1. Shin Nakamura (Center for Quantum Spacetime (CQUeST), Sogang Univ.) Based on S. Kinoshita, S. Mukohyama, S.N. and K. Oda, arXiv:0807.3797 A Holographic Dual of Bjorken Flow

2. Motivation: quark-gluon plasma RHIC: Relativistic Heavy Ion Collider (@ Brookhaven National Laboratory) http://www.bnl.gov/RHIC/inside_1.htm Heavy ion: e.g. 197 Au ~200GeV. Quark-gluon Plasma (QGP) is observed.

3. Also at LHC Similar exp. at FAIR@ GSI NICA@ JINR http://aliceinfo.cern.ch/Public/en/Chapter4/Chapter4Gallery-en.html ALICE ATLAS CMS ALICE

4. QGP Strongly coupled system Time-dependent system Lattice QCD: a first-principle computation However, it is technically difficult to analyze time-dependent systems. (Relativistic) Hydrodynamics This is an effective theory for macroscopic physics. (entropy, temperature, pressure, energy density,….) Information on microscopic physics has been lost. (correlation functions of operators, equation of state, transport coefficients such as viscosity,…) Possible frameworks:

5. Alternative framework: AdS/CFT An advantage: Both macroscopic and microscopic physics can be analyzed within a single framework. This feature may be useful in the analysis of non-equilibrium phenomena, like plasma instability. Plasma instability: seen only in time-dependent systems.

6. However, Construction of time-dependent AdS/CFT itself is a challenge. We construct a holographic dual of Bjorken flow of large-Nc, strongly coupled N=4 SYM plasma. Our work A standard model of the expanding QGP We deal with N=4 SYM instead of QCD.

7. Quark-gluon plasma (QGP) as a one-dimensional expansion http://www.bnl.gov/RHIC/heavy_ion.htm

8. Bjorken flow (Bjorken 1983) (Almost) one-dimensional expansion. We have boost symmetry in the CRR. Relativistically accelerated heavy nuclei After collision Velocity of light Central Rapidity Region (CRR) Time dependence of the physical quantities are written by the proper time. Velocity of light “A standard model” of QGP expansion

9. Local rest frame(LRF) τ=const. y=const. Minkowski spacetime x1x1 t Rapidity Proper-time The fluid looks static on this frame Boost invariance ： y-independence Rindler wedge with Milne coordinates

10. Boost invariance Taken from Fig. 5 in nucl-ex/0603003.

11. Stress tensor on the LRF Local rest frame: The stress tensor is diagonal. Bjorken flow: The stress tensor is diagonal on the Milne coordinates:

12. Hydrodynamics Hydrodynamic equation: We can solve the hydrodynamic equation for the Bjorken flow of conformal fluid, since the system has enough symmetry. Hydrodynamics describes spacetime-evolution of the stress tensor.

13. Solution important T~τ -1/3 Once the parameters (transport coefficients) are given, T μν (τ) is completely determined. expansion w.r.t τ -2/3 But, hydro cannot determine them.

14. AdS/CFT dictionary Bulk on-shell action = Effective action of YM The boundary metric (source) 4d stress tensor Time-dependent geometry Time-evolution of the stress tensor

15. How to obtain the geometry? The bulk geometry is obtained by solving the equations of motion of super-gravity with appropriate boundary data. 5d Einstein gravity with Λ<0 The boundary metric is that of the comoving frame: The 4d stress tensor is diagonal on this frame. Bjorken’s case: We set (the 4d part of) the bulk metric diagonal. (ansatz) This tells our fluid undergoes the Bjorken flow.

16. Time-dependent AdS/CFT Earlier works

17. A time-dependent AdS/CFT A time-dependent geometry that describes Bjorken flow of N=4 SYM fluid was first obtained within a late-time approximation by Janik-Peschanski. Janik-Peschanski, hep-th/0512162 They have used Fefferman-Graham coordinates: stress tensor of YM4d geometry (LRF) boundary condition to 5d Einstein’s equation with Λ<0 geometry as a solution

18. Unfortunately, we cannot solve exactly They employed the late-time approximation: have the structure of We discard the higher-order terms. Janik-Peschanski hep-th/0512162

19. Janik-Peschanski’s result at the leading order Hydrodynamics The statement If we start with unphysical assumption like the obtained geometry is singular: at the point g ττ =0. Regularity of the geometry tells us what the correct physics is.

20. Many success 1st order: Introduction of the shear viscosity: 2nd order: Determination of from the regularity: 3rd order: Determination of the relaxation time from the absence of the power singularity: S.N. and S-J.Sin, hep-th/0607123 Janik, hep-th/0610144 Heller and Janik, hep-th/0703243 For example: same as Kovtun-Son-Starinets

21. But, a serious problem came out. An un-removable logarithmic singularity appears at the third order. (Benincasa-Buchel-Heller-Janik, arXiv:0712.2025) This suggests that the late-time expansion they are using is not consistent.

22. Our work: Formulation without singularity.

23. What is wrong? The location of the horizon (where the problematic singularity appears) is the edge of the Fefferman-Graham (FG) coordinates. Schwarzschild coordinates Only outside the horizon! FG coordinates Static AdS-BH case: This is also the case for the time-dependent solutions.

