Shin Nakamura (Center for Quantum Spacetime (CQUeST) and Hanyang Univ.)

1. A Holographic Dual of QGP Hydrodynamics Shin Nakamura (Center for Quantum Spacetime (CQUeST) and Hanyang Univ.) Refs. : SN, Sang-Jin Sin, JHEP09(2006)020, hep-th/0607123 (Sang-Jin Sin, SN, Sang Pyo Kim, JHEP0612 (2006) 075, hep-th/0610113)

2. Lattice QCD: a first-principle computation. However, there are technical difficulties in the computations if the system has • Finite baryon chemical potential • Time dependence • Large size (e.g. deuterons….) Motivation Many interesting phenomena in QCD lie in the strongly-coupled region. We need non-perturbative methods for analysis. Let us consider time-dependent fluid of YM theories.

3. How about AdS/CFT? Construction of AdS/CFT for time-dependent systems is still a challenge. An example: QGP Quark-gluon plasma (QGP) at RHIC is a strongly coupled, expanding system. • Relativistic hydrodynamics is useful. However, we may be able to obtain something more than hydrodynamics by applying AdS/CFT, in principle.

4. microscopic quantities correlation functions, inter-quark potential, counting of entropy, equation of state,… AdS/CFT Why can AdS/CFT be better than hydrodynamics? Hydrodynamics Time evolution ofmacroscopic quantities temperature, pressure, entropy, energy, transport coefficients (viscosity,…)

5. = conjecture AdS/CFT (Weak version) Classical Supergravity on Maldacena ‘97 4dim. Large-Nc SU(Nc) N=4 Super Yang-Mills at the large ‘t Hooft coupling Strongly interacting quantum YM !!

6. = conjecture AdS/CFT at finite temperature Classical Supergravity on AdS-BH×S5 Witten ‘98 4dim. Large-Nc strongly coupled SU(Nc) N=4 SYM at finite temperature (in the deconfinement phase).

7. The 4d geometry where the YM theory lives. • The energy-momentum tensor of the YM fluid. Described by hydrodynamics. In any case, The bulk geometry is obtained by solving the equations of motion of super-gravity with appropriateboundary conditions.

8. 4d relativistic hydrodynamics boundary conditions Haro-Skenderis-Solodukhin, hep-th/0002330 Solving the SUGRA equation (5d Einstein’s equation with ) 5d Bulk Geometry Time dependent Without viscosity: Janik-Peschanski, hep-th/0512162 Something more than Hydro. In this talk: N=4 SYM fluid with shear viscosity. What we will do

9. The system we consider • (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 (Bjorken ’83) After collision Time dependence of the physical quantities are written by the proper time.

10. Relativistic Hydrodynamics • We take local rest frame (LRF). Our case: • The energy-momentum tensor on this frame: Proper-time Rapidity energy density pressure shear viscosity The bulk viscosity is set to be zero.

11. “Conformal invariance” (or equation of state) Energy-momentum conservation 3 independent components: or 2 independent constraints: Only 1 independent quantity:

12. Stefan-Boltzmann’s low Solution: In the static N=4 SYM system: (dim. analysis) in the slowly varying (late time) region

13. AdS/CFT gives more information. Entropy creation The dissipation creates the entropy. The entropy (per unit volume on the LRF) at the infinitely far future is not determined in this framework. (Integration constant)

14. Now, the energy-momentum tensor is: with in the slowly varying (late time) region.

15. Gravity dual Boost symmetry • 5d metric: asymptotically AdS Solution of 5d Einstein’s eq. with • 4d part of the metric depends only on Boundary cond. Boundary cond. Haro-Skenderis-Solodukhin hep-th/0002330 The Minkowski metric on the LRF The higher-order terms are determined iteratively.

16. Late time approximation The input (hydro.) is valid only at the late time (slowly varying region). We employ the late time approximation: Janik-Peschanski hep-th/0512162 have the structure of We discard the higher-order terms.

17. Solving the Einstein’s equation

18. Solving the Einstein’s equation

19. The metric can be re-summed to be:

20. This is correct up to the order of . The late time geometry This looks to be a black hole with time-dependent horizon.

21. Cf. AdS-Schwarzschild BH In a more standard form,

22. Stefan-Boltzmann: Various quantities from the geometry Entropy creation: Numerical coefficient is given. Integration constant is given. From hydrodynamics: Not only consistent with hydro but also more information in the holographic dual.

23. More about the consistency From gravity: From hydrodynamics: Thermal equiv. Correct relationship from thermodynamics

24. Conclusions • We considered a holographic dual of hydrodynamics of N=4 large-Nc SYM plasma that undergoes Bjorken expansion. • We obtained the bulk geometry in the late time regime. • The holographic dual is not only consistent with the hydrodynamics but also contains more information.

25. Recent progress • (Janik hep-th/0610144) • Relaxation time in the second-order dissipative • hydrodynamics (Heller-Janik hep-th/0703243) Computed by using regularity of the time-dependent geometry.

26. We should appreciate the Newton’s discovery of gravity. Conclusion/Question • Holographic dual “reduces” the problem to that of dynamical black holes. • How much does the gravity “know” the physics of non-equilibrium systems? • Holographic dual may provide us an interesting playground not only for QCD but also for non-equilibrium physics.

27. On the More about the late time limit The position of the horizon: coordinate, the position of the horizon is constant. We are keeping our eyes close to the horizon along the time evolution. The above limit is valid around the horizon. If we take the limit with fixing z, we cannot see the horizon.