Heavy ion collisions and AdS/CFT

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

2. Part 2: Entropy estimates

3. RHIC t < 0 ~ 400

4. 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

5. We’d like to construct a scenario similar to: S ~ 0 S > 0 Our candidate is a collision of two light-like particles which form a black hole. Entropy production in AdS

6. 0 z=z* z Light-like particles in AdS

7. 0 z=z* z Light-like particles in AdS

8. Equations of motion for the metric: Let’s switch to light-like coordinates: Stress tensor of a light-like particle. Then: Light-like particles in AdS

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

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

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

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

13. 0 z Light-like particles in AdS z=z*

14. t=0 Light-like particles in AdS z=z* t The line element we wrote down is a solution anywhere outside the future light-cone of the collision point. x3 x1, x2

15. be the null normal vectors to the surface. Let: and Then, a marginally trapped surface satisfies: 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. ~

16. Horizons A trapped surface is always on or inside an event horizon. The area of the event horizon can only increase The entropy of a black hole is proportional to its area Goal: Find a marginally trapped surface, compute its area, and obtain a lower bound on the entropy of the black hole.

17. Guess: I II Searching for a trapped surface: t t=0 x3 x1, x2 We find y by requiring that the expansion vanishes on this surface.

18. The metric is singular at u=0 and v<0. In order for the metric to be finite we use the coordinate transformation: ! Requiring that it’s light-like, outward pointing and future directing, Searching for a trapped surface: Guess: I II We find y by requiring that the expansion vanishes on this surface. A normal to the surface is given by:

19. The inward pointing null vector is given by: Searching for a trapped surface: Guess: I II We find y by requiring that the expansion vanishes on this surface. A normal to the surface is given by:

20. From symmetry: Searching for a trapped surface: Guess: I II We find y by requiring that the expansion vanishes on this surface. The normals to the surface are given by:

21. 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: I II The normal to the surface is:

22. After some work, we find (using ): With the boundary conditions: Searching for a trapped surface: Guess: I II we can compute the expansion: With and

23. 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 need to solve: With the boundary conditions:

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

25. Horizons A trapped surface is always on or inside an event horizon. The area of the event horizon can only increase The entropy of a black hole is proportional to its area Goal: Find a marginally trapped surface, compute its area, and obtain a lower bound on the entropy of the black hole.

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

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

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

29. So we convert: E=Ebeam=19.7 TeV z*=4.3 fm 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:

30. Naively: But more generally: Recall Converting to boundary quantities We convert: E = Ebeam = 19.7 TeV z* = 4.3 fm

31. Converting to boundary quantities We convert: E = Ebeam = 19.7 TeV z* = 4.3 fm Naively: But more generally: Compare:

32. So that: Converting to boundary quantities We convert: E = Ebeam = 19.7 TeV z* = 4.3 fm

33. Results LHC X 1.6 (PHOBOS, 2003)

34. Analyzing the scaling behavior 0 z

35. N b Off center collisions b

36. Off center collisions N b Npart

37. Off center collisions N/ Npart b Npart

38. Off center collisions

39. z=z* b 0 z Off center collisions

40. Results for off-center collisions

41. “spectators” b For the purpose of this calculation we can “mimic” confinenemnt by setting: Results for off-center collisions In a confining theory the spectators don’t participate in the collisions.

42. Results for off-center collisions

43. References • PHOBOS collaboration nucl-ex/0410022. Multiplicity data. • Aichelburg and Sexl. Gen. Rel. Grav. 2 (1972) 303-312Shock 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/0209003Trapped surface computation in flat space. • Gubser et. al. 0805.1551, Lin et. al 0902.1508, Gubser et. al. 0902.4062Trapped surface computation in AdS space.