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.