Dynamical Spacetimes from Numerical Hydrodynamics

MIT-CTP-4607

Allan Adams,1 Nathan Benjamin,1, 2 Arvin Moghaddam,1, 3 and Wojciech Musial1, 3 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 2Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305 3Center for Theoretical Physics, University of California, Berkeley, Berkeley, CA 94720

We numerically construct dynamical asymptotically-AdS4 metrics by evaluating the ﬂuid/gravity metric on numerical solutions of dissipative hydrodynamics in (2+1) dimensions. The resulting numerical metrics satisfy Einstein’s equations in (3+1) dimensions to high accuracy.

Holography provides a precise relationship between Review of Fluid/Gravity black holes in AdSd+1 and QFTs in d-dimensions at finite density and temperature. When the QFT state lies We begin by recalling the essential features of the in a hydrodynamic regime (i.e. when d-dimensional gra- fluid/gravity correspondence for a (2+1) fluid as given dients are sufficiently small that the stress-tensor may in [1–4]. At equilibrium, a fluid moving with constant 3- µ be expanded as a power-series in derivatives), the dual velocity u at temperature T is dual to an asymptotically spacetime metric may also be so expanded, leading to an AdS4 black brane described by the metric [10] analytic “fluid/gravity” map between solutions of hydro- ds2 = −2u dxµdr−r2f(br) u u dxµdxν +r2P dxµdxν , dynamics in d dimensions and dynamical asymptotically- µ µ ν µν AdSd+1 solutions of the Einstein equations. 3 where b = 4πT is the rescaled inverse temperature, This suggests a simple strategy for constructing nu- P µν =ηµν +uµuν projects onto directions transverse to merical solutions of the Einstein equations in asymptot- µ 1 u , and f(ρ) = 1 − ρ3 is the emblackening factor. It is ically AdSd+1 spacetimes: rather than solve the (d+1)- readily checked that this metric solves the Einstein equa- dimensional Einstein equations numerically, we may nu- tions so long as uµ and T are constant. merically solve d-dimensional hydrodynamics and use the If uµ(x) and T (x) vary in space and time, it follows fluid/gravity map to analytically construct the corre- that this metric is again a solution of the Einstein equa- sponding metric. So long as the fluid flow lies sufficiently tions when expanded to leading (trivial) order in gra- deep in the hydro regime, the hydro equations can be dients. When all gradients are small, it is possible to truncated at a desired order in the gradient expansion; systematically improve the metric order-by-order in gra- the resulting fluid/gravity metric is then an analytic so- dients to construct a solution of the (d+1)-dimensional lution of the Einstein equations up to errors of corre- Einstein equations provided that uµ(x) and T (x) solve sponding order. Na¨ıvely, this should be a much easier the equations of d-dimensional hydrodynamics. calculation, since one need only solve PDEs in 3d, rather Demonstrating this is simplified by working in a gauge than 4d. The main question in principle is whether the in which the nth-order corrections take the form, resulting algorithm is sufficiently robust to small depar- 2 (n) µ 2 µ ν tures from the true hydro solution and numerical errors ds(n) = −h (2uµdx dr + r Pµν dx dx ) to be useful. In practice, this will also serve as an inde- 2 + k(n)u u dxµdxν + j(n) u dxµdxν + α(n)dxµdxν . pendent check of the analytic results in the literature. µ ν ν r µ µν In this paper we construct numerical solutions to 2nd order dissipative relativistic hydrodynamics, evalu- Upon expanding to nth-order in gradients, the rr and ate the fluid/gravity metric on the resulting flows, and µν components of the Einstein equation reduce to linear arXiv:1411.2001v1 [hep-th] 7 Nov 2014 test the satisfaction of the Einstein equations on the equations for the functions h(n), k(n), j(n) and α(n), corresponding dynamical spacetimes. Implementing the 1 d d fluid/gravity map revealed minor typos in the literature r4 h(n)(r, x) = S(n)(r, uµ(x)) r4 dr dr h which we corrected by re-deriving the fluid/gravity ana- d 2 lytic map.[9] The resulting numerical metrics satisfy the − k(n)(r, x) + (1 − 4r3)h(n)(r, x) = S(n)(r, uµ(x)) Einstein equations with great precision. Concretely, we dr r k find that when the flow is hydrodynamic, the 0th order r d 1 d j(n)(r, x) = S(n)(r, uµ(x)) fluid/gravity metric provides good approximation of a so- 2 dr r2 dr ν jν lution to the Einstein equations for a wide range of fluid d 1 d flows within the hydro regime, and that the 1st and 2nd − r4f(r) α(n)(r, x) = S(n) (r, uµ(x)) , dr 2 dr µν αµν order corrections improve the accuracy of this fluid metric appropriately, leading to accurate numerical metrics (n) where the “source” functions S (r, uµ(x)) may be ex- computed at low computational expense. ∗ plicitly computed order-by-order in the gradient expansion and depend only on n or fewer derivatives. These 2 equations are manifestly local in the fluid dimensions, x, Eq (3) of [1]. Since we work with a 2nd order stress ten- so solving them reduces to a series of numerically-simple sor, the resulting conservation equations are third order 1-dimensional integration problems. in time-derivatives of the fluid variables. To avoid spuri- (1) ous solutions and simplify our calculation, we treat the At first order, the sources take simple forms, Sh = 0, µν (1) (1) 1 (1) dissipative stress tensor Π as an independent dynami- S = −4r∇·β, S = − ∂tβi, and Sα = 2rσij, where k ji r ij cal variable whose evolution is determined by the consti- σ = 1 ∇ β + ∇ β − 1 δ ∇·β, β is the fluid velocity ij 2 i j j i 2 ij i tutive relations a la Israel-Stewart. So long as we remain in the local rest frame at position x, β (x) = 0, and i, j i within the hydrodynamic regime, solutions of the result- label the spatial dimensions.