Far-from-equilibrium dynamics of a strongly coupled non-Abelian plasma with non-zero charge density or external magnetic field
PPrepared for submission to JHEP
Far-from-equilibrium dynamics of a strongly coupled non-Abelian plasma with non-zero charge density or external magnetic field
John F. Fuini III, Laurence G. Yaffe
Department of Physics, University of Washington, Seattle WA 98195, USA
E-mail: [email protected] , [email protected] Abstract:
Using holography, we study the evolution of a spatially homogeneous, farfrom equilibrium, strongly coupled N = 4 supersymmetric Yang-Mills plasma with anon-zero charge density or a background magnetic field. This gauge theory problemcorresponds, in the dual gravity description, to an initial value problem in Einstein-Maxwell theory with homogeneous but anisotropic initial conditions. We explore thedependence of the equilibration process on different aspects of the initial departurefrom equilibrium and, while controlling for these dependencies, examine how the equi-libration dynamics are affected by the presence of a non-vanishing charge density or anexternal magnetic field. The equilibration dynamics are remarkably insensitive to theaddition of even large chemical potentials or magnetic fields; the equilibration time isset primarily by the form of the initial departure from equilibrium. For initial devia-tions from equilibrium which are well localized in scale, we formulate a simple modelfor equilibration times which agrees quite well with our results. Keywords: general relativity, gauge-gravity correspondence, quark-gluon plasma
ArXiv ePrint: a r X i v : . [ h e p - t h ] S e p ontents N = 4 SYM in an external field 42.2 Holographic description 92.3 Symmetry specialization 122.4 Asymptotic analysis 142.5 Scaling relations 152.6 Apparent horizon 162.7 Equilibrium solutions 17 The discovery of gauge/gravity duality (or “holography”) has enabled the study ofpreviously intractable problems involving the dynamics of strongly coupled gauge the-ories. In the limit of large gauge group rank N c , and large ‘t Hooft coupling λ , thestrongly coupled quantum dynamics of certain gauge field theories may be mapped,precisely, into classical gravitational dynamics of higher dimensional asymptotically See, for examples, refs. [1–3] and references therein. – 1 –nti-de Sitter (AdS) spacetimes [4–6]. Numerical studies of the resulting gravitationaldynamics can shed light on poorly understood aspects of the quantum dynamics ofstrongly coupled gauge theories.Using the simplest example of gauge/gravity duality, applicable to maximally su-persymmetric SU ( N c ) Yang-Mills theory ( N = 4 SYM), this approach has been appliedto a succession of problems of increasing complexity involving far from equilibrium dy-namics. These include homogeneous isotropization [7–9], colliding shock waves [10–15],and turbulence in two-dimensional fluids [16, 17]. A detailed presentation of the meth-ods used in most of these works is available [18].In this paper, we extend previous work on the dynamics of homogeneous butanisotropic N = 4 SYM plasma [7–9]. We examine the influence on the equilibrationdynamics of a non-zero global U (1) charge density, or a background magnetic field.Inclusion of these effects is motivated by the physics of relativistic heavy ion colli-sions [19–21]. Hydrodynamic modeling of near-central events clearly indicates that thebaryon chemical potential µ B in the mid-rapidity region is significantly smaller thanthe temperature, but not by an enormously large factor at RHIC energies. Hence, itis desirable to understand the sensitivity of the plasma equilibration dynamics to thepresence of a baryon chemical potential and associated non-zero baryon charge density.Similarly, it is clear that large, but transient, electromagnetic fields are generated inheavy ion collisions. A growing body of work [24–28] suggests that electromagneticeffects may play a significant role despite the small value of the fine structure con-stant. Electromagnetic effects on equilibrium QCD properties are also under studyusing lattice gauge theory [29–32].The large N c , strongly coupled N = 4 SYM plasma we study is, of course, only acaricature of a real quark-gluon plasma. But it is a highly instructive caricature whichcorrectly reproduces many qualitative features of QCD plasma (such as Debye screen-ing, finite static correlation lengths, and long distance, low frequency dynamics de-scribed by neutral fluid hydrodynamics). Moreover, in the temperature range relevantfor heavy ion collisions, quantitative comparisons of bulk thermodynamics, screeninglengths, shear viscosity, and other observables show greater similarity between N = 4SYM and QCD than one might reasonably have expected [33, 34]. Since the composi-tion of a plasma depends on the chemical potentials, or associated charge densities, ofits constituents, studying the dependence of the equilibration dynamics on a conservedcharge density provides a simple means to probe the sensitivity of the dynamics to theprecise composition of the non-Abelian plasma. This, in small measure, may help one Inferred values of µ B /T at chemical freeze-out are about 0.15 for RHIC collisions at √ s NN = 200GeV, and roughly 0.005 for LHC heavy ion collisions with √ s NN = 2 . – 2 –auge the degree to which N = 4 SYM plasma properties can be extrapolated to realQCD plasma. At the very least, strongly coupled N = 4 SYM theory provides a highlyinstructive toy model in which one may explore, quantitatively, non-trivial aspects ofnon-equilibrium gauge field dynamics. The remainder of the paper is organized as follows. Section 2 summarizes necessarybackground material. This includes the coupling of an Abelian background gaugefield to a U (1) subgroup of the SU (4) R global symmetry group of N = 4 SYM. This U (1) symmetry may be regarded as analogous to either the baryon number U (1) B orelectromagnetic U (1) EM flavor symmetries of QCD. Turning on a background magneticfield implies an enlargement of the theory under consideration from N = 4 SYM to N = 4 SYM coupled to electromagnetism (which we abbreviate as SYM+EM). Thecombined theory is no longer scale invariant; this has important implications whichwe discuss. This section describes the 5D Einstein-Maxwell theory which provides theholographic description of the states of interest, presents our coordinate ansatz (basedon a null slicing of the geometry), and summarizes relevant portions of the holographicdictionary relating gravitational and dual field theory quantities. This section alsorecords the reduced field equations which emerge from our symmetry specializations,describes the relevant near-boundary asymptotic behavior, and summarizes propertiesof the static equilibrium geometries to which our time dependent solutions asymptoteat late times.The following section 3 briefly describes our numerical methods, which are basedon the strategy presented in ref. [18]. When studying states with a non-zero chargedensity (but no background magnetic field) appropriate numerical methods for asymp-totically AdS Einstein-Maxwell theory are immediate generalizations of methods whichhave previously been found to work well for pure gravity. However, the inclusion ofa background magnetic field induces a trace anomaly in the dual quantum field the-ory which, in the gravitational description, manifests in the appearance of logarithmicterms in the near-boundary behavior of fields. Such non-analytic terms degrade theperformance of spectral methods, on which we rely, and necessitate careful attentionto numerical issues. Section 3 also describes the specifics of our chosen initial data.Results are presented in section 4. We focus on the evolution of the expectationvalue of the stress-energy tensor. We first discuss the sensitivity of the equilibrationdynamics to features in the initial data and, in particular, examine the extent to whichthe evolution shows nonlinear dependence on the initial departure from equilibrium.We find that only disturbances in the geometry originating deep in the bulk, very Previous work examining thermalization in plasmas with non-zero chemical potential (not involv-ing numerical solutions of far from equilibrium geometries) includes refs. [35–38]. – 3 –lose to the horizon, generate significant nonlinearities. This is broadly consistent withearlier work [8, 9]. However, for a very wide variety of initial disturbances, includingones which generate extremely large pressure anisotropies, we find remarkably littlenonlinearity in the equilibration dynamics, often below the part-per-mille level.We then present comparisons of the equilibration dynamics as a function of thecharge density or background magnetic field. We focus on comparisons in which theform of the initial departure form equilibrium and the energy density, or the equilib-rium temperature, is held fixed while either the charge density or magnetic field isvaried. These comparisons reveal surprisingly little sensitivity to the charge density, ormagnetic field, even at early times when the departure from equilibrium is large.We verify the late time approach to the expected equilibrium states, and extractthe leading quasinormal mode (QNM) frequency from the late time relaxation. Quasi-normal mode frequencies extracted from our full nonlinear dynamics are compared,where possible, with independent calculations of QNM frequencies based on a linearizedanalysis around the equilibrium geometry. This provides a useful check on our numericalaccuracy.We define an approximate equilibration time based on the relative deviation of thepressure anisotropy from its equilibrium value, and examine the dependence of thistime on charge density or external magnetic field. Once again, changes in this quantityare largest for initial disturbances which originate very close to the horizon, but theoverall sensitivity of the equilibration time to the charge density or magnetic field isremarkably modest.The final section 5 discusses and attempts to synthesize the implications of ourresults. We present a simple model of equilibration times, for initial disturbanceswhich are well localized in scale, which agrees rather well with our numerical results(but becomes less accurate for disturbances localized extremely close to the horizon).We end with a few concluding remarks. N = 4 SYM in an external field
We study maximally supersymmetric SU ( N c ) Yang-Mills theory ( N = 4 SYM) on fourdimensional Minkowski space when the conserved current for a U (1) subgroup of the As this paper neared completion, we learned of the somewhat related work by A. Buchel, M.Heller, and R. Myers [39]. These authors examine quasinormal mode frequencies in N = 2 ∗ SYMand argue that, in this non-conformal deformation of N = 4 SYM, the longest equilibration times arelargely set by the temperature with little sensitivity to other scales. – 4 – U (4) R global symmetry group either (a) has a non-vanishing charge density, or (b)is coupled to a background Abelian gauge field describing a uniform magnetic field.The embedding of the U (1) symmetry is chosen such that the U (1) commutes with an SU (3) subgroup of the SU (4) R global symmetry.The coupling to the external field has the usual form S = S SYM + (cid:90) d x j α ( x ) A ext α ( x ) , (2.1)where j α ( x ) is the conserved U (1) current normalized such that the four Weyl fermionsof N = 4 SYM have charges { +3 , − , − , − } / √ / √
