Momentum and charge transport in non-relativistic holographic fluids from Hořava gravity

Journal of High Energy Physics, Nov 2016

We study the linearized transport of transverse momentum and charge in a conjectured field theory dual to a black brane solution of Hořava gravity with Lifshitz exponent z = 1. As expected from general hydrodynamic reasoning, we find that both of these quantities are diffusive over distance and time scales larger than the inverse temperature. We compute the diffusion constants and conductivities of transverse momentum and charge, as well the ratio of shear viscosity to entropy density, and find that they differ from their relativistic counterparts. To derive these results, we propose how the holographic dictionary should be modified to deal with the multiple horizons and differing propagation speeds of bulk excitations in Hořava gravity. When possible, as a check on our methods and results, we use the covariant Einstein-Aether formulation of Hořava gravity, along with field redefinitions, to re-derive our results from a relativistic bulk theory.

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Momentum and charge transport in non-relativistic holographic fluids from Hořava gravity

Received: July and charge transport in non-relativistic holographic uids from Horava gravity Richard A. Davison 0 1 3 6 Saso Grozdanov 0 1 3 4 Stefan Janiszewski 0 1 3 5 Matthias Kaminski 0 1 2 3 Tuscaloosa 0 1 3 AL 0 1 3 U.S.A. 0 1 3 Open Access 0 1 3 c The Authors. 0 1 3 0 Victoria , BC, V8W 3P6 , Canada 1 Niels Bohrweg 2 , Leiden 2333 CA , The Netherlands 2 Department of Physics and Astronomy, University of Alabama 3 Cambridge , MA 02138 , U.S.A 4 Instituut-Lorentz for Theoretical Physics, Leiden University 5 Department of Physics and Astronomy, University of Victoria 6 Department of Physics, Harvard University We study the linearized transport of transverse momentum and charge in a conjectured eld theory dual to a black brane solution of Horava gravity with Lifshitz exponent z = 1. As expected from general hydrodynamic reasoning, we these quantities are di usive over distance and time scales larger than the inverse temperature. We compute the di usion constants and conductivities of transverse momentum and charge, as well the ratio of shear viscosity to entropy density, and nd that they di er from their relativistic counterparts. To derive these results, we propose how the holographic dictionary should be modi ed to deal with the multiple horizons and di ering propagation speeds of bulk excitations in Horava gravity. When possible, as a check on our methods and results, we use the covariant Einstein-Aether formulation of Horava gravity, along with eld rede nitions, to re-derive our results from a relativistic bulk theory. holographic; AdS-CFT Correspondence; E ective eld theories; Gauge-gravity correspon- 1 Introduction 2 3 4 Introduction Hydrodynamics and linear response Momentum transport from a Horava black brane Horava black brane solution Black brane excitations Holographic dictionary Hydrodynamic solution of the equation of motion Holographic renormalization and hydrodynamic Green's functions Transport coe cients and susceptibilities Charge transport from a Horava black brane Covariant formalism and eld rede nitions Momentum transport Charge transport Sound waves when = 0 The AdS/CFT correspondence [1] has proven to be an excellent tool with which to study the properties of certain strongly interacting, relativistic quantum eld theories. It has taught us that these eld theories have a robust hydrodynamic limit with a large window of applicability [2{5], and has enabled the calculation of various hydrodynamic properties of these theories, most prominently the ratio of shear viscosity to entropy density [2, 6{20]. For this reason, holography has provided a fertile testing ground for ideas about hydrodynamic descriptions of the quark-gluon plasma and of metals (see e.g. [21, 22]). It is of both fundamental and practical interest to determine whether there are classical gravitational descriptions of strongly interacting eld theories which are not relativistic at zero temperature. See [23] for a recent review of this topic. One proposed class of gravitational duals are non-relativistic solutions of relativistic theories of gravity (general relativity (GR) coupled to appropriate matter content) [24{26]. A second approach [27, 28], which we pursue here, is to work with an intrinsically non-relativistic theory of gravity, like that proposed by Horava in [29]. This theory is not invariant under all spacetime di eomorphisms, and arises as the dynamical theory of Newton-Cartan geometry [30] (to which non-relativistic eld theories naturally couple [31{34]). In this work, we study the linear response of a neutral black brane solution [35] of (3+1)-dimensional Horava gravity. A manifestation of the non-relativistic nature of this state is that the low energy, linearized excitations of di erent elds propagate at di erent speeds, and each eld has its own `sound horizon' (trapped surface) [36]. The causal `universal' horizon of the solution traps modes of arbitrarily high speed, and is the thermodynamic horizon of the solution [35{40]. The solution we study has an asymptotic Lifshitz and space transform identically. It is invariant under spatial rotations and translations in space and time. However, this solution does not have a Lorentz boost symmetry, as this relativistic transformation is not a symmetry of Horava gravity. Moreover, the solution we study has no Galilean boost symmetry. Using the proposed holographic dictionary of [27, 28], with further re nement following [30, 32, 41], we show that there is a simple hydrodynamic description of the linearized transport of both charge density and transverse momentum density over long times and distances in the conjectured dual eld theory. In particular, both of these quantities di use, and have conductivities related to their di usion constants by Einstein relations. This is an important consistency check of the existence of a holographically dual state of this black brane. In terms of bulk quantities, we nd that the charge di usion constant D and the transverse momentum di usion constant D can be neatly expressed as (sound horizon radius); (sound horizon radius); where the relevant speed and sound horizon radius in each case is that of the corresponding dual excitation in the gravitational theory. We note that these constants have the same form as the analogous relativistic formulae, in which case di erent bulk excitations have the same speed and sound horizon radius due to Lorentz invariance. In terms of the temperature of the universal horizon, the di usion constants are given by (3.40) and (4.18). Our results for momentum transport are complementary to those of [42] (which worked with a related, covariant theory1), in which a non-linear hydrodynamic description of transport was derived to leading order in perturbation theory in , one of the coupling