Entropy production from AdS/CFT

Amos Yarom

Together with: S. Gubser and S. Pufu

Overview

x1,x2

x3

x1

x2

S=?

Overview

S=?

E

?

Overview

S

E

AdS/CFT

J. Maldacena

A

E

Overview

S

E

A

E

?

AdS5 spaceds2 = L2=z2

¡¡ dt2+dx2i +dz

2¢

z

0

1

z=z0

x3

x?t

ds2 = L2=z20¡¡ dt2+dx2i

¢

z=z1

x3

x?t

ds2 = L2=z21¡¡ dt2+dx2i

¢

AdS5 spaceds2 = L2=z2

¡¡ dt2+dx2i +dz

2¢

z

0

1

tx3

x?

R¹ º ¡12Rg¹ º ¡

6L2g¹ º =0

Head on collisions in AdS

x3

x?

ds2 = L2=z2¡¡ dt2+dx2i +dz

2¢

0

1z

R¹ º ¡12Rg¹ º ¡

6L2g¹ º = T¹ º

ds2 = ds2AdS5 +Lz©(x? ;z)±(u)du2

t=x3

R¹ º ¡12g¹ ºR ¡

6Lg¹ º

= 8¼G5Ez3

L3±(t ¡ x3)±(z ¡ z¤)±(x? )±t¡ x3¹ ±t¡ x3º

z=z*

Energy ofthe particle

Location of the particle

u=t-x3

v=t+x3

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

x3

x?0

1z

ds2 = L2=z2¡¡ dt2+dx2i +dz

2¢ds2AdS5 = L2=z2

¡¡ dt2+dx2i +dz

2¢Head on collisions in AdS

z=z*

ds2 =ds2AdS5 +Lz©(x? ;z)±(v)dv2

Head on collisions in AdS

x3

x?0

1z

ds2 = ds2AdS5 +Lz©(x? ;z)±(u)du2

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

ds2AdS5 = L2=z2

¡¡ dt2+dx2i +dz

2¢

ds2 =ds2AdS5 +Lz©(x? ;z)

¡±(v)dv2+±(u)du2

¢ds2 =ds2AdS5 +

Lz©(x? ;z)±(v)dv2

Collisions

x3

x?0

1z

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

ds2AdS5 = L2=z2

¡¡ dt2+dx2i +dz

2¢

z=z*

ds2 =ds2AdS5 +Lz©(xi ;x? ;L)

¡±(v)dv2+±(u)du2

¢

Collisions

z=z*

x3

x?0

1z

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )x?

x3

t

z=z*

t=x t=-x

Collisions

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

ds2 =ds2AdS5 +Lz©(xi ;x? ;L)

¡±(v)dv2+±(u)du2

¢x?

x3

t

t=0

z=z*

Collisions

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

ds2 =ds2AdS5 +Lz©(xi ;x? ;L)

¡±(v)dv2+±(u)du2

¢x?

x3

t

t=0

?z=z*

An event horizonIn an asymptotically flat spacetime, an event horizon is the boundary of the region of all events which do not lie in the chronological past of future (null) infinity.

t

x

An event horizonIn an asymptotically flat spacetime, an event horizon is the boundary of the region of all events which do not lie in the chronological past of future (null) infinity.

t

x

Collisions

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

ds2 =ds2AdS5 +Lz©(xi ;x? ;L)

¡±(v)dv2+±(u)du2

¢x?

x3

t

t=0

?z=z*

Penrose, unpublished

Penrose’s trick

x?

x3

t µ= h¹ ºD¹ º̀ = 0

l

µ= h¹ ºD¹ º̀ · 0µ= h¹ ºD¹ º̀

Penrose’s trick

x?

x3

t

A1

A2

A2 ¸

µ= h¹ ºD¹ º̀ = 0

lA0

A1 ¸ A0

Computing the trapped surface

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

ds2 =ds2AdS5 +Lz©(xi ;x? ;L)

¡±(v)dv2+±(u)du2

¢

x?

x3

t

t=0

? v=0u=- (x?,z)

u=0v=- (x?,z)

lx?

Eardley and Giddings, 2002Penrose, unpublished

u=0

v=- (x?,z*)

Computing the trapped surface

x?

x3

t

t=0

? v=0u=- (x?,z)

u=0v=- (x?,z)

l

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

r 2ª = r 2©ª jb=0 (@ª )2 jb=4

Computing the trapped surface

r 2ª = r 2©ª jb=0 (@ª )2 jb=4

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

x2

x1

z

z=L

Computing the trapped surface

ª jb=0 (@ª )2 jb=4

¼µL3

G5

¶1=3(2E z¤)

2=3A0

4 G5

=¡1+O((E z¤)¡ 1)

¢

r 2ª = r 2©

r 2©= ¡ 16¼G5E±(z ¡ z¤)±(x? )

CFT observables

AdS/CFT

J. Maldacena

CFT AdS5

hT¹ º i G¹ º¯¯b

hTL¹ º i =2E z4¤

¼(z2¤ +x2? )3±(u)±

u¹ ±uº

hTR¹ º i =2E z4¤

¼(z2¤ +x2? )3±(v)±

v¹ ±vº

L R

?

