Holography and off-center collisions of localized shock waves

Using numerical holography, we study the collision, at non-zero impact parameter, of bounded, localized distributions of energy density chosen to mimic relativistic heavy ion collisions, in strongly coupled N = 4 supersymmetric Yang-Mills theory. Both longitudinal and transverse dynamics in the dual field theory are properly described. Using the gravitational description we solve 5D Einstein equations, without dimensionality reducing symmetry restrictions, to find the asymptotically anti-de Sitter spacetime geometry. Implications of our results on the understanding of early stages of heavy ion collisions, including the development of transverse radial flow, are discussed.

Introduction.-Therecognition that the quark-gluon plasma (QGP) produced in relativistic heavy ion collisions is strongly coupled [1] has prompted much work using gauge/gravity duality (or "holography") to study aspects of non-equilibrium dynamics in strongly coupled N = 4 supersymmetric Yang-Mills theory (SYM), which may be viewed as a theoretically tractable toy model for real QGP.Work to date has involved various idealizations (boost invariance, planar shocks, imploding shells) which reduce the dimensionality of the computational problem, at the cost of eliminating significant aspects of the real collision dynamics [2][3][4][5][6][7].In this letter, we report the results of the first calculation of this type which does not impose unrealistic dimensionality reducing restrictions.We study the collision, at non-zero impact parameter, of bounded, localized distributions of energy density which mimic colliding relativistic Lorentz-contracted nuclei and form stable incoming projectiles in strongly coupled SYM.
In the dual gravitational description our initial state consists of two localized incoming gravitational waves, in asymptotically anti-de Sitter (AdS) spacetime, which will collide at non-zero impact parameter.The pre-collision geometry contains a trapped surface and the collision results in the formation of a black brane.We numerically solve the full 5D Einstein equations for the geometry after the collision and report on the evolution of the SYM stress-energy tensor T µν .
Gravitational formulation.-Weconstruct initial data for Einstein's equations by superimposing the metric of gravitational shock waves moving in the ±z directions at the speed of light.In Fefferman-Graham coordinates, the metric of a single shock moving in the ±z direction is where x ⊥ ≡ {x, y}, z ∓ ≡ z ∓ t, and The AdS curvature scale has been set to unity.The boundary of the asymptotically AdS spacetime lies at r = ∞.The single-shock metric ( 1) is an exact solution to Einstein's equations for any choice of H ± [8].This geometry represents a state in the dual SYM theory with stress-energy tensor [15] T (and all other components vanishing), where H ± is the 2D transverse Fourier transform of H ± .We choose H ± to be Gaussian, with transverse and longitudinal widths R = 4 and w = 1 2 , respectively, and impact parameter b = 3 4 R x.Hence, the stress tensor (3) describes localized lumps of energy centered about x = ±b/2, y = 0, and z = ±t.Our choice of H ± fixes units in the results presented below; the maximum longitudinally integrated energy density of a single shock has been set to unity.
For early times, wt 1, the Gaussian profiles H ± have negligible overlap and the pre-collision geometry can be constructed from (1) by replacing the last term with the sum of corresponding terms from left and right moving shocks.The resulting metric satisfies Einstein's equations, at early times, up to exponentially small errors.
To evolve the pre-collision geometry forward in time we use the characteristic formulation of gravitational dynamics in asymptotically AdS spacetimes discussed in detail in Ref. [9].Our metric ansatz reads with Greek indices denoting spacetime boundary coordinates, x µ = (t, x, y, z).Near the boundary, µν /r 4 + O(1/r 5 ).The sub-leading coefficients g (4) µν determine the SYM stress tensor, To generate initial data for our characteristic evolution, we numerically transform the pre-collision metric in Fefferman-Graham coordinates to the metric ansatz (5); this require computing the congruence of infalling radial null geodesics [16].We periodically compactify spatial directions (to obtain a finite computational domain) with transverse size L x = L y = 32 and longitudinal length L z = 12.We begin time evolution at t = −2 and evolve to t = 4.Time evolution is performed using a spectral grid of size N x = N y = 39, N z = 145 and N r = 40.Even though our gravitational dynamics is five dimensional, with no symmetry imposed, we are able to perform the time evolution on a 6 core desktop computer with a 14 day runtime.
Results.-In Fig. 1 we plot the energy density T 00 deposited in the interior region.
At sufficiently late times, according to fluid/gravity duality [10], the evolution of the stress tensor should be governed by hydrodynamics.To compare to hydrodynamics, we define the fluid velocity to be the normalized time-like (u µ u µ = −1) future directed (u 0 > 0) eigenvector of the stress tensor, with the proper energy density.Given the flow field u µ and energy density , we construct the hydrodynamic approximation to the stress tensor T µν hydro using the constitutive relations of first order viscous hydrodynamics.
