Some Topics on Black Hole Physics

Rong-Gen Cai （蔡荣根）

Institute of Theoretical PhysicsChinese Academy of Sciences

1. Black Holes in GR

Einstein equations (1915)

12 8R Rg GT

(This equations can be derived from the following action)

4116 [ ] mGS d x gR g S

引力塌缩和致密星

天体 质量 半径 表面引力

太阳 1 1 10^(-6)

白矮星 < 1.4 10^(-2) 10^(-4)

中子星 1-3 10^(-5) 10^(-1)

黑洞 > 3 2GM/c^2 1

Schwarzschild Black Holes: the first exact solution of Einstein equations

(K. Schwarzschild,1873-1916)

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

r rds dt dr r d

真空的爱因斯坦场方程的精确解，描写一个天体的引力场。当这一天体的半径 <2GM, 它就是一个黑洞。

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

r rds dt dr r d Singularity

Horizon

I

Minimal black hole ?

2r GM

Black hole horizon

1/cr M 1/ PM G M

Singularities ? 0R R 2

6

12r

rR R

The geometry near horizon:

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

r rds dt dr r d

2 ( )r r r / 2t r

2 2 2 2 2 22ds d d r d

where

Making a Wick rotation on w, to remove the conical singularityat ,the Euclidean time w has a period In the time coordinate t, the period is

0 2

8 GM

Reissner-Nordstrom Black Hole: gravity coupled to a U(1) gauge field

2 2 1 2 2 22( ) ( )ds f r dt f r dr r d

where 2

2

2

2( ) 1 QMr r

Qtr r

f r

F

horizons:2 2r M M Q

14 F F

The geometry near outer horizon:

2 2 2 2 2 22ds d d r d

where

2( ) / 2t r r r

after a Wick rotation on w, w has the period In the time coordinate t, the Euclidean time it has the period

2

24 /( )r r r

( )

r

r

drf r

Extremal black Holes:

r r | |M Q

2 2 2 2 2 2 22( ) ( )( )ds H r dt H r dr r d

where

In this coordinate, double horizon is at r=0. The geometry near the horizon has the form

0( ) 1 rrH r

220

2 20

2 2 2 2 20 2

rrr r

ds dt dr r d

AdS_2 X S^2

Non-extremal BH Extremal BH

Euclidean Time

period arbitrary

Horizon topology

S^1 x S^2 S^1 x S^2

Horizon Horizon

Kerr-Newman Black Holes:

2 2

2

22 2 2 2 2(1 )Mr Qds dt dr d

where 2 2 2 2cosr a 2 2 22r a Mr Q

1) When a=0, Reissner-Nordstrom black hole solution2) When Q=0. Kerr black hole solution3) When a=Q=0, Schwarzschild black hole solution

Horizons: 0 2 2 2r M M a Q

2 2 2 2 2

2 2

2 2 2 2( 2 ) Sin ( 2 ) Sin[( ) ]sin 2Mr Q a Mr Q ar a d dtd

No-hair theorem of black holes (uniqueness theorem): The most general, asymptotically flat stationary solution of Einstein-Maxwell equations is the Kerr-Newman solution!

Ref: M. Heusler Black Hole Uniqueness Theorem

Cambridge University Press, 1996 (W. Israel)

视界

Kerr-Newman 黑洞

M, J, Q

无毛定理（ No Hair Theorem ）

Cosmological Constant: asymptotically (anti-)de Sitter BHs

12 8R Rg g GT

Cosmological constant

Schwarzschild-(A)dS Black Holes:

2 2 1 2 2 22( ) ( )ds f r dt f r dr r d

223( ) 1 GM r

rf r

When M=0, de Sitter (anti-de Sitter) space:

2 2 1 2 2 22( ) ( )ds f r dt f r dr r d

2

3( ) 1 rf r

Cosmological horizon

IOther solutions?

