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Heavy ion collisions and AdS/CFT Amos Yarom With S. Gubser and S. Pufu.

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Part 2: Entropy estimates

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RHIC t < 0 ~ 400

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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

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Entropy production in AdS S > 0 S ~ 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.

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Light-like particles in AdS z 0 z=z *

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Light-like particles in AdS z 0 z=z *

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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:

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Light-like particles in AdS Equations of motion for the metric: Let’s switch to light-like coordinates: Then: We use an ansatz:

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Light-like particles in AdS The equations of motion for the metric: with the ansatz: reduce to:

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Light-like particles in AdS The solution to: is: where:

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Light-like particles in AdS z 0 z=z *

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Light-like particles in AdS z 0 z=z *

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Light-like particles in AdS z=z * t x3x3 x 1, x 2 t=0 The line element we wrote down is a solution anywhere outside the future light-cone of the collision point.

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Horizons Event 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:

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Horizons A 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

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Searching for a trapped surface: t x3x3 x 1, x 2 t=0 We find by requiring that the expansion vanishes on this surface. Guess: I II

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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:

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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:

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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:

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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:

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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 ):

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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=q c. Then, the boundary conditions turn into algebraic relations between q c and K:

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Searching for a trapped surface: We found a trapped surface: I II Where: with

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Horizons A 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

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Searching for a trapped surface: We found a trapped surface: I II Where: with The area is given by:

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Searching for a trapped surface: We found a trapped surface: I II Its area is: The lower bound on the entropy is:

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Converting to boundary quantities Let’s see what the collision looks like on the boundary. Recall that: So from:

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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=E beam =19.7 TeVz * =4.3 fm

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Converting to boundary quantities We convert: E = E beam = 19.7 TeVz * = 4.3 fm Naively: But more generally: Recall

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Converting to boundary quantities We convert: E = E beam = 19.7 TeVz * = 4.3 fm Naively:But more generally: Compare:

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Converting to boundary quantities We convert: E = E beam = 19.7 TeVz * = 4.3 fm So that:

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LHC X 1.6 Results (PHOBOS, 2003)

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Analyzing the scaling behavior z 0

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Off center collisions b b N

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b N part N

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Off center collisions b N part N/ N part

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Off center collisions

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b z 0 z=z *

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Results for off-center collisions

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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:

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Results for off-center collisions

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References PHOBOS collaboration nucl-ex/ Multiplicity data. Aichelburg and Sexl. Gen. Rel. Grav. 2 (1972) Shock wave geometries in flat space. Hotta et. al. Class. Quant. Grav. 10 (1993) , Stefsos et. al. hep-th/ , Podolsky et. al. gr-qc/ , Horowitz et. al. hep-th/ , Emparan hep-th/ , Kang et. al. hep- th/ Shock wave geometries in AdS space. Penrose, unpublished, Eardley and Giddings, gr-qc/ , Yoshino et. al. gr-qc/ Trapped surface computation in flat space. Gubser et. al , Lin et. al , Gubser et. al Trapped surface computation in AdS space.

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