Operator lifetime and the force-free electrodynamic limit of magnetised holographic plasma
OOperator lifetime and the force-free electrodynamic limitof magnetised holographic plasma
Napat Poovuttikul
1, 2, ∗ and Aruna Rajagopal † Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK University of Iceland, Science Institute, Dunhaga 3, IS-107, Reykjavik, Iceland
Using the framework of higher-form global symmetries, we examine the regime of validityof force-free electrodynamics by evaluating the lifetime of the electric field operator, which isnon-conserved due to screening effects. We focus on a holographic model which has the sameglobal symmetry as that of low energy plasma and obtain the lifetime of (non-conserved)electric flux in a strong magnetic field regime. The lifetime is inversely correlated to themagnetic field strength and thus suppressed in the strong field regime. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] J a n CONTENTS
I. Introduction 2II. The Holographic Model 6A. Linearised solutions in ω/T (cid:28) T (cid:38) ω/ (cid:112) | B | (cid:28) T (cid:46) ω (cid:28) (cid:112) | B | limit 15III. Conclusion 17Acknowledgements 18A. Numerical solution and evaluation of operators lifetime 18B. Frobenius analysis in AdS × R region 19References 21 I. INTRODUCTION
Hydrodynamics [1] is a well-established theoretical framework which universally describes thelong wavelength, low frequency behaviour of interacting systems at finite temperature. Essen-tially, hydrodynamic theory is a description of conserved quantities and the manifestation of thecorresponding symmetries in a system in thermal equilibrium. Theories with widely varying mi-croscopics can have the same macroscopic hydrodynamic description. One possible explanationwhy such a universal description is possible is that all operators except conserved charges haveparametrically short lifetimes compared to the scale of interest and, once the longest-lived non-conserved operator has decayed away, the hydrodynamic description becomes viable (see Fig. 1). While this operator language is more familiar in the context of quantum systems, it is also applicable to classicalsystems via e.g. memory matrix formalism [2, 3]. A more modern introduction may be found in [4].
FIG. 1: A cartoon illustration of the lifetime of operators of a theory that exhibit hydrodynamicbehaviour at late time. Here, there is a parametrically large gap between conserved charges ρ andthe rest. The life time τ of the longest-lived operator, denoted by O set the time scale in whichhydrodynamics becomes applicable.The hydrodynamic framework may be generalised to systems where the conserved charges arethose of a higher-form symmetry [5] which counts the number density of extended objects. A recentexploration of this idea [6] (see also [7–10]) shows that the resulting hydrodynamics of a one-form U (1) charge reproduces the theory of magnetohydrodynamics (MHD) . This should not come asa big surprise. MHD is, after all, a low energy effective theory of plasma where the (dynamical)electric field is screened – the one-form U(1) symmetry associated to electric flux is explicitlybroken. This implies that in, for example, a plasma at zero magnetic field (where the Ohm’s law j = σ E is a good approximation) the electric field has a finite lifetime, δ E ∝ exp( − t/τ E ) ⇐⇒ (cid:104) E i ( − ω ) E j ( ω ) (cid:105) ∼ δ ij ω + i/τ E , (1.1)with the lifetime of the electric field τ E = 1 /σ . The conductivity σ can be computed from firstprinciples. For instance, in quantum electrodynamics, it can be written as [14] σ ∝ Te log e − , (1.2)where e is the electromagnetic coupling. The lifetime of electric field τ ∼ /T , is then much shorterthan the scale t (cid:29) /T (or ω/T (cid:28) T tt and the momentum T ti havealready decayed away, one can expect the hydrodynamic description of a plasma to be governedby ∂ µ T µν = 0 , ∂ µ J µν = 0 . (1.3)The conserved currents T µν and J µν are expressed in terms of energy, momentum, magnetic flux J ti ≡ B i and their conjugates, organised order by order in the gradient expansion. This formulation For the formulation of MHD that closely resembles higher-form symmetry formulation, see e.g. [11–13]. The fact that this quantity has only been computed at the beginning of this century indicates the difficulty of therequired computations. of MHD only requires macroscopic consistency and does not require the introduction of the gaugefield (cid:63)J = F = dA which, due to screening effect, is not a long-lived degree of freedom.This brings us to the central question of the present paper: Is a hydrodynamic description of theform (1.3) applicable in the limit low temperature compared to magnetic flux density T / | B | (cid:28)
1? This question is important if one wants to apply the MHD description to astrophysical plasmaswhere the magnetic field is many orders of magnitude larger than the scale set by the temperature.If one were to naively extrapolate (1.1)-(1.2), the lifetime of the electric field appears to becomearbitrarily long as the temperature decreases. However, there exists a macroscopic description ofplasma in this regime that has been successfully applied. This theory is called force-free electrody-namics or FFE, and has been used extensively in astrophysical setups such as in the magnetosphereof black holes [15, 16], neutron star [17] and solar corona [18] just to name a few. In its conven-tional form, this theory is applied to a system which is magnetically dominated (i.e. | B | > | E | or, covariantly F µν F µν >
0) and whose dynamics is governed by (cid:15) µνρσ F µν F ρσ = 0 , (1.4a) F µν ∇ λ F λν = 0 . (1.4b)Here, the first relation implies that E · B = 0 while the second relation implies that the force j µel F µν , with j µel := ∇ ν F νµ via Maxwell’s equations, acting on plasma vanishes (hence the nameforce-free electrodynamics). More details on the geometric and effective action view point of FFEcan be found in e.g. [19, 20] and [21–24]. One should emphasise that the system of equationsin Eq.