Charge-hyperscaling violating Lifshitz hydrodynamics from black-holes

Journal of High Energy Physics, Dec 2015

Non-equilibrium black hole horizons are considered in scaling theories with generic Lifshitz invariance and an unbroken U(1) symmetry. There is also charge-hyperscaling violation associated with a non-trivial conduction exponent. The boundary stress tensor is computed and renormalized and the associated hydrodynamic equations derived. Upon a non-trivial redefinition of boundary sources associated with the U(1) gauge field, the equations are mapped to the standard non-relativistic hydrodynamics equations coupled to a mass current and an external Newton potential in accordance with the general theory of [43]. The shear viscosity to entropy ratio is the same as in the relativistic case.

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Charge-hyperscaling violating Lifshitz hydrodynamics from black-holes

HJE Charge-hyperscaling violating Lifshitz hydrodynamics Elias Kiritsis 0 1 2 3 4 5 6 7 8 Yoshinori Matsuo 0 1 2 4 5 6 7 8 0 Department of Physics, University of Crete 1 71003 Heraklion , Greece 2 Crete Center for Quantum Complexity and Nanotechnology 3 APC, Astrparticule et Cosmologie, Universite Paris Diderot , CNRS/IN2P3, CEA/Irfu 4 Crete Center for Theoretical Physics 5 Open Access , c The Authors 6 There is also charge- 7 10, rue Alice Domon et Leonie Duquet , 75205 Paris Cedex 13 , France 8 Observatoire de Paris , Sorbonne Paris Cite Non-equilibrium black hole horizons are considered in scaling theories with generic Lifshitz invariance and an unbroken U(1) symmetry. hyperscaling violation associated with a non-trivial conduction exponent. The boundary stress tensor is computed and renormalized and the associated hydrodynamic equations derived. Upon a non-trivial rede nition of boundary sources associated with the U(1) gauge eld, the equations are mapped to the standard non-relativistic hydrodynamics equations coupled to a mass current and an external Newton potential in accordance with the general theory of [43]. The shear viscosity to entropy ratio is the same as in the relativistic case. Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1508.02494 1http://hep.physics.uoc.gr/ kiritsis/. 1 Introduction 3 Hydrodynamic ansatz 4 The rst order solution 2 U(1)-invariant, charge-hyperscaling violating Lifshitz theory 5 Calculation and renormalization of the boundary stress tensor 5.1 Energy and momentum conservation 6 Newton Cartan theory and Milne-boost invariance 7 The entropy current 8 General background gauge eld 8.1 A gauge invariant and Milne-boost invariant stress-energy tensor 8.2 A gauge invariant stress-energy tensor that is not Milne-boost invariant 9 The case of general z 10 Results, interpretation and outlook A Notations B Calculation of the equations of motion at rst order z = 2 B.1 The sound mode B.2 Vector mode B.3 Tensor mode C Calculation of the stress-energy tensor D More on counter terms E Regularity conditions of the gauge eld at the horizon F First order solution for general z G Counter terms for general z { 1 { Introduction The AdS/CFT correspondence [1{4] relates the anti-de Sitter (AdS) space-time to conformal eld theory (CFT) on the boundary. It gives a semiclassical description of strong coupling physics in the dual eld theory in terms of string theory or its low-energy limit: (super) gravity. At nite temperature and in the long wavelength regime, the dual eld theory can be e ectively described by uid mechanics and it can be related to black holes in AdS space-time. The uid/gravity correspondence was rst studied using linear response theory [5{8]. of motion. Subsequently, fully dynamic descriptions were studied using boosted black holes in asymptotically AdS geometries that led to relativistic uid dynamics in the dual CFT [9].1 In this formalism, the uid variables are encoded in the near-equilibrium black hole solution and the uid equations appear as constraints on the solution imposed by the bulk equations Recently, generalizations of the AdS/CFT correspondence to theories with nonrelativistic scaling symmetry have been studied. In particular, many condensed matter systems have critical points with non-relativistic scale invariance [10{14]. Some of these systems have Lifshitz or Schrodinger symmetry [15{17]. Moreover, the hydrodynamics of charge and energy in such systems may be interesting as has been argued recently for the case of cold fermions at unitarity, [18], other strongly correlated systems, [19] and graphene, [20{22]. Holographic techniques have been generalized to geometries with Lifshitz or Schrodinger symmetry in connection with applications to condensed matter systems [23{ 29]. In particular, in [28, 29, 31], all quantum critical holographic scaling theories with a U(1) symmetry respecting translation invariance and spatial rotation invariance were classi ed in terms of three scaling exponents. Two of them (z; ) appear in the metric while another exponent, appears in the pro le of the U(1) gauge eld (it is referred to as in [28, 29, 31]).2 The exponent z is the Lifshitz (dynamical) scaling exponent, and is the hyperscaling-violation exponent, [29, 30]. Even though such theories have been studied intensively many of the aspects are still unclear and, in particular, hydrodynamics with Lifshitz scaling symmetry is not fully understood. More recently, it was found that the boundary theory dual to space-times with Lifshitz asymptotics can be described in terms of the torsional Newton-Cartan gravity theory, which is a novel extension of the Newton-Cartan gravity with a speci c torsion tensor. The application of the Newton-Cartan theory to non-relativistic condensed matter systems (namely the Quantum Hall e ect) was rst discussed in [35]. Interactions between the torsional Newton-Cartan gravity and matter were discussed in [36]. The correspondence between the Lifshitz space-time and boundary torsional Newton-Cartan theory was rst 1A related work was presented in [8]. 2This charge exponent control the anomalous scaling of the charge density, even if it is conserved, has also been introduced independently in [56] and was studied in more detail in [32] and [34]. The reason for the existence of anomalous charge exponent despite conservation is the RG running of the bulk coupling for charged degrees of freedom. { 2 { found in [38, 39] for speci c Lifshitz geometry and further studied in [40{43]. In these works, the correspondence is studied by using the vielbein formalism, in which an appropriate combination of the vielbeins and bulk gauge elds is considered. It turns out to be very useful to use vielbeins to study the boundary theory. This is consistent with the holographic renormalization in the asymptotically Lifshitz space-time, in which the scaling dimension is calculated by using the vielbein [44, 45]. Counter terms in Lifshitz space-time were discussed in generality in [46] by using the Hamilton-Jacobi formalism. The uid/gravity correspondence for non-relativistic uids has been studied in [47, 48] for a special case of the Schrodinger geometry which is related to ordinary AdS by the TsT transformation. In these studies, the non-relativistic uids are obtained by the light HJEP12(05)76 cone reduction of the relativistic uids. The generalization to the charged uid case is studied in [49]. For the Lifshitz space-time, the correspondence to relativistic uids with Lifshitz scaling, in which the velocity eld is de ned by a normalized Lorentz vector, was studied in [50{52]. In these works the anisotropic direction of the Lifshitz symmetry depends on the frame. The uid appears on the surface at nite radius or on the horizon, contrary to the Newton-Cartan theory which appears on the boundary. The hydrodynamics found contains an antisymmetric part in the hydrodynamic stress tensor that contributes a new transport coe cient to the dynamics. In this paper, we consider