[11] ing equations should relax toward solutions of the full Given these sources, the equations above can be inte- equations.[13] We use the satisfaction of the constitutive grated along the radial direction to give [12]: h(1) = 0, relations as a check on our numerical solutions. (1) 1 (1) (1) 3 (1) (1) 1 (1) k = − S , jµ = −r S , and αµν = Sα F (r), µν 4 k jµ r µν Upon fixing Landau gauge (uµΠ =0) and demanding where conformal invariance (Πµ =0), only 2 components of Πµν √ µ −1 2r + 1 1 1 + r + r2 3π remain linearly independent. A convenient parameteri- F (r) = √ Tan−1 √ + log + , zation for these two dissipative variables involves setting 3 3 2 r2 6 Π = (2 + u2 + u2)Π , as presented in [1]. xy x y 2 At second order the sources are somewhat more cum- Πxx = 2(1 + ux) Σ + 2uxuy Π . bersome, so we refer the reader to [1] whose 2nd order sources we have analytically verified modulo minor typos All other components of Πµν may then be determined in (2) terms of regular functions of uµ times Π and Σ. in S and F2, which we find take the values, k The hydro equations then take the form M(u)u ˙ = b(u), 1 1 + 4r3 where u is a vector of our five variables (ux, uy, T ,Σ S(2) = 2S + S − S + F (r) S , k 3 2 5 2r3 6 2 7 and Π), while the 5 × 5 matrix M(u) and the 5-vector 2(1 + r)(1 − 4r3) −1 + 2r + 4r2 + 4r3 b(u) are non-linear functions of u and its spatial (but not F = F (r) + . time) derivatives. To determine the time-derivative of 2 r(1 + r + r2) r(1 + r + r2) our fields at a given point in space, we first evaluate the While the resulting metric satisfies the rr and µν com- required spatial derivatives of our fields, compute M and ponents of the Einstein equations, the rµ components of b, and then numerically solve this 5 × 5 matrix equation. the Einstein equations impose a further set of d condi- Having computed the time-derivatives, we propagate the tions. These constraints are equivalent to the conserva- solution forward in time via standard techniques. tion of the fluid stress tensor built from uµ(x) and T (x) In our numerical computations we fix periodic bound- at the corresponding order in the gradient expansion with ary conditions in both spatial directions with period a specific set of transport coefficients [1,2]. Thus, given L = 1500, with the initial temperature set to a constant 3 µ a solution u (x) and T (x) of the hydro equations of mo- T (0) = 4π . We represent all fields pseudospectrally in a tion, we can simply plug this flow into the fluid-gravity fourier basis of 306×306 plane waves, computing spatial metric equations and, upon solving a set of 1-dimensional derivatives spectrally and propagating the system for- ODEs along the radial direction, construct a numerical ward in time using Matlab’s built-in general-purpose in- metric which satisfies the full Einstein equations to the tegrator, ode45. We focus on two classes of initial condi- appropriate order in the fluid gradients. tions: a superposition of the 1000 lowest-frequency plane waves with randomized amplitudes and phases; and, fol- Numerical Methods lowing [5], a line of 10 vorticity stripes generated by the 2π velocity field uy = cos(5 L x), to which we add small per- Our calculation is naturally divided into two steps. We turbations comprised of the five lowest-frequency plane first construct numerical solutions of the hydrodynamic waves with small random amplitudes and random phases. equations; we then evaluate the fluid/gravity metric on We now turn to evaluating the fluid/gravity metric µ these solution. We estimate our errors by evaluating the gmn(x, r) on a solution u (x) and T (x) of the hydro equa- Einstein equations on the resulting numerical metric. tions. We represent the metric components spectrally. Solving the relativistic hydrodynamic equations is by Along each boundary spatial dimension we again expand now relatively standard. In principle, for constructing in a basis of 306 fourier modes. Along the bulk radial the 2nd order metric we need only solve relativistic hydro dimension we work with the coordinate z = 1/r which at 1st order; in practice, however, 1st order hydro is dy- extends from z = 0 (the AdS boundary) to z = 2 (well namically unstable. We regulate this instability by solv- inside the apparent horizon at z = b). We expand our ing the full 2nd order hydro, using for good measure the fields in a basis of 32 Chebyshev modes along z. Integra- (2) (2) (2) specific transport coefficients derived in the fluid/gravity tion of the equations for h , k and ji is performed analysis. The corresponding stress tensor is given in analytically wherever possible and otherwise spectrally, 3