Reissner-Nordstrom-(A)dS BHs Kerr-(A)dS BHs Kerr-Newman-(A)dS BHs

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

2. Four Laws of Black Hole Mechanics (Ref: J. Bardeen, B. Carter and S. Hawking, CMP 31, 161 (1973))

Kerr Solution:

2 2 2 2 22 2 2 ( )

( )a Sin aSin r a

ds dt dtd

2 2 2 2 22 2 2 2( )r a a Sin

Sin d dr d

where2 2 2r a Cos 2 2 2r a Mr

There are two Killing vectors:

( )a a

t

( )a a

These two Killing vectors obey equations:

; [ : ]a b a b

; ;b b

a b a b

;a b a bb bR ;a b a b

b bR

; [ ; ]a b a b

with conventions: cab acbR R 1

;[ ] 2a

d bc adbcv R v

S

S B S

B

S

;a b a bb bR

Consider an integration for

over a hypersurface S and transfer the volume on the left to an integral over a 2-surface bounding S.S

Note the Komar Integrals:

;18

a bab

S

J d

measured at infinity

;a b a bab b a

S S

d R d

;14

a bab

S

M d

;14(2 )b b a a b

a a b ab

S B

M T T d d

Then we have

where 12 8ab ab abR Rg T

Similarly we have

;18

a b a bb a ab

S B

J T d d

For a stationary black hole, is not normal to the black holehorizon, instead the Killing vector does, where

is the angular velocity.

( )a a

t

a a aH

|tH H

g

g

;14(2 ) 2b b a a b

a a b H H ab

S B

M T T d J d

where;1

8a b

H ab

B

J d

Angular momentum of the black hole

Further, one can express where is the other null

vector orthogonal to , normalized so that and dA is

the surface area element of .

[ ]ab a bd n dA

B

an

1aan

B

;1 14 4

a bab

B B

d dA

where is constant over the horizon;a b

a bn

4(2 ) 2b b aa a b H H

S

M T T d J A

42 H HM J A

For Kerr Black Holes: Smarr Formula

2 4 2 1/ 2

4 2 1/ 2

2 4 2 1/ 2

2 4 2 1/ 2

2 ( ( ) )

( )

2 ( ( ) )

8 ( ( ) )

HH

H

H

H

H

J

M M M J

M J

M M M J

A M M J

where

Integral mass formula

8H HM J A

The Differential Formula: first law

The 0th law k =const.

The 1st law d M=k dA/8πG + J d Ω+Φd Q

The 2nd law d A >0

The 3rd law k ->0

经典黑洞的性质：黑洞力学四定律

（ k, 表面引力，类似于引力加速度） (J.M. Bardeen,B. Carter, S. Hawking, CMP,1973)

Wheeler 问： 假如某个热力学体系掉入黑洞，将导致什么？

热力学第二定律将违背吗？

J. Bekenstein （ 1973 ）：黑洞有热力学熵！ S ~ A, 视界面积

Hawking 不以为然 , 大力反对 Bekenstein 的观点！

可是考虑了黑洞周围的量子力学后， Hawking（ 1974 ， 1975 ）发现黑洞不黑，有热辐射！

12 8 TR Rg G

For Schwarzschild black hole, 3 / 8T c kGM

2KT

4AGS

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

黑洞热力学四定律：

Information Loss Paradox ( 信息丢失佯谬）

S. Hawking, PRD, 1976; a bet established in 1997

3. Black Hole Entropy: Area Formula Refs: a) The path-Integral Approach to Quantum Gravity by S. Hawking, In General Relativity: An Einstein Centenary Survey, eds. S. Hawking and W. Israel, (Cambridge University Press, 1979). b) Euclidean Quantum Gravity by S.W. Hawking, in Recent Developments in Gravitation Cargese Lectures, eds. M. Levy and S. Deser, (Plenum, 1978)

c) Action Integrals and Partition Functions in Quantum Gravity, by G.W.Gibbons and S. Hawking, PRD 15, 2752-2756

The path-integral approach:

The starting point is Feynman’s idea that one can express the amplitude

to go from a state with a metric $g_1$ and matter $\phi_1$ on a surface $S_1$ to a state with a metric $g_2$ and matter $\phi_2$ on a surface $S_2$, as a sum over all field configurations $g$ and $\phi$ which takes the given values on the surface $S_1$ and $S_2$.

