Studying the strongly coupled N=4 plasma

using AdS/CFT

Amos Yarom, Munich

Together with S. Gubser and S. Pufu

AdS/CFT

J. Maldacena

Calculating the stress-energy tensor

T

>> 1 N >> 1

Calculating the stress-energy tensor

• Anti-de-Sitter space.

• Strings in Anti-de-Sitter space.

• The energy momentum tensor via AdS/CFT.

• Results.

ds2 = c2 dx2+c2 dy2+c2 dz2

ds2

dx2

dy2dz2

Flat space

ds2

x

y

dz2= dx2 dy2+ +

z

x c xy c yz c z

+ dw2 - dt2

5d Anti de-Sitter space

ds2 =L2 z-2 (dz2+dx2+dy2+dw2 - dt2)

z

0

+

AdS5 black hole

ds2 =L2 z-2 (dz2/(1-(z/z0)4)+dx2+dy2+dw2 - (1-(z/z0)4) dt2)

z

0

z0

ds2 = gdxdx

Strings in AdSds2 = gdxdx

z

z0

X()

X()

SNG= s ______√g ( X)2 d d

1___20

N=4 SYM plasma via AdS/CFT

AdS/CFT

J. Maldacena

AdS5 CFT

Empty AdS5Vacuum

L4/’2 gYM2 N

L3/2 G5 N2

J. Maldacena hep-th/9711200

T>0

N=4 SYM plasma via AdS/CFT

AdS5 CFT

AdS5 BH Thermal state

L4/’2 gYM2 N

L3/2 G5N2

E. Witten hep-th/9802150

Horizon radius Temperature

Empty AdS5Vacuum

J. Maldacena hep-th/9711200

AdS/CFT

J. Maldacena

Static ‘quarks’ using AdS/CFT

AdS5 CFT

J. Maldacena hep-th/9803002

Massive particle

Endpoint of an open

string on the boundary

z0

z

0

?

SNG

X =0

Moving ‘quarks’ using AdS/CFT

AdS5 CFT

J. Maldacena hep-th/9803002

Massive particle

Endpoint of an open

string on the boundary

z0

z

0

SNG

X =0

?

Moving ‘quarks’ using AdS/CFT

AdS5 CFT

J. Maldacena hep-th/9803002

Massive particle

Endpoint of an open

string on the boundary

z0

z

0

SNG

X =0

Extracting the stress-energy tensor using AdS/CFT

z0

z

0AdS5 CFT

gmn|b <Tmn>

E. Witten hep-th/9802150

z0

z

0

Extracting the stress-energy tensor using AdS/CFT

AdS5 CFT

ds2 = gdx dx

gmn|b <Tmn>

g = gAdS-BH+h

AdS black hole Metric fluctuations

E. Witten hep-th/9802150

The energy momentum tensor

z0

z

0

S = SN G +SE H

SN G =1

2¼®0

Z(g@X@X )1=2d2¾

±S±X

= 0±S±g

= 0D¹ º½¾h½¾= J ¹ º

SE H =1

16¼G5

Z µR +

12L2

¶g1=2d5x

¸ =G5

®0¿ 1

g=gAdS+ h

(Friess, Gubser, Michalogiorgakis, Pufu, hep-th/0607022)

Energy density for v=3/4

Over energy

Under energy

(Gubser, Pufu, AY, ArXiv: 0706.0213, Chesler, Yaffe, ArXiv: 0706.0368)

v=0.75 v=0.58

v=0.25

E = ¡3iK 1v(1+v2)

2¼(K 2? +K

21(1¡ 3v2))

+O(K 0)

D¹ º½¾h½¾= J ¹ º

h½¾=X

n

K nh(n)½¾ Zh½¾eiK X

d3K(2¼)3

(x1;t) ! x1 ¡ vt

Small momentum approximations

(Friess, Gubser, Michalogiorgakis, Pufu, hep-th/0607022)

Small momentum approximations

E = ¡3iK 1v(1+v2)

2¼(K 2? +K

21(1¡ 3v2))

+O(K 0)

1-3v2 < 0 (supersonic)

E(X ) =

(3v(1+v2 )

4¼2X 1

(X 21+(1¡ 3v

2)X 2? )

3=2 inside theMach cone.

0 outside theMach cone.