(1.4) is independent of the microscopic details of the cold plasma, which then stronglyresembles hydrodynamic descriptions. In fact, it turns out that (1.4) can arise in a special limitwhere T (cid:28) (cid:112) | B | of a hydrodynamics description with one-form U (1) symmetry in (1.3), see[6, 23, 24] .The existence of FFE is usually justified by saying that the cold plasma is, on one hand, denseenough to screen the electric field (1.4a) but, on the other hand, dilute enough so that force-freecondition (1.4b) is applicable. This statement can be made more precise in the light of relationsbetween the equations of FFE and hydrodynamics. Thus, we propose a criterion for testing thevalidity of FFE using the lifetime of non-conserved operators – FFE, or equivalently, hydrodynamicdescription of cold plasma in the T (cid:28) (cid:112) | B | limit, is valid when the lifetime of all non-conservedoperators is parametrically shorter than the time scale of interest. A key advantage of this approach Recasting of force-free electrodynamics in the hydrodynamic language also allows the systematic gradient expan-sions [23, 25]. This could serve to classify correction to FFE in order to account for phenomena such as pulsarradio emission where E · B (cid:54) = 0. is that the operator lifetime can be, in principle, computed explicitly from microscopic descriptionand therefore allows one to find the ‘cutoff’ scale where FFE description should break down.Computing the operator lifetime from microscopic description is, however, not always an easytask. In fact, we are not aware of a genuine computation directly from quantum electrodynamics(in the sense of [14]) when both T and B are turned on. To simplify the computations, we shalldemonstrate the validity of FFE in the strongly interacting magnetised plasma with a holographicdual as proposed in [26, 27] where the one-form U (1) global symmetry is taken into account via atwo-form gauge field in the gravity dual. This provides two key advantages. First, the computationof correlation functions boils down to solving simple linearised differential equations (see e.g. [28]).Second, there is strong evidences that charge neutral operators, apart from energy and momentum,have a parametrically short lifetime in this class of theories . Therefore, we shall focus on non-conserved operators in the electromagnetic sector of the theory: the electric flux operators, whoselifetime can be extracted via two-point correlation function as in (1.1). This will provide strongevidence for the validity of FFE limit in a strongly interacting holographic plasma.On the technical side, the computations presented in this note show that there are no quasi-normal modes present in the vicinity of the hydrodynamic regime ω/T (cid:28) ω/ (cid:112) | B | (cid:28) ω/T (cid:28) ω/T ∼ , ω/ (cid:112) | B | (cid:28)
1. Such computation can, however, be done analytically in the simplemodel of [26] thanks to the presence of the BTZ × R bulk geometry in the deep IR [32]. We shouldalso note that the treatment of a (long-lived) non-hydrodynamic modes has been extensively usedto determine the breakdown of hydrodynamic descriptions in the context of QFTs with holographicduals, see e.g. [33–36] .The remainder of this paper is organised as the follows. In section II, we summarise theprocedure involved in the computation of the two-point correlation function in the holographic dualto one-form global symmetry. In section II A, we outline the method for exploring the existence ofdecaying modes in the vicinity of the usual hydrodynamic limit ω/T (cid:28) T / (cid:112) B | (cid:28)
1. Due tothe simplicity of the bulk geometry, we are able to further extend the analysis to arbitrary value of To be more precise, it has been shown in N = 4 supersymmetric Yang-Mills theory, which constitutes the mattersector of the holographic model [26, 27], that there is no long-lived mode besides hydrodynamic modes at any T (cid:54) = 0 and | B | = 0 [29]. A similar conclusion was reached for the same theory in the charge neutral sector at finite non-dynamical magnetic field [30, 31]. ω/T with ω/ (cid:112) | B | (cid:28) T / (cid:112) | B | (cid:28)
1. This is described in section II B. Further open questionsand future directions are discussed in section III.
II. THE HOLOGRAPHIC MODEL
A simple holographic dual to a strongly interacting field theory of matter charged under dy-namical U (1) electromagnetism (that is, the dynamical plasma described by low energy MHD)and formulated in the language of higher-form symmetry was constructed in [26, 27]. We presenta brief review here for completeness. The five-dimensional bulk theory is comprised of Einsteingravity coupled to a two-form bulk gauge field, B µν , and a negative cosmological constant, S = (cid:90) d X √− G (cid:18) R − − L H abc H abc (cid:19) + S bnd − κ (Λ) (cid:90) r =Λ d x √− γ ( n a H aµν )( n b H bµν ) , (2.1)where H = dB and B ab is the bulk 2-form gauge field, Λ is the UV-cutoff, n a is the unit normalto the boundary, and S bnd denotes the Gibbons-Hawking and gravitational counter term. Roughlyspeaking, the two bulk fields G ab and B ab , asymptote to g µν and b µν respectively, which then sourcethe currents, T µν and J µν . (cid:104) T µν (cid:105) ≡ √− g δSδg µν , (cid:104) J µν (cid:105) ≡ √− g δSδb µν (2.2)The generating functional takes the form, Z [ g µν , b µν ] = (cid:28) exp (cid:20) i (cid:90) d x √− g (cid:18) T µν g µν + J µν b µν (cid:19)(cid:21)(cid:29) (2.3)and diffeomorphism invariance and gauge symmetry lead to the following equations, ∇ µ (cid:104) T µν (cid:105) = ( db ) νρσ (cid:104) J ρσ (cid:105) , ∇ µ (cid:104) J µν (cid:105) = 0 . (2.4) H = db is the three-form field strength of the two-form external source. The equilibrium solutionof this holographic model is a domain wall interpolating between an asymptotic AdS geometry inthe UV ( r → ∞ in our convention), and BT Z × R in the near-horizon IR ( r = r h ). It is describedby the following metric and gauge field ds = G ab dX a dX b = − r f ( r ) dt + dr r f ( r ) + e V ( r ) ( dx + dy ) + e W ( r ) dz ,B = h ( r ) dt ∧ dz with (cid:63) H = B dx ∧ dy (2.5)Modulo the subtleties due to the mixed boundary conditions, this is nothing but the hodge dualof the magnetised black brane solution of [32]. The radial coordinate is chosen such that r → ∞ corresponds to the usual asymptotic AdS with f ( r ) = 1 , e V ( r ) = e W ( r ) = r (2.6)in the r → ∞ limit. The BT Z × R solution near the horizon can be written as f ( r ) = 3 (cid:18) − r h r (cid:19) , e V = B√ , e W = 3 r . (2.7)The temperature is set by the horizon radius via 4 πT = r h | f (cid:48) ( r h ) | = 6 r h /L . We set L = 1 forsimplicity. Note also that B is related to the z − component of the ‘physical’ magnetic field B whichdiffers by a prefactor L or the 2-form gauge field coupling in the bulk (e.g. if one were to definethe action with S ∼ (cid:82) (1 /g ) H ). We will keep using B to emphasise its holographic origin butthere is no harm in thinking of it as simply B .One interesting feature of this model is that the leading divergence of B µν in the Fefferman-Graham expansion is logarithmic. Thus, the definition of the source b µν requires mixed boundarycondition