constants of Horava gravity. parameterizes the di erence between the speed of one of the gravitons and the null speed of the boundary metric. While we study only linear (in amplitude) perturbations, we work non-perturbatively in . Our non-perturbative result for the shear agrees with that conjectured in [42]: =s = 22=3=4 , where s is the entropy universal horizon of the Horava solution does not coincide with the Killing horizon of the GR solution in the limit To obtain our results, we must modify the standard prescription for computing twopoint retarded Green's functions in the relativistic case [43{46], due to the existence of multiple horizons. We propose that the linear excitation of a eld should obey ingoing boundary conditions at its sound horizon. In some cases, we are able to check that this 1See section 5 for further comparison between these theories. is a sensible prescription by rstly rewriting Horava gravity in a covariant form (EinsteinAether theory), and then using a eld rede nition invariance of this theory, as well as di eomorphisms, to map the perturbation equations onto those of the Schwarzschild-AdS4 black brane solution of GR. This procedure maps the sound horizon radius of the original Horava solution to the Killing horizon radius of the Schwarzschild-AdS4 black brane. It also provides a natural explanation for the appearance of the speeds of the bulk excitations, rather than the null speed of the boundary metric, in the di usion constants (1.1). We expect that this general principle | that ingoing boundary conditions should be applied at di erent values of r for bulk excitations which travel at di erent speeds | should be valid in Horava gravity beyond these simple cases. nite answers for the transverse momentum density correlators, we performed holographic renormalization by including two counterterms which are invariant under the symmetries of Horava gravity. Upon the mapping to a covariant Einstein-Aether theory, these counterterms coincide with those of the GR calculation. Finally, we exploit the eld rede nition invariance of the covariant form of Horava gravity to identify a special point in the parameter space of the Horava theory (when the to that of the Schwarzschild-AdS4 solution of GR. Therefore, at this special point, the excitation spectrum of the dual eld theory contains a sound mode (5.21) with speed proportional to the spin-2 graviton speed. In the following section we provide a brief overview of linear response in hydrodynamics, and a derivation of the expected forms of the retarded Green's functions for charge density and transverse momentum density. In sections 3 and 4 we study linear perturbations of the Horava black brane solution and derive from this the hydrodynamic forms of the dual Green's functions. The relation between our Horava gravity calculations and those of Einstein-Aether theory are described in section 5, before we conclude in section 6 with a summary of our results and some open questions. Hydrodynamics and linear response In general, a system which is in local thermal equilibrium should have a coarse-grained, hydrodynamic description over long lengths and times, with respect to the scales over which the system is locally equilibrated (in our case, this is the inverse temperature). The hydrodynamic variables are those which vary slowly over these long length and time scales. These are typically the densities of the conserved charges of the system. We are interested in the linear response properties of a U(1) charge density, and the transverse momentum density, in a (2+1)-dimensional, rotationally and translationally charges of the state are its energy, U(1) charge and momentum. The densities of these conserved charges qa obey the following conservation equations @tqa + r~ ~ja = 0; where ~ja is the current density associated with the conserved charge density qa. We will consider the response of states in which both the U(1) charge density and the momentum density have vanishing expectation values. The information about the linear response properties of the state are contained in its two-point retarded Green's functions, which tell us how the expectation values of operators respond to small external sources. The retarded Green's functions of the charge densities and associated current densities can be computed within hydrodynamics using the canonical method of Kadano & Martin [47] (see [48] for a review). Heuristically, this method proceeds in two steps. When a small external source for a conserved charge density is applied at an initial time, the response in the expectation value of the charge density at that time is controlled by the susceptibility .2 This initial change in the expectation value will then evolve in time via the equations of motion (2.1), and the variation of this response at time t, with respect to the initial source, gives the retarded Green's function. To determine the evolution in time of the charge densities, we must supplement the equations (2.1) with constitutive relations for the current densities ~ja in terms of the charges qb. Hydrodynamics is a universal e ective theory, and we therefore construct these relations by writing down all terms containing the conserved charges and their derivatives that are allowed by the symmetries of the system. The relations are written as a derivative expansion, and are a good approximation at long distance and time scales. The microscopic details of the system enter in the values of the coe cients of each term in these derivative expansions. After a Fourier transform in the spatial directions, and using the constitutive relations to replace ~ja with qb, the equations of motion for linear perturbations take the form @tqa(t; k) + Mab(k)qb(t; k) = 0: The two-point retarded Green's functions of the charges are then given by [48] G(!; k) = There has recently been a lot of progress in systematically constructing the full, nonlinear constitutive relations of non-relativistic hydrodynamics [32, 41, 50, 51], and also for Lifshitz hydrodynamics [52{56]. For our purposes this is overkill: by restricting to the linear reponse of parity-invariant theories, simple Kadano -Martin arguments are valid. It can be checked, for example, that imposing parity symmetry on the constitutive relations of [32, 51] reduces the constitutive relations written in terms of Newton-Cartan data to the usual Navier-Stokes equations. Without loss of generality, we will align the y-axis with the direction along which linear perturbations vary in space. The conserved charge densities of our system are the energy density ", the U(1) charge density , and the momentum densities x and y. We also assume that parity is unbroken, and that charge conjugation, under which only the U(1) charge density and current ip sign, is a symmetry of the state. This last condition implies we are studying a state which is not charged under this U(1). 