Gravitational collapse, holography, and hydrodynamics in ... · Paul Chesler 1 Gravitational...
Transcript of Gravitational collapse, holography, and hydrodynamics in ... · Paul Chesler 1 Gravitational...
Paul Chesler
1
Gravitational collapse, holography, and
hydrodynamics in small systems
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
Assumptions:
• Local equilibrium.
• Local isotropy.
Constitutive relations (in local fluid rest frame):
T 00
hydro
= ✏,
T 0ihydro
= 0,
T ijhydro
= p �ij � ⌘⇥riuj +rjui � 2
3
�ijr · u⇤+ . . . .
Dynamics: @µTµ⌫hydro
= 0.}Dissipative stress ⇧µ⌫
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
Assumptions:
• Local equilibrium.
• Local isotropy.
Constitutive relations (in local fluid rest frame):
T 00
hydro
= ✏,
T 0ihydro
= 0,
T ijhydro
= p �ij � ⌘⇥riuj +rjui � 2
3
�ijr · u⇤+ . . . .
Dynamics: @µTµ⌫hydro
= 0.}Dissipative stress ⇧µ⌫
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
Assumptions:
• Local equilibrium.
• Local isotropy.
Constitutive relations (in local fluid rest frame):
T 00
hydro
= ✏,
T 0ihydro
= 0,
T ijhydro
= p �ij � ⌘⇥riuj +rjui � 2
3
�ijr · u⇤+ . . . .
Dynamics: @µTµ⌫hydro
= 0.}Dissipative stress ⇧µ⌫
2
• Definition: Long wavelength, low frequency e↵ective
description of transport of conserved currents.
• Relativistic hydrodynamics:
Hydrodynamics
Macroscopically large system
Assumptions:
• Local equilibrium.
• Local isotropy.
Two remarkable observations from LHC and RHIC
. 1 fm/c
timeliquid
nuclei QGP
?
1. Rapid equilibration in AA collisions:
2. A tiny drop of QGP in pA collisions?
[CMS: 1210.5482], [ALICE: 1212.2001],[ATLAS: 1212.5198], [PHENIX: 1303.1794],[Bzdak, Schenke, Tribedy, Venugopalan: 1304.3403]
Evolution consistent with hydrodynamics.
3
time } R ⇠ 1 fm
R
– Long lived: ⌧m.f.t. � 1/✏
quasiparticle
– Weakly interacting: �m.f.p. � �
de Broglie
– thydro
� ⌧m.f.t.,
– R � �m.f.p.,
4
• Microscopic description: Nearly all practical formulationsof real-time QFT rely on existence of quasiparticles.
• Kinetic theory: Hydrodynamics only applies overtime and length scales
What is the domain of utility of hydrodynamics?
5
Hydrodynamics in extreme conditions
• ✏quasiparticle
⇠ T , �de Broglie
⇠ 1
T .
• Experimentally accessible temperatures:
1
T ⇠ 1 fm.
) Relaxation time: thydro
⇠ 1/✏quasiparticle
.
) pA system size: R ⇠ �de Broglie
.
microscopic
scales
}
• Since �m.f.p. ⇠ 1g4T , natural to expect domain of hydro
to be maximal at strong coupling.
Interesting questions:
1. Is hydro theoretically consistent in such small systems?
2. How big are the smallest drops of liquid?
6
Strongly coupled dynamics and holographic duality
• Equivalence between certain QFTs and theories of of gravityin one higher dimension.
• Weak/strong equivalence:
– Strongly coupled QFT = classical gravity.
– All QFT dynamics — from far-from-eq dynamics to
hydrodynamics — encoded in numerical relativity problem.
• Holography = spherical cow.
– Extra symmetries.
A diverse set of holographic field theories
• Conformal theories
– N = 4 SYM
• Confining theories
– AdS soliton
• Confinement & chiral symmetry breaking
– Sakai-Sugimoto model
• Superconducting theories
– Abelian Higgs model
Very di↵erent ground states.
Universal character of many body systems
• Dual classical equations of motion
RMN � 1
2GMN (R� 2⇤) = 8⇡G
Newton
TMNmatter
.
• TMNmatter
is theory-dependent.
• States with O(N2
c
) entropy dual to black holes.
Universal character of many body systems
• Dual classical equations of motion
RMN � 1
2GMN (R� 2⇤) = 8⇡G
Newton
TMNmatter
.
• TMNmatter
is theory-dependent.
• States with O(N2
c
) entropy dual to black holes.
Universal features of black holes
()universal features of strongly coupled many body systems.