b µν = B µν (Λ) − κ (Λ) (cid:104) J µν (cid:105) , with (cid:104) J µν (cid:105) = −√− Gn α H αµν (2.8)Requiring the source b µν to be independent of the UV cutoff fixes the form of the ‘coupling constant’1 /κ (Λ) which turns out to be logarithmically running. This is a common feature for fields withthis type of near-boundary behaviour where the counterterm also plays the role of the double-tracedeformation [37, 38], see also [26, 27] for a discussion in the present context. Mapping J µν in toa more familiar dynamical field strength via J µν = (cid:15) µνρσ F ρσ , one can see that the double-tracedeformation plays a role similar to the Maxwell term for the dynamical gauge field in the dualQFT with 1 /κ (Λ) as a (logarithmically running) electromagnetic coupling.The finite part of 1 /κ (Λ) plays a crucial role in this setup. While the finite counterterm inthe ordinary bulk Maxwell theory simply results in a contact term in the correlation function, themixed boundary condition for B ab implies the existence of the purely decaying mode ω = − i/τ E that can interfere with the gapless hydrodynamic excitation. This is nothing but the life-timeof the electric flux operator Q E ∼ (cid:82) dS ij J ij which appears in the following correlation function[27, 33] (cid:104) J ij ( t ) J kl (0) (cid:105) ∼ exp ( − it/τ E ) . (2.9)Note that, due to the anisotropy introduced by finite equilibrium magnetic field, the value of τ E depends on which direction of the electric field in consideration. The limit where τ E is small, butfinite, compared to the length scale of interest (set by temperature or magnetic flux density) isof particular interest as it allows one to extract τ E analytically, via a matching procedure thatwe outline below. As argued in the introduction, the lifetime of the electric flux determines thevalidity of MHD and FFE description. A. Linearised solutions in ω/T (cid:28) limit and matching procedure In this section, we outline the computation required to obtain the relaxation time of the electricfield. We focus on the hydrodynamic regime where ω/T (cid:28)
1, and the low temperature limit T / (cid:112) | B | (cid:28)
1. This allows us to solve the bulk equation of motion analytically via a matchingmethod similar to that was employed in [29] (see also [33] for a recent review). We consider thedecay rate of the electric field both along and perpendicular to the equilibrium magnetic fielddenoted by E (cid:107) = J xy and E ⊥ = J xz , J yz respectively.Before proceeding, let us summarise the matching procedure for the ω/T (cid:28) r such that ω/r (cid:28) ω/r ) , which includes the near boundary region. The integration constants of the solution in theouter region are determined by matching the form of inner region solution for intermediate valueof r that connect the two regions together. In our case, this is the region of r close to r h but ωT log f ( r ) (cid:28) ω/r (cid:28) ω/T (cid:28) Similar computation for the holographic theory dual to a system with ordinary(zero-form) U (1) symmetry can befound in e.g. [39, 40].
1. Perturbation parallel to equilibrium magnetic field
As the magnetic field in equilibrium points along the z − direction, we are interested in E (cid:107) = ε zxy (cid:104) J xy (cid:105) . The corresponding bulk perturbation is δB xy which decouples from the metric pertur-bation in the zero wave vector limit. The bulk equation of motion can be written as (cid:0) r f e W − V δB (cid:48) xy (cid:1) (cid:48) + ω r f e W − V δB xy = 0 (2.11)where ( ... ) (cid:48) denotes a derivative w.r.t. the radial coordinate r . The inner region solution for δB xy ,where we substitute the BT Z × R solution for f, V, W , with the ingoing boundary condition canbe written as δB innerxy = c H exp (cid:18) − iω πT log f ( r ) (cid:19) (2.12)The outer region solution can be obtained by considering the solution at linear order in ω/r andone obtains, δB outerxy ( r ) = c − c (cid:32) log Λ − (cid:90) Λ r = r d r e V ( r ) − W ( r ) r f ( r ) (cid:33) = c − c (cid:18) log r − φ ( r ) + e V − W r h f (cid:48) (cid:12)(cid:12)(cid:12) r = r h log f (cid:19) , (2.13)where φ ( r ) is a function regular everywhere in the bulk defined as φ ( r ) = (cid:90) Λ r = r d r e V ( r ) − W ( r ) r f ( r ) − (cid:32) e V ( r ) − W ( r ) r h f (cid:48) ( r ) (cid:33) r = r h f (cid:48) ( r ) f ( r ) − r . This parametrisation allows us to single out leading contributions that dominate when consideringthe solution near r = Λ, where φ ( r ) and log( e − V r f ) vanish, as well as near r ≈ r h where thelog f term dominates. The integration constants c , c in (2.13) are related to the source b µν andthe 2-form current (cid:104) J xy (cid:105) . The precise relations can be obtained via Eq. (2.8) to be (cid:104) J xy (cid:105) = c , b xy = c − (cid:18) log Λ + 1 κ (Λ) (cid:19) c (2.14)Note that, for the source to be independent of the UV cutoff, one requires κ (Λ) − = finite term − log Λ. This is the logarithmically running coupling usually found in a double-trace deformed theoryand resembles the running of electromagnetic coupling as pointed out in [26, 27, 33].For the outer and inner region solutions to match, we consider both solutions in the intermediateregion where we can write the inner solution asexp (cid:18) − iω πT log f (cid:19) ≈ − i ω πT log f + O (cid:16) ωT (cid:17) (2.15)0The matching condition δB innerxy = δB outerxy in this region prompts yield the following algebraicrelations between the boundary quantities b xy , (cid:104) J xy (cid:105) : iω πT c H = (cid:18) B /r h r h f (cid:48) ( r h ) (cid:19) (cid:104) J xy (cid:105) c H = b xy + (cid:20) κ (Λ) + log (cid:18) Λ r h (cid:19) + φ ( r h ) (cid:21) (cid:104) J xy (cid:105) . (2.16)Solving these equations at vanishing source b xy = 0 yields the spectrum of the form ω = − i/τ E (cid:107) where τ E (cid:107) is the lifetime of the electric flux parallel to the equilibrium magnetic field. This is thefirst key result that we advertised earlier, namely τ E (cid:107) = 2 πT B (cid:0) e − r + φ ( r h ) (cid:1) , (2.17)where we write e − r = log(Λ /r h ) + κ (Λ) − which plays the role of renormalised electromagneticcoupling. More details on the T / √B dependence of φ ( r h ) can be found in Appendix A.What does this result tell us about the lifetime of the electric flux operator? While the integral φ ( r h ) can be a dimensionless function of T and B , the renormalised electromagnetic coupling canbe chosen in such a way that e − r (cid:29) φ ( r h ) and e − r T / B (cid:28) ωτ E (cid:107) ∼ ω/T (cid:28)
1. The secondlimit is essential as the matching procedure assumes that ω/T (cid:28)