3. The overall factor of 1 / √ The background U (1) gauge field A ext α ( x ) we take to have the form A ext α ( x ) ≡ µ δ α + B ( x δ α − x δ α ) , (2.2)with µ the chemical potential which, in equilibrium, will be conjugate to the chargedensity j , and B the amplitude of a constant magnetic field pointing in the x direction.Although it should be straightforward to study dynamics when both the charge density j and magnetic field B are non-zero, in this paper we focus for simplicity on the casesof either a non-zero charge density with vanishing magnetic field, j (cid:54) = 0 and B = 0, ornon-zero magnetic field with vanishing charge density, j = 0 and B (cid:54) = 0.With a non-zero magnetic field B in the x direction, changes in the backgroundgauge field under a translation in the x or x directions, or a rotation in the x - x plane, can be compensated by a suitable U (1) gauge transformation. Hence, thetheory retains full spatial translation invariance as well as rotation invariance in the x - x plane.We will be interested in initial states which: ( i ) have non-trivial expectation values (cid:104) T αβ ( x ) (cid:105) and (cid:104) j α ( x ) (cid:105) for the stress-energy tensor and U (1) current density, respectively;( ii ) are invariant under spatial translations as well as O (2) rotations in the x − x plane;and ( iii ) are invariant under the SU (3) R subgroup of the SU (4) R global symmetry whichcommutes with our chosen U (1).Since all N = 4 SYM fields transform in the adjoint representation of the SU ( N c )gauge group, the stress-energy and U (1) current expectation values both scale as O ( N ) We use a mostly-plus Minkowski space metric, η µν ≡ diag( − , +1 , +1 , +1). These charge assignments are 1 / √ – 5 –n the large N c limit. For later convenience, we define a rescaled energy density ε andcharge density ρ , via (cid:104) T (cid:105) ≡ κ ε , (cid:104) j (cid:105) ≡ κ ρ , (2.3)with κ ≡ ( N − / (2 π ) . (2.4) N = 4 SYM is a conformal field theory with a traceless stress-energy tensor. Addinga chemical potential µ introduces a physical scale, but does not modify the microscopicdynamics of the theory and hence does not affect the tracelessness of the stress-energytensor. In contrast, introducing an external magnetic field does affect the microscopicdynamics and, in particular, generates a non-zero trace anomaly, T αα = − κ (cid:0) F ext µν (cid:1) = − κ B . (2.5)The trace anomaly generated by the external magnetic field implies that the the-ory is no longer scale invariant. For example, the ground state energy density, as afunction of magnetic field, need not have the simple form of some pure number times B . This will be seen explicitly below. The trace anomaly implies that there must belogarithmic dependence on a renormalization point. To interpret this dependence, itis appropriate to adopt the perspective that adding an external magnetic field meansthat the theory under consideration has been enlarged — it is now N = 4 SYM coupledto U (1) electromagnetism (SYM+EM). The complete action of the theory is the SYMaction, minimally coupled to the U (1) gauge field, plus the Maxwell action for U (1)gauge field, S SYM+EM = S SYM , min . coupled + S EM , (2.6)with S EM ≡ − (cid:90) d x e F µν . (2.7)The electromagnetic coupling e (having been scaled out of covariant derivatives) ap-pears as an inverse prefactor of the Maxwell action. We regard the electromagneticcoupling e as arbitrarily weak. Hence, quantum fluctuations in the U (1) gauge field We define the external gauge field such that no factor of an electromagnetic gauge coupling appearsin the interaction (2.1), in our U (1) covariant derivatives, or in the trace anomaly (2.5). The coefficientof − F µν in the trace anomaly (2.5) equals the EM beta function coefficient b , given below in eq. (2.9).(Note that the sign of the trace anomaly depends on the metric convention in use.) – 6 –re negligibly small, allowing us to view the EM gauge field as a classical backgroundfield. However, just as in QED, fluctuations in the SYM fields which are electromag-netically charged will cause the electromagnetic coupling e to run with scale. Theassociated renormalization group (RG) equation for the inverse coupling has the usualform, µ ddµ e − ≡ β /e ( e − ) = − b + O ( e ) , (2.8)with the one-loop beta function coefficient b ≡ κ (cid:104) (cid:88) α ( q α f ) + (cid:88) a ( q a s ) (cid:105) = κ . (2.9)Here, q α f = (3 , − , − , − / √ q a s = (2 , , / √ /e ( µ ) = b ln(Λ EM /µ ) + O [ln(ln Λ EM /µ )] , (2.10)with the RG invariant scale Λ EM denoting the Landau pole scale where the (one loopapproximation to the) electromagnetic coupling diverges.The total stress-energy tensor derived from the combined action (2.6) will equal the N = 4 SYM stress-energy tensor, augmented with minimal coupling terms to the EMgauge field, plus the classical Maxwell stress-energy. An essential point, however, is thatwhile the total stress-energy tensor is well-defined, partitioning the stress-energy tensorinto separate SYM and EM contributions is inherently ambiguous, as the individualpieces depend on the renormalization point. We define T αβ tot ≡ T αβ EM ( µ ) + ∆ T αβ SYM ( µ ) , (2.11) In an arbitrary background SU (4) gauge field, the divergence of the SU (4) R current acquires ananomalous contribution, ∂ µ J aµ ∝ d abc F bµν F µνc . This anomaly, when specialized to our chosen U (1)subgroup, is proportional to the sum of the cubes of our fermion charges and is non-zero, (cid:80) α ( q α f ) =8 / √
3. To make the combined SYM+EM theory well defined, one could add to the theory additionalfermions, charged under the U (1) but with no SYM interactions, which would cancel this U (1) anomaly.As we are not concerned with quantum fluctuations in the U (1) gauge field, the presence of this U (1)anomaly (in the absence of compensating spectators) is irrelevant for our purposes. A non-renormalization theorem in supersymmetric N = 4 SYM implies that the short distancebehavior of the current-current correlation cannot depend on the ‘t Hooft coupling λ [41]. Thisimplies that the leading EM beta function coefficient b does not depend on λ , and hence may easilybe evaluated in the λ → – 7 –ith T αβ EM ( µ ) ≡ e ( µ ) (cid:2) F αν F βν − η αβ F µν F µν (cid:3) , (2.12)and ∆ T αβ SYM ( µ ) ≡ T αβ SYM , min . coupled ( µ ) . (2.13)The partitioning (2.11) of the stress-energy tensor puts all quantum corrections otherthan the running of the EM coupling into the SYM contribution ∆ T αβ SYM ( µ ). The scaledependence must, of course, cancel between the two terms because the total stress-energy tensor is a physical quantity. Therefore, the scale dependence in the SYMcontribution to the stress-energy must simply compensate the known running of theinverse electromagnetic coupling (2.8) in the Maxwell stress-energy tensor (2.12), µ ddµ ∆ T αβ SYM ( µ ) = − µ ddµ T αβ EM ( µ ) = b (cid:2) F αν F βν − η αβ F µν F µν (cid:3) . (2.14)Specializing to zero temperature states in a constant static magnetic field B , thescale dependence (2.14) plus dimensional analysis implies that the SYM contributionto the ground state energy density is a non-analytic function of magnetic field, ε ( µ ) = c B − B ln( |B| /µ ) = B ln (cid:2) B ∗ ( µ ) / |B| (cid:3) , (2.15)with c some pure number. (Here and henceforth, when considering physics in a non-zero magnetic field ε ( µ ) ≡ ∆ T ( µ ) /κ denotes the SYM portion of the rescaled energydensity.) In the second form of eq. (2.15), the analytic term has been absorbed bydefining a scale dependent “fiducial” magnetic field amplitude, B ∗ ( µ ) ≡ µ e c . (2.16)Note that the ground state energy acquires a simple quadratic form when the renor-malization point is chosen to scale with the magnetic field, ε ( |B| / ) = c B . Ournumerically determined value for the coefficient c is given below in eq. (2.68).When considering low temperature physics in a background magnetic field, T (cid:28)|B| , it is natural to choose a renormalization point µ = O ( |B| / ), as this is the relevantscale which cuts off long range fluctuations in the charged SYM fields. We will employtwo choices for the renormalization point. One choice is µ = 1 /L , with L the AdScurvature scale (discussed below); this choice is computationally convenient but notphysically significant. We will also report and discuss results with µ = |B| / . For Note that T αβ EM ( µ ) is not the metric variation of some renormalized EM action (whose separationfrom the total action would not be well-defined). Rather, eq. (2.12) is simply defining T αβ EM ( µ ) as theclassical EM stress-energy tensor multiplied by the scale-dependent inverse EM coupling. – 8 –ater convenience, we define abbreviations for the (rescaled) energy density evaluatedat these two renormalization points, ε L ≡ ε (1 /L ) , ε B ≡ ε ( |B| / ) . (2.17) The holographic description of SYM states, within our sector of interest, in the limitof large N c and large ‘t Hooft coupling λ , is given by classical Einstein-Maxwell theoryon 5-dimensional spacetimes which are asymptotically AdS [42]. The 5D bulk actionis S ≡ πG (cid:90) d x √− G (cid:0) R − − L F MN F MN (cid:1) , (2.18)with G ≡ π L /N the 5D Newton gravitational constant, Λ ≡ − /L the cosmologicalconstant, and L the AdS curvature scale. Setting to zero the variation of the actionwith respect to the metric gives the Einstein equation, R KL + (Λ − R ) G KL = 2 L (cid:0) F KM F LM − G KL F MN F MN (cid:1) , (2.19)while varying the bulk gauge field (with F MN ≡ ∇ M A N − ∇ N A M ) gives the usualsourceless Maxwell equation, ∇ K F KL = 0.A 5D Chern-Simons term, A ∧ F ∧ F , could be added to the action (2.18) andwould appear with a known coefficient in a consistent truncation of 10D supergravity.(See, for example, refs. [40, 42].) However, as stated above, in this paper we considersolutions with non-zero chemical potential µ or non-zero magnetic field B , but not both µ and B non-zero. For such solutions, the Chern-Simons term makes no contributionto the dynamics and hence may be neglected.As usual in holography, the expectation value (cid:104) T αβ ( x ) (cid:105) of the stress-energy tensoris determined by the subleading near-boundary behavior of the 5D metric G MN . Theleading near-boundary behavior of the bulk gauge field A M will be fixed by our chosenexternal U (1) gauge field (2.2), while the expectation value (cid:104) j α ( x ) (cid:105) of the U (1) current The coefficient of the Maxwell action may, of course, be set to an arbitrary value by suitablyrescaling the bulk gauge field A M . However, as the on-shell variation of the gravitational action withrespect to the boundary value of the gauge field defines the associated current, such rescaling changesthe normalization of the U (1) current in the holographic description. It will be seen below thatthe coefficient of the Maxwell term in our action (2.18) is correctly chosen so that the U (1) currentnormalization is consistent with our previous charge assignments. If charge assignments are chosen,for example, to be larger by a factor of √
3, then either the Maxwell term in the action (2.18) mustbe multiplied by a factor of 3, or else one must regard the boundary value of the bulk gauge field asequaling √ √ – 9 –ensity is determined by the subleading near-boundary behavior of the bulk gauge field.The precise relations will be shown below.Following ref. [18], we choose a coordinate ansatz, based on generalized Eddington-Finklestein (EF) coordinates, which is natural for gravitational infall problems. Themetric has the general form ds = r L g αβ ( x, r ) dx α dx β − w α ( x ) dx α dr, (2.20)where r is the bulk radial coordinate and x ≡ { x α } , α = 0 , · · · ,
3, denotes the fourremaining spacetime coordinates. The spacetime boundary lies at r = ∞ ; the { x α } may be regarded as coordinates on the spacetime boundary where the dual field theory“lives”. Curves of varying r , with x held fixed, are radially infalling null geodesics,affinely parameterized by r . The one-form (cid:101) w ≡ w α dx α (which is assumed to be time-like) depends only on x , not on r . These infalling coordinates remain regular acrossfuture null horizons.The form of the ansatz (2.20) remains invariant under r -independent diffeomor-phisms, x α → ¯ x α ≡ f α ( x ) , (2.21)as well as radial shifts (with arbitrary x dependence), r → ¯ r ≡ r + λ ( x ) . (2.22)We use the diffeomorphism freedom (2.21) to transform the timelike one-form (cid:101) w to thestandard form − dx (or w α = − δ α ). Our procedure for dealing with the radial shiftinvariance (2.22) is discussed below in subsection 2.6.We are interested in geometries which, at large r , asymptotically approach (thePoincar´e patch of) AdS . This will be the case if g αβ ( x, r ) approaches η αβ as r → ∞ ,with η αβ ≡ diag( − , , ,
1) the usual Minkowski metric tensor. Demanding that themetric and bulk gauge field satisfy the Einstein-Maxwell equations, one may derivethe near-boundary asymptotic behavior of the fields. Using radial gauge, A r = 0, forthe bulk gauge field, and a suitable choice of the radial shift (2.22) (which eliminates O (1 /r ) terms in g αβ ), one finds that for solutions of interest, the metric and gauge fieldhave asymptotic expansions of the form g αβ ( x, r ) ∼ η αβ + (cid:104) g (4) αβ ( x ) + h (4) αβ ( x ) ln rL (cid:105) ( L /r ) + O (cid:2) ( L /r ) (cid:3) , (2.23a) A α ( x, r ) ∼ A ext α ( x ) + A (2) α ( x ) ( L /r ) + O (cid:2) ( L /r ) (cid:3) . (2.23b)– 10 –he coefficient h (4) αβ of the logarithmic term in the metric is only non-zero when thereis an external EM field, h (4) αβ = F αν F βν − η αβ ( F µν F µν + F ν F ν ) . (2.24)For a constant magnetic field in the x direction, (cid:107) h (4) αβ (cid:107) = B diag(+2 , +1 , +1 , − g (4) αβ ( x ) and A (2) α ( x ) cannot be determined solelyfrom a near-boundary analysis of the field equations, and depend on the form of the so-lution throughout the bulk. However, asymptotic analysis does show that (cid:80) i =1 g (4) ii = − F ν F ν . The subleading metric coefficients g (4) αβ ( x ) and h (4) αβ ( x ) encode the expec-tation value of the SYM stress-energy tensor [43, 44]. The appropriate holographicrelation is (cid:104) T µν (cid:105) = κ (cid:110)(cid:101) g (4) µν − η µν tr ( (cid:101) g (4) ) + [ln( µL ) + C ] (cid:101) h (4) µν (cid:111) , (2.25)where (cid:101) g (4) µν ≡ g (4) µν + η µν (cid:16) g (4)00 + h (4)00 (cid:17) , (cid:101) h (4) µν ≡ h (4) µν + η µν h (4)00 , (2.26) κ ≡ L / (4 πG N ) = ( N − / (2 π ), and C is an arbitrary renormalization-schemedependent constant. We adopt a specific value,