the uid/gravity correspondence for Lifshitz geometries and the relation to uids in boundary non-relativistic theories with Newton-Cartan symmetry. We consider black holes in Lifshitz space-time with unbroken U(1) gauge symmetry that are solutions of the Einstein-Maxwell-dilaton (EMD) theories. Although, the geometry has Lifshitz scaling symmetry with dynamical exponent z, the bulk solution has \chargehyperscaling violation"3 due to a nontrivial conduction exponent , associated with the gauge eld and the non-trivial running of the dilaton. We consider the black-hole solution of the theory, boost it using Galilean boosts and then we make all parameters of the solution including the velocities, ~x-dependent. We then proceed with the standard analysis introduced in [9]: we solve the bulk equations of motion order by order in boundary derivatives and compute and renormalize the ( uid) stressenergy tensor. We also calculate the entropy current and consider the thermodynamic relations. What we nd is as follows: The standard stress-energy tensor we obtain from the holographic calculation is expressed in terms of the uid variables: velocity eld vi, energy density E and pressure P , but also contains the (particle number) density n and external source Ai associated to the U(1) symmetry current. It satis es the condition for Lifshitz invariant theories zE = (d 1)P . 3This is distinct from what is called hyperscaling violation in condensed matter physics. Our de nition of charge-hyperscaling violation is based on the existence of scaling but also the existence of anomalous scaling dimensions in the charged sector. In particular, although the scaling dimensions of charge density and conductivity are canonical, the scaling of the charge density and conductivity with temperature is controlled by the conduction exponent , [31{34]. { 3 { By comparing the stress-energy tensor and the constraints in the bulk equations of motion, we nd that the conservation law of the stress-energy tensor is di erent from that of relativistic theories but agrees with that in the Newton-Cartan theory. In [50] the stress-energy tensor required a modi cation (improvement) in order to satisfy the trace Ward identity. Our stress-energy tensor satis es the Ward identity without such a modi cation. It is Milne-boost invariant but is not gauge invariant. The role of the (unbroken) U(1) symmetry in this class of theories is important. It should be noted that this U(1) symmetry is responsible for the Lifshitz background bulk solution. We nd that it behaves very closely to the U(1) mass conservation sis in [50, 51]. Even though the continuity equation and energy conservation equation agree with those in the ordinary non-relativistic uids, the Navier-Stokes equation is di erent from that in the ordinary non-relativistic uids. The e ects of pressure become much larger than other contributions and some terms with velocity eld in the Navier-Stokes equation are absent in our result. These absent terms are replaced by external source-dependent terms associated to the U(1) gauge eld. By rede ning the stress-energy tensor and allowing a (Milne-invariant) Newton potential in our sources, [41]{[43] we can map the uid equations to the standard nonrelativistic uid equations coupled to the torsional Newton-Cartan geometry in the presence of a Newton potential. This is a universal result that we nd interesting and far-reaching. Moreover, there is a stress-energy tensor that is both gauge invariant and Milne-boost invariant, but in this stress tensor the momentum density vanishes. There is also an alternative gauge invariant but Milne-boost non-invariant stress-energy tensor which agrees fully with the standard non-relativistic stress-energy tensor. Our uid can be interpreted as a non-relativistic limit of a uid with Lifshitz scaling symmetry.4 In the ordinary non-relativistic limit of uids, the relativistic energy is separated into that from mass and the non-relativistic internal energy. The nonrelativistic internal energy is much smaller than the mass energy, and hence than the relativistic energy. In ordinary non-relativistic uids the pressure is at the same order to the non-relativistic internal energy and hence is much smaller than the relativistic energy density. However, in our case, pressure and the relativistic energy density are at the same order due to the Lifshitz scaling symmetry, and the energy density is not separated into the mass and the others. For non-relativistic uids with Schrodinger symmetry the uid equations are obtained by introducing the light-cone dimensional reduction in [47{49]. Instead, our non-relativistic limit arises naturally as a rather ordinary limit. 4The non-relativistic limit of the Lifshitz uid is also studied in [54, 55]. { 4 { We nd that the form of the uid equations is independent of the Lifshitz exponent z as well as of the (non-trivial) conduction exponent, . It is only the constitutive relations (equation of state) that depend on these scaling exponents. The entropy satis es the local thermodynamic relation with the energy density and pressure. The divergence of the entropy current is non-negative, compatible with the second law. This paper is organized as follows. In section 2, we introduce the model and its solution of Lifshitz space-time. Then, we rst focus on the case with scaling exponent z = 2. In section 3, we introduce the hydrodynamic ansatz. In section 4, we solve the equations of HJEP12(05)76 motion by using the derivative expansion and obtain the solution to the rst order. In section 5, we calculate the stress-energy tensor on the boundary and study its symmetries and the conservation law. In section 6 we introduce the Newton Cartan geometry and realize it in for the boundary uid in question. In section 7, we investigate the entropy and thermodynamic relation. In section 9, we consider the generalization to general z. Section 10 is devoted to conclusion and discussions. Appendix A contains a list of the variables used in this paper and their de nition. It contains also comparison of variables with two other papers in the literature. More details on the calculation for rst order solution and boundary stress-energy tensor are described in appendix B and C, respectively. In appendix D, we consider the analysis of the general counter terms. In appendix E, we discuss the regularity of the gauge eld at the horizon. More details on the solution and counter terms for general z are discussed in appendices F and G, respectively. 2 U(1)-invariant, charge-hyperscaling violating Lifshitz theory We consider a holographic theory with Lifshitz scaling and an unbroken U(1) global symmetry in d space-time dimensions. The dual (d + 1)-dimensional gravity theory will have a massless U(1) gauge eld A and a dilaton . The bulk action is given by S = 1 where F = dA is the eld strength of the gauge eld, and is a dimensionless coupling constant of the bulk theory. The equations of motion are given by R 2F F 1 d 1 e F 2g 0 = r (e F = = d 1 4 2 1 g e F 2 : 1 2 ); 1 4 e { 5 { This model has the Lifshitz geometry as a solution; ds2 = r2zdt2 + dr2 r2 + X r2(dxi)2; i (2.1) (2.2) (2.3) (2.4) (2.5) with the following gauge eld and dilaton; are related to the parameters of the action (coupling studied in holography in [25, 26] and was generalized in [28]. The metric (2.5) has the Lifshitz scaling symmetry t ! czt ; xi ! cxi ; r ! c 1r ; and no hyperscaling violation ( = 0). However, due to the running of the dilaton, the scaling of the AC conductivity is anomalous and its scaling with temperature or frequency is controlled by the conduction exponent , [31{34]. It is de ned from the solution for At,5 At r z : = (d 1) { 6 { We will call this charge-hyperscaling violation. conduction exponent,6 The solution (2.6) is a solution with charge-hyperscaling violation coming from the although the hyperscaling violation exponent coming from the metric vanishes. This model also has a black hole geometry as a solution [26, 28]; ds2 = r2zf (r)dt2 + dr2 f (r)r2 + X r2(dxi)2; i where solution The Hawking temperature of the black hole is given by The gauge eld and dilaton take almost the same form as in the the zero temperature At = a(rz+d 1 0 rz+d 1); e = r2(1 d) ; but At must vanish at the horizon to be regular. 