Numerical solutions of ads gravity

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 Introduction 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 states in a large-Nc CFT at strong coupling In this talk I will discuss solutions of Rab R gab gab = with planar horizons L In common: applications to HIC & using the ingoing Eddington-Finkelstein coordinates 1/12 Setups event horizon (present from the start) boundary; we demand here that if ds24+1 constant “time” slices L2 = du2 + hµ⌫ (u, x) dxµ dx⌫ u then hµ⌫ (u, x) = ⌘µ⌫ 4⇡GN + hT (x)i · u + µ⌫ !3 L 2⇡ = (for N=4 SYM) Nc initial data (solves constraints) noteZthat in the setups I am going to study d3 x hTtt i = Why the ingoing Eddington-Finkelstein coordinates? Manifestly regular on the horizon + attractive integrations scheme (if no caustics) 1309.1439 [hep-th] Chesler & Yaffe fluid-gravity duality++ 2/12 Isotropization at strong coupling 1202.0981 [hep-th] PRL 108 (2012) 191601: MPH, D Mateos, W van der Schee & D Trancanelli 1304.5172 [hep-th] JHEP 1309, 026 (2013): MPH, D Mateos, W van der Schee & M Triana Holographic isotropization 1202.0981 [hep-th] PRL 108 (2012) 191601: MPH, D Mateos, W van der Schee & D Trancanelli One of the simplest equilibration processes to study holographically is described by ✓ ◆ hTµ⌫ i = diag E, E 1 1 P(t), E + P(t), E + P(t) 3 3 µ⌫ gab = R gab It is identically traceless and conserved EOMs are Rab 2 L Symmetries of the stress tensor lead to a general metric ansatz ds2 = ftt dt2 + 2ftr dtdr + frr dr2 + ⌃2 e 2B dx21 + ⌃2 eB (dx22 + dx23 ) We fix almost all the gauge freedom by adopting (the ingoing EF coordinates) ds2 = 2dtdr Adt2 + ⌃2 e 2B dx21 + ⌃2 eB (dx22 + dx23 ) We can solve Einstein’s equations near the boundary and obtain* B= r ⇢ 00 (3) b4 (t) + b4 (t) + b4 (t) + b4 (t) + r 12r 4r 3/12 with P (t) = N c b4 (t) 8⇡ Equilibration dynamics 1202.0981 [hep-th] PRL 108 (2012) 191601: MPH, D Mateos, W van der Schee & D Trancanelli ds2 = 2dtdr Adt2 + ⌃2 e 2B dx21 + ⌃2 eB (dx22 + dx23 ) initial data: hT00 i = E and B(t = 0, r) absor ption by the h orizo n 3Nc2 r B(t, r) 8⇡ E rh r tT ✓ hTµ⌫ i = diag E, E 1 1 P(t), E + P(t), E + P(t) 3 3 4/12 ◆ µ⌫ Fast relaxation P(t tiso : Checked for circa 103 different n-eq initial conditions tiso ) E  0.1 ! DPêE Tt DPêE 0.4 (RHIC -5 c=0-5%: 0.25 fm ⇥ 500 MeV 0.2= 0.63 ) 0.2 -10 -15 0.4 0.6 0.8 1.0 1.2 0801.4361 [nucl-th] W Broniowski et al 0.0 5/12 0.2 0.4 0.6 0.8 1.0 1.2 Tt Shock wave collisions and hydrodynamization 1305.4919 [hep-th] PRL 111 (2013) 181601: J Casalderrey-Solana, MPH, D Mateos & W van der Schee 1312.2956 [hep-th]: J Casalderrey-Solana, MPH, D Mateos & W van der Schee 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/0512162] R Janik & R Peschanski 1011.3562 [hep-th] P Chesler & L Yaffe t z u 6/12 Gravitational shock wave solutions Janik & Peschanski [hep-th/0512162] Chesler & Yaffe 1011.3562 [hep-th] N c tt zz tz h(t ⌥ z) dual stress tensor: hT i = hT i = ±hT i = 2⇡ t shock wave disturbance moving with the speed of light z ds = (du2 + ⌘µ⌫ dxµ dx⌫ ) + u2 h(x )dx2 u Poincare patch vacuum AdS Solution of Einstein’s equations with the negative CC for any longitudinal profile h(x ) Let’s consider now e=⇢ h(t ± z) = ⇢ exp (in real HIC e ⇠ 1/2 ⇥ (t ± z) /2 u ⇤ But, in a CFT, what matters is: and eCY ⇡ 0.64 corresponds to Pb at RHIC) 7/12 ⇢t 1305.4919 [hep-th] PRL 111 (2013) 181601: J Casalderrey-Solana, MPH, D Mateos & W van der Schee 4 S�⇢ S�⇢ elef t = eCY eright = 0.125 eCY Dynamical crossover N c tt hT i = E 2⇡ „low energy” E�⇢4 „high energy” ⇢t ⇢t ⇢t E�⇢4 ⇢t ⇢z ⇢z ⇢z ⇢z PL �⇢4 loc PL �Eloc for PL �⇢4 deviation from viscous hydro FIG Energy flux for collisions of thick (left) and thin (right) shocks The dotted curves show the location of the maxima of ⇢z ⇢z the flux FIG 3 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 maximum of ⇢t the energy flux ⇢t ⇢t the energy flux in this region is less than 10% of the maxmodel [5] imum incoming flux, as⇢zillustrated by Fig 2(left) At ⇢zlate The thin shocks i times, the velocity of the receding shocks can be read o↵ ⇢z ⇢z 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 3 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 v � 1, as seen in Fig reaches a maximum of about v � 0.88 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 ⇢t ⇢t 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 T loc L loc Hydrodynamization in a shock wave collision 1305.4919 [hep-th] PRL 111 (2013) 181601: J Casalderrey-Solana, MPH, D Mateos & W van der Schee µ⌫ µ ⌫ µ⌫ µ⌫ hT i = {E + P(E)} u u + P(E) ⌘ + ⇧ Hydrodynamics: perfect fluid We use hT µ⌫ iu⌫ = hT tt i hT zz i hT ?? i E0 E0 E0 ⇣ CF T = ⌘ µ⌫ + dissipative E uµ and compare hT zz i and hT ?? i with hydro prediction thyd Thyd = 0.26 dotted: hydro prediction ⇢t at the collision axis (z = 0) Surprise: large anisotropy at the onset of hydrodynamics due to the shear tensor! see also 0906.4426 & 1011.3562 Chesler & Yaffe and 1103.3452 MPH, Janik & Witaszczyk 9/12 ⌘ The nature of hydrodynamics 1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, R A Janik & P Witaszczyk straightforward to check directly that (3.10) transforms homogeneously under Weyl transf mations MPH, R A Janik & P Witaszczyk 1302.0697 [hep-th] PRL 110part (2013) Thus, our final expression for the dissipative of 211602: the stress-energy tensor, up to seco Question: what is the order in derivatives, is nature of hydrodynamic gradient expansion? Hydrodynamic series at high orders Πµν = −ησ µν + ητΠ ⟨ Dσ µν ⟩ + σ µν (∇·u) + κ R⟨µν⟩ − (d − 2)uα Rα⟨µν⟩β uβ d−1 (3 + λ1 σ ⟨µ λ σ ν⟩λ + λ2 σ ⟨µ λ Ων⟩λ + λ3 Ω⟨µ λ Ων⟩λ + The five new constants are τΠ , κ, λ1,2,3 Note that using lowest order relations Πµν = −ησ Idea: use the fluid-gravity duality to compute subsequent gradient terms on-shell Eqs.(3.5) and Dη = −η ∇·u, Eq (3.11) may be rewritten in the form To make it operational, we used the boost-invariant flow Why? d Π = −ησ − τΠ DΠ + Πµν (∇·u) y t = ⌧ cosh y and z = d⌧ − sinh µν µν ⟨ µν ⟩ + κ R⟨µν⟩ − (d − 2)uα Rα⟨µν⟩β uβ (3 λ1 ⟨µ ν⟩λ λ2 ⟨µ ν⟩λ + Π λ Π − Π λ Ω + λ3 Ω⟨µ λ Ων⟩λ η η2 ⌧⌧ 1 rµ u ⌫ ⇠ u @µ = @⌧ and hT i = E(⌧ ) = Nc ⇡ T (⌧ ) ! T (⌧ ) T (⌧ ) ⌧ This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section µ InGradient the linearexpansion regime it actually coincides with Weinemphasi th order solving ODEs in the the Israel-Stewart bulk go theory to 240(6.1) grads however, that one cannot claim that Eq (3.12) 10/12captures all orders in the momentum expans field [13] In the leading order of the gr Note that, in the proper time - rapidity coordinates (1), sulting modes, on the gravity si there is no momentum flow in the stress tensor (2) and duce quasinormal R.to A the Janikscalar & P Witaszczyk 1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, so the flow velocity is trivial and takes the form u = ditional factor of 32 and are dam @⌧ Hydrodynamic