C ≡ − , (2.27)which will make the subsequent explicit expression (2.42a) for the energy density assimple as possible.Inserting expression (2.24) into relation (2.26) shows that (cid:101) h (4) αβ , the coefficient ofthe renormalization point dependent part of the holographic SYM stress-energy, isproportional to the classical EM stress-energy tensor, (cid:101) h (4) αβ = F αν F βν − η αβ F µν F µν , (2.28)or (cid:107) (cid:101) h (4) αβ (cid:107) = B diag(+1 , +1 , +1 , −
1) for a constant magnetic field in the x direc-tion. Using the above relations, one also finds that tr ( (cid:101) g (4) ) = F µν F µν . Since ˜ h (4) istraceless, the holographic relation (2.25) yields the stress-energy trace (cid:104) T αα (cid:105) = − κ tr ( (cid:101) g (4) ) = − κ F µν F µν , (2.29) In Fefferman-Graham (FG) coordinates, for which ds ≡ ( L /ρ ) (cid:2)(cid:101) g αβ (˜ x, ρ ) d ˜ x α d ˜ x β + dρ (cid:3) , one has (cid:101) g αβ (˜ x, ρ ) ∼ η αβ + (cid:2)(cid:101) g (4) αβ (˜ x ) + (cid:101) h (4) αβ ln Lρ (cid:3) ρ + O ( ρ ln ρ ) as ρ →
0. Eq. (2.26) gives the relation betweenthe subleading asymptotic metric coefficients in our infalling EF coordinates and FG coordinates. To perform the required holographic renormalization one must add a counterterm dependinglogarithmically on the UV cutoff. (See, for example, refs. [44–46].) As always, such a logarithmiccounterterm comes with an inevitable finite ambiguity. – 11 –r (cid:104) T αα (cid:105) = − κ B in a constant magnetic field, in agreement with the earlier fieldtheory result (2.5). Similarly, the renormalization point dependence of the stress-energy(2.25) coincides with the QFT result (2.14). Finally, the subleading asymptotic coefficient A (2) α ( x ) for the bulk gauge field en-codes the U (1) current density. One finds (cid:104) j ν (cid:105) = 2 κ A (2) ν . (2.30) As noted earlier, we are interested in studying solutions of Einstein-Maxwell theorywhich are spatially homogeneous. This implies that all metric functions depend onlyon x and r . The arbitrary function λ in the residual radial shift diffeomorphism (2.22)will depend only on x . Henceforth, for convenience, we will use v as a synonym for x ; v is a null time coordinate. (In other words, v = const . surfaces are null slices ofthe geometry.) At the boundary, v coincides with the time t of the dual field theory.We also impose invariance under O (2) rotations in the x - x plane. This impliesthat only the g , g , g , and g = g components of g αβ are non-zero. Our Einstein-Maxwell theory (without a Chern-Simons term) is also invariant under spatial parity,or x → − x reflections, and for simplicity we will also impose parity invariance. Thisrequires the vanishing of g .For the bulk gauge field, the choice of radial gauge, A r = 0, plus our imposedsymmetries imply that A α ( x, r ) = A ext α ( x ) − φ ( v, r ) δ α . (2.31)The corresponding bulk field strength, which is what appears in the field equations,can have a constant ( x and r independent) magnetic field plus a radial electric field, F ( x, r ) = B , F r ( x, r ) = ∂ r φ ( v, r ) ≡ −E ( v, r ) , (2.32)with all other components vanishing.As in ref. [18], it is convenient to rename the non-vanishing metric components as r L g ≡ − A , r L g = r L g ≡ Σ e B , r L g ≡ Σ e − B , (2.33)where A , B , and Σ are functions of v and r . The resulting line element is ds = 2 dv [ dr − A ( v, r ) dv ] + Σ( v, r ) (cid:2) e B ( v,r ) ( dx + dy ) + e − B ( v,r ) dz (cid:3) . (2.34) As in ref. [40], one can also use a comparison of holographic and QFT evaluations of the U (1)anomaly to confirm that the U (1) current normalizations are consistent. – 12 –enceforth, A will always denote the metric function multiplying dv (times − / dx dy dz the spatial volume element), while B characterizes the spatial anisotropy (which shouldnot be confused with the magnetic field amplitude B ).The radial derivative ∂ r is a directional derivative along infalling radial null geodesics.It proves convenient to define a corresponding directional derivative along outward ra-dial null geodesics, d + ≡ ∂ v + A ( v, r ) ∂ r . (2.35)The field equations which result from varying the action (2.18), inserting the abovesymmetry specializations, and re-expressing v -derivatives in terms of the d + modifiedtime derivative (2.35), take a remarkably compact form. The Einstein equations are:Σ (cid:48)(cid:48) + ( B (cid:48) ) Σ = 0 , (2.36a) A (cid:48)(cid:48) − (cid:48) / Σ ) d + Σ + B (cid:48) d + B = + B L e − B Σ − + E L − /L , (2.36b)( d + B ) (cid:48) + (Σ (cid:48) / Σ) d + B + B (cid:48) ( d + Σ) / Σ = − B L e − B Σ − , (2.36c)( d + Σ) (cid:48) / Σ + 2(Σ (cid:48) / Σ ) d + Σ = − B L e − B Σ − − E L + 2 /L , (2.36d) d + ( d + Σ) − A (cid:48) ( d + Σ) + Σ ( d + B ) = 0 , (2.36e)where primes denote radial derivatives, h (cid:48) ≡ ∂ r h . As discussed in ref. [18], theanisotropy function B encodes the essential propagating degrees of freedom. The func-tions Σ and A may be regarded as auxiliary fields, determined by solving eqns. (2.36a)and (2.36b) using data on a single time slice. Information about the time evolution of B is contained in equation (2.36c). Equations (2.36d) and (2.36e) may be viewed asboundary value constraints — if they hold at one value of r , then the other equationsensure that these equations hold at all values of r .Maxwell’s equations reduce to the statements that neither the magnetic field B ,nor the radial electric flux density E Σ , have any radial or temporal variation. In otherwords, B = const . , as already indicated in (2.32), and E ( v, r ) = ρ L Σ − ( v, r ) , (2.37)for some constant ρ which, from eqs. (2.30)–(2.32) plus (2.40) below, one sees is preciselythe U (1) charge density (rescaled by κ ), (cid:104) j (cid:105) = κ ρ . (2.38)The form (2.37) of the radial electric field simply reflects Gauss’ law in 4+1 dimensions,combined with charge conservation and spatial translation invariance, which imply that ρ cannot have any temporal or spatial variation.– 13 –he bulk gauge field A M does not appear in the field equations (except via the fieldstrength), but one may choose to regard A M as satisfying the radial gauge condition, A r = 0, plus the condition that the time component A v vanish at the horizon. Thisfixes the residual r -independent gauge freedom which remains after imposing radialgauge. With these choices, the chemical potential µ is the boundary value of A v inthe late time ( v → ∞ ) equilibrium limit. Equivalently (in radial gauge), the chemicalpotential µ equals the difference between the boundary and horizon values of A v , in theequilibrium geometry. This coincides with the line integral of the radial electric fieldfrom horizon to boundary, µ = lim v →∞ A v (cid:12)(cid:12) ∞ r h = (cid:90) ∞ r h dr E ( ∞ , r ) , (2.39)which gives the work needed to move a unit charge from the boundary to the horizon. Asusual, the charge density and the chemical potential are thermodynamically conjugate.One may consider the chemical potential µ to be a function of the (rescaled) chargedensity ρ , or vice-versa. The choice of perspective (“canonical” vs. “grand canonical”)has no bearing on the dynamics. Asymptotic analysis of these equations is straightforward. We impose a flat boundarygeometry with the requirement that lim r →∞ g αβ ( x, r ) = η αβ , implyinglim r →∞ ( L/r ) A ( v, r ) = , lim r →∞ ( L/r ) Σ( v, r ) = 1 , lim r →∞ B ( v, r ) = 0 , (2.40)for our renamed metric functions. Solutions to Einstein’s equations (2.36) with thisleading behavior may be systematically expanded in integer powers of 1 /r and (fornon-zero magnetic field) logarithms of r . One finds:Σ( v, r ) ∼ L − [ r + λ ( v )] + O [( L/r ) ln rL ] , (2.41a) A ( v, r ) ∼ L − [ r + λ ( v )] − ∂ v λ ( v )+ L (cid:2) a − B ln rL (cid:3) ( L/r ) − L (cid:2) a λ ( v ) + B λ ( v ) (1 − rL ) (cid:3) ( L/r ) + O [( L/r ) ln rL ] , (2.41b) B ( v, r ) ∼ L (cid:2) b ( v ) + B ln rL (cid:3) ( L/r ) + L (cid:2) L ∂ v b ( v ) − b ( v ) λ ( v ) + B λ ( v ) (1 − rL ) (cid:3) ( L/r ) + O [( L/r ) ln rL ] , (2.41c)The constant a and the function b ( v ) cannot be determined just using asymptoticanalysis, and the radial shift λ ( v ) is completely arbitrary. The coefficient a encodes– 14 –he energy density which, due to homogeneity, cannot vary in time, while b ( v ) encodesthe anisotropy in the spatial stress. Using the holographic relation (2.25) and ourconvention (2.27) for defining the stress-energy tensor, one finds (cid:104) T (cid:105) = κ (cid:0) − a + B ln µL (cid:1) , (2.42a) (cid:104) T (cid:105) = (cid:104) T (cid:105) = κ (cid:0) − a + b − B + B ln µL (cid:1) , (2.42b) (cid:104) T (cid:105) = κ (cid:0) − a − b − B ln µL (cid:1) . (2.42c) Consider independent rescaling of the boundary and radial coordinates, x ≡ α (cid:101) x , r ≡ α − γ (cid:101) r , (2.43)with α and γ arbitrary positive numbers. If the metric functions { A, Σ , B } satisfy theEinstein equations (2.36), with asymptotic behavior (2.40), then the rescaled metricfunctions (cid:101) B ( (cid:101) x, (cid:101) r ) ≡ B ( x ( (cid:101) x ) , r ( (cid:101) r )) , (2.44a) (cid:101) Σ( (cid:101) x, (cid:101) r ) ≡ ( α/γ ) Σ( x ( (cid:101) x ) , r ( (cid:101) r )) , (2.44b) (cid:101) A ( (cid:101) x, (cid:101) r ) ≡ ( α/γ ) A ( x ( (cid:101) x ) , r ( (cid:101) r )) , (2.44c)also satisfy the Einstein equations (and our asymptotic conditions) with rescaled pa-rameters (cid:101) L ≡ γ − L , (cid:101)
B ≡ α B , (cid:101) ρ ≡ α ρ . (2.45)The subleading asymptotic coefficients a and b become (cid:101) a ≡ α (cid:2) a − B ln( γ/α ) (cid:3) , (cid:101) b ≡ α (cid:2) b + B ln( γ/α ) (cid:3) . (2.46)Using the holographic relation (2.42) for the stress-energy expectation, one finds that (cid:101) T µν ( (cid:101) µ ) = α T µν ( µ ) , (2.47)with a rescaled renormalization point (cid:101) µ ≡ α µ .If α = γ , then these transformations are just a trivial rescaling of all quantitiesaccording to their dimension. But transformations with α (cid:54) = γ are non-trivial and scalebulk and boundary quantities by different amounts. In particular, transformations with α = 1 but γ (cid:54) = 1 rescale the AdS curvature scale L without affecting the boundarycoordinates or boundary parameters ( B , ρ , or µ ), showing that the value of L has nophysical significance (in the large N c , large λ limit for which classical gravity providesthe dual description). This illustrates, explicitly, the independence of the boundaryfield theory on the AdS curvature scale L .– 15 – igure 1 . With a generic choice of the radial shift λ ( v ) (left panel), the radial position ofthe horizon will change with time. It may be kept fixed (right panel) with a suitable choiceof λ ( v ). With a non-zero homogeneous energy density, the dual geometries of interest will havean apparent horizon at some radial position, r = r h ( v ) [18]. Since we are investigatingnon-equilibrium dynamics, one might expect the horizon position to change significantlybefore ultimately settling down as the system equilibrates. However, as illustrated infig. 1 it is possible, and very convenient, to use the residual radial shift diffeomorphismfreedom (2.22) to place the apparent horizon at a fixed radial position, r h ( v ) ≡ ¯ r h . (2.48)A short exercise [18] shows that the condition for an apparent horizon to be presentat r = ¯ r h is that this location be a zero of the modified time derivative of the spatialscale factor, d + Σ (cid:12)(cid:12) r =¯ r h = 0 . (2.49)This condition serves to fix the radial shift λ ( v ). It is convenient to regard this conditionas a combination of a constraint on initial data (implemented by finding the radialshift λ ( v ) which is needed to satisfy (2.49) at some initial time v ) together with therequirement that the horizon be time-independent, ∂r h /∂v = 0, which requires thatthe time derivative of d + Σ vanish at the apparent horizon. Evaluating this condition,and using the Einstein equations (2.36d, 2.36e) to simplify the result, determines thevalue of the metric function A at the horizon, A (cid:12)(cid:12) r =¯ r h = − L ( d + B ) . (2.50)For the metric to be non-singular on and outside the apparent horizon, the spatial scalefactor Σ must be non-vanishing for r ≥ ¯ r h .– 16 – .7 Equilibrium solutions Given some initial non-equilibrium state of the system, the dynamical evolution shouldasymptotically approach a thermal equilibrium state. In the gravitational description,this implies that the geometry should, at late times, approach some static black branesolution. The specific black brane solution will depend on the values of the conservedenergy and charge densities in the chosen initial state, and on the value of the back-ground magnetic field.