T / √B (cid:29) τ E ∼ /T ( see Fig 2). Naively taking the limit T → ω/T (cid:28) ω/T ∼ , ω/ √B (cid:28)
2. Perturbation perpendicular to equilibrium magnetic field
Unlike the previous case, the perturbation δB xz that corresponds to E ⊥ = (cid:15) yzx (cid:104) J zx (cid:105) is coupledto the metric perturbation. This is manifest in the equations of motion ddr (cid:16) r f e − W δB (cid:48) xz + B ( δG xt ) (cid:17) + ω e − W r f δB xz = 0 ,ddr (cid:0) e V + W ( δG xt ) (cid:48) + 4 B δB xz (cid:1) = 0 , (2.18)where δG µν denotes the metric perturbations. Note that the coupled perturbations { δB xz , δG tx } and { B yz , δG yz } are equivalent due to SO (2) symmetry in the plane perpendicular to the equi-librium magnetic field. Also, the second equation of motion in (2.18) can be written in a totalderivative form dπ tx /dr = 0 with π tx is related to the momentum (cid:104) T tx (cid:105) . Since we are working in1FIG. 2: A sketch of the decay rate (inverse of the lifetime) of the electric field as a function of T / √B , measured in the unit of √B . The high temperature regime (red) depict the result of decayrate at zero magnetic field found in [27, 33] which has the same temperature dependence as in(1.1)-(1.2). In the low temperature regime (blue), however, the operator lifetime becomes thosefound in (2.17).the zero wavevector limit, the conservation of momentum implies that π tx = 0 in Fourier space(which can be shown explicitly using the rx − component of the Einstein equation).The solution for δB xz , δG tx in the outer region can be found by using the property of thebackground geometry combined with the Wronskian method as in [33]. To be more precise, onefirst notes that the time-independent solution of the magnetised black brane can be written in atotal derivative form, which implies the existence of two radially conserved currents. Q = r f ( V (cid:48) − W (cid:48) ) e V + W + 2 B h ( r ) = 0 , (2.19a) Q = e V + W ddr (cid:0) e − V r f (cid:1) − B h ( r ) = sT , (2.19b)where we write the equilibrium ansatz for the gauge field as B = h ( r ) dt ∧ dz with gauge choice h ( r h ) = 0, which, together with the horizon regularity, sets Q = 0. The relation between h ( r ) andthe 3-form field strength is e V − W h (cid:48) = B . (2.19c)More details on obtaining these radially conserved quantities can be found in e.g. [41]. With thisansatz, we can compare (2.18) and (2.19) and find that one of the solutions of (2.18) when ω/r → δB xz = Φ ( r ) = h ( r ) + sT B , δG xt = Ψ ( r ) = − e − V r f . (2.20)2One can use the Wronskian method to find find a pair of solution of (2.18) that are linearlyindependent to { Φ , Ψ } . These solutions areΦ ( r ) = 14 B − (cid:90) ∞ r d r (cid:32) B e W ( r ) Ψ ( r ) r f ( r ) (cid:33) , Ψ ( r ) = Ψ ( r ) (cid:90) ∞ r d r (cid:32) e − W ( r ) r f ( r ) (cid:33) (2.21)As a result, the outer region solution can be written as δB outerxz ( δG xt ) outer − B J xz = c Φ Ψ + c Φ Ψ (2.22)where J xz := ( r f e − W δB (cid:48) xz + B δG tx ) is an integration constant of (2.18) at ω = 0. One cansubstitute the BT Z × R ansatz into the solution in (2.22) to check that Φ , Ψ , are finite at r = r h while Φ is singular. It is convenient to separate out the singular part of Φ in the followingform Φ ( r ) = φ ( r ) − (cid:18) B e W Ψ r f (cid:48) (cid:19) r = r h log f ( r ) (2.23)where φ ( r ) is the integral in (2.21) with the logarithmic divergence subtracted. The boundarycondition where the source for both metric and 2-form gauge field fluctuation vanishes correspondsto the following values of c and c c = J xz B , c = − (cid:18) sT B + h (Λ) + B ˆ κ (Λ) (cid:19) J xz (2.24)One can also check that J xz is identical to the one-point function (cid:104) δJ xz (cid:105) via the definition (2.8).Note also that the ratio c / J xz is finite due to the cancellation of the logarithmic divergence of1 /κ (Λ) and that of the near boundary solution of h ( r ), obtained via (2.19c).Let us also pointed out another way to organise the equations of motion for δB xz . It turns outthat (2.18) can be combined into a single equation of motion that reduces to a total derivativeform at ω = 0. Following the procedure in e.g. [42] and some manipulation, we find (cid:16) [ e V + W (cid:0) e − V r f (cid:1) (cid:48) ] r f e − W δ ˜ B (cid:48) xz (cid:17) (cid:48) + ω r f e W [ e V + W ( e − V r f ) (cid:48) ] δ ˜ B xz = 0 (2.25)where δ ˜ B xz = δB xz / [ e V + W ( e − V r f ) (cid:48) ]. The outer region solution of (2.25) is easily obtained andcan be shown to be identical to those of (2.22).We can now proceed to the inner region solution. This can be found by solving Eq.(2.25) andone find δB innerxz = c H exp (cid:18) − iω πT log f ( r ) (cid:19) . (2.26)3In the intermediate region, we apply the expansion in (2.15). The coefficients c , c are related to c H via (cid:18) − iω πT (cid:19) c H = − (cid:18) B e W Ψ r f (cid:48) (cid:19) r = r h c ,c H = (cid:18) sT B (cid:19) c + φ ( r h ) c , (2.27)Substituting the form of c , c in terms of (cid:104) δJ xz (cid:105) , we can write the relations in a form similar to (cid:104) δJ xy (cid:105) , namely (cid:18) − iω + 1 τ E ⊥ (cid:19) (cid:104) δJ xz (cid:105) = 0 . (2.28)In the case of vanishing sources, we can write c c = − B (cid:16) sT B + h (Λ) + B κ (Λ) (cid:17) and the relaxationtime of the electric field perpendicular to the equilibrium magnetic field is τ E ⊥ = √ πT B Ψ ( r h ) (cid:20) sT B c c + φ ( r h ) (cid:21) (2.29)In contrast to the result at e − r (cid:29) B /T has a very different form. To see this, it is useful to examined thatthe combinations that enter the lifetime as followsΨ ( r h ) ∝ B T , φ ( r h ) ∝ B , c c ∝ B for large1 /κ (Λ) (2.30)with proportionality constants given by some numbers of order O (1). In the limit of large electro-magnetic coupling 1 /κ (Λ) (cid:29) B /T (cid:29)
1, we find that this gives a short lifetime of the form τ E ⊥ ∝ T / B . However, the location of this decaying mode ω = − i/τ E ⊥ lies outside the hydrody-namic regime ω/T (cid:28)
1. Thus, one conclude that there are no modes with long lifetime in thisregime . B. Checking T (cid:38) limit in ω/ (cid:112) | B | (cid:28) regime While the result in the previous section strongly indicated that the electric flux lifetime becomesvery short at extremely low temperature, the simplicity of the holographic model also allows us toextend the analysis beyond the usual hydrodynamic ω/T (cid:28) ω = − i/τ in the Note also that, if one were to perform this analysis for a perturbation in the holographic dual to a theory withzero-form U (1) at T > , µ = 0 (as in [29], see also [33]), one would find a spectrum of the form ω ∼ T . Thissolution is spurious as it lies outside the hydrodynamic regime ω/T (cid:28) ω/T [43]. ω/ (cid:112) | B | regime. Next, we further extend the regime of validity to that of ω/ (cid:112) | B | (cid:28) ω/T . The purpose of the latter is to show that τ E ∝ T / (cid:112) | B | without relying on the ω/T (cid:28)
