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Heavy ion collisions and AdS/CFT. Amos Yarom. With S. Gubser and S. Pufu. Part 2:. Entropy estimates. RHIC. t < 0. ~ 400. Imagine a gas of hadrons at the deconfienment temperature. The entropy per particle is:. S/N ~ 7.5. Thus:. S ~ 37500. RHIC. t > 0. ~ 5000. - PowerPoint PPT Presentation
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Heavy ion collisions and AdS/CFT
Amos Yarom
With S. Gubser and S. Pufu.
Part 2:
Entropy estimates
RHIC
t < 0
~ 400
RHIC
t > 0
~ 5000
S/N ~ 7.5
Imagine a gas of hadrons at the deconfienment temperature. The entropy per particle is:
Thus:
S ~ 37500
Entropy production in AdS
S > 0S ~ 0
We’d like to construct a scenario similar to:
Our candidate is a collision of two light-like particles which form a black hole.
Light-like particles in AdS
z
0
z=z*
Light-like particles in AdS
z
0
z=z*
Light-like particles in AdS
Equations of motion for the metric:
Stress tensor of a light-like particle.
Let’s switch to light-like coordinates:
Then:
Light-like particles in AdSEquations of motion for the metric:
Let’s switch to light-like coordinates:
Then:
We use an ansatz:
Light-like particles in AdSThe equations of motion for the metric:
with the ansatz:
reduce to:
Light-like particles in AdSThe solution to:
is:
where:
Light-like particles in AdS
z
0
z=z*
Light-like particles in AdS
z
0
z=z*
Light-like particles in AdSz=z* t
x3
x1, x2
t=0
The line element we wrote down is a solution anywhere outside the future light-cone of the collision point.
HorizonsEvent horizon: boundary of causal curves reaching future null infinity.
Marginally trapped surface: a 3 dimensional surface for which the outward pointing null vector propagates neither inward nor outward and the other propagates inward.
~
Let: and be the null normal vectors to the surface.
Then, a marginally trapped surface satisfies:
HorizonsA trapped surface is always on or inside an event horizon.
Goal: Find a marginally trapped surface, compute its area, and obtain a lower bound on the entropy of the black hole.
The area of the event horizon can only increase
The entropy of a black hole is proportional to its area
Searching for a trapped surface:t
x3
x1, x2
t=0
We find by requiring that the expansion vanishes on this surface.
Guess:
I
II
Searching for a trapped surface:
Guess:
We find by requiring that the expansion vanishes on this surface. A normal to the surface is given by:
I
II
Requiring that it’s light-like, outward pointing and future directing,
!The metric is singular at u=0 and v<0. In order for the metric to be finite we use the coordinate
transformation:
Searching for a trapped surface:
Guess:
We find by requiring that the expansion vanishes on this surface. A normal to the surface is given by:
I
II
The inward pointing null vector is given by:
Searching for a trapped surface:
Guess:
We find by requiring that the expansion vanishes on this surface. The normals to the surface are given by:
I
II
From symmetry:
Searching for a trapped surface:
Guess:
The normal to the surface is:
I
II
The induced metric should be orthogonal to the normals. To find it, we make the guess:
and determine A, B and C though:
Searching for a trapped surface:
Guess:
With
I
II
and we can compute the expansion:
With the boundary conditions:
After some work, we find (using ):
Searching for a trapped surface:We need to solve:
With the boundary conditions:
The most general, non-singular, solution to the differential equation is:
We denote the boundary by the surface q=qc. Then, the boundary conditions turn into algebraic relations between qc and K:
Searching for a trapped surface:We found a trapped surface:
I
IIWhere:
with
HorizonsA trapped surface is always on or inside an event horizon.
Goal: Find a marginally trapped surface, compute its area, and obtain a lower bound on the entropy of the black hole.
The area of the event horizon can only increase
The entropy of a black hole is proportional to its area
Searching for a trapped surface:We found a trapped surface:
I
IIWhere:
with
The area is given by:
Searching for a trapped surface:We found a trapped surface:
I
II
Its area is:
The lower bound on the entropy is:
Converting to boundary quantities
Let’s see what the collision looks like on the boundary.
Recall that:
So from:
Converting to boundary quantities
Let’s see what the collision looks like on the boundary.
Recall that:
From the form of the metric we find:
So we convert:
E=Ebeam=19.7 TeV z*=4.3 fm
Converting to boundary quantities
We convert:
E = Ebeam = 19.7 TeV z* = 4.3 fm
Naively: But more generally:
Recall
Converting to boundary quantities
We convert:
E = Ebeam = 19.7 TeV z* = 4.3 fm
Naively: But more generally:
Compare:
Converting to boundary quantities
We convert:
E = Ebeam = 19.7 TeV z* = 4.3 fm
So that:
LHC X 1.6Results(PHOBOS, 2003)
Analyzing the scaling behavior
z
0
Off center collisions
b
b
N
Off center collisions
b
Npart
N
Off center collisions
b
Npart
N/ Npart
Off center collisions
Off center collisions
b
z
0
z=z*
Results for off-center collisions
Results for off-center collisions
b
“spectators” In a confining theory the spectators don’t participate in the collisions.
For the purpose of this calculation we can “mimic” confinenemnt by setting:
Results for off-center collisions
References• PHOBOS collaboration nucl-ex/0410022. Multiplicity data.
• Aichelburg and Sexl. Gen. Rel. Grav. 2 (1972) 303-312 Shock wave geometries in flat space.
• Hotta et. al. Class. Quant. Grav. 10 (1993) 307-314, Stefsos et. al. hep-th/9408169, Podolsky et. al. gr-qc/9710049, Horowitz et. al. hep-th/9901012, Emparan hep-th/0104009, Kang et. al. hep-th/0410173. Shock wave geometries in AdS space.
• Penrose, unpublished, Eardley and Giddings, gr-qc/0201034, Yoshino et. al. gr-qc/0209003 Trapped surface computation in flat space.
• Gubser et. al. 0805.1551, Lin et. al 0902.1508, Gubser et. al. 0902.4062 Trapped surface computation in AdS space.
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