5Note that here we use a radial coordinate that is inverse to the one used in [31]. 6Although the conduction exponent is usually referred to as , here we refer to it as in order to avoid confusion with the bulk viscosity . For a general solution the nite part of the conductivity scales as, [32] 0 r and this is controlled by the conduction exponent, . We observe that the temperature dependence although scaling, does not respect the natural dimension of conductivity. This justi es the name charge-hyperscaling violation for the exponent . In the particular case studied here, in view of (2.12) and (2.17) we obtain In this paper, we focus on the case of d = 4. Extensions to other dimensions are expected to be straightforward. 3 Hydrodynamic ansatz of z. Finkelstein coordinates; In this section, we introduce an ansatz for the geometry which describes the physics of uids in the boundary quantum eld theory. We use the method proposed in [9]. We will also x the Lifshitz exponent to be z = 2. In a later section, we will discuss other values In order for the regularity at the horizon to become evident, we change to Eddingtonwhere the null coordinate t+ is de ned by The gauge eld becomes ds2 = r4f dt2+ + 2rdt+dr + r2(dxi)2 ; dt+ = dt + dr r3f : A = a r 5 r 05 dt+ ar2dr ; (2.17) (2.18) HJEP12(05)76 (3.1) (3.2) (3.3) (3.4) (3.5) where we have xed the Ar = 0 gauge in the Fe erman-Graham coordinates. Hereafter, we always take the Eddington-Finkelstein coordinates and t will stand for the null coordinate t+. To implement the hydrodynamic ansatz, we rst boost the black hole geometry. In the case of the ordinary Schwarzschild-AdS5, the boundary eld theory is a relativistic conformal eld theory, and hence the Lorentz boost is employed. The Lifshitz geometry, however, corresponds to the (torsional)-Newton-Cartan theory, [38]{[43]. Therefore, we perform a Galilean boost on the black hole geometry. The metric becomes ds2 = (r4f v2r2)dt2 + 2rdtdr 2r2vidt dxi + r2(dxi)2 : The gauge eld and dilaton are not a ected by the Galilean boost; A = a r 5 r 05 dt ar2dr ; e = r 6: Now, we replace the parameters r0 and vi by slowly varying function r0(x) and vi(x) of the boundary coordinates x . Moreover, we promote a, and the constant part of Ai (which is usually gauged away) to space-time dependent functions. ds2 = (r4f v2(x)r2)dt2 + 2rdtdr 2r2vi(x)dt dxi + r2(dxi)2 (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (4.1) (4.2) (4.3) (4.4) where Ai(x) comes from the constant part of Ai but now is replaced by functions of x . The above are no longer a solution of the equations of motion, and we must introduce additional correction terms; g A = g + h ; = A + a ; = + ' ; where the background elds g , A and are given by (3.6){(3.9). 4 The rst order solution In order to obtain the rst order solution for the hydrodynamic ansatz, we consider the derivative expansion. Then, the equations of motion can be treated as ordinary di erential equations with respect to r, and the correction terms, h , a and ', can be calculated order by order. The di erential equation can be solved at any given point, and we can take the point to be x = 0 without loss of generality. The parameters which are replaced by slowly varying functions can be expanded around x = 0 as )(0) + ; : ; ; The derivative expansion of the equations of motion gives the linear di erential equations for the correction terms h , a and ' at the rst order. The next-to-leading terms in (4.1){ (4.4) are at the rst order and give the source terms in the di erential equations. Solving the (inhomogeneous) linear di erential equations for the correction terms h , a and ', we obtain the rst order solution in the derivative expansion. (See appendix B for more details.) The integration constants generically modify the source terms near the boundary. These contributions can be eliminated by setting the integration constants that { 8 { modify the sources to zero. The rst order solution for the metric is given by ds2 = r4f dt2 + 2rdtdr + r2(dxi vidt)2 3 + 2 r2@ividt2 + r2F ij (dxi vidt)(dxj vj dt) ; where ij is the shear tensor and the function F (r) is given by F (r) = dr Z r 3 r(r5 r 3 and integration constant is chosen such that F (r) ! 0 in r ! 1. The rst order solution of the gauge eld is A = a(x) r 5 r05(x) 3 a(x)r2dr + Ai(x)(dxi vi(x)dt) ; (4.8) and the dilaton has no correction term, ' = 0. Equations of motion implies that (2.9) and its derivatives must be satis ed even after and a is replaced by functions. The solution above must satisfy the following constraints; 1 Cartan geometry. and 5 Calculation and renormalization of the boundary stress tensor In this section, we consider the stress-energy tensor on the boundary. In order to study the asymptotic behavior, we will use the vielbein formalism as this is adapted to the Newton The leading order term of the induced metric is expressed as where = g n n , and g is given by (3.6) and n is the normal vector to dr = 0 surface. On this surface, dr = 0, the vielbeins are given by = r2zf + r2 abe^a e^b = r 2zf 1 v^ v^ + r 2 abe^a e^b ; dx = dt ; v^ r = rt + viri ; e^a dx = dxa vadt ; e^a r = ra : { 9 { (4.5) (4.6) (4.7) (4.9) (4.10) (4.11) (5.1) (5.2) (5.3) (5.4) where Sr = S + Sct : Sct = 1 The variation of the renormalized action is expressed as Sr = Z ^0 v^ + S^a e^a + J^0 A^0 + J^a A^a + O S : Then, the renormalized boundary theory stress-energy tensor and current are given by , v^ and h = e^a e^a will become the basic geometric data of Newton-Cartan geometry that is discussed in the next section. (1; vi) that depends on the velocity vi we introduced in the solution. We also express the gauge eld in this frame; The formulae above specify the vielbeins in a very speci c frame, = (1; 0), v^ = A^0 = v^ A ; A^a = e^a A : As will see in the next section that holographic data above will correspond to an in nite set of Newton-Cartan data, all related by Milne boosts. Because of this the holographic data above will be \Milne boost invariant". In order to renormalize the expectation values, we have to add the counter terms to (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) T b J T r J r!1 r!1 = lim r5T r ; = lim r5Jr ; = S^0v^ S^ae^a ; r = J^0v^ + J^ae^a : T(BY) = 1 ) ; It should be noted that the stress-energy tensor Tb is not a symmetric tensor. It is also not gauge invariant because we work in the vielbein formalism. As we will see later on, when we introduce Milne-boosts, it will be Milne-boost invariant. In section 8 we will de ne di erent stress-energy tensors with di erent properties under gauge transformations and Milne boosts. The stress-energy tensor (5.9) is related to the ordinary Brown-York tensor as T r = T(BY) + J A + T(ct) ; where T(ct) is the counter term contribution and T(BY) is the Brown-York tensor, which can be expressed in terms of the extrinsic curvature K and J is calculated as J = p = e n F ; where n is the unit normal to the boundary. The renormalized stress-energy tensor is obtained from (5.9), (5.10) as a The above expressions show that the stress-energy tensor Tb contains the gauge eld and hence is not gauge invariant. We will discuss several other de nitions of gauge invariant stress-energy tensors in section 8. 