1. Zero temperature
A simple argument for the non-existence of such a slowly decaying mode, is the presence ofLorentz symmetry at zero temperature on the
AdS submanifold in the deep infrared. On theother hand, one can also show this, using matching methods similar to those in [39, 44, 45].To obtain this result, one first realises that the geometry of the magnetised black brane is thatof an interpolation between IR AdS × R and UV AdS . Roughly speaking, the IR geometrystarts to becomes a good approximation as one starts to probe the scale below the magnetic fieldi.e. r ∼ (cid:112) | B | . The inner and outer regions are defined such that they start off from the IR andUV geometry respectively, and extend to cover the overlap region (see Figure 3). This is achievablewhen ω/ (cid:112) | B | (cid:28) ω/r extended from the near horizon limit r → ω/r ∼ ω/ (cid:112) | B | → ω/ (cid:112) | B | (cid:28) ω /r and higher power in ω/r is suppressed, which can be extended toward r (cid:29) (cid:112) | B | as long as the frequency is small.For concreteness, let us demonstrate how this works in the E (cid:107) channel that involves the bulkfield δB xy governed by Eq.(2.11). The solution can be written in the same form as (2.13) evaluated5at zero temperature (i.e. r h = 0). It is worth noting that the singular behaviour near r/ √B → δB outerxy ( r ) = c − c (cid:18) log Λ − ¯ φ ( r ) + B / r (cid:19) + O (cid:18) ω r , ω r log (cid:16) ωr (cid:17)(cid:19) (2.31)where the integration constants can be related to source and response via (2.8). It is worth notingthat the logarithmic divergence appears at order ω . This is can be confirmed via Frobeniusanalysis in AdS region (see e.g. [46]) and AdS × R region (see appendix B). The prefactor of the r − divergence is obtained by evaluating e V ( r ) − W ( r ) /f ( r ) at the horizon r →
0. Here ¯ φ ( r ) is theintegral in (2.13) subtracted by the r − divergent and logarithmic divergent pieces. The resultingintegral evalutated from r = r ∼ √B of the overlapping region to the UV cutoff r = Λ is finiteand its number is not extremely relevant for us as long as one keep e − r large.Next, we consider the inner region solution, which can be obtained by solving (2.11) in the AdS × R region. Upon imposing horizon regularity at r →
0, we find that the inner regionsolution is δB innerxy = c H ζK ( ζ ) , ζ = 3 ωr (2.32)For these two branches of solutions to match, we extend the inner region solution to the regimewhere ζ = ω/r (cid:28)
1. We find that the ‘near boundary’ expansion takes the form δB innerxy = c H (cid:18) γζ + 12 ζ log ζ + ... (cid:19) (2.33)Matching this solution to the outer region, we find that c ∝ ω unlike what happened in theprevious section. Carrying on the matching procedure, we find that the polynomial governing thespectrum only depends on ω and thus rules out the purely imaginary mode ω = − i/τ . The sameargument can also be made for the E ⊥ channel involving δB xz . This is because, the part that isrelevant to the matching procedure only depends on ζ . See appendix B for more details on theform of δB xz in the AdS × R region. T (cid:46) ω (cid:28) (cid:112) | B | limit In this section, we show that the electric flux lifetime can also be obtained regime where ω/T (cid:38) ω/ (cid:112) | B | (cid:28) (cid:112) | B | /T (cid:29)
1. The calculations closely resembles that of the zerotemperature case except that the deep IR geometry is now
BT Z × R instead of AdS × R . Figure4 illustrates this geometry where the AdS joined with the BT Z × R at the ‘boundary’ AdS × R E (cid:107) fluctuations as it is the only channel that containsthe decaying modes in the ω/T (cid:28) T (cid:28) (cid:112) | B | . The inner region, whosesolutions only depends on the ratio ω/r extends from the near horizon limit r → r h (cid:28) √ B to theone where ω/r ∼ ω/ (cid:112) | B | (cid:28)
1, which corresponds to the near boundary region of
BT Z × R geometry, described by AdS × R . The outer region is defined to be the region where ω /r (andhigher powers) is negligible and, therefore, can be extended toward r ∼ (cid:112) | B | in the ω/ (cid:112) | B | (cid:28) AdS to the intermediate AdS × R region has the same form as in (2.31). This is possible only in the limit where √B (cid:29) T so that r/ √B is always much greater than T / √B ∼ r h / √B in this region.The inner region solution in the BT Z × R region can be expressed in terms of a hypergeometricfunction (upon imposing ingoing boundary condition) δB innerxy = c H (cid:18) − r h r (cid:19) − i w / F (cid:18) − i w , − i w , − i w − w ; 1 − r h r (cid:19) (2.34)where w = ω/ (2 πT ) = ω/ r h . Extending this solution in the r (cid:29) r h limit (which is possible dueto r h /r → ω/r →
0) yields the following expansion [47] δB innerxy ∝ c H (cid:20) iωr h r + 14 (cid:16) ω r (cid:17) (cid:18) − γ − ψ (1 − i w / − log (cid:18) r h r (cid:19)(cid:19) + O ( ω ) (cid:21) (2.35)where ψ ( x ) is the digamma function and the constants of proportionality are combinations ofgamma functions that can be absorbed in the definition of c H . The first two terms in [ ... ] are what7important for us. By working to leading order in ω/r (cid:28) AdS × R region, we find the following matching solution c − c log(Λ / √B ) + ¯ φ = c H , (cid:18) B (cid:19) c = iω (cid:18) πT (cid:19) c H (2.36)We can convert c to the source b xy and c as done in the previos sections. Upon taking e − r (cid:29) ¯ φ (so that the solution lies in the regime of validity ω/ √B (cid:28) ω = − i/τ E (cid:107) where τ E (cid:107) is the same as in (2.17). This indicates that the lifetime indeed grows as T / √B increases regardless of the ratio ω/T . III. CONCLUSION