• Fluid/gravity duality: [Blattacharya et al: 0712.2456], [Baier et al: 0712.2451]
– IR behavior of horizons , hydro in QFT:
• Shear viscosity
⌘s = 1
4⇡ quantum & stringycorrections:
– Same in every holographic QFT.[Kovtun, Son & Starinets: hep-th/0405231]
• Non-hydrodynamic relaxation rates:
– Insensitive to choice of QFT.[Fuini & Ya↵e: 1503.07148][Buchel, Heller & Myers: 1503.07114][Janik et al: 1503.07149][Buchel & Day: 1505.05012]
A few universal features of strongly coupled dynamics
• Fluid/gravity duality: [Blattacharya et al: 0712.2456], [Baier et al: 0712.2451]
– IR behavior of horizons , hydro in QFT:
• Shear viscosity
⌘s = 1
4⇡ quantum & stringycorrections:
– Same in every holographic QFT.[Kovtun, Son & Starinets: hep-th/0405231]
• Non-hydrodynamic relaxation rates:
– Insensitive to choice of QFT.[Fuini & Ya↵e: 1503.07148][Buchel, Heller & Myers: 1503.07114][Janik et al: 1503.07149][Buchel & Day: 1505.05012]
A few universal features of strongly coupled dynamics
• Fluid/gravity duality: [Blattacharya et al: 0712.2456], [Baier et al: 0712.2451]
– IR behavior of horizons , hydro in QFT:
• Shear viscosity
⌘s = 1
4⇡ quantum & stringycorrections:
– Same in every holographic QFT.[Kovtun, Son & Starinets: hep-th/0405231]
• Non-hydrodynamic relaxation rates:
– Insensitive to choice of QFT.[Fuini & Ya↵e: 1503.07148][Buchel, Heller & Myers: 1503.07114][Janik et al: 1503.07149][Buchel & Day: 1505.05012]
A few universal features of strongly coupled dynamics
z
T 00
The simplest holographic models of collisions
• Simplest theory to study: N = 4 SYM.
• Gravitational equations of motion:
RMN � 1
2
GMN(R� 2⇤) = 0.
• Collide gravitational shockwaves and make black hole.
• Dual SYM energy density: T 00(t,x?, z) = F (x?, z ⌥ t)
SYM energy
10
Holographic description of QFT (I)Connecting 5d physics to 4d physics
State |�⇥ in QFT � classical field configuration {�i} in gravitational theory.
10
Holographic description of QFT (I)Connecting 5d physics to 4d physics
State |�⇧ in QFT ⇥ classical field configuration {�i} in gravitational theory.
Vacuum
4d QFT
• ⌅0|Tµ� |0⇧ = 0, ⌅0|Jµ|0⇧ = 0, . . .
5d Gravitational description
• Geometry: AdS5 ds2 = r2[�dt2 + dx2] + dr2
r2 .
• Properties:
i. Each slice of constant r is copy of Minkowski space.ii. 4d boundary at r = ⇤.iii. Throat
Deviations in geometry from AdS5 ⇥ excited states in QFT
4d Minkowski Space
19
The ground state and AdS5
QFT
• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,
Gravitational description
• Einstein: RMN � 12gMN (R + 2�) = 0.
• Most symmetric solution: AdS5,
ds2 =L2
u2
��dt2 + dx2 + du2
⇥. (2)
19
The ground state and AdS5
QFT
• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,
Gravitational description
• Einstein: RMN � 12GMN (R + 2�) = 0.
• Most symmetric solution: AdS5,
ds2 =L2
u2
��dt2 + dx2 + du2
⇥. (2)r
ds2 = r2[�dt2 + dx2] +dr2
r2
11
10
Holographic description of QFT (I)Connecting 5d physics to 4d physics
State |�⇥ in QFT � classical field configuration {�i} in gravitational theory.
10
Holographic description of QFT (I)Connecting 5d physics to 4d physics
State |�⇧ in QFT ⇥ classical field configuration {�i} in gravitational theory.
Vacuum
4d QFT
• ⌅0|Tµ� |0⇧ = 0, ⌅0|Jµ|0⇧ = 0, . . .
5d Gravitational description
• Geometry: AdS5 ds2 = r2[�dt2 + dx2] + dr2
r2 .
• Properties:
i. Each slice of constant r is copy of Minkowski space.ii. 4d boundary at r = ⇤.iii. Throat
Deviations in geometry from AdS5 ⇥ excited states in QFT
10
Holographic description of QFT (I)Connecting 5d physics to 4d physics
State |�⇧ in QFT ⇥ classical field configuration {�i} in gravitational theory.
Vacuum
4d QFT
• ⌅0|Tµ� |0⇧ = 0, ⌅0|Jµ|0⇧ = 0, . . .
5d Gravitational description
• Geometry: AdS5: ds2 = r2[�dt2 + dx2] + dr2
r2 .
