Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma
HHorizon formation and far-from-equilibrium isotropizationin supersymmetric Yang-Mills plasma
Paul M. Chesler and Laurence G. Yaffe
Department of Physics, University of Washington, Seattle, WA 98195, USA (Dated: October 24, 2018)Using gauge/gravity duality, we study the creation and evolution of anisotropic, homogeneousstrongly coupled N = 4 supersymmetric Yang-Mills plasma. In the dual gravitational description,this corresponds to horizon formation in a geometry driven to be anisotropic by a time-dependentchange in boundary conditions. Introduction .—The realization that the quark-gluonplasma (QGP) produced at RHIC is strongly coupled[1] has prompted much interest in the study of stronglycoupled non-Abelian plasmas. Hydrodynamic simula-tions of heavy ion collisions have demonstrated that theQGP produced at RHIC is well modeled by near-idealhydrodynamics [2], which is a signature of a stronglycoupled system. The success of hydrodynamic mod-eling of RHIC collisions suggests that the producedplasma locally isotropizes over a time scale τ iso (cid:46) N = 4 supersymmetric Yang-Mills theory (SYM),where one can use gauge/gravity duality to study the the-ory in the limit of large N c and large ’t Hooft coupling λ .This has motivated much work devoted to studying var-ious non-equilibrium properties of thermal SYM plasma.We are interested in exploring the physics of isotropiza-tion in far-from-equilibrium non-Abelian plasmas, in thesimplest setting which allows complete theoretical con-trol. This leads us to focus on the dynamics of ho-mogeneous, but anisotropic, states in strongly coupled,large N c SYM. A conceptually simple way to create non-equilibrium states is to turn on time-dependent back-ground fields coupled to operators of interest. To cre-ate states in which the stress tensor is anisotropic, it isnatural to consider the response of the theory to a time-dependent change in the spatial geometry. For simplic-ity, we limit attention to geometries which have spatialhomogeneity ( i.e. , translation invariance in all spatial di-rections), an O (2) rotation invariance, and a constantspatial volume element. The most general metric satis-fying these conditions may be written as ds = − dt + e B ( t ) d x ⊥ + e − B ( t ) dx || , (1)where x ⊥ ≡ { x , x } .The function B ( t ) describes a time-dependent shear inthe geometry; neglecting (4-dimensional) gravity, B ( t ) is a function one is free to choose arbitrarily. We willchoose B ( t ) to be asymptotically constant as t → ±∞ .We will also choose the initial state to be the SYM vac-uum. A time-dependent geometry will do work on thequantum system. Consequently, the state in the distantfuture will be a non-vacuum state which (when the geom-etry is once again static) will be indistinguishable froma thermal state. During the evolution, because the met-ric (1) changes in an anisotropic fashion, the resultingplasma will also be anisotropic with different pressures( i.e. , stress tensor eigenvalues) in the longitudinal ( x || )and transverse ( x ⊥ ) directions. Spatial translation in-variance implies that no hydrodynamic modes can be ex-cited. Therefore, the non-equilibrium plasma producedby the changing metric (1) provides a nice laboratoryto study the relaxation of non-hydrodynamic degrees offreedom in a far from equilibrium setting. We choose B ( t ) = c [1 − tanh( t/τ )] . (2)For c (cid:54) = 0, this represents a time-dependent rescaling oflengths in transverse directions relative to those in thelongitudinal direction, over a period of order τ . The lackof any other scale in conformally invariant SYM impliesthat the final state energy density will be O ( τ − ). With-out loss of generality we measure all quantities in unitswhere τ = 1. Gravitational description. — Gauge/gravity duality [4]provides a gravitational description of large N c SYM inwhich the 5 d dual geometry is governed by Einstein’sequations with a cosmological constant. Einstein’s equa-tions imply that the boundary metric g B µν ( x ), which char-acterizes the geometry of the