In Fig. 2 we plot the stress tensor components T xx and T zz , and their hydrodynamic approximations T xx hydro and T zz hydro , at the spatial origin x = y = z = 0, as a function of time.At this point the stress tensor is diagonal, u = 0, and T xx and T zz are simply the pressures in the x and z directions.As shown in the figure, the pressures increase dramatically during the collision, reflecting a system which is highly anisotropic and far from equilibrium.After a time t ≈ 1.25 the pressures are well described by viscous hydrodynamics.Remarkably, at this time the transverse pressure T xx is nearly ten times larger than T zz .(This latter observation was also seen in 1 + 1 dimensional flow [2,3].) To quantify the domain in which hydrodynamics is applicable we define a residual measure with ∆T µν ≡ T µν − T µν hydro and p ≡ /3 the average pressure in the local rest frame.The quantity ∆, evaluated in the local fluid rest frame, measures the relative difference between the spatial stress in T µν and T µν hydro .Regions with ∆ 1 are evolving hydrodynamically.
In Fig. 3 we plot ∆ in the transverse plane at proper times τ = 1, 1.25, and 2, and rapidities ξ = 0 and 1.The color scaling is the same in all plots.Focusing first on ξ = 0 (top row), at τ = 1 one sees that ∆ 0.5 in the central region (x, y ≈ 0), and hydrodynamics is not a good description.However, by τ = 1.25 a fluid droplet with ∆ 0.15 has formed at transverse radii x ⊥ ≡ |x ⊥ | 5.3, whose subsequent evolution is well described by hydrodynamics.At τ = 2 the transverse size of the droplet has increased and ∆ < 0.15 for x ⊥ 8.6.Consider now the behavior at rapidity ξ = 1 (bottom row).For small x ⊥ at τ = 1, the system is already evolving hydrodynamically.Moreover, the onset of hydrodynamics occurs earlier for x < 0 than for x > 0. This feature reflects the fact that the receding maxima remain far from equilibrium and non-hydrodynamic, and (as seen in Fig. 1), the maxima with ξ > 0 lies at x > 0.
Interestingly, the inclusion of transverse dynamics seems to hasten the approach to local equilibrium: the equilibration time t hydro ∼ 1.25 is about 30% smaller than was the case in our previous studies [3,9] of planar shock collisions.Recent work [11,12] has found that equilibration time scales of far-from-equilibrium states can be understood, at least semi-quantitatively, in terms of the spectrum of quasinormal modes.Post-collision, a distribution of quasinormal modes will be excited.The decay of these modes controls the approach to equilibrium, with high (spatial) momentum modes decaying  8), is greater than 0.15 have been excised; within these regions the system is behaving non-hydrodynamically.The maximum of |v|, which occurs in the vicinity of the receding maxima, is 0.64.In contrast, the maximum transverse velocity in the x−y plane is 0.3.
faster than low momentum modes.With the inclusion of transverse dynamics, the typical transverse wavevector will be non-zero, presumably leading to a larger average momentum of excited modes than in planar collisions without transverse dynamics.Hence, it is natural to expect the inclusion of transverse dynamics to decrease the time, just as we observe.A striking feature of the the post-collision evolution in Fig. 1 is the appearance of flow in the transverse plane at early times.The early-time acceleration imparted on the transverse flow can have a significant impact on the subsequent transverse expansion.In Fig. 4 we plot the fluid 3-velocity v ≡ u/u 0 in the z−x, z−y, and x−y planes at time t = 4.The color scaling, which indicates |v|, is the same in each plot.The flow lines show the direction of v. Regions in which ∆ > 0.15, and the system is not behaving hydrodynamically, have been excised.Already at time t = 4 and radius x ⊥ ≈ 5 the transverse fluid velocity in the x−y plane has magnitude 0.3.In contrast, the maximum of the longitudinal velocity, which occurs in the neighborhood of the receding maxima, is 0.64.
One sees from Fig. 4 that the fluid velocity in the x−y plane is nearly radial: we see no strong signatures of elliptic flow.This should not be too surprising as the system has not been evolved through the entire hydrodynamic phase of the plasma.Additionally, in the z−x plane the fluid flow is not symmetric about the z axis and the longitudinal flow does not vanish at z = 0.The latter observation is a direct violation of the simplified model of boost invariant flow, where v z = z/t and vanishes at z = 0. Nevertheless, at t = 4 and in the region of space where > 0.6 max( ), the longitudinal flow is roughly described by boost invariant flow at the 20% level or better, with larger deviations appearing at larger rapidities [17].For planar shock collisions, the deviation of the longitudinal fluid velocity from boost invariant flow decreases as Also shown is the approximate form (9) which, for small rapidity, agrees quite well with the full results.
the shock thickness decreases [4, 9] [18].It will be interesting to see if this also holds when transverse dynamics is included.We conclude by discussing the early-time transverse flow predicted in Ref. [13].There, using assumptions of boost invariance and transverse plane rotational symmetry, it is argued that at early times the transverse energy flux is proportional to the gradient of the energy density and grows linearly with time, In Fig. 5 we plot the angular averaged radial flow T 0⊥ ≡ xi ⊥ T 0i , together with the approximation (9), at proper times τ = 1.25 and 2, and rapidities ξ = 0 and 1.The approximation (9) works remarkably well at both times and rapidities, although the agreement is not quite as good at ξ = 1 where the assumption of boost invariance is more strongly violated.It would be interesting to see if the agreement with the approximation (9) improves when the shock thickness decreases.
We thank Wilke van der Schee for discussions of prior work.The work of PC is supported by the Fundamental Laws Initiative of the Center for the Fundamental Laws of Nature at Harvard University.The work of LY was supported, in part, by the U.S. Department of Energy under Grant No. DE-SC0011637.He thanks the Alexander von Humboldt Foundation and the University of Regensburg for their generous support and hospitality during portions of this work.