Schwarzschild
For initial states with vanishing charge density and magnetic field, the bulk geometrywill equilibrate to the 5D AdS-Schwarzschild black brane solution. A standard form ofthis metric is ds = − U (˜ r ) dt + d ˜ r U (˜ r ) + ˜ r L ( dx i ) (2.51)( i = 1 , , U (˜ r ) ≡ ˜ r L − m L ˜ r . (2.52)The radial coordinate ˜ r should not be confused with our Eddington-Finklestein coor-dinate r . The zero of U (˜ r ) determines the horizon location,˜ r h = m / L , (2.53)and the horizon temperature [given by (2 π ) − times the surface gravity at the horizon]is proportional to the horizon radius, πT h = ˜ r h L − = m / /L . (2.54)In our infalling EF coordinates (2.34), this AdS-Schwarzschild solution is described byΣ( r ) = ( r + λ ) /L , A ( r ) = Σ( r ) − m Σ( r ) − , B ( r ) = 0 . (2.55)Using the holographic relation (2.42), one sees that the parameter m is related to the(rescaled) equilibrium energy density ε ≡ (cid:104) T (cid:105) /κ via ε = m L − . (2.56) Reissner-Nordstrom
If the initial state has a non-zero charge density but vanishing magnetic field, then thebulk geometry will equilibrate to a 5D Reissner-Nordstrom (RN) black brane [42]. Thismetric may be written in the form (2.51), with U (˜ r ) ≡ ˜ r L − m L ˜ r + ( ρL ) L ˜ r . (2.57)– 17 – igh - TLow - T ( π T ) ρ / ε / ρ / Figure 2 . The one parameter family of non-extremal equilibrium Reissner-Nordstromcharged black brane solutions (solid line) in the plane of ε/ | ρ | / vs. ( πT ) / | ρ | / . Alsoshown are the high and low temperature asymptotic forms (dashed lines). In the high tem-perature regime, πT (cid:29) ρ / , the curve approaches the Schwarzschild result, ε = ( πT ) .In the low temperature (or near extremal) regime, πT (cid:28) ρ / , the charge density ρ ∼ ρ max (cid:2) − ( ) − / ( πT ) ρ − / (cid:3) and ε/ρ / ∼ ( ) / + ( ) − / ( πT ) ρ − / . The charge density ρ of the Reissner-Nordstrom brane is bounded from above by theextremal charge density ρ max , given by( ρ max L ) = m . (2.58)The relation (2.56) between the energy density and the mass parameter m is unchanged.Hence, the extremal charge density ρ max = ε / .It is convenient to express ρ in terms of the fraction x of the extremal chargedensity, x ≡ ρ/ρ max . (2.59)The horizon radius ˜ r h is given by the outermost positive root of U (˜ r ); explicitly,˜ r h /L = ( m ) / (cid:104) ( − x + i √ − x ) / + ( − x − i √ − x ) / (cid:105) / . (2.60)The horizon radius (divided by L ) varies from m / down to ( m ) / as x varies from0 to 1. The horizon temperature T h is given by πT h L = (˜ r h /L ) (cid:2) − ( m ) / x ( L/ ˜ r h ) (cid:3) . (2.61)The horizon temperature decreases with increasing charge density, and vanishes as thecharge density approaches the extremal value (or x → ε min ( ρ ), which must be a mono-tonically increasing (and convex) function of ρ . This implies that for any given value ofthe energy density, there will be a maximum charge density, corresponding to a groundstate with vanishing temperature. The equilibrium chemical potential µ , thermody-namically conjugate to the charge density ρ , is given by µ = ρ ( L / ˜ r h ) . (2.62)Physically distinct non-extremal solutions may be labeled by the value of one dimen-sionless ratio such as ε/ | ρ | / [or ( πT ) / | ρ | / or ε/ ( πT ) , etc.]. Figure 2 shows a log-logplot of the curve representing these solutions in the plane of ε/ | ρ | / and ( πT ) / | ρ | / .In our infalling EF coordinates, the RN black-brane solution is described byΣ( r ) = ( r + λ ) /L , A ( r ) = Σ( r ) − m Σ( r ) − + ρ L Σ( r ) − , (2.63)and B ( r ) = 0. Magnetic branes
When the magnetic field is non-zero, the bulk geometry will equilibrate to a stationarymagnetic black brane solution. These solutions are not known analytically, but havebeen studied numerically [40, 43]. In our infalling coordinates, the solutions satisfy thestatic specialization of eqs. (2.36). The near-boundary behavior of asymptoticallyAdS solutions is given by the expansions (2.41) (but with no time dependence).The extremal, zero-temperature magnetic brane solution interpolates smoothly be-tween AdS × R near the horizon and AdS near the boundary. In our infallingcoordinates, a series in fractional powers of δr ≡ r − r h describes deviations from theAdS × R geometry near the horizon, A ( r ) = ( δr/L ) (cid:2) η δr γ + O (cid:0) η δr γ (cid:1)(cid:3) , (2.65a)Σ( r ) = ( B L δr ) / (cid:2) − (3+ γ ) η δr γ + O (cid:0) η δr γ (cid:1)(cid:3) , (2.65b) B ( r ) = − ln[27 δr / ( B L )] − (13+2 γ ) η δr γ + O (cid:0) η δr γ (cid:1) , (2.65c) The resulting equations may be written explicitly as: (cid:0) A (cid:48) Σ (cid:1) (cid:48) Σ − = + B L e − B Σ − + E L + 4 /L , (2.64a) (cid:0) A Σ (cid:48) Σ (cid:1) (cid:48) Σ − = − B L e − B Σ − − E L + 2 /L , (2.64b) (cid:0) AB (cid:48) Σ (cid:1) (cid:48) Σ − = − B L e − B Σ − . (2.64c) – 19 –ith γ ≡ − √
57. The constant η cannot be determined from a near-horizonanalysis and must be suitably adjusted after integrating eqs. (2.64) to obtain the desiredboundary geometry. There is a single extremal magnetic brane solution, modulo therescaling transformations (2.43)-(2.45) (which relate solutions with any non-zero valuesof the magnetic field B and curvature scale L ), and radial shift diffeomorphisms (2.22).For non-extremal solutions (with non-zero B but vanishing ρ ), metric functionsnear the horizon have power series expansions in δr ≡ r − r h of the form A ( r ) = a δr L − (cid:2) − (1 − B L s − ) a − δr + O ( δr ) (cid:3) , (2.66a)Σ( r ) = s /β (cid:2) − B L s − ) a − δr + O ( δr ) (cid:3) , (2.66b) B ( r ) = 2 ln β − B L s − a − δr + O ( δr ) . (2.66c)The coefficient a is proportional to the horizon temperature, T = a / (2 πL ) . (2.67)The other two undetermined constants, s and β , which control the horizon values ofthe spatial scale factor and the anisotropy function, must be suitably adjusted afterintegrating eqs. (2.64) to select solutions which have the desired near-boundary behavior(with an isotropic boundary metric). If B (cid:28) T , then the resulting magnetic branegeometry is a small perturbation away from the Schwarzschild black brane (2.55), whileif B (cid:29) T then the geometry may be regarded as interpolating between the BTZ blackbrane ( × R ) near the horizon and AdS near the boundary [40].There is a one parameter family of non-extremal solutions, modulo the rescalingtransformations (2.43)-(2.45) (and radial shift diffeomorphisms). Physically distinctsolutions may be labeled by the value of the dimensionless ratio ε B / B [or ( πT ) / B or ε B / ( πT ) , etc.]. The left panel of figure 3 shows a log-log plot of our numericallydetermined curve representing these solutions in the plane of ε B / B and ( πT ) / B .Extrapolating our lowest temperature numerical results to zero temperature, we findestimates of c ≈ . , B ∗ ( µ ) ≈ . µ , (2.68)for the coefficient c or the equivalent fiducial scale B ∗ defined by eqs. (2.15) and (2.16).If one chooses to measure energy density and magnetic field in units set by thecurvature scale L , then one may traverse the one-parameter family of magnetic branesolutions by varying |B| L for a fixed value of ε L L (or vice versa). The holographicrelation (2.42) shows that these curvature scale dependent quantities are related to theintrinsic dimensionless parameter ε B / B via ε B B = ε L L ( |B| L ) + ln( |B| L ) . (2.69)– 20 – igh - TLow - T ( π T ) ℬ ε ℬ / ℬ ε L L = ε L L = - | ℬ | L - - ε ℬ / ℬ Figure 3 . Left: The one parameter family of non-extremal equilibrium magnetic branesolutions in the plane of ε B / B vs. ( πT ) / B . Also shown are the high and low temperatureasymptotic forms (dashed lines). For high temperatures, πT (cid:29) |B| / , the curve approachesthe Schwarzschild result ε = ( πT ) . For low temperatures, πT (cid:28) |B| / , the form ε B / B ∼ c + c ( πT ) / |B| provides a good fit to our data for c = 0 .
35 and c = 0 .
20. Right: Therelation between the intrinsic parameter ε B / B labeling magnetic brane solutions and thevalue of the magnetic field in curvature scale units, |B| L , for two different fixed values of thecurvature scale energy density, ε L L = ± . This relation between ε B / B and |B| L , for two different fixed values of the curvaturescale energy density ε L L = ± .
75, is plotted in the right panel of fig. 3. Note that twodifferent values of |B| L yield the same value of ε B / B (and hence describe the samephysical solution) when ε L > B ( r )increases (and the scale factor Σ decreases) smoothly as one moves inward from theboundary toward the horizon. Figure 4 (left) plots the resulting anisotropy function B ( r ) for several values of the magnetic field when the energy density at the scale 1 /L is held fixed, ε L L = . (From eq. (2.42a), this is the same as fixing the asymptoticcoefficient a L = − .) The horizon temperatures for this series of solutions, in orderof increasing magnetic field, are given by πT L = 0 . a L ).The right panel of fig. 4 shows a similar set of solutions with increasing magneticfield, but now with the energy density at the scale |B| / held fixed, ε B = 8 L − . Withthe physical parameter ε B held fixed, the horizon value of the anisotropy function isnow monotonically increasing with magnetic field. The temperatures of these solutions(in order of increasing B ) are given by πT L = 1 . L = ℬ L = ℬ L = ℬ L = ℬ L = u B ℬ L = ℬ L = ℬ L = ℬ L = ℬ L = u B Figure 4 . The anisotropy function B ( r ) as a function of inverse radius u ≡ /r for equilibriummagnetic brane solutions with different values of the magnetic field. Left panel: Solutionsat fixed energy density ε L = 0 . L − with B L = 1 .
0, 1.5, 2.5, 3.5, and 4.5. Right panel:Solutions at fixed energy density ε B = 8 . L − with B L = 0 .
0, 1.0, 2.0, 3.0, and 4.0. In allcases, the radial shift λ has been suitably adjusted to fix the horizon radius at u = 1. Notethat the horizon value of the anisotropy function is not a monotonic function of magneticfield at fixed ε L , but is monotonic when ε B is held fixed. We apply the computational strategy presented in ref. [18] to our case of homogeneousisotropization in Einstein-Maxwell theory. For convenience, we choose units in whichthe AdS curvature scale L = 1.Required initial data, on some v = v time slice, consists of an initial choice forthe anisotropy function B ( v , r ) and the radial shift λ ( v ), along with chosen (timeindependent) values of the energy density ε , charge density ρ , and magnetic field B .As noted above, the holographic relation (2.42a) shows that fixing the energy density ε at the scale µ = 1 /L is equivalent to fixing the asymptotic coefficient a . Our choicesfor the initial anisotropy function will be detailed below in subsection 3.3.Given a set of initial data, the linear second order radial ordinary differential equa-tion (ODE) (2.36a) may be integrated to find the spatial scale factor Σ( v , r ). Thetwo leading terms in the asymptotic behavior (2.41a) provide the integration constantsneeded to specify uniquely the desired solution. Next, one solves eq. (2.36d), a linearfirst order radial ODE for d + Σ. The near-boundary asymptotic behavior of this func-tion is d + Σ ∼ ( r + λ ) + a r − + O ( r − ) and homogeneous solutions to eq. (2.36d)behave as r − near the boundary. Hence, the chosen value of the energy density ε – 22 –niquely specifies the desired solution. With B , Σ, and d + Σ determined on the v timeslice, one next solves eq. (2.36c), a linear first order radial ODE for d + B . The desiredasymptotic behavior of this function is d + B ∼ − b r − + O ( r − ) while homogeneoussolutions to eq. (2.36c) behave as r − / near the boundary. So the needed integra-tion constant corresponds to requiring the absence of any such homogeneous solution.Finally, one solves the second order linear ODE (2.36b) to determine A ( v , r ). Homo-geneous solutions are linear or constant functions of r . From the asymptotic behavior(2.41b), one sees that the value of the radial shift λ ( v ) fixes the coefficient of the homo-geneous solution linear in r and provides one of the two needed boundary conditions.The second boundary condition, needed to fix the constant homogeneous solution, isprovided by the horizon stationarity condition (2.50), which determines the value of A on the apparent horizon.Having solved for the modified time derivative d + B ( v , r ), and A ( v , r ), one recon-structs the ordinary time derivative of the anisotropy function via ∂ v B ( v , r ) = d + B ( v , r ) − A ( v , r ) ∂ r B ( v , r ) . (3.1)The time derivative of the radial shift, ∂ v λ ( v ), is extracted from the asymptotic behav-ior (2.41b) of A by evaluating the r → ∞ limit of A − ( r + λ ) . These time derivativesof B and λ provide the information needed to advance in time. Using a standardnumerical integration scheme, one takes a small step forward in time, advancing v to v + ∆ v for some timestep ∆ v . Repeating this process, one progressively determinesthe metric functions on a sequence of equally spaced time slices, v = v k ≡ v + k ∆ v .On each time slice, the asymptotic coefficient b ( v ), needed to determined the stresstensor (2.42), is extracted from the large r behavior of the anisotropy function B . (Inthe presence of a non-zero magnetic field, one extracts b from the large r limit ofa subtracted, rescaled version of the anisotropy function which removes the leadinglogarithmic piece in eq. (2.41c). This is detailed in the next subsection.) We use an inverted radial coordinate u = 1 /r , and arbitrarily choose¯ r h = 1 , (3.2)as our apparent horizon location. This makes our computational domain a fixed radialinterval, 0 ≤ u ≤ u h ≡