2 2 2 1 1 1, , | , ,g S g S

2 2 2 1 1 1, , | , , [ , ]Exp( [ , ])g S g S D g iI g

2 2,g

1 1,g

S_1

S_2

,g

The action in GR:

4 4116 ( 2 ) mGI d x g R d x gL

Equations of motion

12 8R Rg g GT

where1

2mL

ggT

is the energy-momentum tensor of matter fields.

In order to be well-defined for the variation, a Gibbons and Hawkingsurface term has to be added to the action

318 G d x hK C

where C is a term which depends only on the boundary metric h and not on the values of g at the interior points.

4 4 31 116 8( 2 ) mG GI d x g R d x gL d x hK C

and31

08 GC d x hK

Consider a metric which is asymptotically flat in the three spatial directions but not in time

2 1 2 1 2 2 2 22(1 2 ) (1 2 ) ( )t sds M r dt M r dr r d 0 r

If the metric satisfies the vacuum Einstein equations near infinity,then M_s=M_t. Consider a boundary with a fixed radius r_0, one has

3 20(8 4 8 ( ))t sd x hK r M M 0 r dt

For a flat metric, one has 10 02K r

12 2( 2 ) M

t sG GI M M dt dt

Complex spacetime and Euclidean action

(1)For ordinary quantum field theory, make a Wick rotation,

t i

ˆ[ ] ( [ ])Z D Exp I [ ] ( [ ])Z D Exp iI

Where is called Euclidean action, which is greater thanor equal to zero for fields which are real on the Euclidean space defined by the real coordinates. Thus the integral over all such configurations of the field will be exponentially damped and should therefore converge.

I iI

(2) Quantum field theory at finite temperature

To construct a canonical ensemble for a field

2 2 1 1, | , [ ]exp( [ ])t t D iI which expresses the amplitude to propagate from a configuration $\phi_1$ on a surface at time $t_1$ to a configuration $\phi_2$ ona surface at time $t_2$ .

Using the Schrodinger picture,one can also write the amplitude as

2 2 1 1| exp( ( )) |iH t t

1 1, t

2 2, t

Put and and sum overa complete orthonnormal basis of configurations . One has thepartition function

2 1t t i 2 1

n

exp( )nZ E

for the field at a temperature .1/T

One can also express Z as a Euclidean path integral

ˆ[ ] ( [ ])Z D Exp I where the integral is taken over all fields that are real on the Euclideansection and are periodic in the imaginary time coordinate with period.

(3) Apply to quantum gravity

Introducing an imaginary time coordinate , the Euclidean action of gravitational field has the form

it

4 3 41 1016 8

ˆ ( 2 ) ( ) mG GI d x g R d x h K K d x gL (The problem is that the gravitational part of the action is not positive-definite.)

Canonical ensemble for gravitational fields:

One can consider a canonical ensemble for the gravitationalfields contained in a spherical box of radius r_0 at a temperature T, by performing a path integral over all metrics which would fit inside a boundary consisting of a timelike tube of radius r_0 which was periodically identified in the imaginary time direction with period 1/T.

(4) The stationary-phase approximation

* Neglecting the questions of convergence.*Expecting the dominant contribution to the path integral will come from metrics and fields which are near a metric g_0 and $\phi_0$, which are an extremum of the action, i.e. a solution of the classical field equations.*Can expand the action in a Taylor series about the background field

0 0 2ˆ ˆ[ , ] [ , ] [ , ]I g I g I g higher order terms

where

0g g g 0

0 0 2ˆlog [ , ] log [ , ]exp( [ , ])Z I g D g I g

Ignoring the higher order terms results in the stationary-phase approx.

