Numerical solutions of AdS gravity: new lessons about dual equilibration processes at strong coupling

Transcription

1 Numerical solutions of AdS gravity: new lessons about dual equilibration processes at strong coupling Michał P. Heller Universiteit van Amsterdam, the Netherlands & National Centre for Nuclear Research, Poland

2 Introduction

3 Numerical holography numerical relativity + holography = new window on far-from-equilibrium physics. Why interesting? ab initio calculations in a class of interacting quantum field theories. My main motivation will be the creation of quark-gluon plasma in Heavy Ion Collisions. Hence I will consider the Poincare patch of AdS4+1 In this talk I will discuss 3 solutions of R ab 1 2 Rg ab 1/12 states in a large-nc CFT at strong coupling 6 L 2 g ab =0 with planar horizons. In common: applications to HIC & using the ingoing Eddington-Finkelstein coordinates.

4 Setups event horizon (present from the start) constant time slices boundary; we demand here that if ds = L2 u 2 then du 2 + h µ (u, x) dx µ dx h µ (u, x) = µ + 4 G N! L 3 ht µ (x)i u = 2 2 N 2 c (for N=4 SYM) initial data (solves constraints) Z note that in the setups I am going to study d 3 x ht tt i = 1. Why the ingoing Eddington-Finkelstein coordinates? Manifestly regular on the horizon + attractive integrations scheme (if no caustics). fluid-gravity duality [hep-th] Chesler & Yaffe 2/12

5 Isotropization at strong coupling [hep-th] PRL 108 (2012) : MPH, D. Mateos, W. van der Schee & D. Trancanelli [hep-th] JHEP 1309, 026 (2013): MPH, D. Mateos, W. van der Schee & M. Triana

6 Holographic isotropization [hep-th] PRL 108 (2012) : MPH, D. Mateos, W. van der Schee & D. Trancanelli One of the simplest equilibration processes to study holographically is described by ht µ i = diag E, 1 3 E 2 3 P(t), 1 3 E P(t), 1 3 E P(t) It is identically traceless and conserved. EOMs are. 1 R ab 2 Rg ab Symmetries of the stress tensor lead to a general metric ansatz 6 L 2 g ab =0 ds 2 = f tt dt 2 +2f tr dtdr + f rr dr e 2B dx e B (dx dx 2 3) We fix almost all the gauge freedom by adopting (the ingoing EF coordinates) ds 2 =2dtdr Adt e 2B dx e B (dx dx 2 3) µ We can solve Einstein s equations near the boundary and obtain* B = 1 r 4 b 4 (t)+ 1 r b0 4(t)+ 2 12r 6 b00 4(t)+ 1 4r 3 b(3) 4 (t)+... with P (t) = N c 2 b 4 (t) 3/12

7 Equilibration dynamics [hep-th] PRL 108 (2012) : MPH, D. Mateos, W. van der Schee & D. Trancanelli ds 2 =2dtdr Adt e 2B dx e B (dx dx 2 3) initial data: ht 00 i = E and B(t =0,r) absorption by the horizon 3N 2 c 8 2 E r4 B(t, r) r h r tt ht µ i = diag E, 1 3 E 2 3 P(t), 1 3 E P(t), 1 3 E P(t) µ 4/12

8 Fast relaxation t iso : P(t t iso ) E apple 0.1 Checked for circa 10 3 different n-eq initial conditions! (RHIC c=0-5%: ) 0.25 fm 500 MeV = [nucl-th] W. Broniowski et al. 5/12

9 Shock wave collisions and hydrodynamization [hep-th] PRL 111 (2013) : J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee [hep-th]: J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee

10 Towards a holographic heavy ion collision general issue: which holographic initial conditions are closest to the experiment? practical viewpoint: collide two lumps of matter moving at relativistic speeds. [hep-th/ ] R. Janik & R. Peschanski [hep-th] P. Chesler & L. Yaffe 0 t z u 6/12

11 Gravitational shock wave solutions Janik & Peschanski [hep-th/ ] Chesler & Yaffe [hep-th] ds 2 = 1 u 2 (du2 + µ dx µ dx )+u 2 h(x )dx 2 Poincare patch vacuum AdS dual stress tensor: ht tt i = ht zz i = ±ht tz i = N c 2 h(t z) 2 2 t shock wave disturbance moving with the speed of light z 0 Solution of Einstein s equations with the negative CC for any longitudinal profile h(x ) u Let s consider now h(t ± z) = 4 exp (t ± z) 2 /2 2. But, in a CFT, what matters is: e = (in real HIC e 1/2 and e CY 0.64 corresponds to Pb at RHIC) 7/12