24. Better coordinates? boundary future event horizon past event horizon singularity dynamical apparent horizon Eddington-Finkelstein coordinates trapped region un-trapped region Cf. Bhattacharyya-Hubeny-Minwalla-Rangamani (0712.2456) Bhattacharyya et. al. (0803.2526, 0806.0006)

25. Eddington-Finkelstein coordinates Static AdS-BH: The trapped region and the un-trapped region are on the same coordinate patch. At least for the static case,

26. Our proposal Construct the dual geometry on the EF coordinates. You may say, coordinate transformation does not remove the singularity. where the potential singularity appears The coordinate transformation from FG coordinates to EF coordinates is singular at the “horizon”. What we will do is not merely a coordinate transformation.

27. Our parametrization Parametrization of the dual geometry: We assume a, b, c depend only on τand r, because of the symmetry. boundary metric: Differential equations of a, b, cThe 5d Einstein’s eq. The boundary condition: at r= ∞.

28. Late-time approximation It is very difficult to obtain the exact solution. We introduce a late-time approximation by making an analogy with what Janik-Peschanski did on the FG coordinates. Janik-Peschanski: τ -2/3 expansion with zτ -1/3 = v fixed. Now, r ~ z -1 (around the boundary). Let us employ τ -2/3 expansion with rτ 1/3 = u fixed. Our late-time approximation

29. More explicitly, We solve the differential equations for a(τ,u), b(τ,u), c(τ,u) order by order: (similar for b and c) zeroth orderfirst ordersecond order (u=rτ 1/3 )

30. The zeroth-order solution This is u This reproduces the correct zeroth-order stress tensor of the Bjorken flow. We have an apparent horizon. trapped region if u<w. The location of the apparent horizon: u=w+O(τ -2/3 ) r = w τ -1/3

31. Location of the apparent horizon The location of the apparent horizon is given by “expansion” Lie derivative along the null direction volume element of the 3d surface : un-trapped region : trapped region normalization

32. The (event) horizon is necessary We have a physical singularity at the origin. However, this is hidden by the apparent horizon at u=w hence the event horizon (outside it). OK, from the viewpoint of the cosmic censorship hypothesis. Not a naked singularity.

33. The first-order solution gauge degree of freedom c 1 is regular at u=w, only when

34. Regularity of c 1 is necessary. We can show : a regular space-like unit vector Riemann tensor projected onto a regular orthonormal basis This has to be regular to make the geometry regular. (projected onto a local Minkowski)

35. What is this value? Gubser-Klebanov-Peet, hep-th/9602135 First law of thermodynamics Our definition and result: Combine all of them: The famous ratio by Kovtun-Son-Starinets (2004)

36. Second-order results: We have obtained the solution explicitly, but it is too much complicated to exhibit here. From the regularity of the geometry, “relaxation time” is uniquely determined. consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al. 2nd-order transport coefficient

38. Second-order results: We have obtained the solution explicitly, but it is too much complicated to exhibit here. From the regularity of the geometry, “relaxation time” is uniquely determined. consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al. 2nd-order transport coefficient

39. All-order results n-th order Einstein’s equation: Diff eq. for b n = source which contains only k(<n)-th order metric “Regular enough” to show the regularity of b n and its arbitrary-order derivatives (except at the origin). We can show the regularity by induction.

40. All-order results n-th order Einstein’s equation: Diff eq. for a n = source which contains only k(<n)-th order metric and b n “Regular enough” to show the regularity of a n and its arbitrary-order derivatives (except at the origin).

41. All-order results n-th order Einstein’s equation: Diff eq. for c n = source which contains only k(<n)-th order metric and a n, b n Not always regular enough! Unique choice of the transport coefficients. n-th order transport coefficients comes here linearly

42. All-order results There is no un-removable logarithmic singularity found on the FG coordinates. We can make the geometry regular at all orders by choosing appropriate values of the transport coefficients. Our model is totally consistent and healthy!

43. Area of the apparent horizon This is consistent with the time evolution of the entropy density to the first order. There is some discrepancy at the second order. However, it does not mean inconsistency immediately. From Hydro. greater!

44. What we have done: We constructed a consistent gravity dual of the Bjorken flow for the first time. (cf. Heller-Loganayagam-Spalinski-Surowka-Vazquez, arXiv:0805.3774) Our model is a concrete well-defined example of time-dependent AdS/CFT based on a well-controled approximation.

45. Hydrodynamics Time evolution of the stress tensor hydrodynamic equation (energy-momentum conservation) equation of state (conformal invariance) transport coefficients Our model 5d Einstein’s eq. at the vicinity of the boundary Reguarity around the horizon Related to local thermal equilibrium

46. Discussion The definition of the late-time approximation is a bit artificial. (τ -2/3 expansion with rτ 1/3 = u fixed.) Is this a unique choice? Can we derive it purely from gravity? What is the physical meaning? Is this related to the choice of the vacuum?

47. Discussion At this stage, the connection among our method and other methods are not clear. Kubo formula: Quasi normal modes In this case, we impose the “ingoing boundary condition” at the horizon. regularity?

48. Comparison with Veronicka’s work Veronica’s workOur work Derivative expansion w.r.t 4d coordinates by fixing r. Derivative expansion w.r.t 4d coordinates by fixing u= rτ 1/3. The leading order is already time-dependent const. + higher derivative (static) Full Minkowski spacetimeRindler wedge Expansion around a regular exact solution The leading-order metric is not an exact solution