spacetime boundary, is dy-namically unconstrained. The specification of the bound-ary metric serves as a boundary condition for the 5 d Ein-stein equations. This reflects the fact that the dual fieldtheory (which resides on the boundary) does not back-react on the boundary geometry, whereas the boundarygeometry can influence the dynamics of the field theory.We consider a 5 d geometry which coincides with AdS in the distant past. This geometry is dual to the vacuumof SYM. A time dependent boundary metric g B µν ( x ) willcreate gravitational radiation which propagates from theboundary into the bulk. This infalling gravitational radi-ation will lead to the formation of a horizon , which acts a r X i v : . [ h e p - t h ] J u l as an absorber of gravitational radiation — any radiationwhich passes through the horizon cannot escape back tothe boundary. At late times when the boundary geom-etry is no longer changing, the bulk geometry outsidethe horizon will relax and asymptotically become static.This is the gravitational description of thermalization inSYM.Diffeomorphism and translation invariance allows oneto chose the metric ansatz ds = − A dv + Σ (cid:2) e B d x ⊥ + e − B dx || (cid:3) + 2 dr dv , (3)where A , B , and Σ are all functions of the radial coor-dinate r and time v only. Infalling radial null geodesicshave constant values of v (as well as x ⊥ and x || ). Out-going radial null geodesics satisfy dr/dv = A . At theboundary, located at r = ∞ , the coordinate v coincideswith the boundary time t . The geometry in the bulk at v > t > r → r + f ( v ) , where f ( v ) isan arbitrary function.With a metric of the form (3), Einstein’s equations maybe reduced to the following set of differential equations:0 = Σ ( ˙Σ) (cid:48) + 2Σ (cid:48) ˙Σ − , (4a)0 = Σ ( ˙ B ) (cid:48) + (cid:0) Σ (cid:48) ˙ B + B (cid:48) ˙Σ (cid:1) , (4b)0 = A (cid:48)(cid:48) + 3 B (cid:48) ˙ B − (cid:48) ˙Σ / Σ + 4 , (4c)0 = ¨Σ + (cid:0) ˙ B Σ − A (cid:48) ˙Σ (cid:1) , (4d)0 = Σ (cid:48)(cid:48) + B (cid:48) Σ , (4e)where, for any function h ( r, v ), h (cid:48) ≡ ∂ r h, ˙ h ≡ ∂ v h + A ∂ r h . (5)Eqs. (4d) and (4e) are constraint equations; the radialderivative of Eq. (4d) and the time derivative of Eq. (4e)are implied by Eqs. (4a)–(4c).The above set of differential equations must be solvedsubject to boundary conditions imposed at r = ∞ . Therequisite condition is simply that the boundary metric g B µν ( x ) coincide with our choice (1) of the 4 d geometry.In particular, we must havelim r →∞ Σ( r, v ) /r ≡ , lim r →∞ B ( r, v ) ≡ B ( v ) . (6)One may fix the residual diffeomorphism invariance bydemanding thatlim r →∞ (cid:2) A ( r, v ) − r (cid:3) /r = 0 . (7)These boundary conditions, plus initial data satisfyingthe constraint (4e) on some v = const . slice, uniquelyspecify the subsequent evolution of the geometry.Given a solution to Einstein’s equations, the SYMstress tensor is determined by the near-boundary be-havior of the 5 d metric [5] . If S G denotes the gravi-tational action, then the SYM stress tensor is given by T µν ( x ) = (2 / (cid:112) − g B ( x )) δS G /δg B µν ( x ) . Near the boundary one may solve Einstein’s equationswith a power series expansion in r . Specifically, A , B andΣ have asymptotic expansions of the form A ( r, v ) = (cid:88) n =0 [ a n ( v ) + α n ( v ) log r ] r − n , (8a) B ( r, v ) = (cid:88) n =0 [ b n ( v ) + β n ( v ) log r ] r − n , (8b)Σ( r, v ) = (cid:88) n =0 [ s n ( v ) + σ n ( v ) log r ] r − n . (8c)The boundary conditions (6) and (7) imply that b ( v ) ≡ B ( v ), s ( v ) ≡ a ( v ) ≡
1, and a ( v ) ≡