• Properties:
i. Each slice of constant r is copy of Minkowski space.ii. 4d boundary at r = ⇤.iii. Throat
Deviations in geometry from AdS5 ⇥ excited states in QFT
4d Minkowski Space
19
The ground state and AdS5
QFT
• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,
Gravitational description
• Einstein: RMN � 12gMN (R + 2�) = 0.
• Most symmetric solution: AdS5,
ds2 =L2
u2
��dt2 + dx2 + du2
⇥. (2)
19
The ground state and AdS5
QFT
• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,
Gravitational description
• Einstein: RMN � 12GMN (R + 2�) = 0.
• Most symmetric solution: AdS5,
ds2 =L2
u2
��dt2 + dx2 + du2
⇥. (2)r
ds2 = r2[�dt2 + dx2] +dr2
r2
11
}
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
Analogous classical EM problem: image charges
conductor
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
Analogous classical EM problem: image charges
conductor
Jµinduced
Image problem from electrodynamics
Excited states and expectation values
“Image” Tµ⌫
12
}
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
Analogous classical EM problem: image charges
conductor
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
Analogous classical EM problem: image charges
conductor
Jµinduced
Image problem from electrodynamics
Excited states and expectation values
“Image” Tµ⌫
12
}
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
Analogous classical EM problem: image charges
conductor
7
Connecting 5d dynamics to 4d dynamics
Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)
Object deforms 5d geometry.
Analogous classical EM problem: image charges
conductor
Jµinduced
Image problem from electrodynamics
Excited states and expectation values
“Image” Tµ⌫
[de Haro, Solodukhin & Skenderis: hep-th/0002230]
Reinterpret Tµ⌫ ! hTµ⌫i
12
5
What should the geometry correspondingto a liquid look like?
Local fluid rest frame:
• T
µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.
• Local temperature T (x) ⇠ ✏(x)
1/4.
• Local entropy s(x) ⇠ T (x)
3.
• Dynamics: @µTµ⌫
= 0
5
What should the geometry correspondingto a liquid look like?
Local fluid rest frame:
• T
µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.
• Local temperature T (x) ⇠ ✏(x)
1/4.
• Local entropy s(x) ⇠ T (x)
3.
• Dynamics: @µTµ⌫
= 0
5
What should the geometry correspondingto a liquid look like?
Local fluid rest frame:
• T
µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.
• Local temperature T (x) ⇠ ✏(x)
1/4.
• Local entropy s(x) ⇠ T (x)
3.
• Dynamics: @µTµ⌫
= 0
5
What should the geometry correspondingto a liquid look like?
Local fluid rest frame:
• T
µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.
• Local temperature T (x) ⇠ ✏(x)
1/4.
• Local entropy s(x) ⇠ T (x)
3.
• Dynamics: @µTµ⌫
= 0
Gravitational description:
• Dynamics: R
MN
� 12GMN
(R+ 2⇤).
• Local geometry: AdS-Schwarzschild black brane
ds
2= r
2[�f(x, r)dt
2+dx
2]+
dr
2
r
2f(x, r)
+ gradients, f = 1�⇣
rh(x)r
⌘4.
[Bhattacharyya, Hubeny,Minwalla, Rangamani: 0712.2456]
5
What should the geometry correspondingto a liquid look like?
Local fluid rest frame:
• T
µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.
• Local temperature T (x) ⇠ ✏(x)
1/4.
• Local entropy s(x) ⇠ T (x)
3.
• Dynamics: @µTµ⌫
= 0
Gravitational description:
• Dynamics: R
MN
� 12GMN
(R+ 2⇤).
• Local geometry: AdS-Schwarzschild black brane
ds
2= r
2[�f(x, r)dt
2+dx
2]+
dr
2
r
2f(x, r)
+ gradients, f = 1�⇣
rh(x)r
⌘4.
[Bhattacharyya, Hubeny,Minwalla, Rangamani: 0712.2456]
timespace
Black hole
“Holographic image”energy density
Colliding particles
Gravitational models of heavy ion collisions
14
timespace
Black hole
“Holographic image”energy density
Colliding particles
liquidtime
Gravitational models of heavy ion collisions
14
timespace
Black hole
“Holographic image”energy density
Colliding particles
liquidtime
Gravitational models of heavy ion collisions
5D gravity: a challenging numerical problem
14
Numerical relativity in a box
6
Gau
ge/g
ravi
tydual
ity
Mal
dac
ena
conje
cture
(Mal
dac
ena:
hep
-th/9
7112
00)
•C
onje
ctur
edeq
uiva
lenc
ebe
twee
nst
ring
sin
asym
ptot
ical
lyA
dS5�
S5an
d4d
N=
4SY
M.
•D
omai
nof
utili
ty:
Larg
eN
c,�
SYM⇥⇤
clas
sica
lapp
roxi
mat
ions
.