Zero-loop One-loop

Some Examples

(1) Schwarzschild BH

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

r rds d dr r d

Introducing the Euclidean time

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

r rds dt dr r d

Introducing the coordinate 1/ 24 (1 2 / )x GM GM r

2 2 2 2 2 2 2 2 2 22( / 4 ) ( / 4 )ds x GM d r G M dx r d

8 GM

it

The Euclidean section of the Schwarzschild solution is periodic inthe Euclidean time coordinate tau, the boundary at radius r_0 hasthe topology S^1xS^2, and the metric will be the stationary phase point in the path integral for the partition function of a canonicalensemble at temperature T=1/beta. The action is

12I M

(2) Reissner-Nordstrom BHs 18mL F F

12

ˆ ( )I M Q where /Q r

(3) Kerr Black Holes

The Euclidean section of the Kerr metric provided that the massM is real and the angular momentum J is imaginary. In this casethe metric will be periodic in the frame that co-rotates with the horizon, i .e. the point is identified with

( , , , )r

( , , , )r i

12I M

Black Hole Entropy

21 12 16

ˆLog GZ I M

Schwarzschild BHs

According to the formulas

LogE Z LogS E Z

2 24 / 4S G M A G E M

Kerr-Newman BHs

1ˆ ( )I M Q

/Q r

According to T= /2

where

LogZ=- /W T

i ii

W M TS C

12 ( )W M Q W M TS Q J

/ 4S A G 4 2GM A Q J

4. Black Holes in anti-de Sitter Spaces

* For a Schwarzschild BH, its Hawking is ,Therefore its heat capacity

It is thermodynamic unstable.

* The lifetime of a Schwarzschild BH,

* If

1/ 8T M

28MTC M

4dMdt AT

2 3t G M

15t by 1210M Kg

Put a BH in a cavity

3434S M aVT

4E M aVT

where2 71 1

15 8 2( )b f sa n n n

set 5 1/ 4/ , (1/ 3 )( / )x M E y aV E

2 3/ 4(1 )f x y x [0,1]x

2 3/ 4(1 )f x y x

(1) When y > 1.4266, there are no turning points and the maximal value of f is at x=0.

(2) When 1.0144=y_c < y < 1.4266, there are two turning points: a local minimum for x < 4/5; and a local maximum for x >4/5. However, a global maximum of f is still at x=0.

(3)when y< y_c, there is a global maximal with x > x_c=0.97702.

0.2 0.4 0.6 0.8 1

1.1

1.2

1.3

1.4

1.5

0.2 0.4 0.6 0.8 1

0.85

0.875

0.9

0.925

0.95

0.975

0.2 0.4 0.6 0.8 10.98

1.02

1.04

1.06

1.08

1.1

1.12

Black holes in AdS space: embedded BHs to AdS space

i) AdS space Anti de Sitter space 0

(1) Negative energy density

(2) Closed timelike curves

(3) Not globally hyperbolic

(4) Ground states of some gauged supergravities

(5) The positive mass theorem holds, the AdS is stable

1 3 x RS

Why AdS?

AdS Space: SO(2,D-1) with topology D-1

1 x RS

2 2 2 2 20 1 1D Dx x x x l

2 2 2 2 2 2 1 2 22(1 / ) (1 / ) Dds r l dt r l r d

Why AdS BHs?

(1) AdS BHs are quite different from their conterparts in asymptotically flat spacetime or de Sitter spacetime.

(2) Horizon topology may not be S^2 (S. Hawking, 1972)

(3) AdS/CFT correspondence (J. Maldacena, 1997) AdS BHs/thermal CFTs (E. Witten, 1998)

ii) AdS BHs and Hawking-Page phase transition Ref: S. Hawking and D. Page, CMP 87, 577 (1983)) E. Witten, Adv. Theor.Math.Phys.2:505-532,1998

Consider the action

2

4 361 116 8( )G Gl

I d x g R d x hK

has solutions

2 2 1 2 2 22( ) ( )ds f r dt f r dr r d

(1) AdS space 2

21 rl

f

the time coordinate has period

Euclidean AdS space

Thermal AdS space with temperature T=1/beta:

Local temperature

2 2 1 2 2 22( ) ( )ds f r d f r dr r d

1 1/ 2( ) /locT r f T f

' 2 l

Thermal radiation gas in AdS

2 00( ) ( 4 )T p T f

Where

The energy density

2 490( ) ...p T gT

2 4 2 430 gT f r

The total energy is finite.