fluid in the hydro regime, with visible transfer of spectral weight from high to low wave numbers in a classic inverse cascade. Figures 2 and 3 plot Eloc(t) and E¯(t) respectively as a function of time for the 0th, 1st and 2nd order metrics. On the left are the results for the random-wave initial condition; on the right, the vorticity stripe. As is apparent, the 0th order metric is already a good approximation, the 1st order metric further reduces this error considerably, with the 2nd order corrections giving us an extremely accurate numerical solution of the bulk Einstein equations. At the bottom of each plot is a measure of the satisfaction of the constitutive relations. Errors in the constitutive FIG. 1: Snapshots of the vorticity along the flow for both relations are strongly correlated with errors in the bulk classes of initial conditions at time 1000, 2000 and 3000. Top: metric, as they should: the constitutive relations arise weakly-perturbed vorticity stripe initial conditions rapidly de- from the asymptotic radial constraint equations of the cay via a classic two-stream instability into a slowly-relaxing bulk Einstein equations. quasi-turbulent state. Bottom: random-wave initial condi- To summarize, the fluid/gravity correspondence pro- tions gradually fade away. Red indicates positive vorticity, blue negative and black zero. vides a novel, robust and fast algorithm for constructing non-equilibrium asymptotically-AdS numerical solutions of the Einstein equations. As we have seen, this approach with the required boundary conditions imposed by sub- is quantitatively effective even when the dynamics drive (2) the system nonlinear and turbulent, so long as typical traction. The αij equation, however, must be solved as a 2-point boundary value problem due to the necessity gradients remain bounded so that the hydrodynamic ex- of imposing one boundary condition (regularity) at the pansion is reliable. To go beyond the hydro regime, or to horizon and another at the boundary, at both of which test the fluid/gravity correspondence as one approaches points the equation is degenerate. This is done via a the hydro regime, requires direct numerical solution of single inversion of the boundary-blocked linear operator the Einstein equations, as in [5–7]. Nonetheless, the ease appearing in the α(2) equation which is then multiplied and efficiency of this approach makes it a useful tool for (n) gravitational questions within the hydro regime. against the bordered source Sα . The Einstein tensor is then evaluated on the resulting metric in the same spectral basis, with time derivatives of relevant GR tensors Acknowledgments th computed using a 4 order finite differencing scheme. We thank Lincoln Carr, Paul Chesler, Ethan Dyer, Jae 1 Given an exact solution, Emn ≡ Gmn − 2 Λgmn would Hoon Lee, Luis Lehner, Hong Liu, R. Loganayagam and vanish point-wise. We thus use Emn evaluated on the Mark Van Raamsdonk for helpful discussions. AA thanks fluid metric as a measure of the accuracy of our solution. the organizers of the “Cosmology and Complexity 2012” More precisely, we estimate the maximum local error, and “Relativistic hydrodynamics and the gauge-gravity µν duality at the Technion” workshops where this work has Eloc(t) = max |E | , V,µ,ν been discussed, and the Aspen Center for Physics for hos- pitality. The work of AA is supported by the U.S. De- where V is the spatial computational domain, as well as partment of Energy under grant Contract Number DE- the global RMS error, SC00012567. The work of NB, AM and WM was sup- √ R g (Em )2 ported by MIT’s Undergraduate Research Opportunities ¯ 2 V m E(t) = R √ . Program. V g The value of these observables represents an estimate of the error in our numerical metric. Note that these quanti- ties scale as two powers of space-time gradients at leading [1] Mark Van Raamsdonk. Black Hole Dynamics From At- order in the gradient expansion for the metric. mospheric Science. JHEP, 0805:106, 2008, 0802.3224. [2] Sayantani Bhattacharyya, Veronika E Hubeny, Shiraz Results and Conclusions Minwalla, and Mukund Rangamani. Nonlinear Fluid Dy- namics from Gravity. JHEP, 0802:045, 2008, 0712.2456. Figure 1 displays the vorticity of the fluid at three mo- [3] Sayantani Bhattacharyya, R. Loganayagam, Ipsita Man- ments along the numerical evolution of two typical fluid dal, Shiraz Minwalla, and Ankit Sharma. Conformal flows, one from each class of initial conditions. The re- Nonlinear Fluid Dynamics from Gravity in Arbitrary Di- sulting flows behave much as one expects of a normal 2d mensions. JHEP, 0812:116, 2008, 0809.4272. 4