Bas
icid
ea:
•4d
field
theo
ryliv
eson
the
boun
dary
ofhi
gher
dim
ensi
onal
curv
edsp
ace.
•O
bjec
tsex
isti
ngin
high
erdi
men
sion
s:st
ring
s,bl
ack
hole
s,E
Mfie
lds
...
•O
bjec
tsex
isti
ngin
QFT
:qua
rks,
plas
mas
,con
serv
edcu
rren
ts..
.
5th
dim
ensi
on
bla
ckbra
ne
Dyn
amic
sin
5den
codes
dyn
amic
sin
fiel
dth
eory
.
{ �GMN
Asymptotically AdS:
• ds
2 ! r
2⌘µ⌫dx
µdx
⌫+
dr2
r2 .
• Time-like boundary at r = 1.
) must impose BCs at r = 1.
• Saving grace: infinite redshift.
15
�GMN
Numerical relativity in a box
6
Gau
ge/g
ravi
tydual
ity
Mal
dac
ena
conje
cture
(Mal
dac
ena:
hep
-th/9
7112
00)
•C
onje
ctur
edeq
uiva
lenc
ebe
twee
nst
ring
sin
asym
ptot
ical
lyA
dS5�
S5an
d4d
N=
4SY
M.
•D
omai
nof
utili
ty:
Larg
eN
c,�
SYM⇥⇤
clas
sica
lapp
roxi
mat
ions
.
Bas
icid
ea:
•4d
field
theo
ryliv
eson
the
boun
dary
ofhi
gher
dim
ensi
onal
curv
edsp
ace.
•O
bjec
tsex
isti
ngin
high
erdi
men
sion
s:st
ring
s,bl
ack
hole
s,E
Mfie
lds
...
•O
bjec
tsex
isti
ngin
QFT
:qua
rks,
plas
mas
,con
serv
edcu
rren
ts..
.
5th
dim
ensi
on
bla
ckbra
ne
Dyn
amic
sin
5den
codes
dyn
amic
sin
fiel
dth
eory
.
{ �GMN
Asymptotically AdS:
• ds
2 ! r
2⌘µ⌫dx
µdx
⌫+
dr2
r2 .
• Time-like boundary at r = 1.
) must impose BCs at r = 1.
• Saving grace: infinite redshift.
15
�GMN
Numerical relativity in a box
6
Gau
ge/g
ravi
tydual
ity
Mal
dac
ena
conje
cture
(Mal
dac
ena:
hep
-th/9
7112
00)
•C
onje
ctur
edeq
uiva
lenc
ebe
twee
nst
ring
sin
asym
ptot
ical
lyA
dS5�
S5an
d4d
N=
4SY
M.
•D
omai
nof
utili
ty:
Larg
eN
c,�
SYM⇥⇤
clas
sica
lapp
roxi
mat
ions
.
Bas
icid
ea:
•4d
field
theo
ryliv
eson
the
boun
dary
ofhi
gher
dim
ensi
onal
curv
edsp
ace.
•O
bjec
tsex
isti
ngin
high
erdi
men
sion
s:st
ring
s,bl
ack
hole
s,E
Mfie
lds
...
•O
bjec
tsex
isti
ngin
QFT
:qua
rks,
plas
mas
,con
serv
edcu
rren
ts..
.
5th
dim
ensi
on
bla
ckbra
ne
Dyn
amic
sin
5den
codes
dyn
amic
sin
fiel
dth
eory
.
{ �GMN
Asymptotically AdS:
• ds
2 ! r
2⌘µ⌫dx
µdx
⌫+
dr2
r2 .
• Time-like boundary at r = 1.
) must impose BCs at r = 1.
• Saving grace: infinite redshift.
15
�GMN
Numerical relativity in a box
6
Gau
ge/g
ravi
tydual
ity
Mal
dac
ena
conje
cture
(Mal
dac
ena:
hep
-th/9
7112
00)
•C
onje
ctur
edeq
uiva
lenc
ebe
twee
nst
ring
sin
asym
ptot
ical
lyA
dS5�
S5an
d4d
N=
4SY
M.
•D
omai
nof
utili
ty:
Larg
eN
c,�
SYM⇥⇤
clas
sica
lapp
roxi
mat
ions
.
Bas
icid
ea:
•4d
field
theo
ryliv
eson
the
boun
dary
ofhi
gher
dim
ensi
onal
curv
edsp
ace.
•O
bjec
tsex
isti
ngin
high
erdi
men
sion
s:st
ring
s,bl
ack
hole
s,E
Mfie
lds
...
•O
bjec
tsex
isti
ngin
QFT
:qua
rks,
plas
mas
,con
serv
edcu
rren
ts..
.