1-3v2 > 0 (subsonic)

E(X ) =3v(1+v2)

8¼2X 1

(X 21 +(1¡ 3v2)X

2? )

3=2

(Gubser, Pufu, AY, ArXiv: 0706.0213)

E =

¡3K 2

1v2(K 2

? (2+v2) +2K 2

1(1+v2)

2¼(K 2? +K

21(1¡ 3v2))2

¡3iK 1v(1+v2)

2¼(K 2? +K

21(1¡ 3v2))

+O(K 1)

E =¡3iK 1v(1+v2) +O(K 2)

2¼(K 2? +K

21(1¡ 3v2) ¡ ivK 2K 1)

+O(K 1)

Small momentum approximations

(Gubser, Pufu, AY, ArXiv: 0706.0213)

E =¡3iK 1v(1+v2) +O(K 2)

2¼(K 2? +K

21(1¡ 3v2) ¡ ivK 2K 1)

+O(K 1)

¡@2t +@

2x(c

2s +¡ s@t)

¢E = sources

cs2=1/3

s=1/3

Small momentum approximations

(Gubser, Pufu, AY, ArXiv: 0706.0213)

Energy density for v=3/4

0

v=0.75 v=0.58

v=0.25

Large momentum approximations

E(X ) = ¡ vX 1(5¡ 11v2)X 2

1 +(1¡ v2)(5¡ 8v2)X 2

?

72(1¡ v2)5=2³

X 21

1¡ v2 +X2?

´5=2

(Gubser, Pufu hep-th: 0703090 AY, hep-th: 0703095)

E(X ) = ¡ vX 1(5¡ 11v2)X 2

1 +(1¡ v2)(5¡ 8v2)X 2

?

72(1¡ v2)5=2³

X 21

1¡ v2 +X2?

´5=2

Large momentum approximations

(Gubser, Pufu hep-th: 0703090 AY, hep-th: 0703095)

The Poynting vector

V=0.25

S1 S?

V=0.58

V=0.75

(Gubser, Pufu, AY, ArXiv: 0706.4307)

Small momentum asymptotics

Sound Waves ?

S1 = ¡ iK 1(1+v2)

2¼(K 2 ¡ 3K 21v2)

+ i1

2¼K 1+O(K 0)

K 21v

2¼(K 2 ¡ 3K 21v2)2

+3K 4

1v2(1+v2)

2¼(K 2 ¡ 3K 21v2)2

+K 2

8¼K 21v¡

18¼v

S1 = ¡ iK 1(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)

+4v+O(K )

2¼(K 2 ¡ 4iK 1v)

(Gubser, Pufu, AY, ArXiv: 0706.4307)

Small momentum asymptotics

S1 = ¡ iK 1(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)

+4v+O(K )

2¼(K 2 ¡ 4iK 1v)Z

4v2¼(K 2 ¡ 4iK 1v)

eiK Xd3K(2¼)3

= ¡X 1

8¼2X 3e¡ 2v(X 1+X )

X 1

X 3e¡ 2v(X 1+X ) »

8<

:

X 1X 3 e¡ 4vX 1 jX j À 1; X 1 > 0

X 1X 3 e

¡ vX 2?

X 1 jX j À 1; X 1 <0

(Gubser, Pufu, AY, ArXiv: 0706.4307)

The poynting vector

V=0.25

S1 S?

V=0.58

V=0.75

(Gubser, Pufu, AY, ArXiv: 0706.4307)

Energy analysis

S2 = ¡ iK 2(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)

+O(K )

2¼(K 2 ¡ 4iK 1v)

S3 = ¡ iK 3(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)

+O(K )

2¼(K 2 ¡ 4iK 1v)

S1 = ¡ iK 1(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2 ¡ iK 2K 1v)

+4v+O(K )

2¼(K 2 ¡ 4iK 1v)

² = ¡3iK 1v(1+v2) +O(K 2)

2¼(K 2 ¡ 3K 21v2+iK 2K 1v)

_² ¡ @iSi = ¡@²@¿

Zd3x

limK ! 0

(iK 1v² ¡ iK iSi ) = F 0

= F 0

@¹ T ¹ 0 = f 0

(Friess, Gubser, Michalogiorgakis, Pufu, hep-th/0607022, Gubser, Pufu, AY, ArXiv: 0706.0213, 0706.4307)

z0

z

0 F

_² +@iSi = ¡@²@¿

Z

d3x

limK ! 0

(iK 1v² ¡ iK iSi ) = F 0 =v2

2¼

= F 0

(Herzog, Karch, Kovtun, Kozcaz, Yaffe, hep-th: 0605158, Gubser, hep-th: 0605182)

Energy analysis(Friess, Gubser, Michalogiorgakis, Pufu, hep-th/0607022, Gubser, Pufu, AY, ArXiv: 0706.0213, 0706.4307)

+ limK ! 0

(iK 1v² ¡ iK iSi )¯¯sound

limK ! 0

(iK 1v² ¡ iK iSi ) = F 0limK ! 0

(iK 1v² ¡ iK iSi )¯¯wake

_² +@iSi = ¡@²@¿

Z

d3x

=v2

2¼

= F 0jwake+F 0jsound

F 0jwake : F 0jsound = ¡ 1 : 1+v2

= F 0

Energy analysis(Friess, Gubser, Michalogiorgakis, Pufu, hep-th/0607022, Gubser, Pufu, AY, ArXiv: 0706.0213, 0706.4307)

F 0jwake : F 0jsound = ¡ 1 : 1+v2

S1

Energy analysis(Friess, Gubser, Michalogiorgakis, Pufu, hep-th/0607022, Gubser, Pufu, AY, ArXiv: 0706.0213, 0706.4307)

Summary• AdS/CFT enables us to obtain the energy

momentum tensor of the plasma at all scales.