1. We use a 4th order Runge-Kutta method (described inref. [18]) for time integration. This requires four integrations of our radial ODEs pertime step, but yields much better accuracy, for a given timestep ∆ v , than a lower ordermethod. – 23 –o integrate the radial ODEs (2.36a–2.36d), we have used both traditional short-range finite difference approximations, and spectral methods [47]. In the latter ap-proach, one implicitly represents the radial dependence of functions as a (truncated)series of Chebyshev polynomials. Explicitly, functions are represented by the list oftheir values on a discrete, finite collocation grid consisting of the points u k = (cid:104) π kM − (cid:105) , k = 0 , · · · , M − , (3.3)and derivatives are represented by (dense) M × M matrices acting on the finite list offunction values. The (truncated) spectral expansion converts each ODE into a straight-forward linear algebra problem. Boundary conditions are simply encoded into the firstor last rows of the resulting matrix [47].Although there are subtleties (described momentarily) in applying spectral methodsto our problem, we have found the use of spectral methods to be clearly superior tofinite difference approximations, yielding both faster computation and more accurateresults. Using an M point discretization of the computation domain, short-range finitedifference methods have errors which scale as an inverse power of M while spectralmethods, in favorable cases, produce errors which fall exponentially with increasing M .Spectral methods presume that one is approximating functions which are regularand well-behaved on the computational domain. However, our radial ODEs have reg-ular singular points at u = 0 or r = ∞ (due to the r growth of the scale factor nearthe boundary), and our functions Σ, d + Σ, and A all diverge at the u = 0 endpoint.Therefore, we define subtracted functions in which the known singular near-boundarybehavior is removed. To minimize loss of numerical precision in spectral approxima-tions of derivatives, it is also helpful to rescale the subtracted functions so that theydo not vanish faster than linearly as u →
0. If the magnetic field is zero, so no log-arithmic terms are present in the near-boundary behavior (2.41), then our subtractedfunctions are analytic in a neighborhood of the u ∈ [0 ,
1] radial interval and spectralmethods converge exponentially. With a non-zero magnetic field, logarithmic termsare unavoidably present, showing that u = 0 is a branch point of the metric functions.This degrades convergence of the spectral series, leading to power-law convergence at arate which depends on the behavior of the leading non-analyticity. Consequently, it isdesirable to subtract logarithmic terms to as high an order as is practicable. We chose– 24 –o introduce subtracted/rescaled functions (denoted with a subscript ‘s’) via:Σ( r ) = ( r + λ ) + r − Σ s ( r ) , (3.4a) A ( r ) = ( r + λ ) − B ln r (cid:2) r − − λ r − + 3 λ r − − λ r − (cid:3) + A s ( r ) . (3.4b) d + Σ( r ) = Σ( r ) − B ln r (cid:2) r − − λ r − + 3 λ r − − λ r − (cid:3) + ( d + Σ) s ( r ) , (3.4c) d + B ( r ) = − B ln r (cid:2) r − − λ r − + 6 λ r − − λ r − (cid:3) + r − ( d + B ) s ( r ) . (3.4d)All our numerical work is performed using these subtracted/rescaled functions; wedirectly solve for Σ s , ( d + Σ) s , ( d + B ) s , and A s . The expressions (3.4) are used toreconstruct the original functions when needed. We also use a subtracted/rescaledanisotropy function B s ( r ), introduced via B ( r ) = B ln r (cid:2) r − − λ r − + 10 λ r − (cid:3) + r − B s ( r ) . (3.5)This removes leading logarithmic terms and introduces a convenient rescaling. Hence-forth, B s will be referred to as the subtracted anisotropy function.In the above subtractions, the series in 1 /r multiplying each logarithm are justtruncated expansions of ( r + λ ) − k for k = 2, 3 or 4. The choice to truncate thesebinomial series was arbitrary, but we found that our numerics were sufficiently well-behaved with the above subtractions. At higher orders in 1 /r , additional terms appearwhich involve the asymptotic coefficient b and its (a-priori unknown) time derivatives,as well as higher powers of logarithms. As indicated above, one must select the value of the energy density (or asymptoticcoefficient a ) and the initial value of the radial profile of the anisotropy function, B ( v , r ). And one must choose the value of the magnetic field B or charge density ρ .For the charged case, with vanishing magnetic field, physics can only depend on thedimensionless combination of charge and energy densities ρ/ε / , so a (positive) valueof ε may be chosen arbitrarily without loss of generality. Given a choice of ε , possiblevalues of the charge density ρ are limited by the extremality bound | ρ | ≤ ρ max = ε / .For the initial anisotropy function, we chose to focus on Gaussian profiles. In the λ = 0 frame, B ( v , r ) = A e − ( r − r ) /σ . (3.6) Note that ( d + Σ) s (cid:54) = d + (Σ s ), and likewise for ( d + B ) s ; these are just names for the sub-tracted/rescaled functions for d + Σ and d + B , respectively. – 25 –e investigate the dependence of results on the parameters of the Gaussian (amplitude A , width σ , and mean position r ) in the first part of section 4. Motivated by the factthat in our coordinates lines of varying r , at fixed v , are radially infalling geodesics, wewill refer to this initial Gaussian as a “pulse” of initial anisotropy.For the magnetic case, as discussed above, the breaking of scale invariance impliesthe presence of logarithmic terms in the asymptotic behavior of the anisotropy function.We simply add the log terms shown in eq. (3.5) to the Gaussian (3.6). With vanishingradial shift, λ = 0, our chosen initial anisotropy function takes the form B ( v , r ) = A e − ( r − r ) /σ + B r − ln r . (3.7)Arguably, a more natural choice for the magnetic case initial data might be to adda Gaussian to the full equilibrium solution for the anisotropy function in the chosenmagnetic field. This could be seen as nicely paralleling the charged case (in whichthe equilibrium solution has vanishing anisotropy). Nevertheless, we will stick withour somewhat arbitrary choice (3.7), which is an adjustable initial pulse added tothe correct asymptotics. As will be seen, the Gaussian pulse will nearly always bethe dominant portion of the deviation from equilibrium and the driving force of theresulting anisotropy in the boundary stress. We doubt that differing choices in theprecise form of the slowly varying function to which the Gaussian pulse is added wouldimpact, in any significant way, the characteristic equilibration times or other significantfeatures of the results presented below.After choosing the initial anisotropy function (in the λ = 0 frame) the initial valueof the radial shift, λ ( v ), is adjusted to fix the location of the apparent horizon, asdiscussed in section 2.6.It should be noted that, in all cases (charged, uncharged, magnetized) it is quitepossible to select physically inconsistent initial data. This happens when the initialanisotropy, for a given energy density, is so large that no apparent horizon shields acoordinate singularity produced by a vanishing scale factor Σ. This sets a natural limiton the amplitude of the initial pulse which is meaningful to study. Before presenting results for equilibration in charged or magnetized plasmas, we firstdiscuss general features of the time evolution in the uncharged, unmagnetized case and For results from an exploration of a broader range of initial anisotropy profiles, in the case ofvanishing charge density and magnetic field, see ref. [9]. – 26 –xamine the sensitivity of results to specific features in the initial data. As noted above,we choose a Gaussian profile (3.6) for the initial anisotropy function, with an adjustableamplitude, width, and mean position. Typical evolution of our subtracted/rescaledanisotropy function, B s ( v, u ) ≡ u − B ( v, u ), is shown on the left in figure 5. One seesthe initial pulse profile on the back side of the plot at v = 0. The figure clearly showsthe influence of the pulse propagating outward and reflecting off the boundary at u = 0.The outgoing portion of the pulse essentially propagates along an outgoing radial nullgeodesic which, in our coordinates, near the boundary are 45 ◦ lines at constant valuesof u + v . The influence of the anisotropy pulse, after the reflection, largely falls throughthe horizon along an ingoing radial null geodesic which is instantaneous in our null timecoordinate v . The asymptotic coefficient b , which equals the slope of B s at u = 0,controls the anisotropy in the stress tensor,∆ P ≡ (cid:104) T (cid:105) + (cid:104) T (cid:105) − (cid:104) T (cid:105) , (4.1)with ∆ P /κ = 3 b . Hence, the reflection of the pulse off the boundary directly producesthe pressure anisotropy ∆ P in the boundary theory. The time dependence of therelative pressure anisotropy, defined as ∆ P / ( κ(cid:15) ), is plotted on the right in figure 5.As shown in the figure, at late times the anisotropy function approaches zero, asrequired for equilibration to the isotropic Schwarzschild black brane solution. At suffi-ciently late times, when the departure from equilibrium is small, the evolution shouldbe well described by a linearized approximation to the full nonlinear dynamics. Thelinearized dynamics of infinitesimal perturbations away from equilibrium may be repre-sented as a sum of quasinormal modes (QNM), which are eigenfunctions of the linearizeddynamics with complex frequencies, φ ( t ) = Re( A e − iωt ) with Im ω <
0. The lowestquasinormal mode (for which − Im ω is minimal) dominates the late time approach toequilibrium. For the Schwarzschild black brane, quasinormal mode frequencies havebeen previously evaluated by Starinets [48]. From the late time behavior of our fullnonlinear evolution, it is straightforward to extract an estimate of the lowest quasinor-mal mode frequency. Comparing with the independent results of ref. [48] provides auseful test of the accuracy of our numerics. Fitting the late time (4 (cid:46) v ε / (cid:46) | A | e (Im ω ) v cos[(Re ω ) v + φ ], yields an estimate of the lowest QNM frequency ω whichagrees with ref. [48] to five digits, ω/ ( πT ) ≈ . − . i .We will see the same vanishing of the anisotropy function at late times for thecase of charged plasmas, whose gravitational duals equilibrate to an isotropic Reissner-Nordstrom black brane solution. For the magnetic case, however, at late times there Note that, for unmagnetized plasma, the energy density is three times the average pressure, ε = (cid:104) T ii (cid:105) ≡ P . – 27 – .2 0.4 0.6 0.8 1.0 v ε / - - Δ / κε Figure 5 . Left: Rescaled anisotropy function B s = u − B , for a typical case of equilibrationto the Schwarzschild black brane, as a function of inverse bulk radius u and time v . Initialpulse parameters are A = 5 × − , r = 4 and σ = , with ε = L − . The (rescaled) energydensity ε is used to the set the scale for time. One sees that the effect of the initial Gaussianpulse propagates outward, essentially along an outgoing radial null geodesic, reflects off theboundary, and then largely falls through the horizon (along an ingoing radial null geodesicwhich is instantaneous in v ). After one “bounce”, the anisotropy rapidly approaches zero.Right: The corresponding relative pressure anisotropy, ∆ P /κε ≡ ( T + T − T ) /κε ,induced in the boundary field theory, as a function of time. Note how the peaks of the pressureanisotropy correspond directly to the reflection of the anisotropy pulse off the boundary. is a non-zero profile for the anisotropy function, reflecting the spatial anisotropy ofequilibrium magnetic brane solutions.We now turn to an examination of the dependence of the pressure anisotropy on theparameters of the initial Gaussian (3.6). Of particular interest will be the dependenceof the response on the amplitude and position of the initial pulse. Less interesting isthe dependence on the width of the pulse, which affects the duration of the reflection offthe boundary (and also produces changes more naturally associated with the positionof the pulse).In figure 6, we compare the pressure anisotropies created by the pulse shown infig. 5, and an otherwise identical pulse with half the amplitude. Visually, one sees thatthe smaller amplitude pulse produces roughly half the pressure anisotropy as does thelarger pulse, but with a virtually identical time course. The peak pressure anisotropy(divided by energy density), for both pulses is over 4, significantly larger than unity.Hence both initial pulses represent large departures from equilibrium. Given the highlynonlinear nature of the Einstein equations, one might have expected to see clear signs of– 28 – .2 0.4 0.6 0.8 v ε / - - Δ / κε v ε / - × - - × - × - × - NL Figure 6 . Left: The pressure anisotropy ∆
P/κε produced by the same initial pulse shownin fig. 5 (blue curve), overlaid with the pressure anisotropy produced by an initial pulse withhalf the amplitude (purple curve). Halving the initial amplitude roughly halves the inducedpressure anisotropy. Right: The “nonlinearity” (NL), defined as the difference between thepressure anisotropy produced by the larger pulse and twice the pressure anisotropy producedby the half amplitude pulse. nonlinearity in the dependence of the pressure anisotropy on the initial pulses. However,even for these pulses producing large departures from equilibrium, the amplitude of thepeaks in the induced pressure anisotropy are nearly linear in the amplitude of the initialGaussian pulse.The right hand panel of figure 6 makes this comparison quantitative. This showsthe nonlinearity (NL) defined as the difference between the pressure anisotropy ∆ P /κε of the larger initial pulse and twice the pressure anisotropy produced by the halvedinitial pulse. Compared to the pressure anisotropies themselves, the relative size of thenonlinearity is roughly one part in 10 . This suggests that the dynamics, as probed bythese initial pulses, are surprisingly close to a linear dynamical system.In asymptotically AdS gravitational solutions, deviations of the geometry fromthat of pure AdS space necessarily vanish as one approaches the boundary. Hence, onemight expect nearly linear dynamics to be evident for initial pulses which are localizedsufficiently close to the boundary, while anticipating much larger nonlinearities forinitial pulses localized closer to the horizon. To test this expectation, we used a verylarge Gaussian — the same initial Gaussian profile which generated fig. 5 but with theamplitude increased by a factor of 40 — and then examined the resulting evolutionwhen the mean position of the initial Gaussian was progressively shifted deeper intothe bulk. Figure 7 compares the evolution for mean positions r = { , , , , } , in the For the chosen values of position and width of the Gaussian, plus energy density ε = L − , thisamplitude is close to the upper limit set by demanding the existence of an apparent horizon. – 29 – .2 0.4 0.6 0.8 1. u B s