5th
dim
ensi
on
bla
ckbra
ne
Dyn
amic
sin
5den
codes
dyn
amic
sin
fiel
dth
eory
.
{ �GMN
Asymptotically AdS:
• ds
2 ! r
2⌘µ⌫dx
µdx
⌫+
dr2
r2 .
• Time-like boundary at r = 1.
) must impose BCs at r = 1.
• Saving grace: infinite redshift.
15
�GMN
Numerical relativity in a box
6
Gau
ge/g
ravi
tydual
ity
Mal
dac
ena
conje
cture
(Mal
dac
ena:
hep
-th/9
7112
00)
•C
onje
ctur
edeq
uiva
lenc
ebe
twee
nst
ring
sin
asym
ptot
ical
lyA
dS5�
S5an
d4d
N=
4SY
M.
•D
omai
nof
utili
ty:
Larg
eN
c,�
SYM⇥⇤
clas
sica
lapp
roxi
mat
ions
.
Bas
icid
ea:
•4d
field
theo
ryliv
eson
the
boun
dary
ofhi
gher
dim
ensi
onal
curv
edsp
ace.
•O
bjec
tsex
isti
ngin
high
erdi
men
sion
s:st
ring
s,bl
ack
hole
s,E
Mfie
lds
...
•O
bjec
tsex
isti
ngin
QFT
:qua
rks,
plas
mas
,con
serv
edcu
rren
ts..
.
5th
dim
ensi
on
bla
ckbra
ne
Dyn
amic
sin
5den
codes
dyn
amic
sin
fiel
dth
eory
.
{ �GMN
Asymptotically AdS:
• ds
2 ! r
2⌘µ⌫dx
µdx
⌫+
dr2
r2 .
• Time-like boundary at r = 1.
) must impose BCs at r = 1.
• Saving grace: infinite redshift.
15
�GMN
Numerical relativity in a box
6
Gau
ge/g
ravi
tydual
ity
Mal
dac
ena
conje
cture
(Mal
dac
ena:
hep
-th/9
7112
00)
•C
onje
ctur
edeq
uiva
lenc
ebe
twee
nst
ring
sin
asym
ptot
ical
lyA
dS5�
S5an
d4d
N=
4SY
M.
•D
omai
nof
utili
ty:
Larg
eN
c,�
SYM⇥⇤
clas
sica
lapp
roxi
mat
ions
.
Bas
icid
ea:
•4d
field
theo
ryliv
eson
the
boun
dary
ofhi
gher
dim
ensi
onal
curv
edsp
ace.
•O
bjec
tsex
isti
ngin
high
erdi
men
sion
s:st
ring
s,bl
ack
hole
s,E
Mfie
lds
...
•O
bjec
tsex
isti
ngin
QFT
:qua
rks,
plas
mas
,con
serv
edcu
rren
ts..
.
5th
dim
ensi
on
bla
ckbra
ne
Dyn
amic
sin
5den
codes
dyn
amic
sin
fiel
dth
eory
.
{ �GMN
Challenges:
• Must impose BCs at r = 1.
– Imposing BCs at r = finite
yields unstable evolution.
• Einstein singular at r = 1.
Asymptotically AdS:
• ds
2 ! r
2⌘µ⌫dx
µdx
⌫+
dr2
r2 .
• Time-like boundary at r = 1.
) must impose BCs at r = 1.
• Saving grace: infinite redshift.
15
Characteristic formulation of PDEs
Example: 1 + 1D wave equation
• Equation of motion:
gµ⌫@µ@⌫� = (�@2⌧ + @2
r )� = f(�).
• Null coordinate: t = ⌧ � r
• Equation of motion:
(2@r@t + @2r )� = f(�).
• Integral form:
@t� =
1
2
Z r
0dr0[f(�)� @2
r�] + C(v).
16
Characteristic formulation of PDEs
Example: 1 + 1D wave equation
• Equation of motion:
gµ⌫@µ@⌫� = (�@2⌧ + @2
r )� = f(�).
• Null coordinate: t = ⌧ � r
• Equation of motion:
(2@r@t + @2r )� = f(�).
• Integral form:
@t� =
1
2
Z r
0dr0[f(�)� @2
r�] + C(v).
Benefits:
1. Easier implementation
of BCs
2. Easier to deal with
singularities.
16
Einstein’s equations in characteristic form
black brane
radialdirection
time-like boundary
17
Metric ansatz:
ds
2= �Adt
2+ 2Fidx
idt+ 2drdt+ ⌃
2gijdx
idx
j, det gij = 1.
Infalling null geodesics: t = const., x = const.