constituent relations lead, then, to [14] Upon including viscous c gradient expanded energy density of the form tain a further nontrivial powerl 1/n ✏n ✓ ◆ X 2 00 1 12/3 n ↵ ✏2 ✏ = Nc ⇡ T 4/3= ✏(⌧ ✏2 +) ✏⇠ ✏n✏(⌧ ) , (T(4)1 rµ u⌫ ⇠ ⌧ 2/3 )✏ ⇠ ⌧ qnm exp ( i 2/3 + 4/3 + ⌧ ⌧ ⌧ n=2 Explicit gravity calculation fo where the choice of ✏2 sets an overall energy scale, in par!qnm = 3.1195 2.7467, ↵qnm ticular for the quasinormal frequencies (7) and 9) The prefactor was chosen to match the N = super YangatThelarge orders frequency !qnm agrees w Mills theory at large-Nc and strong coupling In the folat we lowchoose orders factorial growth of gradient lowest nonhydrodynamic scalar lowing, the units by setting ✏2 = ⇡ was calculated before in [14], w behavior is different with order Large-⌧ expansion of the energy density in powers ofcontributions ↵qnm is a new 1/2n result specific t ⌧ 2/3 , as in (4), is equivalent to the hydrodynamic gra(2⇡n) 1/n ⇠ of the expanding · n black dient expansion and arises from expressing gradients of (n!)cations SupplementaleMaterial for furt velocity (rµ u⌫ ⇠ ⌧ ) in units of the e↵ective temperalowing, we will be able to repro ture (T ⇠ ✏1/4 ⇠ ⌧ 1/3 ) The value of the coefficient ✏3 from the large order behavior of is related to the shear viscosity ⌘, whereas ✏4 is a sum Large order behavior of h of two transport coefficients: relaxation time ⌧⇧ and the density Numerical implementa so-called [10] Higher order contributions to the enlined in [11, 12] allow for efficien ergy density are expected to be linear combinations of so namic series given by (4), up to far unidentified transport coefficients Note also that the one is e↵ectively solving a set expansion (4) is sensitive to both linear and nonlinear from Einstein’s equations) at ea gradient terms methods we iteratively solved th As explained in [11, 12] (see also Supplemental maevidence terial), that hydrodynamic expansion a zero radius ofexpansion convergence! time reconstructing higher order contributions to thehas energy density the order 240, i.e up to the term (4) can be obtained by solving Einstein’s equations with of our knowledge this is the first a negative cosmological constant 11/12 for the metric ansatz of Hydrodynamic series at high orders First Summary Summary ! DPêE 1202.0981 [hep-th] PRL 108 (2012) 191601: 0.4 Tt 0.2 0.4 0.6 0.8 1.0 1.2 MPH, D Mateos, W van0.2 der Schee & D Trancanelli DPêE -5 -10 0.0 -15 E�⇢4 ⇢t ⇢t ⇢z PL �⇢4 0.4 0.6 0.8 1.0 1.2 Tt -0.2 -20 -25 0.2 ✏n 1/n ✏2 time predicted by the full and is the di↵erence between the isotropization -0.4 E�⇢4 Figure (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 ⇢z (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 ⇢t ⇢t ⇢z ⇢z 1305.4919 [hep-th] PRL 111 (2013) 181601: P �⇢ J Casalderrey-Solana, MPH, D Mateos & W van der Schee P �⇢ T 1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, R A Janik & P Witaszczyk T – 16 – 12/12 http://goo.gl/BBYh7J extra ... creation of quark-gluon plasma in Heavy Ion Collisions Hence I will consider the Poincare patch of AdS4 +1 states in a large-Nc CFT at strong coupling In this talk I will discuss solutions of Rab... moving with the speed of light z ds = (du2 + ⌘µ⌫ dxµ dx⌫ ) + u2 h(x )dx2 u Poincare patch vacuum AdS Solution of Einstein’s equations with the negative CC for any longitudinal profile h(x ) Let’s... maximum of about v � 0.88 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