• A sonic boom and wake exist.

• The ratio of energy going into sound to energy going into the wake is 1+v2:-1.

The energy momentum tensor

D¹ º½¾h½¾= T ¹ º

hmn =

0

BBBB@

h00 h01 h02 h03 h04h10 h11 h12 h13 h14h20 h21 h22 h23 h24h30 h31 h32 h33 h34h40 h41 h42 h43 h44

1

CCCCA

hmn =

0

BBBB@

h00 h01 h02 h03 0h10 h11 h12 h13 0h20 h21 h22 h23 0h30 h31 h32 h33 00 0 0 0 0

1

CCCCA

hmn =

0

BBBB@

h00 h01 h02 0 0h10 h11 h12 0 0h20 h21 h22 0 00 0 0 h33 00 0 0 0 0

1

CCCCA

Gauge choiceCylindrical symmetry

µz3@zz¡ 3g(z)@z ¡

µk2 ¡

v2k21g(z)

¶¶©T = ze¡ ik1»(z)

©T =12v2

Ã

¡ h11+2µk1k?

¶2h22+

µkk?

¶2h33

!Tensor modesVector modes

µ©V1©V2

¶=

0

@12v

³h01 ¡ k1

k?h02

´

12v2

³¡ h11+

³k1k?¡ k?

k1

´h12+h22

´

1

A

(x1;t) ! x1 ¡ vtZh½¾eikx

d3k(2¼)3

The energy momentum tensor

µz3@zz¡ 3g(z)@z ¡

µk2 ¡

v2k21g

¶¶©T = ze¡ ik1»(z)

¡@2z +K V@z +VV

¢µ©V1©V2

¶= ~SV

zge¡ ik1»

©T =12v2

Ã

¡ h11+2µk1k?

¶2h22+

µkk?

¶2h33

!

VV =k2

g

á g gv2

¡ k1k

¢2

¡ 1 v2¡ k1k

¢2

!

K V =

á 3z 00 ¡ 3

z +g0

g

!~SV =

µ11

¶

Tensor modes

Vector modesµ©V1©V2

¶=

0

@12v

³h01 ¡ k1

k?h02

´

12v2

³¡ h11+

³k1k?¡ k?

k¡

´h12+h22

´

1

A

+ first order constraint

The energy momentum tensor

µz3@zz¡ 3g(z)@z ¡

µk2 ¡

v2k2¡g

¶¶©T = ze¡ ik¡ »(z)

¡@2z +K V@z +VV

¢µ©V1©V2

¶= ~SV

zge¡ ik¡ »¡

@2z +K S@z +VS¢

0

BB@

©S1©S2©S3©S4

1

CCA = ~SS

zge¡ ik¡ »

Tensor modes

Vector modes

+ first order constraint

Scalar modes

+ 3 first order constraints

D¹ º½¾h½¾= T ¹ º

Large momentum approximations

E =v2(1¡ v2)K 2

1 ¡ (2+v2)(K 2

1(1¡ v2) +K 2

? )

24pK 21(1¡ v2) +K

2?

+i¼vK 12v2(1¡ v2)K 2

1 +(5¡ 11v2)(K 2

1(1¡ v2) +K 2

? )18(K 2

1(1¡ v2) +K2? )

2

+O(K ¡ 3)

+O(X 0)

E(X ) =

¡ vX 1(5¡ 11v2)X 2

1 +(1¡ v2)(5¡ 8v2)X 2

?

72(1¡ v2)5=2³

X 21

1¡ v2 +X2?

´5=2

X 21 +(1+v

2)X 2?

12¼2p1¡ v2

³X 21

1¡ v2 +X2?

´3

Large momentum approximations

E =v2(1¡ v2)K 2

1 ¡ (2+v2)(K 2

1(1¡ v2) +K 2

? )

24pK 21(1¡ v2) +K

2?

+i¼vK 12v2(1¡ v2)K 2

1 +(5¡ 11v2)(K 2

1(1¡ v2) +K 2

? )18(K 2

1(1¡ v2) +K2? )

2

+O(K ¡ 3)

+O(X 0)

E(X ) =

¡ vX 1(5¡ 11v2)X 2

1 +(1¡ v2)(5¡ 8v2)X 2

?

72(1¡ v2)5=2³

X 21

1¡ v2 +X2?

´5=2

X 21 +(1+v

2)X 2?

12¼2p1¡ v2

³X 21

1¡ v2 +X2?

´3