Shin Nakamura (Center for Quantum Spacetime (CQUeST) , Sogang Univ.)

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

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

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

4. 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,…) QGP • Strongly coupled system • Time-dependent system 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 deal with N=4 SYM instead of QCD. Our work We construct a holographic dual of Bjorken flow of large-Nc, strongly coupled N=4 SYM plasma. A standard model of the expanding QGP

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

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

9. t x1 Local rest frame(LRF) Rapidity y=const. Proper-time τ=const. Boost invariance： y-independence The fluid looks static on this frame Minkowski spacetime Rindler wedge with Milne coordinates

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

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

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

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

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

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

16. Time-dependent AdS/CFT Earlier works

17. 4d geometry (LRF) stress tensor of YM boundary condition to 5d Einstein’s equation with Λ<0 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: geometry as a solution

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

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 gττ=0. Regularity of the geometry tells us what the correct physics is.

20. Many success For example: • 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 same as Kovtun-Son-Starinets Janik, hep-th/0610144 Heller and Janik, hep-th/0703243

21. But, a serious problem came out. • An un-removablelogarithmic 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. Static AdS-BH case: FG coordinates Schwarzschild coordinates What is wrong? The location of the horizon (where the problematic singularity appears) is the edge of the Fefferman-Graham (FG) coordinates. Onlyoutsidethe horizon! This is also the case for the time-dependent solutions.

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

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

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

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

28. Janik-Peschanski: τ-2/3 expansion with zτ-1/3 = v fixed. Let us employ τ-2/3 expansion with rτ1/3 = u fixed. Our late-time approximation 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. Now, r ~ z-1 (around the boundary).

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 order first order second order (u=rτ1/3)

30. 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 zeroth-order solution r = w τ-1/3 The location of the apparent horizon: u=w+O(τ-2/3)

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

32. Not a naked singularity. 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.

33. c1 is regular at u=w, only when The first-order solution gauge degree of freedom

34. : a regular space-like unit vector Regularity of c1 is necessary. We can show Riemann tensor projected onto a regular orthonormal basis (projected onto a local Minkowski) This has to be regular to make the geometry regular.

35. Gubser-Klebanov-Peet, hep-th/9602135 First law of thermodynamics Our definition and result: Combine all of them: What is this value? 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. n-th order Einstein’s equation: Diff eq. for bn = source which contains only k(<n)-th order metric “Regular enough” to show the regularity of bn and its arbitrary-order derivatives (except at the origin). All-order results We can show the regularity by induction.

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

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

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. greater! Area of the apparent horizon • This isconsistent 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.

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

46. Discussion • The definition of the late-time approximation • is a bit artificial. (τ-2/3 expansion with rτ1/3 = 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. regularity? 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.

48. const. + higher derivative (static) The leading order is already time-dependent Comparison with Veronicka’s work Veronica’s work Our work Derivative expansion w.r.t 4d coordinates by fixing r. Derivative expansion w.r.t 4d coordinates by fixing u= rτ1/3. Full Minkowski spacetime Rindler wedge Expansion around a regular exact solution The leading-order metric is not an exact solution