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FIG. 2: Log base 10 error of the 0th (blue), 1st (red) and 2nd (green) order fluid/gravity metrics as measured by the maximum absolute value of the Einstein equation as a function of time. Left: random-wave initial conditions. Right: weakly-perturbed vorticity stripe initial conditions. The accuracy of the metric is excellent in both cases, converging as expected at subsequent orders. Note that the large increase in error around times 700 and 1000 for the bottom graph correspond to moments where the fluid gradients become unusually large. The bottom plots indicate the convergence of the constitutive relations as a function of time along the corresponding flow.

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FIG. 3: Log base 10 error of the 0th (blue), 1st (red) and 2nd (green) order ﬂuid/gravity metrics as measured by the average RMS value of the Einstein equation as a function of time. Left: random-wave initial conditions. Right: weakly-perturbed vorticity stripe initial conditions.

[4] Veronika E. Hubeny, Shiraz Minwalla, and Mukund viscous hydrodynamics, conformal invariance, and holog- Rangamani. The fluid/gravity correspondence. 2011, raphy. JHEP, 0804:100, 2008, 0712.2451. 1107.5780. [9] We thank Jae-Hoon Lee and Mark Van Raamsdonk for [5] Allan Adams, Paul M. Chesler, and Hong Liu. Holo- discussions on this point. graphic turbulence. Phys.Rev.Lett., 112:151602, 2014, [10] Throughout this paper, m, n ∈ {0, .., 3} denote bulk 3+1 1307.7267. indices, µ, ν ∈ {0, 1, 2} denote boundary 2+1 indices, [6] Paul M. Chesler and Laurence G. Yaffe. Horizon forma- while i, j ∈ {1, 2} denote boundary spatial indices. tion and far-from-equilibrium isotropization in supersym- [11] Note that we have fixed to a local co-moving frame, which metric Yang-Mills plasma. Phys.Rev.Lett., 102:211601, simplifies the calculations; it is straightforward to pro- 2009, 0812.2053. mote these relations to covariant expressions. [7] Paul M. Chesler and Laurence G. Yaffe. Holography [12] Appropriate boundary conditions follow from regularity, and colliding gravitational shock waves in asymptoti- renormalizability and our choice of gauge, as explained cally AdS5 spacetime. Phys.Rev.Lett., 106:021601, 2011, in [2]. 1011.3562. [13] See for example [8] and references therein. [8] Rudolf Baier, Paul Romatschke, Dam Thanh Son, An- drei O. Starinets, and Mikhail A. Stephanov. Relativistic