FIG. 1 :
FIG.1:The energy density T 00 (top) and energy flux |T 0i | (bottom), at four different times, in the plane y = 0. Streamlines in the lower plots denote the direction of energy flux.At the initial time t = −2 the shocks are at z = ±2.The non-zero impact parameter in the x-direction is apparent.The shocks move in the ±z direction at the speed of light and collide at t = z = 0.After the collision the remnants of the initial shocks, which remain close to the lightcone z = ±t, are significantly attenuated in amplitude with the extracted energy deposited in the interior region.Note the appearance of transverse flow at positive times.

(FIG. 2 :
FIG.2: Stress tensor components T xx and T zz at the spatial origin, x = y = z = 0, as a function of time.Dashed lines denote the viscous hydrodynamic approximation.Around t = 0 the system is highly anisotropic and far from equilibrium.Nevertheless, at this point, the system begins to evolve hydrodynamically at t ≈ 1.25.

FIG. 3 :
FIG.3:The residual ∆ in the transverse plane, at several proper times τ rapidities ξ. with ∆ 1 are evolving hydrodynamically.At ξ = 0 (top row) the central region becomes hydrodynamic at τ ≈ 1.25, whereas at ξ = 1 (bottom row) hydrodynamic behavior of the central region has already begun by τ ≈ 1.At ξ = 1, hydrodynamic behavior first sets in at x < 0. This feature reflects the fact that the receding maxima remain far from equilibrium and non-hydrodynamic, and the maxima with ξ > 0 lies at x > 0.

6 FIG. 4 :
FIG. 4: The fluid 3-velocity |v| at time t = 4, in the z−x, z−y, and x−y planes.Streamlines denote the direction of v. Regions in which the residual ∆, defined in Eq. (8), is greater than 0.15 have been excised; within these regions the system is behaving non-hydrodynamically.The maximum of |v|, which occurs in the vicinity of the receding maxima, is 0.64.In contrast, the maximum transverse velocity in the x−y plane is 0.3.

2 ξFIG. 5 :
FIG. 5:The average lab-frame radial energy flux, as a function of x ⊥ , at two values each of proper time and rapidity.Also shown is the approximate form (9) which, for small rapidity, agrees quite well with the full results.