2Such susceptibilities are also sometimes referred to as `thermodynamic transport coe cients', e.g. [49]. We begin with the constitutive relation for the longitudinal U(1) charge current density jy. The goal is to write down, to linear order in perturbations, the most general derivative expansion of the charges that is consistent with the symmetries above. To leading order in the derivative expansion, only one term is allowed jy = momentum density x is equally simple jyx = The ellipsis denotes higher order terms in the derivative expansion. The constant D is a transport coe cient that is not xed by this analysis but depends upon microscopic details of the theory. The linearized constitutive relation for the longitudinal current jyx of the transverse In this case, it is parity symmetry (under which x ! x) that restricts the form of the right hand side. D is a transport coe cient which, in general, is unrelated to D . Combining the linearized constitutive relations with the conservation equations (2.1), we nd that linearized perturbations of both the charge density and the transverse momentum density obey a di usion equation 2 = 0; D r2 x = 0; and that the transport coe cients D and D are the di usion constants of charge density and transverse momentum density, respectively. Di usion constants have dimensions of Green's functions of the conserved charges From the di usion equations (2.6), we can use (2.3) to compute the hydrodynamic G (!; k) = G x x(!; k) = @vx vx=0 denote the static susceptibilities of the conserved charge densities where the chemical potential is the source for the charge density, and the velocity vx is the source for the transverse momentum density. has units of mass density. The two-point retarded Green's functions of the associated current densities are xed by Ward identities to be Gjyjy (!; k) = G (!; k); G jy (!; k) = Gjy (!; k) = G (!; k); Gjyxjyx (!; k) = G x x(!; k); G xjyx (!; k) = Gjyx x(!; k) = up to contact terms. In the long time (dc) limit, we de ne the linear response conductivities of U(1) charge, and of transverse momentum as Im kli!m0 Gjyjy (!; k) = Im kli!m0 Gjyx jyx (!; k) = respectively. The rst of these corresponds to the usual de nition of the conductivity via Ohm's law, and the second corresponds to the usual de nition of the shear viscosity (see e.g. [57]). The conductivities are xed in terms of the di usion constants by the Einstein relations (2.10), which follow simply from the form of the Green's functions (2.7). From now on we will refer to these conductivities by their conventional names of the electrical conductivity and the shear viscosity, respectively. We have refrained from a full discussion of non-relativistic [51] or Lifshitz [52{55] hydrodynamics and have presented only the elements which are relevant for our holographic computation. We have shown that transverse momentum and charge both di use, regardless of whether the system is relativistic or not. We note that the presence of an additional conserved particle number charge will not alter our conclusions, as it cannot enter the linearized constitutive relations (2.4) and (2.5) due to symmetry reasons. In the following sections, we will show that the Green's functions of the strongly interacting state purportedly dual to a Horava gravity black brane are of the hydrodynamic form (2.7), and will derive explicit expressions for the transport coe cients D and D (or equivalently and ) of this state. Momentum transport from a Horava black brane Horava gravity [29] is a non-relativistic quantum theory of gravity that breaks the local Lorentz covariance between space and time enjoyed by GR. We are interested in the low energy, classical regime of Horava gravity, whose degrees of freedom, GIJ , N I and N , are the components of the ADM decomposition of a spacetime metric gXY gXY dxX dxY = N 2dt2 + GIJ dxI + N I dt GIJ is the spatial metric on slices of constant global time t; N is the lapse function, which encodes the normal distance between the leaves of the foliation by t; and N I is the shift vector, which identi es events with the same spatial coordinates on di erent time slices.3 In (3+1)-dimensions, the two derivative bulk action of Horava gravity is SH = (1 + )K2 + (1 + )(R 3Our notation is that indices X; Y : : : are bulk spacetime indices with x0 t, while I; J : : : are bulk spatial indices covering both the bulk radial direction x r and the transverse directions shared with the is the extrinsic curvature of slices of constant t, K is its trace, and R and G are the Ricci scalar and the determinant of the spatial metric, respectively. Indices are raised and lowered with GIJ and GIJ , and rI is the covariant derivative with respect to the spatial metric. In addition to the cosmological constant and the gravitational constant GH (which has length dimension 2), there are three new coupling constants ( ; ; ) allowed by the less restrictive symmetries of Horava gravity. These dimensionless constants must satisfy 2(1 + ), and In comparison with the full spacetime di eomorphism invariance of GR, Horava gravity is only invariant under the di eomorphisms that preserve the foliation by slices of constant t. These are the spatial di eomorphisms xI ! x~I (t; xJ ) and reparametrizations of the global time t ! t~(t). In particular, spatially dependent time di eomorphisms are not symmetries of Horava gravity. Horava black brane solution For the case = 0, and with cosmological constant 3, there is an asymptotically AdS black brane solution to Horava gravity [35] with GIJ = BB N = NI = @ r3 consistent with the aforementioned constraints, and is smoothly connected to the numerical solutions of [35]. We have checked (to leading order in ) that, when = 0, this solution has smooth corrections. The corresponding spacetime metric (3.1) of this solution is asymptotically AdS, and the boundary metric has a \null speed" of 1. This is a choice of units, and all speeds in the formulae that follow are in units of this null speed. normal distance between slices of constant t, vanishes [36{40]. In Horava gravity, causal signals propagate only forward in global time t. The leaves of the asymptotic temporal Therefore, events at r > rh can only signal to larger r, and can have no causal in uence on those at r rh. This causal event horizon traps modes of any speed, and is interpreted as the thermodynamic horizon of the solution [35]. The Killing horizon of the solution is at rk rh=(1 + p 1 + )1=3. Its physical significance is that it is the trapped surface for modes of unit speed.4 While the null speed of 4The locations of the various trapped surfaces, or sound horizons, for di erent speeds can be determined by examining the Killing horizon of an e ective metric, as will be explicitly demonstrated in section 5. horizon of the spin-2 graviton rs, and the universal horizon rh, are trapped surfaces for waves with speed s = 1, s2 = p horizon rs can be inside or outside of the Killing horizon rk. and s0 ! 