Schematic form of Einstein’s equations:
�@
2r +Q⌃[g]
�⌃ = 0,
�@
2r + PF [g,⌃]@r +QF [g,⌃]
�F = SF [g,⌃],
(@r +Q⌃[g,⌃])˙
⌃ = S⌃[g,⌃, F ],
⇣@r +Q ˙g[g,⌃]
⌘˙
g = S ˙g[g,⌃, F,˙
⌃],
@
2rA = SA[g,⌃, F,
˙
⌃,
˙
g],
with
˙
h ⌘ @th+
12A@rh.
Einstein’s equations in characteristic form
black brane
radialdirection
time-like boundary
Never solve more than linear ODE!17
Metric ansatz:
ds
2= �Adt
2+ 2Fidx
idt+ 2drdt+ ⌃
2gijdx
idx
j, det gij = 1.
Infalling null geodesics: t = const., x = const.
Schematic form of Einstein’s equations:
�@
2r +Q⌃[g]
�⌃ = 0,
�@
2r + PF [g,⌃]@r +QF [g,⌃]
�F = SF [g,⌃],
(@r +Q⌃[g,⌃])˙
⌃ = S⌃[g,⌃, F ],
⇣@r +Q ˙g[g,⌃]
⌘˙
g = S ˙g[g,⌃, F,˙
⌃],
@
2rA = SA[g,⌃, F,
˙
⌃,
˙
g],
with
˙
h ⌘ @th+
12A@rh.
A few (of many) technical details
18
• Horizon excision & residual di↵eomorphism invariance.
• Field redefinitions to ameliorate boundary singularities.
• Discretize using pseudo-spectral methods.
• Employ domain decomposition in radial direction.
• Filtering.
• Parallelization.
• Choice of units: max(initial energy density) = 1.
• Total runtime: 14 days on 6-core desktop computer.
Results animated
Energy density T 00
“proton” “nucleus”
[PC: 1506.02209]
Results animated
Energy density T 00
“proton” “nucleus”
[PC: 1506.02209]
z
x
t = �1.125 t = 0 t = 1.125 t = 2.25
Energy
Momentum
x?
Results illustrated
z
x?
A tiny drop of liquid
Energy at t = 1.5
Temperature
x?
z
20%
15%
Stress at x? = z = 0
How small of a droplet?
• E↵ective temperature
T�1
e↵
⌘ @seq
@✏eq
��✏eq=✏
.
• Result: RTe↵
⇡ 1.
Rapid equilibration?
• Result: thydro
Te↵
⇡ 0.3.
Rel. Mag. of 1
st
order corrections
Rel. Mag. of 2
nd
order corrections
z
x?
Stress at x? = z = 0
Hydrodynamics in extreme conditions (II)
Rel. Mag. of 1
st
order corrections
Rel. Mag. of 2
nd
order corrections
z
x?
Stress at x? = z = 0
} thydro
� ⌧m.f.t. �
1
T,
R � �m.f.p. �
1
T,
Common lore:
Lesson:
Hydrodynamics in extreme conditions (II)
Rel. Mag. of 1
st
order corrections
Rel. Mag. of 2
nd
order corrections
z
x?
Stress at x? = z = 0
} thydro
� ⌧m.f.t. �
1
T,
R � �m.f.p. �
1
T,
Common lore:
Lesson:
Hydrodynamics in extreme conditions (II)
Rel. Mag. of 1
st
order corrections
Rel. Mag. of 2
nd
order corrections
z
x?
Stress at x? = z = 0
} thydro
� ⌧m.f.t. �
1
T,
R � �m.f.p. �
1
T,
Common lore:
Lesson:
No theoretical inconstancy with hydro in pA.
Hydrodynamics in extreme conditions (II)
Universality and critical gravitational collapse
�
Critical gravitational collapse: [Choptuik: 1993]
• Vary dimensionless parameter p ⌘ �Eprobe
.
• Must exists critical p = pc which for p < pcno black hole forms.
) Hydrodynamic evolution for p > pc.
) No hydrodynamic evolution for p < pc.
Universality and critical gravitational collapse
�
Critical gravitational collapse: [Choptuik: 1993]
• Vary dimensionless parameter p ⌘ �Eprobe
.
• Must exists critical p = pc which for p < pcno black hole forms.
) Hydrodynamic evolution for p > pc.
) No hydrodynamic evolution for p < pc.
Universal gravitational dynamics as p ! pc (w.r.t. initial conditions)
• Self-similar geometry, scalings
– Entropy production: �S ⇠ (p� pc)3� , � ⇠ 0.4.
• What are universal dynamics in dual QFT?
• Interesting to look at low energy dynamics where
signs of hydrodynamic behavior turn o↵.
Universality and critical gravitational collapse
�
Critical gravitational collapse: [Choptuik: 1993]
• Vary dimensionless parameter p ⌘ �Eprobe
.
• Must exists critical p = pc which for p < pcno black hole forms.