(2) AdS BH solution

2

221 GM r

r lf

One horizon is located at f(r)=0. The inverse temperature

2

2 2

4

3

l r

l r

r

0r

The maximum value

03

lr

0 2 / 3l

The Euclidean action

2

4 361 116 8

ˆ ( )M MG GlI d x g R d x hK

(1)

(2) The reference background: thermal AdS space

2 2 1 2: R x S , : S x SM M

R boundarya) Take a finite boundary at R,b) R->infinity

1/ 2 1/ 20( ) ( , 0) ( , 0)loc R f R M f R M

AdS BH thermal AdS space

2

438

ˆR

BH Glr

I d x g

2

438

0

ˆR

AdS GlI d x g

2 2 2

2 2

( )

(3 )ˆ r l r

G r lI

ˆlog Z = - I

logE Z M

log / 4S E Z A G

For larger BHs:3 2 2/32 , r GMl A M

The density of states:

The partition of function:

converges; the CE is well-defined.

2/3( ) exp[ ]N M M

( ) exp[ / ]Z N M M T dM

For Schwarzschild BHs:

The partition function diverges; the CE is pathological!

2( ) exp[ ]N M M

0.5 1 1.5 2

-0.8

-0.6

-0.4

-0.2

0.2

2 2 2

2 2

( )

(3 )ˆ r l r

G r lI

ˆ 0I

ˆ 0I

If r_+ < lSmall BHs

If r_+ > lLarge BHs

I F

The thermal radiation in AdS:

4 390log ( / )Z g l

4 4 330E gT l

4 4 390F gT l

r

0

1

0 1r r

C > 0

C < 0

1 0 / 3r l r l

HP transition

iii) Topological BHs in AdS space

Consider the following action:

11 116 8( 2 )n n

G GS d x g R d x hK 2

( 1)( 2)

2

n n

l

12 0R Rg g

The metric ansatz:2 2 1 2 2 2( ) ( ) kds f r dt f r dr r d

2 i jk ijd dx dx ( 2)( 3)R d d k

One has the solution:2

3 2nn

w M rr l

f k

216( 2) ( ) , ( ) nG

n n Volw Vol d x

Without loss of generality, One can take I) k=1; ii) k=0;

iii) k=-1.

ii) The case of k=0

2 2 1 2 2 2( ) ( ) kds f r dt f r dr r d 2

3 2 - nn

w M rr l

f

horizon:

temperature:

3nnr w M

radius

betaC >0

24( 1)

ln r

AdS soliton?2 2 1 2 2 2 2 2( ) ( ) ( )ds f r dt f r dr r dx dy dz

x it t i 2 2 2 2 1 2 2 2 2( ) ( ) ( )ds r dt f r d f r dr r dy dz

If has a period ,the solution is regular everywhere. The solution has a minimal mass (negative)! (Horowitz and Myers, 1998)

2 2 1 2 2 21( ) ( )ds f r dt f r dr r d

iii) The case of k=-1:

2

3 2 -1 nn

w M rr l

f

(1) Even when M=0, a horizon exists, r_+=l “massless black hole”.

(2) Hawking temperature

T > 0

2 2

2

( 1) ( 3)

4

n r n kl

l rT

3

1/ 231

( 3) / 2322 1

( )

( )n

n

ncrit n

nn lcrit n n w

r r l

M

“negative mass BHs”

(3)when , the BH has two horizons when M>0, the BH has only one horizon.