1. 2. v ε / - - Δ / κε
1. 2. v ε / - - u B s
1. 2. v ε / - Δ / κε
1. 2. v ε / u B s
1. 2. v ε / - Δ / κε
1. 2. v ε / u B s
1. 2. v ε / Δ / κε
1. 2. v ε / × - × - × - NL u B s
1. 2. v ε / Δ / κε
1. 2. v ε / × - × - NL Figure 7 . Comparisons of initial anisotropy functions (left column), induced pressureanisotropy (middle column), and nonlinearity (right column) defined as in fig. 6, for a seriesof five Gaussian initial anisotropy functions differing only in their depth in the bulk. Fromtop to bottom, the mean position of the initial pulse, in the λ = 0 frame, is r = { , , , , } .In all cases the energy density is ε = L − . The plots in the first row come from the sameinitial data as in fig. 5, but with the amplitude increased by a factor of 40 ( A = 0 . r = 4, σ = ). The left column shows the initial anisotropy function scaled by u − and plottedas a function of the inverse radial coordinate u , after adjusting the radial shift λ to fix theapparent horizon at u = 1. – 30 –rame with radial shift λ = 0. In all cases, the energy density was held fixed at a value( ε = L − ) which puts the equilibrium horizon position at r = 1. In other words, theonly change in the five cases shown in fig. 7 is the radial position of the initial Gaussiananisotropy function (3.6), viewed as a function of r .Each row of fig. 7 displays results for one of these five cases. In each row, theleft hand panel shows the initial anisotropy function, but plotted as a function ofthe inverse radial coordinate u , after adjusting the radial shift λ to fix the apparenthorizon at u = 1, as discussed in sec. 2.6. The middle panels show the resulting pressureanisotropy as a function of time, and the rightmost panels display the nonlinearity (NL),again defined as the difference between the pressure anisotropy of the given initial pulseand twice the pressure anisotropy after halving the initial amplitude.From the middle column of plots, one sees that the magnitude of the pressureanisotropy decreases significantly as the initial pulse is moved deeper into the bulk.Moreover, both the time it takes for the effect of the pulse to reach the boundary, andthe width of the resulting peaks in the pressure anisotropy, grow with increasing depthof the initial pulse. This reflects the usual holographic mapping between bulk andboundary: phenomena deeper in the bulk correspond to lower energy or longer timescales in the boundary field theory.Turning to the nonlinearity plots in the right hand column, one sees that themagnitude of the nonlinearity also decreases as the pulse moves deeper into the bulk.Dividing the peak nonlinearity by the peak pressure anisotropy gives a relative measureof nonlinearity. This is about 1 × − for the top row, 3 × − for the middle row,and 3 × − for the bottom row. So in this comparison, as the initial pulse is pusheddeeper into the bulk, the relative nonlinearity decreases systematically.This comparison does not, however, imply that nonlinearities are never significant.The amplitude of the Gaussian pulse in the initial anisotropy function was kept fixedin fig. 7, resulting in a decreasing size of the induced pressure anisotropy as the pulsemoves deeper into the bulk. While the first two rows of the figure show pressureanisotropies which are large departures from equilibrium, ∆ P /κε (cid:29)
1, the final rowswith ∆ P /κε (cid:28) A = 5 × − , r = 4, σ = ), while the lower row shows thepressure anisotropy and nonlinearity of a pulse with the same shape as the last rowof fig. 7, but with larger amplitude ( A = 2 . r = , σ = ). The energy densityremains fixed, ε = L − . The peak pressure anisotropy is similar in the two cases,– 31 – .5 1.0 1.5 2.0 2.5 v ε / - - Δ / κε v ε / - × - × - × - NL v ε / - Δ / κε v ε / - - - NL Figure 8 . Top row: pressure anisotropy (left) and nonlinearity (right), defined as in fig. 6,as a function of time for the initial the pulse which created fig. 5. ( A = 5 × − , r = 4, σ = ). Bottom row: corresponding plots of pressure anisotropy and nonlinearity for a pulselocated deeper in the bulk ( A = 2 . r = , σ = ) with amplitude adjusted to produce asimilar peak pressure anisotropy. In both cases the energy density is ε = L − . Substantialnonlinearity is present for this case, where an initial pulse deep in the bulk has sufficientamplitude to produce a large departure from equilibrium. and corresponds to a large departure from equilibrium. For the latter case of a pulsedeep in the bulk, large enough to induce a far from equilibrium pressure anisotropy, thenonlinearity is significant, much larger than the previous examples. However, even forthis case, the size of the nonlinearity relative to the peak pressure anisotropy is onlyabout 5%, NL / (∆ P /κε ) ≈ . The data shown in fig. 7 inspire several further questions. Looking down the middlecolumn of the figure, one sees that the onset of the response (i.e., the time of the firstpeak in the pressure anisotropy) increases as the initial pulse moves deeper into thebulk but seems, perhaps, to be approaching a maximum value. Is this really true, orcan one craft initial data for which the onset of the response is much greater? As shown We have also examined the level of nonlinearity by comparing the pressure anisotropy resultingfrom a sum of two different Gaussians to the sum of anisotropies induced by the individual Gaussians.The results were comparable to those discussed above and do not warrant separate discussion. – 32 – .2 0.4 0.6 0.8 1.0 u B s v ε / - - Δ / κε v ε / - - - Figure 9 . Initial anisotropy function (left), induced pressure anisotropy (middle), and non-linearity (right) for a narrow “deep pulse” ( A = , r = , σ = ) localized closer to thehorizon than the deepest pulses of fig. 7. The energy density remains fixed at the same value, ε = L − . Relative to the previous case of fig.8, the induced pressure response has a delayedonset, but is otherwise very similar. on the left panels of the lower rows of the figure, when the average position r of theGaussian pulse is moved into the bulk, an increasingly large portion of the Gaussianends up lying behind the apparent horizon. And when plotted as a function of ourcomputational coordinate u = (¯ r − λ ) − (with ¯ r the λ = 0 frame radial coordinate),initial pulses with small values of r are only moderately localized near the horizon —even though these pulses had constant widths when viewed as functions of r . As maybe seen by comparing the left and middle columns of fig. 7, it is the leading edge ofthe anisotropy pulse (the region of near-maximal slope) which produces the first largeresponse in the boundary anisotropy. Looking at the last two rows of the figure, onemay question whether we are doing an adequate job exploring the response from initialdisturbances which are localized close to the horizon. Will initial pulses which are morestrongly localized near the horizon show significantly greater nonlinearity?Figures 9 and 10 show results of an effort to explore these questions. Fig. 9 showsthe initial anisotropy function, along with the resulting pressure anisotropy and non-linearity, for a significantly narrower “deep pulse” ( A = , r = , σ = ). Andfig. 10 shows a 3D plot of the time dependent anisotropy function, plus the inducedpressure anisotropy, for an extremely narrow deep pulse ( A = , r = 1, σ = ). The energy density in both cases remains fixed, ε = L − . For the narrowest pulse, theamplitude A = is near the upper limit which can be studied without destabilizingthe horizon. In both figures 9 and 10, the peak pressure anisotropy ∆ P /κε is largecompared to unity, showing that these pulses are producing far from equilibrium initialstates.As seen in these figures, pulses which are more sharply localized very near thehorizon do lead to a delayed onset in the resulting pressure anisotropy, occurring at This width is at the limit of what our numerics could do using a 240 point spectral grid. – 33 – v ε / - Δ / κε v ε / - - - Figure 10 . Rescaled anisotropy function B s (left) as a function of inverted radius u and time v , for an extremely narrow pulse sitting at the horizon ( A = , r = 1, σ = ). Resultingpressure anisotropy (middle) and nonlinearity (right) as a function of time. Energy density ε = L − . The outward movement of the pulse toward the boundary clearly resembles thebehavior of outgoing null geodesics originating very close to an event horizon, which can“hug” the horizon for extended periods before eventually escaping. vε / ≈ .
75 for the case of fig. 9 and vε / ≈ B s ,one sees that the outward movement of the pulse toward the boundary resembles thebehavior of outgoing null geodesics originating very close to an event horizon, which“hug” the horizon for extended periods before eventually escaping. One may wonderif the onset of the response in the pressure anisotropy could be delayed indefinitely bygoing to narrower and narrower initial pulses localized at the apparent horizon. We donot have a firm analytic argument, but doubt that this is possible if one simultaneouslydemands that the amplitude of the response remain bounded away from zero. The relative nonlinearity for the case of a narrow deep pulse shown in fig. 9 is quitesmall, about half a percent. But for the extremely narrow pulse with near maximalamplitude of fig. 10, the nonlinearity, relative to the pressure anisotropy, reaches the10% level. Linearization of the dynamics about equilibrium must provide an accurateapproximation to the full nonlinear dynamics when the deviation of the geometry fromthe equilibrium Schwarzschild black brane solution is sufficiently small, as will be trueat sufficiently late times. For asymptotically anti-de Sitter geometries, where metricfunctions have the asymptotic forms (2.41), this will also be the case for initial datainvolving perturbations localized sufficiently close to the boundary. (See refs. [8, 9, 49] This expectation reflects the diverging redshift of late emerging outgoing geodesics in the geomet-ric optics picture, and is consistent with the gapped spectrum of quasinormal mode frequencies fortranslationally invariant perturbations. – 34 –or related discussion.) It should be noted that reasonably good agreement betweenlinearized dynamics and full nonlinear evolution was previously reported in ref. [8] andfurther explored in ref. [9]. In these works, the authors found agreement at a 20% levelbetween the linearized and full dynamics for a variety of initial anisotropy profiles. Ourresults examining, more systematically, the dependence of the relative nonlinearity onthe parameters of our initial Gaussian anisotropy function complement and extend thisearlier work. Overall, despite prior knowledge of refs. [8, 9], we are still surprised by thesmall, often extremely small, levels of nonlinearity which we find even at early timeswhen the induced pressure anisotropy is maximal, for initial perturbations localizeddeep in the bulk and producing large departures from equilibrium.
We now turn to the equilibration of charged plasmas (by which we mean SYM plasmaswith a non-zero density of the global U (1) conserved charge — not a plasma in whichelectromagnetic forces are included in the dynamics and Coulomb repulsion plays asignificant role). As noted earlier in section 2.7, the bulk geometry should equilibrate toa non-singular Reissner-Nordstrom black brane solution provided the charge and energydensities satisfy the extremality bound (2.58), ρ < ρ max = ε / . However, for values ofthe charge density near ρ max , we find that sufficiently large initial metric perturbationscan destroy the apparent horizon. We expect that such initial data are unphysical, notrepresenting SYM initial states which could be produced by an operational proceduresuch as turning on time dependent external fields (which correspond to time dependentboundary data in the holographic description). In any case, we limit our attention toinitial perturbations for which an apparent horizon is present at all times. We find thatif one suitably decreases the amplitude of the initial departure from equilibrium whileincreasing the charge density, one can approach ρ max from below while maintaining theexistence of an apparent horizon.Figure 11 (left) compares the time dependence of the pressure anisotropy whichresults from initial data consisting of precisely the same Gaussian initial anisotropyfunction B ( v , r ) (in the λ = 0 frame) and energy density as in fig. 5, and a chargedensity ρ equal to 0, 20, 40, 60, or 80% of the extremal density ρ max . The immediatelyobvious qualitative result is that the five different curves are so close together than theycannot be visually distinguished! Varying the charge density (at fixed energy densityand fixed initial anisotropy function) has stunningly little impact on the subsequenttime evolution. This is quantified in the right panel of fig. 11 which plots the differ-ence in the pressure anisotropy ∆ P /κε between the cases of ρ = 0 . ρ max and ρ = 0.Comparing the scales on the right and left hand plots, one sees that for this initialanisotropy function the sensitivity to the charge density is less than one part in 10 .– 35 – % % % % % v ε / - - Δ / κε % vs. 0 % v ε / - - - δ ( Δ / κε ) Figure 11 . Left: time dependence of the pressure anisotropy (relative to κε ) for values of thecharge density ρ which are 0, 20, 40, 60, or 80% of the extremal density ρ max . The differentcurves are virtually indistinguishable. The initial anisotropy function B ( v , r ) and energydensity ε are the same as in fig. 5. Right: difference in ∆ P /κε between ρ = 0 . ρ max and ρ = 0. In the plots of fig. 11, we used the fourth root of the (rescaled) energy density, ε / ,to set the scale for time. Since the energy density was held constant in the comparisonsof fig. 11, this was a simple and convenient choice. For the five cases shown in thefigure, ε/ ( πT ) = 0 .
75, 0 .
76 , 0 .
79, 0 .
86, and 1 .