0. Substitut-ing the above expansions into Einstein’s equations andsolving the resulting equations order by order in r , onefinds that there is one undetermined coefficient, b ( v ).All other coefficients are determined by the boundarygeometry, Einstein’s equations, and b ( v ) [10].By substituting the above series expansions into thevariation of the on-shell gravitational action, one maycompute the expectation value of the stress tensor interms of the expansion coefficients. This procedure hasbeen carried out in Ref. [5], so we simply quote the re-sults. In terms of the expansion coefficients, the SYMstress tensor reads T µν = ( N c / π ) diag( −E , P ⊥ , P ⊥ , P || ) , (9)where (with b ( k )0 ≡ ∂ kv b ): −E = a + (cid:104) b (1)0 ) + 14( b (2)0 ) − b (1)0 b (3)0 (cid:105) , (10a) P ⊥ = − a + b + (cid:104) b (1)0 ) − b (1)0 ) b (2)0 + 10( b (2)0 ) + 4 b (1)0 b (3)0 + 64 b (4)0 (cid:105) , (10b) P || = − a − b + (cid:104) b (1)0 ) + 936( b (1)0 ) b (2)0 + 10( b (2)0 ) + 4 b (1)0 b (3)0 − b (4)0 (cid:105) . (10c) Numerics. —One may solve the Einstein equations(4a)–(4c) for the time derivatives ˙Σ, ˙ B , and A (cid:48)(cid:48) . DefineΘ( r, v ) ≡ (cid:90) ∞ r dw (cid:2) Σ( w, v ) − h ( w, v ) (cid:3) − H ( r, v ) , (11a)Φ( r, v ) ≡ (cid:90) ∞ r dw (cid:104) w, v ) B (cid:48) ( w, v ) Σ( w, v ) − / − h ( w, v ) (cid:105) − H ( r, v ) , (11b)where H n is an indefinite (radial) integral of h n , h n = H (cid:48) n . (12)Then Eqs. (4a)–(4c) are solved by˙Σ = −
2Θ Σ − , (13a)˙ B = − ΦΣ − / , (13b) A (cid:48)(cid:48) = − −
24Θ Σ (cid:48) Σ − + Φ B (cid:48) Σ − / . (13c)The functions h n ( r, v ) are not constrained by Einstein’sequations — their presence inside the integrands ofEq. (11) are compensated by the subtraction of theirintegrals H n ( r, v ). However, since we have chosen theupper limit of integration in Eq. (11) to be r = ∞ , thefunctions h n ( r, v ) must be suitably chosen so that theintegrals (11) are convergent. The simplest choice to ac-complish this is to set h ( r, v ) equal to the asymptoticexpansion of Σ( r, v ) up to order 1 /r k , for some k > h ( r, v ) equal to the asymptotic expansion of2Θ( r, v ) B (cid:48) ( r, v ) / Σ( r, v ) / up to order 1 /r k . In our nu-merical solutions reported below, we use k ≥
4. Thischoice makes the large r contribution to the integrals inEq. (11) quite small. As the coefficients of the series ex-pansions (8) only depend on b ( v ) and b ( v ) and their v derivatives, this choice determines h n ( r, v ) in terms ofone unknown function b ( v ).With the subtraction functions h n specified by theaforementioned asymptotic expansions, integrating Eq.(12) fixes the compensating integrals H n up to an in-tegration constant which is an arbitrary function of v .Integrating Eq. (13c) for A ( r, v ) introduces two further( v dependent) constants of integration. The most directroute for fixing these constants of integration is to matchthe large r behavior of the expressions (13a) and (13b)and the integrated version of Eq. (13c) to the correspond-ing expressions obtained from the series expansions (8).This fixes all integration constants in terms of b and b .Our algorithm for solving the initial value problemwith time dependent boundary conditions is as fol-lows. Given an initial geometry defined by B ( r, v ),one knows b ( v ). Integrating the constraint equation(4e), with the fully determined asymptotic behavior (8c),yields Σ( r, v ). From this information, one can com-pute A ( r, v ) by integrating Eq. (13c). With A ( r, v ), B ( r, v ) and Σ( r, v ) known, one can then compute thetime derivative ∂ v B ( r, v ) from Eq. (13b) and step for-ward in time, B ( r, v + ∆ v ) ≈ B ( r, v ) + ∂ v B ( r, v ) ∆ v . (14)Repeating the above process using this updated profileof B determines Σ and A at time v + ∆ v , from whichone computes ∂ v B for the next time step. For an initialgeometry corresponding to the SYM vacuum, plus thechoice (2) of boundary data, one has B ( r, −∞ ) = c , Σ( r, −∞ ) = r , A ( r, −∞ ) = r . (15)An important practical matter is fixing the computa-tion domain in r — how far into the bulk does one wantto compute the geometry? If a horizon forms, then onemay excise the geometry inside the horizon as this re-gion is causally disconnected from the geometry outsidethe horizon. Furthermore, one must excise the geome-try to avoid singularities behind horizons [6] . To per-form the excision, one first identifies the location of an apparent horizon (an outermost marginally trapped sur-face) which, if it exists, must lie inside a true horizon[7] . We have chosen to make the incision slightly insidethe location of the apparent horizon. For the metric (3),the location r h ( v ) of the apparent horizon is given by˙Σ( r h ( v ) , v ) = 0 or, from Eq. (13a), Θ( r h ( v ) , v ) = 0 . Results and Discussion. —Fig. 1 shows a plot of theenergy density and transverse and longitudinal pressuresproduced by the changing boundary geometry (1), with c = 2. These quantities begin at zero in the distant pastwhen the system is in its vacuum state, and at late timesapproach thermal equilibrium values given by T µν eq = ( π N c T /