0critM M

Euclidean action

2

( ) 1 2 3

16 = ( )

BH BG

Vol n ncritGl

I I I

r kl r M

The background:0 for k=0,1

for k=-1crit

M

M M

( ) 24, / 4Vol n

crit GE M M S r A G

0I

The heat capacity:

n-2 2 2+

2 2

24

4 r ( 1) ( 3)

( 1) ( 3)

C= for k=0

= for k=-1

n

n

nw

n r n lw n r n l

r

iv) BTZ black hole and its higher dimensional generalization

3-dim.gravity: Einstein tensor ~ Riemann tensor since Weyle tensor=0 (1) The Einstein equations still hold (2) GR is dynamically trivial (3) The vacuum solution is flat (4) The localized source has effect only on the globally geometry (5) A static point source forms a cone

(Deser, Jackiw and ‘t Hooft, Ann. Phys(NY), 1983)

When a cosmological constant is present

for a static point source at the origin, one has the solution (Deser and Jackiw, Ann. Phys. (NY), 1984)

where

12 8R Rg g GT

2 2 2 2 2 2( ) ( )( )ds N R dt R dR R d

1 4Gm

2 2

2 2

2 2 1 2 2 2(1 ) (1 )r rds dt dr r d

If r r2 2 2 2 1 2 2 2 2(1 ) (1 )ds r dt r dr r d

2 (1 ) deficit angle:

40 0

2 20 0 0 0

( / ) ( / ) 4( / ) ( / ) [( / ) ( / ) ]

( ) ( )=R R R R

R R R R R R R R RN R R

BTZ Black Hole (Banados, Teitelboim and Zanelli, PRL, 1992)

Where

The BTZ BH has two horizons

provided M>0 and J< Ml .

2 2 2 2 2 2 2( ) ( ) ( ( ) )ds N r dt N r dr r N r dt d

22

2 2 2

2 16 4( ) 8 ( )J Jrl r r

N r M N r

2 2 24 (1 1 ( / ) )r Ml J ML

If J=0:

(1) AdS space, if –8M=1, in global coordinates

(2) AdS space, if M=0, in Poincare coordinates (massless BTZ black hole since r=0 is a null singularity)

Therefore there is a gap between the BTZ BH and AdS space!

2 2 2 2 2 2 1 2 2 2( 8 / ) ( 8 / )ds M r l dt M r l dr r d

1/ 8 0 ?M

Some Remarks:

(1) BTZ BH is locally equivalent to a 3-dim. Ads Space, but not globally.

(2) BTZ BH appears naturally in SUGRA (4-dim. BH with 4 charges and 5-dim. With 3 charges)

(3) The statistical degrees of freedom of the BTZ BH i) Carlip’s method (1994, gr-qc/9405070, 9409052) ii) Strominger’s method (1997, hep-th/9712251)

The higher dim. generalization of BTZ BHs:

Constant curvature case: k = -1 and M=0.

This is not the counterparts of the BTZ BH in higher dims.

What is the counterparts?

As an example, consider the 5-dim. Case

There are 15 Killing vectors: 7 rotations and 8 boosts

2 2

3 2 3 2

2 2 1 2 2 22 2( ) ( )n nGM GMr r

kr l r lds k dt k dr r d

2 2

2 2

2 2 1 2 2 21( 1 ) ( 1 )r r

l lds dt dr r d

2 2 2 2 2 2 20 1 2 3 4 5x x x x x x l

Consider a boost vector: its norm

4 5 5 4( )rl x x

2 2 2 2 24 5( ) /r x x l

2 25 4

2 25 4

2 25 4

0

0

0

x x

x x

x x

There are two special cases:

(1) , it is a null surface

(2) When , it is a hyperloloid

pS

fS

pH

fH0x

The killing vector is spacelike in the Region contained-in-between S_f andS_p;is null at S_f and S_p;is timelike in the causal future of S_fand in the causal past of S_p;

Remarks: (1) Identifying the points along the orbit of \xi, another one-dim. manifold becomes compact and isomorphic to S^1. (2) The region \xi^2 <0 has a pathological chronological structure and therefore must be cut off from the physical spacetime. (3) Identifying the points along a boosting Killing vector to construct a black hole; if along a rotating Killing vector, results in a conical singularity. (4) topological structure: M_4 x S^1 usual case (Sch. BH): R_2 x S^3 (5) This is a constant curvature BH since we start from an AdS.

