entropy production from ads/cft
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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Penrose, unpublished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Eardley and Giddings, 2002Penrose, unpublished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
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