The higher-form symmetry viewpoint of magnetohydrodynamics and its low temperature incar-nation, the force-free electrodynamics, leads to new insights. The central focus of the present workwas to established the absence of long-lived non-conserved operators. In turn, this indicates thevalidity of a hydrodynamic description at low temperature and strong magnetic field. The ques-tion of whether the only operators that govern the deep IR dynamics are the conserved charges isimportant and ought to be asked before any quantitative attempt is made to study hydrodynamicproperties (such as shear viscosity etc). All non-conserved operators must decay much faster thanthe scale of interest if a hydrodynamic interpretation is to be meaningful.We work with a holographic model which shares the same global symmetry as that of the plasma,namely only the energy, momentum and magnetic flux commute with the Hamiltonian. The modelis simple enough for the lifetime of electric flux to be determined by classical bulk dynamics andthe precise question is whether or not the electric flux is sufficiently long-lived to interfere withhydrodynamic modes. Due to the anisotropy of the system in the presence of a strong expectationvalue of magnetic field, the lifetime of the electric field depends on its orientation. Our results canbe summarised as follows • For electric flux E (cid:107) parallel to the magnetic field, the lifetime has a strong dependence onthe double-trace coupling κ which plays a role similar to the renormalised electromagneticcoupling. In the extreme limit of e − r (cid:29) | B | / T , the lifetime can be large enough to bedetectable by the analytic computation in both the ‘usual’ hydrodynamic regime ω/T (cid:28) ω/ √ B (cid:28) ω/T may remains finite. Wefound that the lifetime becomes shorter as one decreases the ratio of T / (cid:112) | B | . The latterindicates that the lifetime will become extremely short in the extremely strong magnetic8field regime T / (cid:112) | B | (cid:28) ω/ (cid:112) | B | (cid:28) • For the component of electric flux E ⊥ perpendicular to the magnetic field, we find that thereis no pole in the vicinity of ω/T (cid:28)
1. The dependence of the lifetime on the renormalisedelectromagnetic coupling disappears as one approaches the strong magnetic field limit.We also performed a consistency check at T → ω/ (cid:112) | B | (cid:28)
1. In this regime, the modes that indicate (potentially) long lifetime of E (cid:107) disappear from the low energy spectrum as anticipated.These computations are basic checks on the validity of FFE description. In the holographiccontext, it would be interesting to check if all the accessible non-conserved operator truly have aparametrically short lifetime as well as confirming the low energy spectrum predicted by force-freeelectrodynamics (and its subsequent derivative corrections). Extraction of FFE effective actionfrom gravity akin to [48–50] or the full constitutive relation as in [51–53] would be desirable as adefinitive proof of FFE description in the dynamically magnetised black brane geometry. Last butnot least, it would be very interesting to investigate operators lifetime in (weakly coupled) quantumelectrodynamics at finite T and B to better understand FFE and its limitations in a system moredirectly connected to astrophysical plasma than the strongly coupled holographic model consideredhere. ACKNOWLEDGEMENTS
We would like to thank Jay Armas, Saˇso Grozdanov, Nabil Iqbal, Kieran Macfarlane, WatseSybesma and L´arus Thorlacius for helpful discussions and comments. We are particularly grate-ful to S. Grozdanov, N. Iqbal and L. Thorlacius for commenting on the manuscript. The workof N. P. was supported by Icelandic Research Fund grant 163422-052 and STFC grant numberST/T000708/1. The work of A.R was supported in part by the Icelandic Research Fund undergrant 195970-052 and by the University of Iceland Research Fund.
Appendix A: Numerical solution and evaluation of operators lifetime
In this section, remarks on the evaluation of the electric flux are elaborated. The numericalbackground solution for this geometry can be constructed in the same way as [32] using shootingmethod. The solution is a one-parameter family characterised by B /T which allows us the freedom9to choose r h = 1 , r h f (cid:48) ( r h ) = 1 (or equivalently T = 1 / π ). It is also convenient to set V ( r h ) = W ( r h ) = 0 which results in the UV boundary metric of the formlim r →∞ ds = r (cid:0) − dt + v ( dx + dy ) + wdz (cid:1) + dr r (A1)Upon rescaling of spatial coordinates { dx, dy, dz } → { dx/ √ v, dy/ √ v, dz/ √ w } , we recover thedesired background solutions. Note also that the physical magnetic flux is related to the inputparameter (that produced the metric in (A1)) by B physical = B input /v . A small caveat of thismethod is that one cannot find a smooth solution beyond B input (cid:38) √ / T / √B = (4 π (cid:112) B input /v ) − ≈ .
05. This is most likely an artifact of the presentednumerical method as there exists a smooth solution in the zero temperature limit corresponding tothe
AdS × R geometry in the deep IR. We should also note that this is a sufficiently low energytemperature as the entropy becomes sufficiently close to s ∝ T obtained from BT Z × R geometry(c.f. [26, 32]). The background is generated for r from [1 + 10 − , ] and varying the (numerical)cutoffs within this order of magnitude does not change the obtained numerical results.Let us also remark on the the numerical value of the renormalised electromagnetic coupling e − r = log(Λ /r h ) + κ (Λ) − . This quantity strongly influences both the thermodynamics and lowenergy spectrum [26, 27, 33] of the model. In particular a small value of e − r would result in thespeed of sound becoming imaginary [26]. Another way to see that this quantity should be large isto write it in terms of a renormalisation group independent scale M ∗ that denotes the energy scaleof a Landau pole [27] i.e. e − r ∼ log( M (cid:63) /T ) where M (cid:63) (cid:29) T . We take this to be the largest scale inthe problem–much larger than the accessible value of √B /T .Numerical value of the integral for φ ( r h ) in (2.17) is shown in Figure 5. For a larger temperature(when φ ( r h ) ≈ O (1)), the lifetime can be sensibly approximated to be τ E (cid:107) ≈ π ( T / B ) e − r . As T / √B decreases, the lifetime becomes shorter and, if we are to extrapolate the fitting function φ ∼ B T log B T to even lower temperature where e − r (cid:38) φ , it will escape the regime of the validityof small ω/T, ω/ √B expansions. In this scenario, one shall conclude that there are no long-livedmodes that can interfere with the low energy excitations. Appendix B: Frobenius analysis in
AdS × R region Consider the equation of motion for δB xy in the intermediate AdS × R region: δB (cid:48)(cid:48) xy ( r ) + 3 r δB (cid:48) xy ( r ) + ω r δB xy ( r ) = 0 (B1)0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● FIG. 5: Numerical evaluation of φ ( r h ) in (2.17) as a function of T / √B . The black dots denotethe numerical evaluation while the red line denotes the fitting function for small T / √B as φ ≈ − (0 . B T log(5 . B /T ). For high temperatures, the value of φ ( r h ) is approximatelyconstant around 0 .