5.1 Energy and momentum conservation We now consider the conservation of the stress-energy tensor. In the Newton-Cartan theory (that is described in more detail in the next section), the conservation law takes a slightly di erent form from standard relativistic cases. It cannot be expressed in a uni ed form in terms of the space-time stress-energy tensor, and we have to introduce the energy vector Eb , momentum density Pb and stress tensor Tb , which are de ned by (5.15) (5.16) (5.17) (5.18) (5.19) (5.20) (5.21) (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) Then, the conservation of energy and momentum is given by (see for example [36, 41, 43]) From the rst order solution, (5.16){(5.19), the energy vector Eb , momentum vector (1form) Pb and stress tensor Tb are given by Eb0 = Pb0 = Tb ij = 3 1 The other components of Tb condition becomes , namely Tb 00 and Tb 0i vanish. The Lifshitz-scaling invariance in terms of the stress-energy tensor Tb , or equivalently z v^ Tb + e^a e^aTb = 0 : zEb0 Tb ii = 0 : As was already noted, the momentum density contains a contribution from the external source A , and hence is not gauge invariant. The external source A dependence originates from the non-standard de nition of the stress-energy tensor. Introducing an appropriate rede nition of the stress-energy tensor, we will obtain a gauge invariant momentum density, which is related to the velocity eld vi. We will discuss this in section 8. 1 2 Using (5.25){(5.27), the energy conservation (5.23) becomes 0 = and its leading order terms give 5 2 which is the constraint (4.10). The momentum conservation (5.24) becomes 1 1 j a 1 1 1 a v j a : (5.32) The leading order terms give 1 a Aj @ivj + 1 1 j a 1 a v j a (5.33) which is a combination of (4.9) and (4.11). The next-to-leading order terms of the conservation law provide the constraints at second order. In order to calculate the solution to second order, we need to introduce the derivative expansion of the correction terms, for example, (5.28) (5.29) (5.30) (5.31) g = g + h(1) + 2h(2) + ; (5.34) where g is given by (3.6) and h(1) is the correction terms which we calculated in the previous section. The expansion parameter is that of the derivative expansion, @ = O( ). In order to calculate the second order solution, we further introduce the second order correction terms h(2). However, as the correction terms do not contribute to the constraint at rst order, these second order correction terms do not contribute to the constraint at the second order. Therefore, we do not need to take h(2) into account to study the second order constraints. The background metric g does not consist only of O( 0) contributions, but also contains higher order corrections. The higher order corrections in g are included in x-dependent parameters r0, vi, a, and Ai. They can be expanded as where r0(0), etc. are the leading order terms which we studied in the previous section, and satis es the constraints at the rst order. The higher order terms, for example r0(1), do not contribute the rst order terms, but must satisfy the second order constraints. After some algebra, the second order constraint equation for the gauge eld gives Together with the rst order constraint, it can be expressed as where is the heat conductivity and is the shear viscosity whose values are Eb0 = E ; Ebi = E v i Pbi = nAi ; Tb ij = P ij ij ; = 1 3 8 G r0 ; = 1 3 16 G r0 : (5.35) (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) (5.43) (5.44) (5.45) (5.46) (5.47) (5.48) This implies that there are no additional terms in this constraint at second order. In a similar fashion, from the spatial component of the constraints in Einstein equation we obtain 0 = 1 1 j a 1 1 From the temporal component, we obtain (5.42) agrees with (5.30), and an appropriate combination of (5.40) and (5.41) gives (5.32). From the stress-energy tensor and the current, we can read o the energy density E , charge density n and pressure P as E = 3 In terms of these quantities, the energy ow, momentum density and stress tensor are expressed as The Lifshitz invariance condition now becomes the following form; In terms of the above uid variables and transport coe cients, the uid equations take ij ij Eq. (5.58) can also be expressed as where the current is given by v^ is de ned in (5.4) and the eld strength is de ned as F = dA with In terms of the temperature (note that here z = 2) uid variables and transport coe cients can be expressed as and The scaling dimension under the Lifshitz scaling is given as T = 4 = nv^ : A = Ai(dxi vidt) : 1 2 1 2 (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) (5.55) (5.56) (5.57) (5.58) (5.59) (5.60) (5.61) (5.62) (5.63) (5.64) (5.65) HJEP12(05)76 For comparison, the ordinary non-relativistic uid equations are given by ij ij where (5.63) gives conservation of energy, the Navier-Stokes equation (5.64) comes from conservation of momentum, and the continuity equation (5.65) implies the conservation of mass density. Eqs. (5.57) and (5.59) agree with the energy conservation (5.63) and continuity equation (5.65), respectively. However, (5.58) is di erent from the Navier-Stokes equation (5.64). Before we proceed further we must clarify the role of Newton Cartan theory. The Galilei data ( ; h ) are constant under the covariant derivative; HJEP12(05)76 r = 0 ; r h = 0 : Contrary to Einstein gravity, (6.2) does not uniquely x the Galilei connection. In order to determine the connection, we must introduce a contravariant vector v (not to be confused with velocities) and a two-form B . The vector v satis es the normalization condition By using the vector v , we also de ne the spatial covariant (symmetric) metric h , which satis es h v = 0 ; h h = P v : Then, the Newton-Cartan connection is expressed as In general, the Newton-Cartan connection (6.5) has torsion, The curvature is de ned via the commutator of the covariant derivative and given by Newton Cartan theory and Milne-boost invariance We brie y review here the Newton-Cartan theory [64, 65]. We rst introduce the Galilei metric of d-dimensional Galilei space-time, which consists of a 1-form and a contravariant symmetric tensor h of rank (d 1). The 1-form de nes the time direction of the Galilei space-time, and h gives the spatial inverse metric. They satisfy the orthogonality condition, It should be noted that v has no relation to the uid velocity, and is in general, di erent from the uid velocity vector v^ , although v is referred to as the velocity eld sometimes in the Newton-Cartan literature. Here, v is the inverse timelike vielbein to be distinguished in general from the uid velocity eld. 1 2 h (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) (6.7) (6.8) 1 2 R If we impose the Newtonian condition gauge eld; B = dB. are (h ; ; v ; B ). where [ ] and ( ) in the indices stand for the antisymmetric part and symmetric part, respectively, we obtain the condition dB = 0. Then, B is (locally) the eld strength of a To summarize the Newton-Cartan data that determine a given Newton-Cartan frame h However di erent Newton-Cartan data may describe the same physics. To see this we introduce the concept of the Milne boost (see [66]), which is an internal symmetry of the Newton-Cartan theory. Here, we focus on the torsion free cases, since for our solution on the gravity side, (5.1), (5.2) we have = (1; 0) and this give zero torsion in (6.6).7 We introduced v and B to de ne the Newton-Cartan connection. Two pairs (v ; B ) and (v0 ; B0 ) are physically the same if they give the same Newton-Cartan connection. The torsionless part of the Newton-Cartan connection is invariant under the following (Milne boost) transformation; v ! v0 = v + h V ; B ! B0 = B + P V dx 1 2 h V V dx ; h0 = h ( P + P )V + h V V (6.9) (6.10) (6.11) HJEP12(05)76 where V is a vector which parametrizes the Milne-boost transformation. It should be noted that all non-trivial degrees of freedom of v can be absorbed into B by using the Milne boost. The normal direction to the timeslice is xed by the normalization condition v = 1 and the other directions are freely transformed by the Milne boost (6.9). Therefore, we may choose an arbitrary but appropriately normalized inverse timelike vielbein v . Now, we will relate the Newton-Cartan data to the vielbeins (5.3) and (5.4) which we introduced in the induced metric on the boundary (5.1) and (5.2). The timelike vielbein is simply identi ed to that in the Newton-Cartan theory. The inverse spacelike vielbein e^a should also be identi ed with that in the Newton-Cartan theory, which implies that the inverse spatial metric in Newton-Cartan theory is expressed in terms of e^a as h = e^a e^a : (6.12) In the frame we use it is the unit matrix in the spatial directions. For the leftover NewtonCartan data (v ; B ) there are many choices related by Milne boosts. By using the Milne boost (6.9), we can choose a special \frame" in which the inverse timelike vielbein equals the uid velocity v0 = v^ .8 We will call this Newton-Cartan frame the \holographic frame" from now on. It remains to identify the gauge eld B in this frame. This is facilitated by comparing our equation in (5.60) with the one derived in [36] in Newton Cartan theory (equation (5.17) of that paper). This gives the following identi cation, B = A with A = ( viAi; Ai) ; Bt = 7The Milne boost in torsional cases is discussed in [36, 37]. 