03 for ρ = 0, 20, 40, 60 and 80% of ρ max ,respectively. And, for comparison, the values of the equilibrium chemical potentialscorresponding to these charge densities are given by µ/T = 0, 0.34, 0.73, 1.26, and2.21, respectively.If the initial anisotropy pulse begins deeper in the bulk, then the sensitivity tothe charge density is larger. Figure 12 shows a comparison of pressure anisotropies fordiffering charge densities, now using the deep pulse initial anisotropy function whoseradial profile has the shape shown in fig. 9. As seen on the left panel, the amplitude ofthe response increases significantly as the charge density varies from 0 to 80% of ρ max .However, the time course of the equilibration (e.g., the times of the first or second peaksin the response, or the zero-crossing between these peaks) is only modestly affected,with changes of 3% percent or less.The fact that the sensitivity to the charge density is greatest for pulses close tothe horizon is to be expected. In the equilibrium geometry (2.57), one sees that asthe radius ˜ r increases from the horizon, the influence of the charge density decreases One might guess that the pulse used in the bottom row of fig. 8 would exhibit greater sensitivityto charge since it had a larger nonlinearity than the deep pulse of fig. 9. However, the latter pulseexhibits much greater sensitivity to charge. – 36 – % % % % % v ε / - - - - Δ / κε % vs. 0 % v ε / - δ ( Δ / κε ) Figure 12 . Left: comparison of pressure anisotropies produced by different charge densities,up to 80% of extremality, when the initial anisotropy function is a “deep pulse” ( A = , r = , σ = ) with ε = L − . Although the amplitude of the response grows, as shown,with increasing charge density, it is striking how little the time course of the response varies.Right: difference in pressure anisotropy ∆ P /κε between ρ = 0 . ρ max and ρ = 0. rapidly relative to the other terms in the metric. Only near the horizon, and close toextremality, does the charge density produce an O (1) effect on the equilibrium geometry.At the beginning of this work, we expected that one interesting outcome wouldbe information on the change in equilibration time produced by varying the plasmacharge density. By “equilibration time”, we mean some rough but useful measure ofwhen the departure from equilibrium is no longer substantial. To make this a bit morequantitative we adopt, somewhat arbitrarily, the criterion∆ P ( t ) /κε ≤ . , (4.2)for all times t > t eq , as indicating that the system is near equilibrium at time t eq .Looking at the left panels of figures 11 and 12, it is clear that the effect of thecharge density on any reasonable measure of equilibration time can be summarizedeasily: there is very little effect! Even in fig. 12, where the sensitivity to charge densityis the largest we have found with horizon-preserving initial data, the time t peak ofthe initial response peak and the approximate equilibration time t eq both vary by lessthan 5%.Since figures 11 and 12 plot time in units set by the energy density, the high degreeof insensitivity of the relaxation time course to the charge density seen in these figures Achieving good numerical accuracy becomes increasingly difficult as one pushes toward extremal-ity, where the equilibrium solution bifurcates. Investigating very near extremal behavior more carefullyis an interesting topic we leave to future work. – 37 – % % v π T - - Δ / κε Figure 13 . Comparison of the pressure anisotropies produced by two different charge densi-ties, 0% and 80% of extremality, when holding fixed the equilibrium temperature, πT = 1 /L .The energy densities are 0.75 and 2.68, respectively. The initial anisotropy function is same asin fig. 5 ( A = 5 × − , r = 4, σ = ). With time is plotted in units of inverse temperature,the relaxation time course shows negligible sensitivity to the charge density (although theamplitude of the response varies significantly). might lead one to think that the total energy density is playing a special role in settingthe time scale of relaxation. But it should be borne in mind that these figures showcomparisons in which, by design, both the initial anisotropy function (in the λ = 0frame) and the total energy density have been held fixed. Because the ratio of energydensity to temperature (to the fourth power) varies significantly with increasing chargedensity, it is natural to ask whether the degree of (in)sensitivity of the relaxation dy-namics to the charge density is substantially different if one holds fixed the equilibriumtemperature instead of the energy density. Figure 13 shows such a comparison. Plottedare the pressure anisotropies resulting from the same initial anisotropy function usedin figs. 5 and 11, and charge densities of either 0 or 80% of extremality, but now withthe energy density in either case suitably adjusted to fix the equilibrium temperature, πT = 1 /L . One again sees a significant change in the amplitude of the response withincreasing charge density, but now with the temperature held fixed, increasing chargedensity decreases the amplitude of the response. Nevertheless, with time now plottedin units of ( πT ) − , one again sees negligible ( ≈ . ρ max .Performing the same constant temperature response comparison using the deeppulse initial anisotropy function (whose radial profile is shown in fig. 9), we find alarger — but still quite small — variation in the time course, approximately 2%, as thecharge density varies from zero to 80% of ρ max .We have also examined the degree of nonlinearity in the above examples of equili-– 38 – harged ( ρ (cid:54) = 0, B = 0) ρ/ρ max Re λ/ε / Im λ/ε / λ/πT Linearized λ/ε / . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i . ± . − . ± . . − . i . − . i Table 1 . Lowest quasinormal mode frequency for charge densities ranging from 0 up to 80%of extremality. The second and third columns show results (with estimated uncertainties)for the real and imaginary part of the QNM frequency in units of ε / , obtained from fittingthe late time behavior of our full nonlinear evolution. The fourth column shows these sameresults converted to units of πT . (Fractional uncertainties are the same as in the precedingcolumns.) The rightmost column shows results from an independent analysis of the linearizedsmall fluctuation equations by Janiszewski and Kaminski [50]. brating charged plasmas. The results for the relative size of the nonlinearity are quitesimilar to our earlier results for equilibrating uncharged plasmas. Because of this, wewill refrain from presenting explicit nonlinearity plots for charged plasmas.Finally, as noted earlier, at sufficiently late times the relaxation must be accuratelydescribed by a superposition of quasinormal modes (eigenfunctions of the linearizeddynamics about the equilibrium solution). Extracting the leading quasinormal modefrequency by fitting the late time (4 (cid:46) vε / (cid:46)
20) behavior of our calculated pressureanisotropy to a decaying, oscillating exponential, as described in the previous section,yields the results shown in table 1. The second and third columns (with uncertainties)show our estimates for the real and imaginary parts of the leading QNM frequency inunits of ε / , while the fourth column (without uncertainties) shows our estimates con-verted to units of πT . The rightmost column shows independent results of Janiszewskiand Kaminski [50] obtained by analyzing the linearized small fluctuation equationsabout the RN black brane solution. The agreement is a satisfying confirmation of ournumerical accuracy. Interestingly, the imaginary part of the lowest QNM frequencyvaries by over 15% between ρ = 0 and ρ = 0 . ρ max . This is enormously larger thanthe part in 10 sensitivity seen in fig. 11, and substantially bigger than the largest( ≈ We now present results of an analogous investigation of equilibration in plasmas (withvanishing charge density) in a homogeneous magnetic field B . Our discussion willparallel, as much as possible, the previous treatment of charged plasmas. But thebreaking of scale invariance by the magnetic field, discussed in section 2.1, producessome notable differences. One change, seen in section 2.4, is that the anisotropy function B ( v, r ) must now contain logarithmic terms in its near boundary behavior. We chooseour initial anisotropy function to have the form (3.7) in which an adjustable Gaussianis added to the required leading logarithmic term. As in the previous discussion ofcharged plasmas, we will keep fixed the parameters of the Gaussian part of the initialanisotropy function B ( v , r ) as we dial up the external magnetic field. We will also holdfixed the energy density ε L defined at a renormalization point µ = 1 /L . As shown by theholographic relation (2.42), this is the same as holding fixed the asymptotic coefficient a . In the following plots, axis labels involving energy density ε , without any explicitindication of scale, will denote the energy density evaluated at the curvature scale, ε (1 /L ) = ε L . Similarly, the pressure anisotropy ∆ P should be understood as ∆ P (1 /L )unless otherwise indicated explicitly.Instead of keeping ε L fixed as the magnetic field B is varied, one could chooseto hold fixed the energy density ε B defined at the scale set by the magnetic field, µ = |B| / . Since the AdS curvature radius L is not a physical scale present in the dualQFT, fixing ε B instead of ε L is arguably more natural. However, for computationalreasons it is easier to hold fixed ε L as the magnetic field is increased. The issue isthat accurate numerical calculations become progressively more difficult the deeper onepenetrates into the high-field/low-temperature regime, T / |B| (cid:28)
1. (This is analogousto the difficulty of approaching extremality in the charged case, where the horizontemperature also vanishes.) By holding fixed ε L instead of ε B , we are able to performscans in B which avoid dipping too deeply into the very low temperature region.Another difference concerns the definition of pressure (or stress) anisotropy. Withour choice (2.27) for fixing the scheme dependent ambiguity in the stress-energy tensor,the resulting holographic relation (2.42), when evaluated at µ = 1 /L , puts the traceanomaly solely in the transverse components of the stress, T and T . So the pressureanisotropy (4.1), defined as the difference between transverse and longitudinal stress,when evaluated at µ = 1 /L has a “kinematic” contribution of − κ B plus a “dynamic”contribution of 3 κ b ( v ). In presenting results below, we will omit the uninteresting– 40 – .2 0.4 0.6 0.8 1.0 1.2 1.4 v ε / - - Δ dyn / κε Figure 14 . Left: The subtracted/rescaled anisotropy function, B s ( v, u ), as a function oftime v (in units of ε − / L ) and inverse radial depth u , for the case of B L = 1 .
0, with initialGaussian parameters chosen to match the initial pulse which generated fig. 5 ( A = 5 × − , r = 4, σ = ), and energy density ε L = L − . At late times the anisotropy functionapproaches the non-trivial profile of the equilibrium magnetic brane solution discussed insection 2.7. Right: Corresponding evolution of the dynamical contribution to the relativepressure anisotropy , ∆ P dyn /κε L , with both ∆ P and ε L evaluated at the scale 1 /L . Thelate time limit of the pressure anisotropy is non-zero, but too small to be easily visible,lim v →∞ ∆ P dyn (1 /L ) /κε L = 0 . static kinematic contribution, and just plot the dynamic contribution∆ P dyn ≡ ( T + T ) − T + κ B , (4.3)(relative to the energy density), evaluated at the renormalization point µ = 1 /L . A further difference comes from the fact that equilibrium magnetic brane solu-tions are intrinsically anisotropic. The anisotropy function B does not vanish at latetimes, but rather settles down to the profile of the equilibrium magnetic brane solu-tion discussed in section 2.7. This is illustrated in figure 14, which shows the (sub-tracted/rescaled) anisotropy function B s as a function of time v and inverse radialdepth u . One sees similar features as in fig. 5: the initial pulse propagates outward,reflects off the boundary, and largely disappears into the horizon. But in addition onealso sees that the anisotropy function is approaching the non-trivial profile of a staticmagnetic brane solution. Examining the holographic relation (2.42), one sees that simply shifting the renormalization pointto µ = e / /L would accomplish the same removal of this uninteresting kinematic contribution to thepressure anisotropy. – 41 – L = ℬ L = ℬ L = ℬ L = ℬ L = v ε / - - Δ dyn / κε Figure 15 . Time dependence (in units of ε − / L ) of the dynamical pressure anisotropy, atthe scale of 1 /L , for values of background magnetic field ranging from 0 to 4 /L . The energydensity (at the curvature scale) is held fixed, ε L = L − , and the parameters of the initialGaussian pulse in the anisotropy are the same as in fig. 14. Correspondingly, the pressure anisotropy in the dual field theory asymptoticallyapproaches a non-zero constant. To examine equilibration, it is the difference betweenthe pressure anisotropy and its asymptotic value which is of interest. As a measure of(near) equilibration, the condition (4.2) is naturally replaced by[∆ P ( t ) − ∆ P ( ∞ )] / ( κε L ) ≤ . , (4.4)for all times t > t eq .The time dependence of the resulting (dynamical contribution to the) pressureanisotropy is shown in fig. 15 for a series of magnetic fields, B L = 0, 1, 2, 3, and 4.The energy density is held fixed at ε L = L − and the parameters of the Gaussianpulse in the initial anisotropy function are those of the pulse which generated fig. 5( A = 5 × − , r = 4, σ = ). For the five cases shown, the ratios of magnetic fieldto the equilibrium temperature (squared) are given by B /T = 0, 13.0, 30.2, 30.5, and26.3 for B L = 0, 1, 2, 3 and 4, respectively. Note that, at this fixed value of ε L L , B /T is not monotonic as a function of B L . The energy densities at the intrinsic scaleset by the magnetic field for this series of solutions are given by ε B / B = ∞ , 0 .
75, 0 . .
36, and 0 .
39, respectively.Differences in the late time values of the pressure anisotropy are obvious in fig. 15. The tiny positive late time pressure anisotropy barely visible in the B L = 1 curve is a consequence – 42 – L = ℬ L = ℬ L = ℬ L = v π T - - Δ dyn / κε Figure 16 . Time dependence (in units of ( πT ) − ) of the dynamical pressure anisotropy forvalues of background magnetic field ranging from 0 to 2 . /L . The temperature is held fixed, πT L = 1, and the parameters of the initial Gaussian pulse in the anisotropy are the same asin fig. 14. The values of ε B / B for these solutions, in order of increasing field, are ∞ , 1.2,0.51, and 0.39, respectively. One sees very little sensitivity to the magnetic field in the timecourse of the response. Small temporal variations are also evident after v ≈ .