) Hydrodynamic evolution for p > pc.
) No hydrodynamic evolution for p < pc.
Universal gravitational dynamics as p ! pc (w.r.t. initial conditions)
• Self-similar geometry, scalings
– Entropy production: �S ⇠ (p� pc)3� , � ⇠ 0.4.
• What are universal dynamics in dual QFT?
• Interesting to look at low energy dynamics where
signs of hydrodynamic behavior turn o↵.
theory-dependent
Thank you
t
T 00
z
T 00
t z
Planar shock collisions
Key observations:
1. Sensitivity to shock profilenear lightcone.
[Casalderrrey-Solana et al: 1305.4919]
2. Hydrodynamic flow insidethe lightcone.
[PC & Yaffe: 1011.3562]
3. Hydrodynamic flow isinsensitive to shock profile.
[PC, van der Schee & Kilbertus]
thick width w
narrow width w
Universal hydrodynamic flow
ξ/ξFWHM
-1 -0.5 0 0.5 1
ϵ/ϵ(ξ=
0)
0
0.2
0.4
0.6
0.8
1
w = wo
to 6wo
Key points:
• Initial hydro data well described by
boost invariant uµand
✏(⇠, w)|⌧=const. = A(w)f(⇠
⇠FWHM(w)).
• Insensitive to functional form of shockwave.
properenergy✏
fluidvelocityu⌧
[PC,van der Schee, Kilbertus, to appear shortly]
ξ-1 0 1
ϵ/µ4
0
0.05
0.1
0.15
0.2
0.25
ξ-1 0 1
uτ
0.999
1
1.001
1.002
1.003
1.004w = 7w
o
w = wo
w = 7wo
w = wo
ξ-1 0 1
ϵ/µ4
0
0.05
0.1
0.15
0.2
0.25
ξ-1 0 1
uτ
0.999
1
1.001
1.002
1.003
1.004w = 7w
o
w = wo
w = 7wo
w = wo
rapidity ⇠ rapidity ⇠
Hydro variables at fixed proper time ⌧ = const.
Decreasing probe width
probe
target
How to make a small droplet of liquid
Viscous flow at LHC C. Gale, S. Jeon, B.Schenke, P.Tribedy, R.Venugopalan, PRL110, 012302 (2013)
10-5
10-4
10-3
0 500 1000 1500 2000 2500
P(d
Ng/d
y)
dNg/dy
Glasma centrality selection
0-5
%
5-1
0%
10-2
0%
20-3
0%
30-4
0%
40-5
0%
50-6
0%
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12
P(b
)
b [fm]
Distribution of b in 20-30% central bin
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
ATLAS 20-30%, EP
η/s =0.2
v2 v3 v4 v5
Hydro evolution
MUSIC
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50
⟨vn
2⟩1
/2
centrality percentile
η/s = 0.2ALICE data vn{2}, pT>0.2 GeV v2
v3 v4 v5
Experimental data:ATLAS collaboration, Phys. Rev. C 86, 014907 (2012)ALICE collaboration, Phys. Rev. Lett. 107, 032301 (2011)
Björn Schenke (BNL) BNL, March 2013 32/45
Viscous flow at RHIC and LHC C. Gale, S. Jeon, B.Schenke,P.Tribedy, R.Venugopalan, PRL110, 012302 (2013)
RHIC �/s = 0.12
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
RHIC 200GeV, 30-40%
open: PHENIX
filled: STAR prelim.
v2 v3 v4 v5
LHC �/s = 0.2
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
ATLAS 30-40%, EP v2 v3 v4 v5
Experimental data:A. Adare et al. (PHENIX Collaboration), Phys.Rev.Lett. 107, 252301 (2011)Y. Pandit (STAR Collaboration), Quark Matter 2012, (2012)ATLAS collaboration, Phys. Rev. C 86, 014907 (2012)
Lower effective ⌘/s at RHIC than at LHC needed to describe dataHints at increasing ⌘/s with increasing temperatureAnalysis at more energies can be used to gain information on (⌘/s)(T )
Björn Schenke (BNL) RHIC AGS Users’ Meeting 2013, BNL 19/28
Viscous flow at RHIC and LHC C. Gale, S. Jeon, B.Schenke,P.Tribedy, R.Venugopalan, PRL110, 012302 (2013)
RHIC �/s = 0.12
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
RHIC 200GeV, 30-40%
open: PHENIX
filled: STAR prelim.