1, respectively. Depending on the value of , the spin-2 sound NI = 0, and N = 1, one speeds squared of these modes are the asymptotic metric at the boundary is 1, this is not the speed at which excitations of generic elds travel in Horava gravity. In contrast to GR, Horava gravity has more than one nds a spin-2 graviton and an additional spin-0 graviton. The s22 = 1 + ; s20 = respectively [58]. Our background (3.4) also supports multiple gravitons, and we will see shortly that the most important of these, for our purposes, travels at speed s2. This is nite and therefore has a sound horizon (the trapped surface for modes of this speed) at a radius rs, outside the universal horizon. When = 0, s0 ! 1, and a mode of this speed has a sound horizon which coincides with the universal horizon. A schematic location of the various horizons is shown in gure 1. Black brane excitations To determine the linear response of the conserved momentum density x of our purported dual theory, we will study linearized excitations of the shift Nx(t; r; y) around the black brane solution. We will shortly outline in more detail the holographic dictionary that we use to explicitly identify the sources. Nx(t; r; y) couples to both Gyx(t; r; y) and Grx(t; r; y). After choosing the gauge motion for these linearized excitations are kp1 + r3r3 (!g(r) + kn(r)) + i(r3 h !rh3n0(r) = 0; krh6r (!g(r) + kn(r)) + (r3 rh3)( 2n0(r) + rn00(r)) = 0; p1 + (2r5 + rh3r2) + i!rh6 (!g(r) + kn(r)) ikp1 + r3r3n0(r) + (1 + )(2r6 h 3rh3r3 + rh6)g00(r) = 0; where we have performed a Fourier transform with respect to the global time t and the spatial direction y Gyx(t; r; y) Nx(t; r; y) We leave it implicit that g and n both depend upon ! and k. Only one of the two second order equations of motion is linearly independent (because of the residual di eomorphism invariance after our gauge choice Grx = 0), and we can make this manifest by working directly with the variable This eld is invariant under the gauge freedom and obeys the second order equation i z4( 5 + z3) + ( 2 + z3)2(1 + 2z3) 3z3 + z6)2 + i z4( 10 + 6z3 + z6) i (z) + z( 2 + z3) 2q2( 1 + z3)(2 + z3( 3 00(z) = 0; where we have de ned a rescaled radial coordinate z, frequency and wavenumber q as 21=3r 2r1h=3 k; 21=3p1 + This rescaling manifestly removes from the equation of motion, as it entered only in the combination with ! that we have de ned as . The rescaled radial coordinateordinate response dynamics in this sector are completely independent of the coupling constant Holographic dictionary To extract the linear response correlators of the dual eld theory, we will follow a similar procedure as in the relativistic case, e.g. [2, 43]. Firstly, we must solve the di erential equation (3.9) subject to two boundary conditions. Our rst boundary condition is to To identify the sources in Horava gravity, we will use a re ned holographic dictionary rst presented in [27, 28]. This was originally motivated by the understanding of nonrelativistic symmetry groups of [24, 59], and now has a more rigorous formulation in terms of the Newton-Cartan geometry of [30, 32]. The relation to Newton-Cartan geometry is most clearly illustrated by comparing the large c ! 1 limits of [28] and [41]. In [41], the most general spacetime metric is written as which is satis ed by the ADM decomposition (3.1) for5 nX nY + hXY ; nX = ( N; 0); hXY = In addition to this timelike vector nX and the degenerate symmetric \metric" hXY , Newton-Cartan geometry contains a \velocity" eld vX and an \inverse metric" hXY that are de ned to obey hXY vY = 0; X = 1; hXY nY = 0; which implies that they can be expressed in terms of the ADM X = hXY = We are now in position to express the sources of the eld theory in terms of the boundary values of the Horava elds by using the de nition of Newton-Cartan sources found in [32] (see also [33]): n0 is the source for energy density,6 h is the source for the stress tensor, and v is the source for momentum density. The barred notation is due to the fact that the sources should be varied arbitrarily, while the elds v must obey the constraints (3.13). The explicit relation between variations of barred sources and unbarred elds are given in [32], but for our background they are expressed in terms of the bulk hij = r2 Gij jr=0; vi = N ijr=0; where the powers of r are needed to strip o the leading behavior of the bulk elds as we approach the boundary at r ! 0. The sources of stress yx and momentum density x are hyx = r2 Gyxjr=0 = r2 Gyxjr=0 vx = N xjr=0 = r2 Nxjr=0 respectively. This agrees with the discussion of boundary conditions at r ! 0 above. In the notation of section 2, yx is equal jyx , the longitudinal component of the current associated with the conserved charge density the bulk elds in terms of eld theory sources, we need to apply another boundary condition in order to solve the second order equation of motion (3.9). In the relativistic case, to 5To make this identi cation unambiguous, powers of c need to be reinstated in the ADM expansion, 6The lack of spatial components ni renders us unable to calculate the energy current. determine the retarded Green's function of the dual eld theory, one must impose ingoing boundary conditions at the black brane horizon [2, 43]. Heuristically, this is because the retarded Green's function is the causal response function in the eld theory, and causality in the bulk implies that nothing should come out of the black hole. The situation is more subtle in Horava gravity; we must take care as there are multiple horizons. In fact, the equation of motion (3.9) has singular points at both the spin-2 sound horizon and the universal horizon. By studying the characteristic exponents near each singular point, we nd that it is only possible to impose ingoing boundary conditions at the spin2 horizon, which is the outermost singular point. We therefore choose this location to impose ingoing (in global time) boundary conditions. These boundary conditions, as well as the identi cation of sources, will be further justi ed in section 5 via a mapping to a covariant calculation. After imposing these boundary conditions, we will determine the dual Green's functions from the on-shell action of Horava gravity, as in the relativistic case. This step requires an appropriate holographic renormalization to obtain a nite answer, as will be explained shortly. Hydrodynamic solution of the equation of motion The equation of motion (3.9) cannot be solved analytically in general. It can be solved analytically in a perturbative expansion at small frequencies and wavenumbers. This is su cient for our purposes as we are ultimately interested in the dual Green's function in this hydrodynamic limit. Anticipating the existence of a di usive excitation, we will perform a perturbative expansion in small As explained above, we rst impose ingoing boundary conditions at the spin-2 horizon z = 1 (z) = (1 z ! 