8Here, \frame" is di erent from the coordinate frame but a special point in the internal space of the Milne boost symmetry. No coordinate transformation is needed to take this \frame". where Ai appears in (5.16){(5.19). We conclude, that in the \holographic frame" the Newton-Cartan data are A 0 0 0 1 v ! v^ = (1; ~v) (6.14) together with (6.13). In any other frame, and h remain invariant, but v and B change by the Milne boosts, (6.9) We would like now to change to a more canonical ( at) frame that is appropriate for HJEP12(05)76 non-relativistic physics in particular the standard Navier-Stokes equation. We will call this the \Newton frame" and it is determined by v = (1; ~0). We will go from the holographic frame to the Newton frame by a Milne boost with parameter V = (0; ~v). In the Newton frame we therefore obtain a new gauge eld that we call Ae using (6.9) It should be noted that v^ does not transform under the Milne boost. The inverse vielbein v transforms under the Milne boost, but the uid velocity v^ is Milne-boost invariant. We will de ne also a class of gauge elds that are Milne-boost invariant. It is simple to show that for any Milne-invariant vector X , that is normalized: X = 1, the following gauge eld Bb = B + h X dx h X X dx ; is Milne-boost invariant as can be directly veri ed using the transformations in (6.9){(6.11). We choose as such a vector the uid velocity vector, X = v^ , which satis es v^ = 1, to de ne the invariant gauge eld as in (6.16) Binv = B + h v^ dx h v^ v^ dx : 1 2 1 2 We can evaluate Binv in the Newton frame (6.15) to nd that 1 2 Binv = Ae + h v^ dx h v^ v^ dx = Ae + vidxi 2 1 v2dt = A : As Binv is Milne-boost invariant its evaluation in the holographic frame, will also give the same result. We conclude that the gauge eld A in the holographic frame, given in (6.13) is Milne-boost invariant. Next, we consider the Navier-Stokes equation. We have already found the NavierStokes equation in the holographic frame in (5.58) to have the form Fi J where F = dA and where as we have shown above, all quantities that enter are Milneboost invariant. However this does not look like the usual non-relativistic Navier-Stokes (6.15) (6.16) (6.17) (6.18) (6.19) equation (5.64) because we are not in the Newton frame. To do this we must rewrite it using the gauge eld Ae in the Newton frame, (6.15). We directly compute using (5.58), (5.61) and (6.15). Substituting this equation in (6.19) we obtain Fei J which is the conventional Navier Stokes equation albeit in the presence of an external force. To bring this equation to an even more familiar form we follow [43], and choose the gauge eld A such that This implies Then, (6.21) is expressed as v^ = t + h Aet = A This expression agrees with the Navier-Stokes equation with the external force Fi; the force being the gravitational force from the Newton potential Fi = n@i e. Therefore, (6.21) can be interpreted as the Navier-Stokes equation in the non-trivial Newton potential; e = Aet = fected by the Newton potential. The conservation of total energy is obtained by appropriate combination of (5.57) and (6.21), and can be expressed in terms of the Newton potential (6.26) as ne E + P + ne v i ij v j This is consistent with the energy conservation of uids in the Newton potential; for timeindependent Newton potential @t e = 0, the energy conservation is expressed as ne E + P + ne v i ij v j (6.28) The stress-energy tensor (5.16){(5.19) is now rewritten as 1 2 nv2 1 2 nv2 Tb0i = nvi ; Tbij = P ij E ne + E + P ne + 1 2 nv2 ; 1 2 ij + nvivj : 1 2 nv2 1 2 nv2 (6.20) (6.21) (6.22) (6.23) (6.24) (6.25) (6.26) (6.29) (6.30) (6.31) (6.32) This is the ordinary stress-energy tensor for non-relativistic uids with the Newton potential terms. The conservation of total energy (6.27) and Navier-Stokes equation (6.24) can be expressed in terms of this stress-energy tensor as 7 The entropy current The holographic entropy current JS is de ned by the dual of the (d 1)-dimensional volume form on the time slice on the horizon; The entropy current JS can be expressed in terms of the normal vector n as 1 d JS 1 dx 2 ^ ^ dx d : J S = p h n 4G n0 ; T = 4 r0 ; (6.33) (7.1) (7.2) (7.3) (7.4) (7.5) (7.6) (7.7) (7.8) where the normal vector at the horizon is given by to rst order in the derivative expansion.9 is obtained as S = r r0(x) ; JS0 = J Si = Then, the holographic entropy current for the Lifshitz space-time with d = 4 and z = 2 Comparing this expression with (5.25) and (5.43), we nd that the entropy current satis es T JS = Eb + P v^ = T b v^ + P v^ ; where the Hawking temperature T is given by to rst order in the derivative expansion. of the entropy current is calculated as We can easily check that the entropy current satis es the second law. The divergence S = 1 4G 1 12G 9The horizon radius has corrections at higher order in the derivative expansion. Since these correction terms appear from the second order, it does not contribute to the entropy current at the rst order. The uid has the following properties; The stress-energy tensor has a form similar to that of non-relativistic uids, and is expressed in terms of the uid variables: the velocity eld vi, the energy density E , the pressure P and the charge density n. It also contains the external gauge eld A. At rst order, it has as transport coe cients the thermal conductivity and shear viscosity while the bulk viscosity is zero due to the Lifshitz scaling symmetry. All the above variables and transport coe cients are functions of temperature T . The (particle number) density appears associated to the external gauge eld. The stress-energy tensor calculated directly from the bulk solution agrees with that of nonrelativistic uids except for the terms where the density appears. The terms which contain the density are di erent from those of a non-relativistic uid. In particular, the momentum density T 0i is proportional to Ai and vanishes for Ai = 0. The conservation law of the stress-energy tensor is not given in the form of its covariant derivative, but is that appropriate to a boundary Newton-Cartan theory. It is expressed in terms of the energy ow E , momentum density Pi and stress tensor T ij . The stress-energy tensor satis es the Ward identity of the scale invariance 1)P without any modi cation. In terms of the uid variables, the conservation law takes the form of the uid equations: the energy conservation, continuity equation and the Navier-Stokes equation. The uid equations are similar to the non-relativistic uid equations. The continuity equation and energy conservation equation agree with that for standard nonrelativistic uids. However, the Navier-Stokes equation is di erent from that for ordinary non-relativistic uids. It does not contain the velocity eld except the terms appearing in the shear tensor. It contains the coupling to the external gauge eld instead. A