69. The value of φ ( r h ) at lowest achievable temperature is at φ ( r h ) = − . ζ = 3 ω/r and redefine δB xy = ζc ( ζ ). It follows that c ( ζ ) is the solutionof the Bessel equation of order 1, which has a regular singular point at ζ = 0. The near-boundary r → ∞ , or equivalently ζ →
0, akin to the Fefferman-Graham expansion in the usual holographicrenormalisation, can be written as δB xy ( ζ ) = c M P ( ζ ) + (cid:16) c M + h log ζ (cid:17) P ( ζ ) (B2a)where c M , c M are integration constants and P i ( ζ ) are regular polynomials of the following form P = 1 + ∞ (cid:88) n =1 p [ n ]1 ζ n , P = ζ (cid:32) ∞ (cid:88) n =1 p [ n ]2 ζ n (cid:33) (B2b)Similar to the usual procedure in the holographic renormalisation [54], all the coefficients p [ n ]1 , p [ n ]2 , h except p [2]1 , which can be set to zero without loss of generality [46], can be obtainedrecursively. The important piece of information here is the coefficient h = 1 which can be obtainedby recursively solving the equation (B1). Another easy way to see this is to recast (B1) as theBessel equation of order 1 as pointed out earlier. Then, using the fact that the Bessel functions1 K ( ζ ) and I ( ζ ) are two independent solutions of such equation and, for small ζ they admit thefollowing asymptotic expansions (see e.g. § I ( ζ ) = ζ ζ
16 + O ( ζ ) , K ( ζ ) = (cid:18) γ + log ζ (cid:19) I ( ζ ) + 1 ζ (B3)will result in the series expansions of the solution in AdS × R region in (B2a).A similar procedure can also be applied for E ⊥ using Eq.(2.25). Substituting δ ˜ B xz = ζ c ( ζ ),one finds that it obeys the Bessel equation of order 2 whose ζ (cid:28) ζ . [1] L. D. Landau and E. M. Lifshitz, Fluid Mechanics . Butterworth-Heinemann, 2nd ed., 1987.[2] R. W. Zwanzig,
Statistical mechanics of irreversibility . Lectures on Theoretical Physics Volume 3, 139(Interscience, 1961), 1961.[3] D. Forster,
Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions . PerseusBooks, 1995.[4] S. A. Hartnoll and D. M. Hofman, “Locally Critical Resistivities from Umklapp Scattering,”
Phys.Rev. Lett. (2012) 241601, arXiv:1201.3917 [hep-th] .[5] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,”
JHEP (2015) 172, arXiv:1412.5148 [hep-th] .[6] S. Grozdanov, D. M. Hofman, and N. Iqbal, “Generalized global symmetries and dissipativemagnetohydrodynamics,” Phys. Rev. D no. 9, (2017) 096003, arXiv:1610.07392 [hep-th] .[7] D. Schubring, “Dissipative String Fluids,” Phys. Rev. D no. 4, (2015) 043518, arXiv:1412.3135[hep-th] .[8] J. Hernandez and P. Kovtun, “Relativistic magnetohydrodynamics,” JHEP (2017) 001, arXiv:1703.08757 [hep-th] .[9] J. Armas and A. Jain, “Magnetohydrodynamics as superfluidity,” Phys. Rev. Lett. no. 14, (2019)141603, arXiv:1808.01939 [hep-th] .[10] J. Armas and A. Jain, “One-form superfluids & magnetohydrodynamics,”
JHEP (2020) 041, arXiv:1811.04913 [hep-th] .[11] W. G. Dixon, Special relativity: the foundation of macroscopic physics . CUP Archive, 1982.[12] A. M. Anile,
Relativistic fluids and magneto-fluids: With applications in astrophysics and plasmaphysics . Cambridge University Press, 2005.[13] S. S. Komissarov, “A Godunov-type scheme for relativistic magnetohydrodynamics,”
Monthly Noticesof the Royal Astronomical Society no. 2, (02, 1999) 343–366.[14] P. B. Arnold, G. D. Moore, and L. G. Yaffe, “Transport coefficients in high temperature gaugetheories. 1. Leading log results,”
JHEP (2000) 001, arXiv:hep-ph/0010177 . [15] R. Blandford and R. Znajek, “Electromagnetic extractions of energy from Kerr black holes,” Mon.Not. Roy. Astron. Soc. (1977) 433–456.[16] S. Komissarov, “Electrodynamics of black hole magnetospheres,”
Mon. Not. Roy. Astron. Soc. (2004) 407, arXiv:astro-ph/0402403 .[17] P. Goldreich and W. H. Julian, “Pulsar Electrodynamics,”
Astrophys. J. (Aug., 1969) 869.[18] T. Wiegelmann and T. Sakurai, “Solar Force-free Magnetic Fields,”
Living Rev. Sol. Phys. (2012) 5, arXiv:1208.4693 [astro-ph.SR] .[19] S. E. Gralla and T. Jacobson, “Spacetime approach to force-free magnetospheres,” Mon. Not. Roy.Astron. Soc. no. 3, (2014) 2500–2534, arXiv:1401.6159 [astro-ph.HE] .[20] G. Comp`ere, S. E. Gralla, and A. Lupsasca, “Force-Free Foliations,”