8) diag(3 , , , , (16)where T is the final equilibrium temperature. Non-monotonic behavior is seen when the boundary geometrychanges most rapidly around time zero [11]. FIG. 1: Energy density, longitudinal and transverse pressure,all divided by N / π , as a function of time for c = 2. Fig. 2 displays the congruence of outgoing radial nullgeodesics, for c = 2. The surface coloring shows A/r .In the SYM vacuum ( i.e. , at early times) this quantityequals 1, while at late times A/r = 1 − ( r h /r ) . Ex-cised from the plot is a region of the geometry behindthe apparent horizon. In the SYM vacuum, outgoinggeodesics are given by 1 /r + v/ . , and appear asstraight lines in the early part of Fig. 2 . In the vicin-ity of v = 0, when the boundary geometry is changingrapidly and producing infalling gravitational radiation,the geodesic congruence changes dramatically from thezero temperature form to a finite temperature form. Asis evident from the figure, at late times some outgoinggeodesics do escape to the boundary, while others fallinto the bulk and never escape. Separating the ‘escap-ing’ and ‘plunging’ geodesics is one geodesic which doesneither — this geodesic, shown as the black line in Fig. 2,defines the true event horizon of the geometry.Fig. 3 plots the area of the apparent and true eventhorizons, again for c = 2. Nearly all growth of the ap-parent horizon area occurs in the interval − < v < FIG. 2: The congruence of outgoing radial null geodesics.The surface coloring displays
A/r . The excised region isbeyond the apparent horizon, which is shown by the dashedgreen line. The geodesic shown as a solid black line is theevent horizon; it separates geodesics which escape to theboundary from those which cannot escape.FIG. 3: Area elements of the true event horizon and theapparent horizon as a function of time. | c | τ T τ iso T τ iso /τ T and isotropizationtime τ iso (in units of T − or τ ), for various values of c . Theisotropization time τ iso is the time at which the pressuresdeviate from their equilibrium values by less than 10%. In contrast, the area of the true horizon grows in the dis-tant past long before the boundary geometry is signifi-cantly perturbed. This is a reflection of the global natureof event horizons — the location of the event horizon de- pends on the entire history of the geometry. It has beenargued [8] that it is the area element of the apparenthorizon, pulled back to the boundary along v = const . infalling null geodesics, which should be identified withthe entropy density (times 4 G N ) in the dual field theory.Table I shows, for various values of c , the final equilib-rium temperature T and a measure of the isotropizationtime τ iso . (These quantities only depend on | c | .) Wedefine τ iso as the time when the transverse and longi-tudinal pressures equal their final values to within 10%.When | c | (cid:38)
2, we find that τ iso ≈ τ , while for | c | (cid:46) τ iso ≈ . /T . Since τ iso is only a few times longer thanthe time scale τ over which the boundary geometry (1) ischanging, this measure of isotropization time should bestbe viewed as an upper bound on isotropization times as-sociated with plasma dynamics alone. Nevertheless, itis interesting to note that τ iso ≈ . /T corresponds to atime of fm/c when T = 350 MeV, similar to estimates ofthermalization times inferred from hydrodynamic mod-eling of RHIC collisions [3].This work was supported in part by the U.S. Depart-ment of Energy under Grant No. DE-FG02-96ER40956. [1] E. V. Shuryak, Nucl. Phys. A750 , 64 (2005), hep-ph/0405066.[2] M. Luzum and P. Romatschke, Phys. Rev.