related property of the associated stress tensor is that it is gauge non-invariant but Milne-boost invariant. The absence of the velocity elds in our Navier-Stokes equation can be explained as follows. For an ordinary uid, the pressure is comparable to the non-relativistic energy which does not include the mass energy, and hence it is much smaller than the relativistic energy density. In our case, the pressure is of the same order as the energy density because of the Lifshitz scaling symmetry. For this to happen the uid equations must be di erent and this is what we nd, namely, the contribution from pressure is much larger than that in ordinary uids, and then, the terms with velocity elds becomes negligible compared to the pressure. This is the reason that some terms with velocity elds in the Navier-Stokes equation are absent. These terms are replaced by the external source which is identi ed with the gauge eld in the Newton-Cartan theory. We may do some rede nitions of the stress-energy tensor and by identifying the gauge eld A to the gauge eld in Newton-Cartan theory, which takes the form of A = Ae + vidxi 1 v2dt our Navier-Stokes equation agrees with the standard nonrelativistic Navier-Stokes equation. In such a case the stress-energy tensor is gauge invariant but Milne-boost non-invariant. Finally there is a de nition of the stressenergy tensor that is both Milne-boost and gauge invariant. Since the gauge eld in the Newton-Cartan theory is a generalization of the Newtonian gravity theory, a general external gauge eld gives a uid in a non-trivial gravitational potential. The Newton potential appears in the gauge eld as e = Aet. The Navier-Stokes equation we obtain agrees with the ordinary Navier-Stokes equation in the presence of an external (gravitational) force. The gauge eld does not contribute to the conservation of (internal) energy density E . The conservation of total energy can be obtained from the conservation of E and the Navier-Stokes equation and agrees with that for ordinary non-relativistic uid in a non-trivial Newton potential. The entropy density is de ned in terms of the horizon area and satis es the local thermodynamic relation with energy density and pressure. The divergence of the entropy current is non-negative, which is consistent with the second law. The form of the uid equations is independent of the Lifshitz exponent z as well as of the conduction exponent . This dependence appears at rst order, inside the various state functions and therefore only in the constitutive relations. There are several obvious interesting questions that remain unanswered by our work. The rst is the extension of our results to hydrodynamics in the presence of hyperscaling violation in the metric ( 6 = 0). This is under way. A naive guess would be that the hydrodynamics would be a dimensional reduction of the one found here, along the lines described in [29, 59]. In particular in [29] it was shown that Lifshitz solutions with hyperscaling violation can be obtained as suitable dimensional reductions of higher-dimensional Lifshitz invariant theories without hyperscaling violation. The associated reduction of the hydrodynamics will provide equations similar to the ones here but with a non-zero bulk viscosity. This needs to be veri ed. actively pursued in [60]. A further extension involves Lifshitz geometries with broken U(1) symmetry. This is An interesting question in relation to the above is: what is the appropriate hydrodynamics for QFTs that are RG Flows that interpolate between relativistic and nonrelativistic theories. To motivate the answer to this question, we consider rst non-Lorentz invariant (but rotationally invariant) ows between Lorentz invariant xed points,11 [62], but where the velocity of light in the IR is di erent for that in the UV. In such a case, the hydrodynamics of this theory, is relativistic, but with a speed of light that is temperature dependent. This example suggests that in an (Lorentz-violating) RG ow from a CFT (with an unbroken U(1) symmetry that is used to drive the breaking of Lorentz invariance) to an 11The fact that the speed o light can vary on branes was pointed out rst in [61]. of the associated theories. necessary. Acknowledgments IR non-relativistic scaling (rotational invariant) geometry at an arbitrary temperature, the hydrodynamics will be again of the relativistic form (but with a general equation of state) and with a speed of light c(T ) that is again temperature dependent. In the IR, c(T ! 0) = 1 and the hydrodynamics reduces to the one found here with the U(1) symmetry becoming the mass-related symmetry. This is nothing else than the standard non-relativistic limit12 of the the relativistic hydrodynamics while all thermodynamic functions and transport coe cients are smooth functions of T (if no phase transition exists at nite T ). Otherwise they follow the standard behavior at phase transitions. A more general breaking of Lorentz invariance during a RG ow must involve higher form elds of tensors in the bulk, or a multitude of vector elds and the details of the RG ow become complicated. It is important that such ows are analyzed as they hold the key to understanding general non-relativistic ows as well as generalized hydrodynamics Finally a more detailed study of the above issues in the absence of U(1) symmetry is We would like to thank A. Mukhopadhyay for discussions. We especially thank J. Hartong and N. Obers for participating in early stages of this work and for extensive and illuminating discussions on the topics presented here. This work was supported in part by European Union's Seventh Framework Programme under grant agreements (FP7-REGPOT-2012-2013-1) no. 316165, the EU program \Thales" MIS 375734 and was also co nanced by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program \Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) under \Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes." A Notations Since in the topic treated in this paper there are a lots of variables involved and many rede nitions, we present here a list of the variables and their de nitions with comments when necessary. We also present a translation dictionary to the variables used by Jensen, [37] and Hartong et al., [43]. In table 1, we present the relation between the variables in this paper and those in [35] and [41]. In table 2, we present the correspondence of the uid variables in this paper and those in [35]. Variables de ned on the gravity side. d is the dimension of the space-time boundary, and the dimension of the boundary 12This is expected to happen along the lines presented in [63] although this needs to be veri ed. vi: boost parameter introduced into the (static) black hole geometry in (3.4). r0: the horizon radius which is de ned in (2.14). a: the coe cient of the rz+d 1 term of the gauge eld, introduced in (2.6). Note that a 6= a dx . A : the constant part of the gauge eld, which is de ned in (3.8). This corresponds to the Milne boost-invariant gauge eld in the Newton-Cartan theory Binv, or equivalently, B in the holographic frame v = v^ . In this paper, it is sometimes expressed as the 1-form A = A dx . v^ : the timelike inverse vielbein on the boundary which is de ned up to the factor r 2zf 1, or equivalently, de ned in (5.2) and (5.4) and given by v^ = (1; vi) in our model. This corresponds to the velocity vector eld of the uid. The holographic frame of the boundary Newton-Cartan geometry is de ned by v = v^ . : the timelike vielbein on the boundary which is de ned up to the factor of r2zf , or equivalently, de ned in (5.1) and (5.3) and given by = (1; 0) in our model. This corresponds to the timelike unit normal which de nes the time direction in the Newton-Cartan theory. It is automatically invariant under the Milne boost. e^a : the spacelike vielbein on the boundary which is de ned up to the factor of r2, or equivalently, de ned in (5.1) and (5.3), and given by ea dx = dxa vadt in our model. This is invariant under the Milne boost and equals to the spacelike vielbein in the Newton-Cartan theory if we take the holographic frame v = v^ . e^a : the spacelike inverse vielbein on the boundary which is de ned up to the factor of r 2, or equivalently, de ned in (5.2) and (5.4), and given by ea @ This corresponds to the spacelike inverse vielbein. It is automatically invariant under the Milne boost. Variables in the (boundary) Newton-Cartan theory. v : the timelike inverse vielbein in Newton-Cartan theory. This notation is introduced above (6.3) to de ne the Galilei connection. The timelike inverse vielbein is not invariant under the Milne boost but it is covariant. In the literature it is sometimes called \velocity" but must be distinguished from the velocity of the uid. h : the induced covariant metric on the time-slice. It is de ned by (6.4) and given by h = diag(0; 1; 1; 1) for a 4-dim space-time in the Newton frame. It is not invariant under the Milne boost but it is covariant. ea : the spacelike vielbein, which is introduced in (8.37){(8.38). It is given by ea = diag(0; 1; 1; 1) for a 4-dim space-time. It satis es h = ea ea. ea : the spacelike vielbein, which is introduced in (8.37){(8.38). It is given by ea = diag(0; 1; 1; 1) for a 4-dim space-time in the Newton frame. Since it is invariant under a Milne boost, it is equal to e^a . h A M v^ u v^ e^ a { e^ a A ( e; 0) n T 0 e At Mt v u i i Mi from the relation of the metric and gauge eld in gravity side. In [43], v^ is de ned such that spacial components of the associated Milne invariant gauge eld vanish. We can also de ne such velocity eld in our notation, v^ Ab . If we identify this combination to v^ in [43], we obtain another correspondence. In this case, no variables in [43] correspond to the Milne boost invariants in this paper. B : the gauge eld in the Newton-Cartan theory. This notation is introduced below (6.8). It is not invariant under the Milne boost but it is covariant. Ae : the gauge eld in the Newton-Cartan theory in Newton frame v = (1; ~0). This notation is introduced in (6.15). It is not invariant under the Milne boost but it is covariant. Bb: a Milne boost invariant combination for the gauge eld. It is de ned in (6.16), with arbitrary but appropriately normalized Milne-invariant vector X . Binv: the Milne boost invariant combination Bb for X = v^ . It is de ned in (6.17). vidxi In terms of the gauge eld in the Newton frame Ae, it is expressed as Binv = Ae + 1 v2dt. It also equals to the gauge eld in the holographic frame A. 2 VEVs and uid variables. T(nr) : the stress-energy tensor without counter terms on dr = 0 surface. To be exact, the stress-energy tensor is the coe cient of O(r 5) term of this tensor (O(r z 3 for general z). It appears in (C.7){(C.10). Tr : the renormalized stress-energy tensor on dr = 0 surface or its regular part in the section in which we are discussing only on the boundary uid. This is introduced in (5.9) and the boundary stress-energy tensor is given by the coe cients of O(r 5) terms. T b : the boundary stress-energy tensor, or its regular part in the section in which we are discussing only the boundary uid. This is de ned in (5.9). T T : it is de ned by (8.15), Tb = T + J A J A . This de nition is used to de ne Milne-boost-invariants for a general background of A. It is both Milne-boost invariant and gauge invariant. : de ned by (8.30), T = Tb J Ae + J Ae gives the standard non-relativistic uids' stress-energy tensor. It consists of the physical energy vector, physical momentum density, and physical stress tensor. It is gauge invariant but not Milne-boost invariant. J : the current without counter terms de ned in (5.10). It is regular even without the counter terms. It corresponds to the mass current in a non-relativistic theory. Eb : de ned by T b vector. It is not gauge invariant. v^ in (5.20). It corresponds to the (Milne boost invariant) energy Pb : de ned by Tb e^ae^a in (5.21). It corresponds to the (Milne boost-invariant) momentum density. It is di erent from the physical momentum density P , which is not invariant under the Milne boost. It is not gauge invariant. Tb : de ned by Tb (e^ae^a )(e^b e^b ) in (5.22). It corresponds to the (Milne boost invarivector. It is gauge invariant. T v^ in (8.17). It corresponds to the (Milne boost invariant) energy P : de ned by T e^ae^a in (8.18). It corresponds to the (Milne boost-invariant) momentum density. It is di erent from the physical momentum density P , which is not invariant under the Milne boost. It is gauge invariant. : de ned by T (e^ae^a )(e^b e^b ) in (8.19). It corresponds to the (Milne boost invariT v in (8.36). It corresponds to the physical energy vector, which T T E : de ned by P : de ned by T ant) stress tensor. It is gauge invariant. contains a contribution from the mass density. eaea in (8.37). It corresponds to the physical momentum density, which contains a contribution from the mass density. : de ned by T (eaea )(eb eb ) in (8.38). It corresponds to the physical stress tensor, which contains a contribution from the mass density. is introduced in (5.43) and equals E 0. E : (Milne boost invariant) energy density, or equivalently, internal energy density. It P : the pressure. It can be read o from the stress tensor and introduced in (5.43). n: it is de ned by 1=a. It is introduced in (5.43). It corresponds to the particle number density, or equivalently, the mass density. T : the temperature. It can be calculated as the Hawking temperature of the black hole (2.15). S case. time-slice at the horizon. J : the entropy current, which is de ned from the volume form in (7.2) on the s: the entropy density, which is de ned in (7.10). It equals to JS0 in the non-relativistic E T variables do not contain contributions from the external source. energy tensor, or the uid equations. stress-energy tensor, or uid equations. e: the Newton potential, which is e = Aet. It is introduced around (6.24). : the shear viscosity. It is introduced in (5.47) and can be read o from the stress: the heat conductivity. It is introduced in (5.45), and can be read o from the elds and constants in gravity side. g : the metric. It is introduced in (2.1). A: the gauge eld. It is introduced in (2.1). : the dilaton. It is introduced in (2.1). G: the Newton constant. It is introduced in (2.1). A^: the gauge eld with local lorentz indices. It is de ned in (5.5). J^: the current with local lorentz indices. It is de ned in (5.8). h : the correction terms for the metric. It is introduced in (3.10). a : the correction terms for the gauge eld. It is introduced in (3.11). ': the correction terms for the dilaton. It is introduced in (3.12). R : the Ricci tensor. It appears rst in (2.1). : the cosmological constant. It appears rst in (2.1). T (bulk): the energy-momentum tensor in the bulk. It appears in (B.11). T(BY): the Brown-York tensor which is de ned by (5.14), T(BY) = 8 1G ( K K ). n : the normal vector to the boundary or horizon. It appears rst in (5.15). K : the extrinsic curvature on the boundary. It appears in (5.14). To be precise, it is de ned on constant but nite r surface. : the induced metric on the boundary. It is introduced in (5.1). To be precise, it is de ned on constant but nite r surface. Calculation of the equations of motion at rst order z = 2 In this section, we calculate the rst order solution in derivative expansion around (3.6){ (3.9). The coordinate can be chosen such that vi(x) = 0 at any given point, therefore we may take vi(0) = 0 without loss of generality. Although we will work in vi(0) = 0 coordinates, the solution for vi(0) 6= 0 can be obtained by boosting the solution uniformly. From now on we work at the point x = 0, but we omit the subscript \(0)", hereafter. Here, we take the following gauge conditions; grr = 0 ; gr / v^ ; Tr[g 1h] = 0 ; ar = 0 ; (B.1) The correction terms can be classi ed by using the SO(3) symmetry along the spatial directions. The equations of motion are separated into that for scalar (sound mode), vector mode and tensor mode. The sound mode consists of the following components and vector mode and the tensor mode is the traceless part of the metric htt htr hii at ' ; hti ai ; hij : To simplify the di erential equations, we rede ne the correction terms for the metric as follows htt htr hti hxx hij gtt = r4( f + htt); gtr = gti = r2hti 2 htr; gij = r2( ij + hxx ij + hij ) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) where hxx is the trace part and hij is the traceless part in xi-components. The gange condition gives additional constraints; and the (r; r)- and (r; i)-components of the correction terms must vanish. We de ne The other correction terms a and ' are similar to the de nitions (3.11) and (3.12), but ar is eliminated by the gauge condition. 