studying the strongly coupled n=4 plasma using ads/cft
DESCRIPTION
Studying the strongly coupled N=4 plasma using AdS/CFT. Amos Yarom, Munich. Together with S. Gubser and S. Pufu. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A. AdS/CFT. J. Maldacena. Calculating the stress-energy tensor. T . - PowerPoint PPT PresentationTRANSCRIPT
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