Phys. Rev. D no. 12, (2016)124012, arXiv:1606.06727 [math-ph] .[21] T. Uchida, “Theory of force-free electromagnetic fields. i. general theory,” Phys. Rev. E (Aug,1997) 2181–2197. https://link.aps.org/doi/10.1103/PhysRevE.56.2181 .[22] C. Thompson and O. Blaes, “Magnetohydrodynamics in the extreme relativistic limit,” Phys. Rev. D (1998) 3219–3234.[23] S. E. Gralla and N. Iqbal, “Effective Field Theory of Force-Free Electrodynamics,” Phys. Rev. D no. 10, (2019) 105004, arXiv:1811.07438 [hep-th] .[24] P. Glorioso and D. T. Son, “Effective field theory of magnetohydrodynamics from generalized globalsymmetries,” arXiv:1811.04879 [hep-th] .[25] B. Benenowski and N. Poovuttikul, “Classification of magnetohydrodynamic transport at strongmagnetic field,” arXiv:1911.05554 [hep-th] .[26] S. Grozdanov and N. Poovuttikul, “Generalised global symmetries in holography:magnetohydrodynamic waves in a strongly interacting plasma,” JHEP (2019) 141, arXiv:1707.04182 [hep-th] .[27] D. M. Hofman and N. Iqbal, “Generalized global symmetries and holography,” SciPost Phys. no. 1,(2018) 005, arXiv:1707.08577 [hep-th] .[28] D. T. Son and A. O. Starinets, “Minkowski space correlators in AdS / CFT correspondence: Recipeand applications,” JHEP (2002) 042, arXiv:hep-th/0205051 .[29] P. Kovtun, D. T. Son, and A. O. Starinets, “Holography and hydrodynamics: Diffusion on stretchedhorizons,” JHEP (2003) 064, arXiv:hep-th/0309213 .[30] J. F. Fuini and L. G. Yaffe, “Far-from-equilibrium dynamics of a strongly coupled non-Abelianplasma with non-zero charge density or external magnetic field,” JHEP (2015) 116, arXiv:1503.07148 [hep-th] .[31] S. Janiszewski and M. Kaminski, “Quasinormal modes of magnetic and electric black branes versusfar from equilibrium anisotropic fluids,” Phys. Rev. D no. 2, (2016) 025006, arXiv:1508.06993[hep-th] . [32] E. D’Hoker and P. Kraus, “Magnetic Brane Solutions in AdS,” JHEP (2009) 088, arXiv:0908.3875 [hep-th] .[33] S. Grozdanov, A. Lucas, and N. Poovuttikul, “Holography and hydrodynamics with weakly brokensymmetries,” Phys. Rev. D no. 8, (2019) 086012, arXiv:1810.10016 [hep-th] .[34] R. A. Davison and B. Gout´eraux, “Momentum dissipation and effective theories of coherent andincoherent transport,” JHEP (2015) 039, arXiv:1411.1062 [hep-th] .[35] C.-F. Chen and A. Lucas, “Origin of the Drude peak and of zero sound in probe brane holography,” Phys. Lett. B (2017) 569–574, arXiv:1709.01520 [hep-th] .[36] R. A. Davison, S. A. Gentle, and B. Gout´eraux, “Impact of irrelevant deformations onthermodynamics and transport in holographic quantum critical states,”
Phys. Rev. D no. 8,(2019) 086020, arXiv:1812.11060 [hep-th] .[37] E. Witten, “Multitrace operators, boundary conditions, and AdS / CFT correspondence,” arXiv:hep-th/0112258 .[38] M. Berkooz, A. Sever, and A. Shomer, “’Double trace’ deformations, boundary conditions andspace-time singularities,”
JHEP (2002) 034, arXiv:hep-th/0112264 .[39] R. A. Davison and A. Parnachev, “Hydrodynamics of cold holographic matter,” JHEP (2013) 100, arXiv:1303.6334 [hep-th] .[40] U. Moitra, S. K. Sake, and S. P. Trivedi, “Near-Extremal Fluid Mechanics,” arXiv:2005.00016[hep-th] .[41] S. S. Gubser and A. Nellore, “Ground states of holographic superconductors,” Phys. Rev. D (2009)105007, arXiv:0908.1972 [hep-th] .[42] R. A. Davison, B. Gout´eraux, and S. A. Hartnoll, “Incoherent transport in clean quantum criticalmetals,” JHEP (2015) 112, arXiv:1507.07137 [hep-th] .[43] P. K. Kovtun and A. O. Starinets, “Quasinormal modes and holography,” Phys. Rev. D (2005)086009, arXiv:hep-th/0506184 .[44] E. D’Hoker, P. Kraus, and A. Shah, “RG Flow of Magnetic Brane Correlators,” JHEP (2011) 039, arXiv:1012.5072 [hep-th] .[45] E. D’Hoker and P. Kraus, “Magnetic Field Induced Quantum Criticality via new AsymptoticallyAdS Solutions,”
Class. Quant. Grav. (2010) 215022, arXiv:1006.2573 [hep-th] .[46] P. Kovtun and A. Starinets, “Thermal spectral functions of strongly coupled N=4 supersymmetricYang-Mills theory,” Phys. Rev. Lett. (2006) 131601, arXiv:hep-th/0602059 .[47] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables . Dover, New York, ninth dover printing, tenth gpo printing ed., 1964.[48] D. Nickel and D. T. Son, “Deconstructing holographic liquids,”
New J. Phys. (2011) 075010, arXiv:1009.3094 [hep-th] .[49] P. Glorioso, M. Crossley, and H. Liu, “A prescription for holographic Schwinger-Keldysh contour innon-equilibrium systems,” arXiv:1812.08785 [hep-th] . [50] J. de Boer, M. P. Heller, and N. Pinzani-Fokeeva, “Holographic Schwinger-Keldysh effective fieldtheories,” JHEP (2019) 188, arXiv:1812.06093 [hep-th] .[51] S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, “Nonlinear Fluid Dynamics fromGravity,” JHEP (2008) 045, arXiv:0712.2456 [hep-th] .[52] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam, and P. Surowka,“Hydrodynamics from charged black branes,” JHEP (2011) 094, arXiv:0809.2596 [hep-th] .[53] J. Erdmenger, M. Haack, M. Kaminski, and A. Yarom, “Fluid dynamics of R-charged black holes,” JHEP (2009) 055, arXiv:0809.2488 [hep-th] .[54] S. de Haro, S. N. Solodukhin, and K. Skenderis, “Holographic reconstruction of space-time andrenormalization in the AdS / CFT correspondence,” Commun. Math. Phys. (2001) 595–622, arXiv:hep-th/0002230 .[55] C. Bender and S. Orszag,