v2 v3 v4 v5
LHC �/s = 0.2
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
ATLAS 30-40%, EP v2 v3 v4 v5
Experimental data:A. Adare et al. (PHENIX Collaboration), Phys.Rev.Lett. 107, 252301 (2011)Y. Pandit (STAR Collaboration), Quark Matter 2012, (2012)ATLAS collaboration, Phys. Rev. C 86, 014907 (2012)
Lower effective ⌘/s at RHIC than at LHC needed to describe dataHints at increasing ⌘/s with increasing temperatureAnalysis at more energies can be used to gain information on (⌘/s)(T )
Björn Schenke (BNL) RHIC AGS Users’ Meeting 2013, BNL 19/28
Viscous flow at RHIC and LHC C. Gale, S. Jeon, B.Schenke,P.Tribedy, R.Venugopalan, PRL110, 012302 (2013)
RHIC �/s = 0.12
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
RHIC 200GeV, 30-40%
open: PHENIX
filled: STAR prelim.
v2 v3 v4 v5
LHC �/s = 0.2
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
⟨vn2⟩1
/2
pT [GeV]
ATLAS 30-40%, EP v2 v3 v4 v5
Experimental data:A. Adare et al. (PHENIX Collaboration), Phys.Rev.Lett. 107, 252301 (2011)Y. Pandit (STAR Collaboration), Quark Matter 2012, (2012)ATLAS collaboration, Phys. Rev. C 86, 014907 (2012)
Lower effective ⌘/s at RHIC than at LHC needed to describe dataHints at increasing ⌘/s with increasing temperatureAnalysis at more energies can be used to gain information on (⌘/s)(T )
Björn Schenke (BNL) RHIC AGS Users’ Meeting 2013, BNL 19/28
η/s = 0.12η/s = 0.2
Figure 1: Model calculations compared to measurements of the harmonic decomposition of azimuthalcorrelations produced in heavy-ion collisions. The left panel shows model calculations and data for v
n
vs.collision centrality in Pb+Pb collisions at
psNN = 2.76 TeV. The right panel shows similar studies for
the p
T
dependence of v
n
in 200 GeV Au+Au collisions. The comparison of the two energies providesinsight on the temperature dependence of ⌘/s.
determination of these two unknowns is aided by measurements of multiple flow observables sensitiveto medium properties in di↵erent stages of the evolution [15, 35, 36]. Due to the large event-by-eventfluctuations in the initial state collision geometry, in each collision the created matter follows a di↵erentcollective expansion with its own set of flow harmonics (magnitude v
n
and phases �n
). Experimentalobservables describing harmonic flow can be generally given by the joint probability distribution of themagnitudes v
n
and phases �n
of flow harmonics:
p(vn
, v
m
, ..., �n
, �m
, ...) =1
Nevts
dNevts
dv
n
dv
m
. . . d�n
d�m
. (1)
Specific examples include the probability distribution of individual harmonics p(vn
), flow de-correlationin transverse and longitudinal directions, and correlations of amplitudes or phases between di↵erentharmonics (p(v
n
, v
m
) or p(�n
, �m
)). The latter are best accessed through measurements of correlationswith three or more particles. The joint probability distribution (1) can be fully characterized experimentallyby measuring the complete set of moments recently identified in Ref. [37]. With the added detail providedby these measurements, hydrodynamic models can be fine-tuned and over-constrained, thus refiningour understanding of the space-time picture and medium properties of the heavy-ion collisions. Initialmeasurements of some of these observables [38–40] and comparison to hydrodynamic models [25, 41–43]already provided unprecedented insights on the nature of the initial density fluctuations and dynamicsof the collective evolution. However, at this point none of the state-of-the-art hydrodynamic modelsproperly accounts for the dynamical fluctuations generated during the evolution by thermal noise [44–47]– future quantitative work will need to address these, too.
Precision determination of key QGP parameters
The agreement between the model and the data shown in Figure 1 suggests that the essential featuresof the dynamic evolution of heavy-ion collisions are well described by current models. These modelcalculations depend on a significant number of parameters that are presently poorly constrained byfundamental theory, and a reliable determination of the QGP properties requires a systematic explorationof the full parameter space. An example of such an exploration [48,49] is shown in Figure 2 where theshape of the QCD EOS is treated as a free parameter. The left panel shows a random sample of thethousands of possible Equations of State, constrained only by results on the velocity of sound obtained byperturbative QCD at asymptotically high temperature and by lattice QCD at the crossover transition
13
[Gale et al: 1209.6330]
[Kovtun, Son, Starinets: hep-th/0309213],
[Buchel, Liu: hep-th/0311175],
[Benincasa, Buchel, Naryshkin: hep-th/0610145].
RHICLHC
[Arnold, Moore, Ya↵e:hep-ph/0010177]
Applying holographic duality to heavy ion collisions
28
All holographic theories have
⌘
s=
1
4⇡+ finite coupling corrections.
Weak coupling:
⌘s ⇠ ⌧
m.f.t.✏quasiparticle =5.12
g(T )
4log 2.42/g(T )