3. These are produced by therelaxation of the initial non-Gaussian background profile of the anisotropy functionto the correct equilibrium form. These small variations at relatively late times wouldbe absent if we had constructed our initial anisotropy function by adding a Gaussianperturbation to the equilibrium solution (instead of merely adding a Gaussian to theleading asymptotic term). Given our choice of initial data, there are two distincttime scales in the equilibration shown in figs. 14 and 15, the first associated with theboundary reflection and subsequent infall of the Gaussian pulse, and the second withthe time it takes for the background anisotropy profile to reach its equilibrium form.It is the latter which is responsible for the late time variations; fortunately, there isrelatively little ambiguity in separating the two contributions to the dynamics.Once again, a notable feature of in the comparison of fig. 15 is the similarity in thetime dependence of the pressure anisotropy during the pulse-driven period of signifi-cant departure from equilibrium (0 . (cid:46) vε / L (cid:46) . t eq of our removal of the static kinematic part of the anisotropy; the final value of the total pressureanisotropy, at the scale 1 /L , monotonically decreases with increasing B L in this series of solutions. – 43 – L = ℬ L = ℬ L = ℬ L = ℬ L = v ε / - - Δ dyn / κε Figure 17 . Time dependence (in units of ε − / L ) of the relative pressure anisotropy for valuesof background magnetic field ranging from 0 to 2 . /L , for the same deep pulse ( A = , r = , σ = ) used in fig. 9 and fig. 12. The energy density ε L = L − is held fixed. clearly does not vary much with magnetic field, and neither does t peak for these rela-tively near boundary pulses. This insensitivity result relies, of course, on the constancyof the initial Gaussian perturbation in the anisotropy function, and also on our choiceto hold fixed the energy density at the scale 1 /L .Fig. 16 compares the response, for different values of magnetic field, when one holdsfixed the equilibrium temperature T (as well as the initial Gaussian perturbation),instead of fixing the energy density ε L . With time now plotted in units of ( πT ) − , onealso sees remarkable similarity in the time dependence of the response, with the timesof the first, second, or third peaks in the pressure anisotropy varying by less than 0.3%as B /T varies from 0 to 26.7.Sensitivity to the magnetic field is significantly larger when the initial pulse is placedvery close to the horizon. This is shown in fig. 17, which plots the time dependence ofthe dynamical pressure anisotropy for magnetic field values B L = 0, 1, 1.5, 2, and 2.5,using the same “deep pulse” Gaussian parameters ( A = , r = , σ = ), and fixedenergy density ε L = L − , which generated fig. 9. For this series of solutions we have |B| /T = 0, 13.0, 22.85, 30.16, and 31.95, and ε B / B = ∞ , 0.75, 0.43, 0.36, and 0.35,respectively.With this deep initial pulse, differences in the time course of the response are muchmore pronounced. Larger magnetic fields greatly suppress the size of the pulse-drivenpeaks in the pressure anisotropy (for a fixed amplitude of the initial Gaussian), and lead– 44 – agnetic ( B (cid:54) = 0, ρ = 0) B /T ε B /T P /κT ∆ P /κT Re λ/ε / B Im λ/ε / B λ/ ( πT )0 73.06 24 .
35 0 3 . ± . − . ± . . − . i . − .
13 3 . ± . − . ± .
06 3 . − . i . − .
60 3 . ± . − . ± .
06 3 . − . i . − .
76 3 . ± . − . ± .
01 3 . − . i . − .
36 3 . ± . − . ± .
04 3 . − . i − .
35 5 .
00 3 . ± . − . ± .
03 3 . − . i − .
21 39 .
97 2 . ± . − . ± .
03 3 . − . i Table 2 . Equilibrium energy density, average pressure, and pressure anisotropy, plus lowestquasinormal mode frequency, for various values of the external magnetic field. Reportedvalues of energy densities and pressures are evaluated at a renormalization point µ = |B| / ,not at the (physically irrelevant) curvature scale. The pressure anisotropy is the completevalue, including the − B kinematic contribution which was removed in figs. 15 – 17. Resultsfor the lowest quasinormal mode frequency are reported both in units of ε / B (middle column,with uncertainties), and as well as in units of πT (final column). to increasingly large and negative final values for the anisotropy. For the lowest curvewith B L = 2 .
5, the contribution of the Gaussian pulse is completely swamped by thecontribution from the relaxation of the background profile of the anisotropy function tothe correct equilibrium form. Excluding this curve, the time t peak of the first peak, aswell the rough equilibration time t eq , characterizing the portion of the evolution arisingfrom the Gaussian pulse, vary at most by 20% as the magnetic field ranges from 0 to2 /L . This is the largest difference in the relaxation time course we have seen in ourexploration of magnetized plasmas with Gaussian initial perturbations.A constant temperature comparison (not plotted), analogous to fig. 16 but usingthe same deep pulse as in figs. 9 and 17, shows variations in the time course of up to15% as |B| /T ranges from 0 to 22 — beyond which the response of the pulse cannotbe clearly distinguished from the background evolution.To conclude this section, we report in table 2 equilibrium properties, plus ourestimates for the lowest quasinormal mode frequency, for values of magnetic field whichextend from small fields well into the strong field regime. The ratio B /T ranges fromzero to just over 30. The equilibrium energy density, pressure, and pressure anisotropyare given in units of T , and have been converted from the µ = 1 /L renormalizationpoint used in our calculations (and above presentation) to the intrinsic scale µ = |B| / .Results for the lowest quasinormal mode frequency are given both in units of ε / B andin units of πT . These estimates are the result of fitting the late time (4 (cid:46) v ε / L (cid:46) B /T ranges from zero up to 30, monotonically increasing with increasing field whenmeasured in units of πT , but slightly increasing and then decreasing when measuredin units of ε / B . The above results show that to a good (often extremely good) level of accuracy:1. the pressure anisotropy response is a linear functional of the initial anisotropypulse profile;2. the time course of the response, measured in units set by the energy density, is insensitive to the charge density or background magnetic field when the pulseprofile and the energy density are held fixed;3. the time course of the response, measured in units set by the equilibrium tem-perature, is insensitive to the charge density or background magnetic field whenthe pulse profile and equilibrium temperature are held fixed.How can one synthesize these observations? Consider some feature in the time courseof the response, such as the time of the first (or second) peak in the pressure anisotropy,or the approximate equilibration time discussed above. For simplicity, we will focus onthe time t peak of the first pressure anisotropy peak. This time must be some functionof the equilibrium state parameters (energy density plus charge density or magneticfield), as well as the Gaussian pulse parameters (depth, width, and amplitude).Consider first the charged case. The response time t peak is a function of the energydensity ε , the fraction x of the extremal charge density, and the Gaussian pulse param-eters r , σ , and A . But since the temperature decreases monotonically with increasingcharge density (for fixed energy density) one may equally well regard the equilibriumstate as labeled by the energy density ε and temperature T , and write t peak /L = f ( εL , T L, r /L, σ/L, A ) , (5.1)for some function f of the indicated arguments. We have written all quantities indimensionless form using, in effect, our computational units. Our results on the degree Specifically, ε L in the magnetic case. – 46 –f nonlinearity imply that the function f is nearly independent of the last argument,the Gaussian amplitude A . Only for our narrowest pulse, located right at the horizon,did the relative nonlinearity reach 10%. Away from this corner of parameter space, thenonlinearity was substantially smaller, rapidly falling to much less than a percent asthe initial pulse becomes less localized at the horizon. So, to a good approximation,one can regard the function f as being independent of A .For narrow pulses, σ (cid:28) r , the dependence of the response time t peak on the pulsewidth is negligible; there is a smooth σ → σ centered at some depth r , one should expect that thetime of the first peak in the response will be most similar to the corresponding responsetime for a narrower pulse located not at the depth r , but rather at a depth of r + nσ for some positive O (1) multiplier n . At the same level of accuracy determined by thedegree of nonlinearity, one should be able to merge the dependence of the response time t peak on the depth r and width σ of the initial pulse into a single effective depth r eff given by r + nσ . (The accuracy of this simplification will be discussed below.) Hence,the above functional dependence for t peak can be replaced by a simpler form, t peak /L ≈ g ( εL , T L, r eff /L ) , (5.2)for some function g . Now, the results of section 4.2 (including figs. 11 and 12 andassociated discussion) show that the time course of the pressure anisotropy response hasremarkably little dependence on the charge density when comparisons are made holdingfixed the initial pulse parameters and the energy density. Since varying the chargedensity at fixed energy density is, as noted above, equivalent to varying the equilibriumtemperature, this implies that the function g describing the response time (5.2) isnearly independent of the second argument, T L . At the same time, the comparisonsat fixed temperature also discussed in section 4.2 (fig. 13 and associated discussion)show that the time course of the pressure anisotropy response also has remarkably littledependence on the charge density when the initial pulse parameters and the equilibriumtemperature are held fixed. This implies that the function g describing the responsetime (5.2) is nearly independent of the first argument, εL . Hence, at a level of accuracydetermined by the minimal level of nonlinearity, and the minimal dependence on chargedensity in comparisons at both constant energy density and constant temperature, the– 47 –esponse time t peak must be a function of only the effective depth of the initial pulse, t peak /L ≈ h ( r eff /L ) , (5.3)for some function h . Finally, this function must be consistent with the scaling relationsdiscussed in section 2.5, which imply that L /r scales in the same fashion as a distance(or time) in the boundary theory. Consequently, the dependence of the response timeon the effective depth must have the form t peak ≈ CL /r eff , (5.4)for some dimensionless constant C .A similar line of reasoning is applicable to the magnetic case. Since we used ε L ≡ ε ( L ) and B L as parameters labeling the equilibrium magnetic brane geometry in ourcomparisons of magnetic plasma response, it is convenient to view the response time t peak as depending on these parameters plus the Gaussian pulse parameters, t peak /L = f ( ε L L , B L , r /L, σ/L, A ) , (5.5)for some function f . Once again, the observed near-linearity of the response allows usto simplify this to the form t peak /L ≈ g ( ε L L , B L , r eff /L ) , (5.6)for some function g . The near-independence of the time course of the response on themagnetic field B , for fixed ε L and a fixed initial pulse, implies that the function g isnearly independent of its second argument. Because ε L does not transform homoge-neously under the scaling relations (2.43)-(2.45) (due to the use of the curvature scale L instead of a physical scale in the dual QFT for setting the renormalization point)consistency with the scaling relations requires that the function g be independent ofits first argument and depend inversely on the third. So, just as for the charged case, t peak ≈ CL /r eff , (5.7)for some dimensionless constant C .Figure 18 compares this simple model with a sample of our data for neutral,charged, and magnetized plasmas. The left panel shows data for relatively narrowpulses of width σ = 1 /
20, while the right panel shows data from rather wide pulses withwidth σ = 1 /
2. The abscissa for both panels is the inverse effective depth u eff ≡ /r eff .(Data points at the largest values of u eff shown in these plots come from pulses centeredvery near the horizon.) In both panels, rather good agreement with the simple model– 48 – ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ σ = / u eff L t peak L ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ σ = / u eff L t peak L Figure 18 . Time of the first peak in the pressure anisotropy response as a function of theinverse effective depth of the pulse, u eff ≡ /r eff . The left panel shows results for narrowpulses with σ = 1 /
20, while the right panel shows results from wide pulses with σ = 1 / B L = 1 .
5. The green straight line showsthe prediction of our simple model (5.7) with n = 2 . C = 2 . is found when the multiplier n in the effective depth r eff ≡ r + nσ is chosen to be 2.5,and the coefficient C ≈ . . (5.8)For both sets of data, the model is least accurate for pulses located very close to thehorizon. Accuracy for the case of magnetized plasma is a bit worse than for charged orneutral plasma. But for all cases, even in the near-horizon regime, this simple modelworks at about the 20% level or better. As pulses move away from the horizon, theaccuracy rapidly improves. In weakly coupled plasmas, adding a conserved charge density to the system (e.g., flavorcharge in a QCD plasma) significantly changes screening lengths. When the associatedchemical potential is of order πT , the relative change in the Debye screening length is O (1) [51]. Such changes in screening lengths significantly affect transport coefficientslike the viscosity [52]. Lattice studies [53] of the effect of a baryon chemical potential inthe deconfined phase, when the system is not asymptotically weakly coupled, claim tofind measurable sensitivity in the Debye screening length, comparable to perturbativeestimates, but with quite large error bars. Lattice QCD studies of magnetoresponse– 49 –29–32] also find substantial changes in thermodynamics when the magnetic field energydensity becomes large compared to T .Consequently, when this work on strongly coupled N = 4 SYM plasma was initi-ated, we expected to find significant changes in equilibration dynamics when a conservedcharge density is added to the plasma, or when the system is placed in a backgroundmagnetic field. The most notable result we have found is that this expectation waswrong. At least within the range of charge densities we studied, up to 80% of ex-tremality, the equilibration dynamics is remarkably insensitive to the presence of aconserved charge density. Additionally, magnetic fields which are well into the strongfield region, B /T (cid:29)
1, induce almost no change in the equilibration time course.Efforts to use results of holographic calculations in strongly coupled N = 4 SYM asthe basis for predictions about real heavy ion collisions [21] are inevitably hampered byour limited understanding of the effect on the relevant dynamics of changing the theoryfrom real QCD to a supersymmetric Yang-Mills model theory. From this perspective,the insensitivity of the equilibration dynamics to the charge density is reassuring, asthis provides an example where changes in the plasma constituents have very littleimpact on the overall dynamics.A further notable feature in our results is the remarkably small degree of nonlin-earity in the dynamics governing the pressure anisotropy. Despite the fact that one issolving the highly nonlinear Einstein equations, the dependence of the induced pres-sure anisotropy on the initial anisotropy function is surprisingly close to linear. Wefind deviations from linearity of at most ≈ Acknowledgments
We are grateful to Han-Chih Chang, Michal Heller, Stefan Janiszewski, MatthiasKaminski, Andreas Karch, Per Kraus, Julian Sonner, and Mikhail Stephanov for help-ful discussions. This work was supported, in part, by the U.S. Department of Energyunder Grant No. DE-SC0011637. We also thank the University of Regensburg and theAlexander von Humboldt Foundation for their generous support and hospitality duringa portion of this work.
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