CFT observables

AdS/CFT

J. Maldacena

AdS5 CFT

hT¹ º iG¹ º¯¯b

hT¹ º i =2EL4

¼(L2+x2? )3±(u)±

u¹ ±uº

hT¹ º i =2EL4

¼(L2+x2? )3±(v)±

v¹ ±vº

AdS5 CFT

Blackhole

Thermalstate

A04G5

=¼µL3

G5

¶1=3(2E z¤)

2=3

CFT observables

AdS/CFT

J. Maldacena

AdS5 CFT

S A/4G5

S ¸

S ¸ ¼µL3

G5

¶1=3(2E z¤)

2=3 sinh¡ 1 ¯

¯p1+¯2

¯ = b=2z¤

(Comparison with Lin and Shuryak, 2009)

S ¸ ¼µL3

G5

¶1=3(2E z¤)

2=3

AdS5 CFT

S A/4G5

=¼µL3

G5

¶1=3(2E z¤)

2=3

Comapring to QCD

S ¸A04G5

?

L3

G5» 1:9

QCD vrs. CFT

S ¸ ¼µL3

G5

¶1=3(2E z¤)

2=3

?

= ² =3¼3

16L3

G5T4

?

ZhTttid3x = E

hT¹ º i =2E z4¤

¼(z2¤ +x2? )3±(u)±

u¹ ±uº

QCD vrs. CFT

S ¸ ¼µL3

G5

¶1=3(2E z¤)

2=3

L3

G5» 1:9

ZhTttix2? d

3x = Ez2¤

² /1

1+ej~x¡ R j=a±(u)±u¹ ±

uº

ZhTtt ix2? d

3x = Ea

sLi5(e¡ R=a)L i3(e¡ R=a)

ZhTttid3x = E

?

= 19.7 TeV =

=E(4.3fm)2=

?

hT¹ º i =2E z4¤

¼(z2¤ +x2? )3±(u)±

u¹ ±uº

QCD vrs. CFT

ZhTttid3x = E

ZhTttix2? d

3x = Ez2¤

² /1

1+ej~x¡ R=a±(u)±u¹ ±

uº

ZhTtt ix2? d

3x = Ea

sLi5(e¡ R=a)L i3(e¡ R=a)

ZhTttid3x = E = 19.7 TeV =

=E(4.3fm)2=

S ¸ ¼µL3

G5

¶1=3(2E L)2=3

L3

G5» 1:9

?

QCD vrs. CFT

S ¸ ¼µL3

G5

¶1=3(2E L)2=3

L3

G5» 1:9

E=19.7 TeV

z*2=(4.3 fm)2

S ¸ 35000µ p

sN N200GeV

¶2=3

7.5 Ncharged » S

Ncharged ¸ 4700µ p

sN N200GeV

¶2=3?

(Pal and Pratt nucl-th/0308077)

A head-on collision(PHOBOS, 2003)

LHC£ 1.6

Ncharged ¸ 4700µ p

sN N200GeV

¶2=3

Slicing AdS space

z << L

z >> L

AdS/CFT

J. Maldacena

zUV < z < zIR » 1/(0.2 GeV)1/(2 GeV) »

Geometry of the trapped surface

x3

t

t=0

v=0u=- (x?,z)

u=0v=- (x?,z)

z=z*

Geometry of the trapped surface

Geometry of the trapped surface

A0 » E2/3

Geometry of the trapped surface

z << L

z >> L

1/L << E << UV

A0 » E2/3

E >> UV

A0 » E1/3

Head-on collisions(PHOBOS, 2003)

LHC£ 1.6

(Sliced AdS)

£ 0.8

Off center collisions

S ¸ ¼µL3

G5

¶1=3(2E z¤)

2=3 sinh¡ 1 ¯

¯p1+¯2

2z*

Ncharged ¸ 4700sinh¡ 1(¯(Np))

¯(Np)p1+¯(Np)2

Off center collisions

Ncharged ¸ 4700sinh¡ 1(¯(Np))

¯(Np)p1+¯(Np)2

Off center collisions

QCD: CFT:

S ¸ ¼µL3

G5

¶1=3(2E z¤)

2=3 sinh¡ 1 ¯

¯p1+¯2

E !(Npart)2£ 197

E

Off center collisions

E !(Npart)2£ 197

E

Summary

LHC£ 1.6£ 0.8

Summary

Summary

E !(Npart)2£ 197

E

Thank you

Summary

LHC£ 1.6£ 0.8

Thank you