The Penrose diagram:

In the region , introducing six diemnsionless local coordinates2 0 ( , )iy

Here with the restriction –1 <y^2 <1. and - < <iy

Introducing local “spherical” coordinates :

It does not cover the full outer region of black holes!

The coordinates covering the entire exterior of BH:

Dual Field Theory: Stress-energy tensor

Starting from the action:

The surface counterterm

Resulting in the quasi-local stress-energy tensor of the gravitational field

Take a surface with a fixed $r$ as the boundary, one has

The boundary metric, on which dual CFT resides

Using the relation

One has

(1) Independent of r_+,

(2) Vanishing trace,

(3) No Casimir energy

(For details, see for example, R.G.Cai , PLB 544, 176 (2002); PLB552, 66 (2003))

v) AdS Black Hole in string/M theory

Type II SUGRA:

2

10 2 2 21 13122

10 2 21 12 42 2! 2 4!

( 4( ) )

+ d ( ...)

IIAS d x ge R H

x g F F

2

10 2 2 21 13122

10 2 2 21 1 13 52 2 3! 2 5!

( 4( ) )

+ d ( ( ) ...)

IIBS d x ge R H

x g F F

Self-dual

NS sector

RR sector

11-dim. SUGRA:

2

11 21 111 42 4!2

( )S d x g R F

Dp-brane solution:

Take the near-horizon limit (decoupling limit)

707

7

7

2 1/ 2 2 2 1/ 2 2 2 28

(3 ) / 4

5 2 7 2 8

( ) ( )

1

1

(2 ) [(7 ) / 2] 2 (2 )

p

p

pp s s

p

p p

p

r

r

C g Nl

r

pp s s

ds H fdt dx H dr r d

e H

f

H

C p g l

2 2 ( 3) / 2(2 ) ' fixed ' 0

/ ' fixed (r=2 ' )

p pYM sg g

u r

D3-brane case:

2 2

2 2

2 2 2 1 2 2 23 5d ( )r R

R rs fdt dx f dr R d

AdS_5 S^5

where

Entropy S ~N^2 SU(N) SYM theory

2 43 s sR C g Nl

IIB theory on AdS_5 x S^5 N=4 SU(N) SYM

(J. Maldacena, 1997)

AdS/CFT Correspondence

M2-brane:

606

626

2 2/3 2 2 1/3 2 2 211 2 7

6 2 62 2 11

( ) ( )

1

1 =32

r

r

r

r

ds H fdt dx H dr r d

f

H r N l

Near horizon geometry: AdS_4 x S^7

2 2 211 4 2 7 BH +ds AdS r d

M5-brane:

Near horizon geometry: AdS_7 x S^4

303

35

2 1/3 2 2 2/3 2 2 211 5 4

3 35 5 113

( ) ( )

1

1 =

r

r

rr

ds H fdt dx H dr r d

f

H r N l

2 2 211 7 5 4 BH +ds AdS r d

BTZ BH and 5-dim. BH in string theory

Consider the D1+D5+W system:

2

202

20

6

2 1/ 2 1/ 2 2 2 21 5

1/ 2 1/ 2 2 1/ 2 1/ 2 2 21 5 11 1 5 31

2 2 20 1,5 1,5

1 1 52

[ (cosh sinh ) ]

+ ( )

where / 1 sinh

sinh(2 ) s s

m m

drK

r

r

Vr

g l

ds H H dt dz K dt dz

H H dx H H r d

K r r H

N N

2

02

2 2

52

2

sinh(2 )

sinh(2 ) s s

s s

r

g l

RVm mg l

N

Near-horizon geometry:

2 2 41 5 3 BH + ds BTZ r r d T

Thanks!