0. To nd the perturbative solutions, we expand (z) = and solve order by order in , imposing regularity at the horizon of (z = 1) at each order. At leading order in the small hydrodynamic expansion, the solution is 1(z) = C1 + C2z3: We can identify these constants in terms of the near-boundary expansions of the fundamental elds using the de nition (3.8). Note that while the normalization of these elds is given by the sources g0 and n0, and that g1, g2, n1, and n2 are determined in terms of these sources by the near boundary equations of motion, g3 and n3 are un xed by near boundary analysis, and are related to the expectation values of the elds dual to g and n. Comparing the expansion (3.20) to the leading order hydrodynamic solution yields 1(z) = Moving to the rst subleading order in the perturbative expansion, 2 obeys the same equation of motion as 1. Identifying the constants in terms of the near-boundary expansions of the fundamental elds, we nd that 2(z) = g3(0) + qn03(0) : Recall that in addition to the second order equation we are solving for , there is also a rst-order constraint equation (3.6) for g and n. This constraint equation places further restrictions on the allowed near-boundary expansions (3.20). In particular it requires that g3(0) = 0; g30(0) = To determine the Green's functions to leading order, we do not need to explicitly solve the equations of motion at higher order in . However, we do need to identify the subleading coe cients n3(0) and n03(0) in terms of the boundary values of the elds. To do this, it is 4(z) here. This results in n3(0) = n03(0) = Holographic renormalization and hydrodynamic Green's functions The nal step in our calculation of the retarded Green's functions in the hydrodynamic limit is to evaluate the on-shell gravitational action at the boundary r ! 0. As in the usual relativistic case, we must supplement the action (3.2) with two kinds of terms to obtain the correct answer. Firstly, we have to add a Gibbons-Hawking-like boundary term such that the variation problem is well-de ned. As shown in [35], the Horava-Gibbons-Hawking the current case. It can be written as: SHGH = where H is the determinant of Hij (the induced spatial metric of the boundary), and 2 is the trace of its extrinsic curvature, as embedded in the bulk slices of constant t. Note Secondly, we must renormalize the boundary action so that it is nite. To do this, we supplement the action with counterterms of boundary geometric objects that are invariant under foliation-preserving di eomorphisms (the symmetries of the theory). A nite answer is obtained by including the following two counterterms SHCT;1 = SHCT;2 = nite, and is of the form where 2Kij is the extrinsic curvature of the slices of constant t, as embedded in the boundThe result of including all of these terms is that the on-shell quadratic action is now + n0( !; k)Gng(!; k)g0(!; k) + n0( !; k)Gnn(!; k)n0(!; k) ; (3.29) where we have explicitly reinstated the dependence upon ! and k. Equipped with the onshell action, we now determine the retarded Green's functions of the operators x and jyx using the standard relativistic prescription. This prescription is that the retarded Green's functions GAB(!; k) are given by GAB(!; k) = 2G'A'B (!; k); where 'A is dual to the operator A. We identi ed the elds dual to x and jyx ( yx) in (3.16) as n and g, respectively. One can roughly think of (3.30) as varying a generating functional, provided by the on-shell gravity action, with respect to the sources n0 and g0. Transport coe cients and susceptibilities Using the solution of the linearized equation of motion in the small ! and k limit, and the prescription (3.30), the hydrodynamic retarded Green's functions of the operators in the dual eld theory are (up to contact terms independent of ! and k) G x; x(!; k) = Gjxy ; x(!; k) = 8 GH 21=3rh2 i! 8 GH 21=3rh2 i! G x;jxy (!; k) = Gjxy ; x(!; k); Gjxy ;jxy (!; k) = 8 GH 21=3rh2 i! 21=33 21=33 21=33 These Green's functions have the characteristic form associated with di usive transport, in agreement with that expected based upon hydrodynamic considerations. The hydrodynamic Green's functions (2.7) and (2.9) have two free parameters that are determined by microscopic details of the speci c theory: the momentum susceptibility the momentum di usion constant D . For the eld theory purportedly dual to our solution of Horava gravity, these take the values Our Green's functions obey the Einstein relation (2.10) and thus the eld theory viscosity (or conductivity of transverse momentum) is 8 GH rh 3; 3 21=3 rh: 24=3 T 22=3 The di usion constant agrees with the expression (1.1) given in the introduction. The Horava black brane solution (3.4) obeys a rst law-like relation [35] d = T ds; where is the ADM energy density of the solution, and the entropy density s and temperature T are properties of the universal horizon In terms of these thermodynamic quantities, the momentum susceptibility is s = T = should have units of mass density. The expression (3.39) is telling us that the rather than the null speed of the boundary metric. of energy of the uid, we nd where it is again apparent that s2 = p is the natural speed of the system. The ratio of viscosity to entropy density of the system takes the -independent value This result was conjectured in [42], based upon a perturbative calculation to leading order in , and we have provided an explicit veri cation of it. In light of the evidence above that appears only when multiplying the null speed of the asymptotic metric, it is perhaps ~ = kB = 1). Finally, we note that there is no continuity with the GR results in the when we express quantities in terms of thermodynamic variables. In this limit, the universal horizon of the Horava solution, which determines the thermodynamic quantities, does not coincide with the spin-2 sound horizon, which determines, for example, . In GR, the thermodynamic and sound horizons coincide. Charge transport from a Horava black brane We will now address the transport of a conserved U(1) charge in the eld theory. Holographically, the situation is identical to the relativistic case, e.g. [2]. The source of a U(1) charge and current is a U(1) background potential. These eld theory sources are simply dual to the leading near-boundary term (which is r0 for our black brane) of a bulk U(1) gauge potential: the bulk gauge transformations that preserve the radial gauge choice eld theory potential source. The subleading near-boundary term (in this case r1) of the bulk U(1) gauge potential encodes the expectation values of the conserved U(1) charge and current. A U(1) gauge potential is comprised of two parts: a spatial vector potential AI , and a scalar potential . These transform as AI ! AI gauge transformations, where is the spacetime dependent gauge parameter. In GR these two potentials combine to form a spacetime vector, but under the less restrictive symmetries of Horava gravity they are separately well-de ned geometric objects. minimal gauge invariant action of these U(1) potentials which is invariant under foliation preserving di eomorphisms is SHEM = 1 Z F IJ FIJ F JI NJ where the magnetic and electric eld strengths are FIJ @J AI and EI @tAI , respectively; c is the speed of electromagnetic waves; and 0 is the vacuum permeability, which gives the overall normalization of the action. From the point of view of the action (4.1), the speed c is a coupling constant. The combined action of (3.2) and (4.1) still has the black brane solution (3.4), with AI = 0 and this corresponds to a eld theory state with zero density of the global U(1) charge. We will concentrate on longitudinal perturbations of the gauge potential. These are dual to charge density and longitudinal current density perturbations of the eld theory. It is these operators which should exhibit interesting physics in the hydrodynamic limit, as we outlined in section 2. The equations of motion for the longitudinal linear uctuations Ay(r; t; y) d!dke i!t+ikya(r); d!dke i!t+iky (r); k2 (r) + k!a(r) 00(r) = 0; ! 