2 3 4ds dz dz dz dz dz

A four dimensional de Sitter space is a hyperboloidembedded in a five dimensional Minkowski space!

TopologyAnother one:

2 2 2 2 2 2 2 2 2cosh ( / )[ sin ( sin )]ds dt l t l d d d

In the global coordinates:

2 2 2 2 2 2 1 2 2 2(1 / ) (1 / )ds r l dt r l dr r d

Cosmological constant

1) Cosmological horizon thermodynamics:

1/ 2T l 2/ 4 pS A l

2) Asymptotically de Sitter Space: for example, SdS space

2 2 2 2 2 2 1 2 2 2(1 2 / / ) (1 2 / / )ds m r r l dt m r r l dr r d

(G. Gibbons and S. Hawking,1977)

In the static coordinates:

3) D-Bound (R. Bousso, 2001):

Entropy bound of a system in de Sitter space

20( ) / 4m pS A A l

This is consistent with the Bekenstein Bound ！

D-Bound and Bekenstein Bound:

(R.G. Cai, Y.S. Myung and N. Ohta, CQG 18 (2001) 5429)

Neutral system Charged system in 4 dim. (S. Hod, J. Bekenstein, B. Linet,2000)

Consider a charged system in de Sitter space

When M=Q=0, a pure de Sitter space has a cosmological horizon and entropy

D-Bound leads to

On the other hand, the cosmological horizon obeys

Consider the large cosmological horizon limit:

One has, up to the leading order,

The D-Bound gives

3. Holography in AdS and dS Spaces

AdS Spaces: 12 0R g R g

2 2

2 2

2 2 1 2 2 2(1 ) (1 )r rl l

ds dt dr r d

AdS/CFT Correspondence (J. Maldacena, 1997)

A well-known example: 5 4AdS / CFT

IIB superstring on N=4 SYM on the boundary of AdS_5

55AdS x S

(E. Witten)

(A. Polyakov) (I. Klebanov) (S. Gubser)

AdS/CFT Correspondence:

AdS/CFT Correspondence:

where has two interpretations: on the gravity side, these fields correspond to boundary data or boundary values, for the bulk fields which propagate in the AdS space.

on the field theory side, these fields correspond to external source currents coupled to various CFT operators.

Holography in dS Space (A. Strominger, 2000)

Quantum Gravity in dS

Euclidean CFT on the Boundary of dS Space

2 2 2 2 2 2 2 2 2cosh ( / )[ sin ( sin )]ds dt l t l d d d

准德西特相 德西特时空

标准大爆炸宇宙模型引力描写：

全息描写：

暴胀 未来

CFT1 QFT CFT2

重整化群的流动

我们的宇宙有个全息图吗？

2 2 2exp[2 / ]ds dt t l dx

(1) Holography for AdS Black Holes

Cardy-Verlinde Formula for higher dimensional CFTs (J. Cardy, 1986, E. Verlinde, 2000)

The CFTs reside on

Consider an (n+2)-dimensional Schwarzschild-adS black hole

where

Some thermodynamic quantities:

The boundary metric:

Thus, one has the thermodynamic quantities of CFTs:

It is easy to obtain

and to verify:

4) Friedmann equations and first law of thermodynamics

Schwarzschild-de Sitter Black Holes:

Black hole horizon and cosmological horizon:

First law:

Friedmann-Robertson-Walker Universe:

22 2 2 2 2 2 2 2

2( )( sin )

1

drds dt a t r d r d

kr

1) k = -1 open

2) k = 0 flat

3) k =1 closed

Friedmann Equations:

Where:

Our goal :

(R.G. Cai and S.P. Kim, hep-th/0501055 (JHEP 02 (2005) 050))

22 2 2 2 2 2 2 2

2( )( sin )

1

drds dt a t r d r d

kr

Horizons in FRW Universe:

Particle Horizon:

Event Horizon:

Apparent Horizon:

Apply the first law to the apparent horizon:

Make two ansatzes:

The only problem is to get dE

Suppose that the perfect fluid is the source, then

The energy-supply vector is: The work density is:

Then, the amount of energy crossing the apparent horizon within the time interval dt

（ S. A. Hayward, 1997,1998)

By using the continuity equation:

What does it tell us:

Classical General relativity Thermodynamics of Spacetime

Quantum gravity Theory Statistical Physics of Spacetime

?

T. Jacobson, Phys. Rev. Lett. 75 (1995) 1260

Thermodynamics of Spacetime: The Einstein Equation of State

Thanks !