12 [hep-th] PRL 111 (2013) : 3 J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee 4 4 S S elef t = 2 ecy eright = ecy Dynamical crossover 2 N c tt ht i = E 2 2 low energy E 4 high energy 2 E 4 PL 4 loc PL Eloc for PL 4 deviation from viscous hydro FIG. 2. Energy flux for collisions of thick (left) and thin (right) shocks. The dotted curves show the location of the maxima of the flux. FIG thick (left) and thin (right) shocks. The white areas indicate the The grey areas indicate regions where hydrodynamics deviates by more than 100%. Th the maxima of the energy flux, as in Fig. 2. maximum of the energy flux the energy flux in this region is less than 10% of the maxmodel [5]. imum incoming flux, asillustrated by Fig. 2(left). At late The thin shocks i times, the velocity of the receding shocks can be read o P In this case the sh P from thecoalesce same figure as the inverse slopehydro of the dotted only at midshocks and explode hydroapplicable their shap FIG P E for thick (left) and thin (right) shocks. The white areas indicate the vacuum regions outsidealthough the light cone. grey areasconstant indicate regions where hydrodynamics deviates by more than 100%. times The dotted curves indicate the location of line. This(similar isthe not in time, but at late it dynamically to the Landau picture) rapidities and late enough!!! the maxima of the energy flux, as in Fig. 2. v 1, as seen in Fig. reaches a maximum of about v The validity of the energythat flux in this region is lesscoupling than 10% of the maxmodel [5]. cation in their shape Dispels the myth strong necessarily leads to immediate stopping* imum incomingdescription flux, as illustrated by Fig. 2(left).be At late the hydrodynamic can seen 3(left) thinfig. shocks illustrate the transparency scenario. Thein times, the velocity of the receding shocks can be read8/12 o trails right behind t In this case the shocks pass through each other and, T 4 T loc L loc 4

13 Hydrodynamization in a shock wave collision [hep-th] PRL 111 (2013) : J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee = µ +... Hydrodynamics: ht µ i = {E + P(E)} u µ u + P(E) µ + µ perfect fluid dissipative We use ht µ iu = E u µ and compare ht zz iand ht?? i with hydro prediction. ht tt i 3 E 0 ht zz i E 0 ht?? i E 0 CFT dotted: hydro prediction t hyd T hyd =0.26 t at the collision axis (z = 0) Surprise: large anisotropy at the onset of hydrodynamics due to the shear tensor! see also & Chesler & Yaffe and MPH, Janik & Witaszczyk 9/12

14 The nature of hydrodynamics [hep-th] PRL 110 (2013) : MPH, R. A. Janik & P. Witaszczyk

15 Hydrodynamic series at high orders [hep-th] PRL 110 (2013) : Question: what is the nature of hydrodynamic gradient expansion? Π µν = ησ µν [ + ητ Π Dσ µν + 1 ] d 1 σµν ( u) + λ 1 σ µ λσ ν λ + λ 2 σ µ λω ν λ + λ 3 Ω µ λω ν λ.+... MPH, R. A. Janik & P. Witaszczyk Idea: use the fluid-gravity duality to compute subsequent gradient terms on-shell. To make it operational, we used the boost-invariant flow. Why? t = cosh y and z = sinh y ] + κ [R µν (d 2)u α R α µν β u β u µ and ht i = E( ) = 3 8 N 2 c 2 T ( ) 4 1 T ( ) r µu 1 T ( ) 1! Gradient expansion solving ODEs in the bulk go to 240 th order in grads. 10/12

16 Hydrodynamic series at high orders [hep-th] PRL 110 (2013) : MPH, R. A. Janik & P. Witaszczyk n 2 1/n = 3 8 N 2 c 2 1 T 00 = ( ) 4/3 1X / /3 n=2 n ( 2/3 ) n (T 1 r µ u 2/3 ) at low orders behavior is different at large orders factorial growth of gradient contributions with order (n!) 1/n (2 n)1/2n e n First evidence that hydrodynamic expansion has a zero radius of convergence! 11/12

17 Summary

18 Summary! DPêE [hep-th] PRL 108 (2012) : 0.4 Tt MPH, D. Mateos, W. van0.2 der Schee & D. Trancanelli DPêE E 4 PL Tt n 1/n 2 time predicted by the full and is the di erence between the isotropization E 4 Figure 5. (Top) tiso the linear equations. The height of each bar in the histogram indicates the number of initial states for which the evolution yielded values in the corresponding bin. The total number of initial states is more than 800. We see both that holographic isotropization proceeds quickly, at most over a time scale set by the inverse temperature, and that the linearized Einstein s equations correctly reproduce the isotropization time with a 20% accuracy in most cases. Note that the histogram is based on a di erent sample of initial states than those originally considered in [1]. In particular, we incorporated the binary search algorithm absent in [1] and were stricter about the maximum violation of the constraint that we allowed. (Botom) Close inspection of one of the few profiles for which the linearized approximation seemingly fails by more than 20% ( tiso /tiso = 0.5) shows that it is the imperfect isotropization criterium PL 4 which leads to the mismatch rather than the failure of the linear approximation. Indeed, the left plot shows that, on the scale of the initial anisotropy, the linear result yields a good approximation. However, the isotropization criterium makes no reference to this scale, and results in a 50% di erence in the isotropization times, indicated by the arrows on the right plot. See [9] for a related discussion of subtleties involved in defining the thermalization (or more accurately hydrodynamization) time in a similar setup. + hydrodynamization [hep-th] PRL 111 (2013) : P J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee P T [hep-th] PRL 110 (2013) : MPH, R. A. Janik & P. Witaszczyk 4 T /12

19 extra