0(r) = 0; r3)2 + (1 + )k2r6 e00(r) = 0: where we have not explicitly written that both a(r) and (r) are also functions of ! and k, only one is linearly independent. This is due to the residual U(1) gauge symmetry and can be made manifest by working with the gauge invariant eld !a(r) + k (r): This is a multiple of the electric eld component Ey, and obeys the second order equation 3ip1 + r2) + k2(1 + )r6(!rh6 + 3ip1 + r2(rh3 r3)2(!(2rh6! 3ip1 + r2(rh3 + 2r3)) + k2(1 + )r6)i e(r) r3)) + k2(1 + )r6) a00(r) = 0; To obtain the correlators of the dual eld theory operators, we will follow a similar procedure as in the previous section. The equation of motion (4.5) has singular points both r = 1=3 : The latter of these is the sound horizon for excitations of speed c. Following our logic in the previous section, we expect this to be the relevant horizon for the linear response calculation. Indeed, this is the only singular point of the equation of motion at which ingoing boundary conditions may be imposed, and is also the outermost singular point. After imposing ingoing boundary conditions at r , we expand the remaining part of the e(r) = 1 , as before. In terms of the near-boundary expansions of the original elds the lowest order solutions are e1 = k 0(0) + rk 1(0); e2 = !a0(0) + k 00(0) + !a1(0) + k 01(0) r: is nite and has the form Identifying 0 and a0 as the sources of tion (3.30) to obtain the Green's functions 0( !; k)G (!; k) 0(!; k) + 0( !; k)G a(!; k)a0(!; k) + a0( !; k)Ga (!; k) 0(!; k)+a0( !; k)Gaa(!; k)a0(!; k) : and jy, respectively, we use the prescripSolving the constraint equation near the boundary imposes that a1(0) = 0; a01(0) = And nally, demanding regularity of e3 and e4 at the sound horizon xes 1(0) = 01(0) = k 00(0) + !a0(0) : To determine the Green's functions of the dual operators and jy, we again need to evaluate the action on this solution. Unlike in the previous section, we do not need to add any boundary terms or counterterms to the bulk action (4.1). This r ! 0 boundary action GqR;q(!; k) = GqR;jy (!; k) = GjRy;jy (!; k) = GjRy;q(!; k) = GqR;jy (!; k); From the Einstein relation, we can then extract the electrical conductivity obeys the relation (1.1). bulk excitations Note that the bulk electromagnetic wave speed c and the corresponding sound horizon radius r appear naturally in these formulae. As previously advertised, the di usion constant As a function of the temperature, the di usion constant depends on the speeds of both D = 1=3 2 T in the hydrodynamic limit. These Green's functions have the characteristic form (2.7) and (2.9) associated with the di usive transport of a conserved charge density. In this case, it is the U(1) charge density. By comparing the hydrodynamic results to our holographic results (4.15), we can extract the following expressions for the U(1) charge susceptibility and di usion constant D in the eld theory state dual to the black brane solution of Horava gravity D = cr = 1=3 : while the dimensionless ratio of momentum and charge di usion constants depends on = 3 21=3 relativistic result [60]. Covariant formalism and eld rede nitions Until this point, we have worked solely with the formalism of Horava gravity. To better understand our results, and to give further justi cation for our choice of boundary conditions and holographic renormalization counterterms, it is instructive to express Horava gravity in a generally covariant way in terms of Einstein-Aether theory [61{63]. This is a theory of a spacetime metric coupled to a dynamical, timelike `aether' vector uX of unit norm. The two-derivative action of Einstein-Aether theory is [35] SAE = c3r~ X uY r~ Y uX + c4uX r~ X uY uZ r~ Z uY ; where g is the determinant of the full spacetime metric gXY , r~ is its covariant derivative, and R~ is its Ricci scalar. This action is invariant under the full coordinate di eomorphism symmetry of general relativity. For a hypersurface orthogonal uX (like that arising in Horava gravity), not all aether terms in the action are linearly independent, and one of To map onto Horava gravity, one performs partial gauge xing by choosing the time coordinate such that uX = N Xt . This is possible when uX is hypersurface orthogonal. The action (5.1) is then equal to the Horava action, as long as the spacetime metric is decomposed in the ADM form (3.1). The coupling constants in each action are related in the following way = 1 + We have been studying the Momentum transport We will now exploit this mapping to relate our Horava gravity calculations to a more conventional holographic calculation: linear perturbations around the Schwarzschild-AdS4 black brane solution of the Einstein-Hilbert action with a cosmological constant. The metric of this solution is ds~2 = r3=rs3 dt~2 + r3=rs3) and with aether vector given by By a temporal di eomorphism, we may transform this Schwarzschild black brane into the on the metric and the aether eld, where we have chosen the constant After this eld rede nition, the solution takes the form = s22 = 1 + . ds2 = ut = where we have relabelled rh = 21=3rs. unfamiliar form ds^2 = We now perform a non-trivial eld rede nition u^r = 0: This solution is identical to the Einstein-Aether formulation of our Horava gravity solution (3.4). This is not a coincidence. Under the eld rede nitions (5.7), the EinsteinAether action maps to itself, but with GAE ! GAE = and with di erent values of the c2 = hypersurface orthogonal Einstein-Aether theory to Horava gravity using (5.2), we conclude that the net e ect of these coordinate transformations and eld rede nitions is to convert the GR action into the Horava gravity action with 6= 0, = 0. This explains why the series of coordinate transformations and eld rede nitions we explicitly performed above turns the Schwarzschild-AdS solution into our Horava gravity solutions, but since the Horava solution has = 0 and is independent of , we can set = 0 and utilize the mapping above. The mapping is much more powerful than this. Not only is our background solution of Horava gravity independent of , but the equations of motion for linearized perturbations of the transverse gravitons around this state are also independent of . This implies that, after suitable eld rede nitions and coordinate transformations, the GR equations of motion for linearized transverse perturbations around the Schwarzschild-AdS solution are equivalent to the linearized Horava equations of motion (3.6) around the solution (3.4). To be explicit, they are identical after identifying hXY , the perturbations of the Schwarzschild-AdS spacetime (5.3), with and then replacing As the Schwarzschild-AdS calculation is very well-understood, we can use this equivalence to shed light on the results, and the calculational details, of our Horava gravity computations in section 3. Firstly, we note that the black brane Killing horizon of the Schwarzschild solution maps to the spin-2 sound horizon rs of the Horava gravity solution. Therefore, our imposition of ingoing boundary conditions at this sound horizon in the Horava formulation of the calculation is equivalent to imposing them at the Killing horizon of the Schwarzschild metric. Secondly, the counter-terms (3.28) we added to obtain a nite on-shell Horava action are equivalent to those of the relativistic case [65] after the appropriate transformations. These give justi cation for our choice of boundary conditions and counter-terms in the Horava formulation of the problem. Under the mapping described above, the coe cients n0; n3; g0; g3 in the near-boundary expansions (3.20) of the Horava gravity elds map in a trivial way to the corresponding cohyx = ht~x = hrx = e cients of the near-boundary expansions of the elds hxt; hxy in the Schwarzschild-AdS calculation. There should therefore be a close relationship between the Green's functions obtained from Horava gravity and those obtained from the Schwarzschild-AdS black brane in [60]. The Green's functions obtained from the Horava calculation should be those of the relativistic case, in which the black brane Killing horizon radius is replaced by the spin-2 sound horizon radius, and the relativistic speed of light is replaced by the spin-2 speed s2 = p null speed of the Schwarzschild-AdS solution (5.3) is given by p 1 + . This is precisely 1 + . The latter of these conditions is because, after the mapping, the asymptotic what we found in section 3.6. Finally, we note that we are not aware of any way of mapping thermodynamic quantities using this procedure. Although the Schwarzschild-AdS black brane Killing horizon maps to the spin-2 horizon of the Horava gravity solution, this is not the causal horizon of the theory, as there may be excitations that travel at speeds greater than s2. This means that there is no direct way of mapping thermodynamic quantities associated with the horizon, such as the entropy and temperature, between the solutions. A covariant expression of the location of the causal horizon in Einstein-Aether theory is given by uX X = 0, where gXY and g^XY above [38{40]. Charge transport It is also possible to recast our calculation of charge di usion in section 4 in terms of a covariant Einstein-Aether theory. The electromagnetic action (4.1) can be written in the Einstein-Aether formalism as [66] SEAM = 1 Z where FXY = r~ X AY r~ Y AX . In terms of the coupling constants used in (5.12), the speed of electromagnetic waves is The action (5.12) di ers from the usual covariant Maxwell action due to the second term proportional to . We can again take advantage of eld rede nitions and coordinate transformations to make this situation more intuitive. Under the eld rede nitions (where g^AB = gAB u^A = uA F^AB = F AB; the action (5.12) transforms to itself, but with di erent values of the coupling constants and 0. This eld rede nition is the inverse of that in (5.7). to be the speed of light squared c2 = = c2; the new metric of the black brane solution is ds^2 = 3 c2 1 2 rr3 + 1 h which can be brought into the nicer diagonal form ds~2 = f (r) = 1 with a radially dependent temporal di eomorphism. This new metric is an \e ective metric" for the electromagnetic modes, meaning that the sound horizon for modes of speed c in the Horava theory is now a Killing horizon.7 The null speed in the asymptotic region of the new metric is c. Although this is not the Schwarzschild-AdS solution, it gives an intuitive reason for why it is the speed of light c and the sound horizon r for light, rather than the null speed of the asymptotic metric and the universal horizon, which appear and light both move at the same speed, this reduces to the Schwarzschild-AdS solution. After the eld rede nition, the new coupling constants in the action are ^0 = c 0; ^ = and so the action of the rede ned elds is still not the Maxwell action. Sound waves when = 0 Finally, we will comment brie y on longitudinal linearized perturbations around our Horava This will tell us, amongst other things, about the longitudinal transport of momentum in the dual eld theory. There is one limit in which these transport properties are relatively simple to deduce. In section 5.1, we described how to relate our = 0 Horava equations of motion to those of perturbations around the Schwarzschild-AdS solution of GR. For the transverse linearized perturbations of Horava gravity, the equations of motion are independent of and thus we could perform this mapping for any . The longitudinal perturbation equations are -dependent and so generically we cannot use this mapping. However, in the special of motion for perturbations hXY of the Schwarzschild-AdS metric (5.3) are transformed 7For c3 = (1 c2 + )=(1 + ) and adequate ut and ur, equation (5.17) is a solution. It would be intersting to investigate this solution further. ht~t~ = hxx = hyy = hrr = hrt~ = hyt~ = hyr = into the Horava equations by identifying 2 (1 + ) N N F (r) = p followed by the replacement (5.11). We have not written either set of longitudinal equations explicitly as they are very lengthy. As the near-boundary expansions of each eld map in a trivial way, we can painlessly determine the longitudinal, hydrodynamic quasi-normal modes of the Horava solution (3.4), where we de ne a quasi-normal mode as a solution that is ingoing at the sound horizon horizon radius with the spin-2 horizon radius, and the asymptotic null speed with p and whose leading term vanishes at the asymptotic boundary. By replacing the Killing in the Schwarzschild-AdS results [67], we nd the dispersion relations ! = These are the dispersion relations of the sound waves in the eld theory dual of Horava are most naturally expressed in terms of the speed of the spin-2 graviton, rather than the null speed of the boundary metric. The attenuation coe cient may be rewritten as to be more complicated, as there will be another graviton excitation. In [42], the dual hydrodynamics of the solution (3.4) was studied (perturbatively in ) within Einstein-Aether theory and found to be consistent with relativistic hydrodynamics to rst order in the derivative expansion. In our Horava theory calculation, this is only the not solutions of Horava gravity in a global time. We have used the non-relativistic holographic duality conjectured in [27, 28], and reinforced in the language of [30, 32], to study the collective transport properties of a dual to a black brane solution of Horava gravity. In agreement with the general principles Grr + 2F (r) Nr + F (r)2 2 (1 + ) N N Nr + F (r) 2 (1 + ) N N Gyr + F (r) Ny; of hydrodynamics outlined in section 2, we have shown that both charge and transverse momentum di use, and calculated the di usion constants and conductivities associated with these processes, in equations (3.35), (3.36), (4.16) and (4.17). The ratio (3.41) of entropy density to viscosity is independent of and is increased by a factor of 22=3 compared to the relativistic case, in agreement with the conjecture of [42]. Geometrically, this factor stems from the fact that the thermodynamic horizon does not coincide with the relevant trapped surface, i.e. the spin-2 sound horizon. From the point of view of the gravitational theory, we have derived new results for the hydrodynamic quasi-normal modes of the solution (3.4) of Horava gravity. We have outlined how the canonical method for determining linear response properties from a dual theory of relativistic gravity should be modi ed for the non-relativistic case of Horava gravity. The major di erence is that, in Horava gravity, di erent excitations (e.g. di erent gravitons, the photon) have di erent speeds, and these speeds are di erent from the null speed of the asymptotic metric. In GR, Lorentz invariance xes these speeds to all be the same. In Horava gravity, these di erent excitations each have their own independent sound horizon, which speci es the region from which the excitation cannot escape. In principle, these are independent from the universal horizon, the causal horizon of the spacetime itself. We propose that calculation of the two point retarded Green's function of the dual eld theory requires one to impose ingoing boundary conditions at the sound horizon of the dual excitation. We have focused here on the most tractable Horava gravity calculations: the di usive excitations around the analytic black brane solution (3.4). These examples make it easiest to identify the fundamental di erences with the relativistic calculations of two-point functions, and to identify how the canonical methods should be modi ed. Indeed, in some of these cases we had the luxury of using the mapping to GR described in section 5 to check that our proposed modi cations are sensible. In general, such a simple mapping is not possible and one should then directly use the procedure we have outlined for Horava gravity. There is one important feature of Horava gravity that does not a ect the examples we have studied here: the existence of new graviton excitations due to the reduced di eomorphism symmetry. These appear to be present in the 6= 0 calculation of longitudinal transport, leading to a complicated set of coupled equations for linearized excitations, in contrast to the single equation when is therefore to complete our understanding of the dynamics in the longitudinal sector, and to identify the nature and consequences of this new degree of freedom in the eld theory. Due to the existence of this mode, we expect the hydrodynamics of the longitudinal sector to be signi cantly di erent from that of the relativistic case. A second important generalization of this work is to study Horava theories with which is not unusual in the low energy limit of many-body systems. Given the recent holography-inspired success of applying hydrodynamic techniques to explain the transport properties of relativistic many-body systems, we are hopeful that the non-relativistic generalization will be similarly useful. For example, in relativistic hydrodynamics, energy cannot ow independently of momentum, and thus the thermoelectric conductivity matrix 6= 0. depends only on one transport coe cient [68]. This will not generically be the case for a system without Lorentz symmetry, and it is worthwhile verifying this directly from the dual gravitational perspective. As mentioned in the dictionary discussion of section 3.2, the Horava gravity theory (3.2) seems unable to encode the source of the energy current, as there is no bulk eld corresponding to the spatial components ni of the Newton-Cartan clock vector. To restore this source and allow calculation of correlators of the energy current will require the study of a more general form of Horava gravity, a task explored in [30]. There is undoubtedly fruitful work to be pursued in this direction. Although we have determined the counter-terms necessary to holographically renormalize the Horava action for the di usive channel, a complete description of the process is still lacking, although progress was made in [58]. A reasonable guiding principle is the analog of the GR procedure: counter-terms should be constructed out of geometrically invariant boundary data. For Horava gravity, this means terms invariant under foliation vector Nr generically does not vanish. Therefore, in addition to the boundary data of the induced spatial metric Gij , the induced shift vector Ni, and lapse N , the boundary value of Nr should be included in possible counter-terms. In the relativistic case, the calculation of the dual linear response conductivities can be greatly simpli ed using `membrane paradigm' techniques, that relate them to properties of the black brane horizon [68{71]. It may be possible to generalize these techniques to Horava gravity, in which case we expect the properties of the sound horizon to play the corresponding role. A more challenging task is to determine the full non-linear response of the black brane and interpret it in terms of the non-linear hydrodynamics of a dual theory. Some work in this direction was undertaken in [42]. Finally, it would be interesting to construct and study charged black brane solutions of engineer a charged solution of Horava gravity by starting with the Reissner-Nordstrom-AdS solution of GR and using the mappings described in section 5. Acknowledgments We are very grateful to Ste en Klug for his collaboration during the initial stages of this project. We thank Christopher Eling and Jelle Hartong for useful discussions and remarks on the draft of this paper. The work of R.A.D. is supported by the Gordon and Betty Moore Foundation EPiQS Initiative through Grant GBMF#4306. S.G. is supported in part by a VICI grant of the Netherlands Organisation for Scienti c Research (NWO), and by the Netherlands Organisation for Scienti c Research/Ministry of Science and Education (NWO/OCW). The work of S.J. is supported in part by NSERC of Canada. M.K. thanks the Instituut-Lorentz and Koenraad Schaalm for hospitality during important phases of Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE]. hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE]. hydrodynamics. 2. 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Richard A. Davison, Sašo Grozdanov, Stefan Janiszewski. Momentum and charge transport in non-relativistic holographic fluids from Hořava gravity, Journal of High Energy Physics, 2016, 170, DOI: 10.1007/JHEP11(2016)170