2htr + 3hxx = 0 h1 = 1 2 hxx : Some components of the equations of motion do not become di erential equations for the correction terms, but give the constraints on the parameters. From the Einstein equation we obtain n R = 8 Gn T (bulk) where n is the normal vector and is the induced metric on r =const. surfaces. In fact the above equation contains no correction terms. For the sound mode, by contracting the (B.11) with v^ = (1; vi), we obtain the following constraint; 0 = p = z + d 2(z 1 1) 1=2 : Equation (B.12) must be satis ed for arbitrary r, and hence the rst and second terms must vanish independently, From (2.3), the r-component does not contain correction terms and gives another where constraint; where the rst equation originates from (2.4) and the second from the t-component of (2.3). The third to fth equations are the (t; t)-, (t; r)- and (r; r)-components of (2.2). The last After substituting the above constraints to the equations of motions for the sound mode, we obtain the following di erential equations; 0 = 2p6a0t(r) a 0 = ra0t0(r) 0 = 6p6r5h01(r) + 6p6r05h01(r) + p6r5h0tt(r) + 5p6r4htt(r) + r6'00(r) rr05'00(r) + 6r5'0(r) r05'0(r) + 30r4'(r) 2p6r2@ivi 4a0t(r) + 30ar5h01(r) + 5p6a0r5'0(r) + 3ar6h0t0t(r) + 30ar5h0tt(r) + 60ar4htt(r) + 20p6ar4'(r) 12ar2@ivi + 8a0t(r) 72ar5h01(r) 18ar05h01(r) 240ar4h1(r) 0 = 3a + 8a0t(r) + 20p6ar4'(r) 240ar4h1(r) + 60ar4htt(r) 0 = 0 = 12h01 + p6'0 3rh010 2a0t(r) ar2 3r4h010(r) + 3r05h010(r) r 36r3h01(r) + 21r05h01(r) r2 120r2h1(r) (B.11) (B.12) (B.13) (B.14) (B.15) (B.16) (B.17) (B.18) (B.19) (B.20) (B.21) (B.22) HJEP12(05)76 (10), and at(0) are integration constants and h2 is given by h2(r) = 5r04 0 q 37 2 5 1 2 0 ; 2; r5 A 5r4 C2 G23;;13 r 5 4 p185 ; 110 13 + p185 equation is the trace part of the spatial component of (2.2). The above equations are not independent but an appropriate combination gives the constraints, which we have already imposed, and hence becomes trivial. We rst impose the constraints to the parameters and then solve the di erential equations. The solution for the sound modes is htt = p 2 6 5 5 1 3 h1 = 5 0 r5 1 6r05 r5 1 0 r5 1 r5 h02(r) 4 + 0 5 r5 1 a 15h(10) + p6r5 (0) 15r5h2(r) p r56 h(10) + 3 r 3 2 h2(r) + r 3 2 rh02(r) (B.23) (B.24) (B.25) (B.26) (B.27) (B.28) (B.29) (B.30) (B.32) (B.33) where pFq and Gpm;q;n are hypergeometric function and Meijer G-function, respectively [58], and C1 and C2 are integration constants. B.2 Vector mode As for the sound modes, spatial component of (B.11) gives a constraint a) : Since this constraint must be satis ed at arbitrary r, the rst and second terms must vanish independently, Then, the equations of motion for the vector modes are 0 = r 0 = r2 5 5 0 r 5 r 05 a0i0(r) + 5ar7h0ti(r) + 7r05 + ar2 r 5 0 r 5 h0t0i(r) + 4ar r 5 0 2r5 a0i(r) (B.31) r 5 h0ti(r) 2 r 5 r 05 a0i(r) 0 = 2a0i + 4ar3h0ti + ar4h0t0i where the rst equation is the xi-component of (2.3) and the others are the (t; xi)- and (r; xi)-components of (2.2), respectively. The solution for the vector modes is hti = Z dr ht(i0) 3r05 Z 2 2 a ai + 2r3 where ht(i0) and ai(0) are integration constants. The function a1 is given by and is expanded around r = r0 as In order for the solution to be regular at r = r0, we we must take and for the rest we obtain a1 = 3C3 375r07(r C3 = hti = ai = ai(0) r5 r3 0 a2 ht(i0) + ai(0) r5 + r 2 0 3 5 Tensor mode The equation of motion for the tensor mode is given by 0 = 2( 6r5 + r05)h0ij + 2( r6 + rr05)h0i0j There are no constraints for the tensor mode. The solution is where hij = ij Z r2dr r5 r 0 5 + C4 Z dr r(r5 r05) 2 Regularity at r0 implies C4 = r03. We nally obtain hij = ij Z (r3 r(r5 (B.34) (B.35) (B.36) (B.37) (B.38) (B.39) (B.40) (B.41) (B.42) (B.43) (B.44) 8 GT(BY)00 = 3 8 GT(BY)i0 = 8 GT(BY)0i = 8 GT(BY)ij = 4 ij + 3 2r7 vi + 1 1 2r5 4r05vi + @ir05 + O(r 7) ; 2r5 r0 ij + O(r 7) ; 3 16 GJ 0 = 16 GJ i = 2 1 ar5 + ar10 r05 + O(r 11) ; vi + ar10 r05vi + O(r 11) : to the rst order in the derivative expansion. The current is given by From (5.13) we can obtain the non-renormalized part of Tr as 8 GTr(nr)00 = 4 + 2r05 8 GTr(nr)i0 = 8 GTr(nr)0i = 1 r5 1 8 GTr(nr)ij = 4 ij + 2 0 1 ar5 Ai + O(r 6) ; 1 viAi + O(r 6) ; + O(r 6) ; 1 r5 21 r05 ij 1 3 1 viAj + O(r 6) : the terms of O(r 5) become nite. Those of O(r0) diverge at the boundary, r ! 1, but can be subtracted by introducing the boundary cosmological constant term. We can further introduce a boundary counter term proportional to A2; Sct = 1 16 G Z d x 4 p This induces a counter term for the stress-energy tensor 8 + C + Ce Calculation of the stress-energy tensor Here, we calculate the stress-energy tensor. For the solution (4.5), the Brown-York tensor (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) 1 r5 Since the volume form on the boundary behaves as r5 ; 1 1 5 2 1 2r5 0 r 5 8 GTr(ct) (4 C) + C The coe cient C in the counter term can be xed by the regularity condition for the operator dual to the dilaton scalar . The vacuum expectation value of the operator O is C 5 16 G r0 ; P = 1 + C 5 16 G r0 : O = lim r5Or r!1 Or = p 1 Sr = n r : Or (nr) = 1 16 G 6 r 3 r05 ! 2 r5 + O(r 6) : (C.14) (C.15) (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) (C.22) (C.23) (C.24) (C.25) (C.26) (C.27) HJEP12(05)76 Subtracting the counterterm, the renormalized stress-energy tensor becomes 1 1 1 1 1 5 0 1 viAi ; 2 0 1 vivj Aj + 1 3 ; (1 + C)r05 ij 1 3 1 viAj t : From the equations above we can read the energy density E and pressure P Tb00 = Tb0i = Tbij = r05 ij 3 5 2 0 r 1 viAi ; 2 0 1 a 1 vivj Aj + 1 3 ; 1 viAj ; For our rst order solution, the non-renormalized expectation value is calculated as The regular term is at O(r 5) while the counter terms becomes Or (ct) = 1 5 16 G 2 C e B B = 1 16 G 6 C + p 5 6 C r05 + O(r 6) : The renormalized expectation value is Or = 1 16 G 6 (1 C) r 3 2 (1 2C) r05 r5 ! + O(r 6) : To obtain a nite O , we must take C = 1, as the rst term is divergent at the boundary. We obtain and Moreover the renormalized stress tensor (5.16){(5.19) satis es O 1 16 G 1)P : D More on counter terms We can also consider higher order terms of A for the counter terms. Due to the constraint on a and , A always appears with the factor of e =2. Then, general counter terms is d x 4 p X c 5 2 e A A ; where the boundary cosmological constant term is xed such that the boundary stressenergy tensor becomes nite. Then, the stress-energy tensor becomes Tbij = 1 1 8 G a Ai ; 1 2 + 2 1 X c 2 0 1 0 1 viAi ; 1 vivj Aj + 1 3 j 1 + X c 1 3 The dual operator to the dilaton is calculated from Or = 1 16 G 6 1 X c 1 X c ! r5 # 0 r5 + O(r 6) : In order to regularize O = limr!1 r5Or, the coe cient c must satisfy (C.28) (C.29) (D.1) (D.2) (D.3) (D.4) (D.5) (D.6) (D.7) Then, the stress-energy tensor is same to (5.16){(5.19). E Regularity conditions of the gauge eld at the horizon If the guage eld A has non-zero At at the horizon, it becomes singular at the horizon. It can be easily seen by taking the wick rotation and by considering the Polyakov loop wrapping on the time circle. Although the horizon is a point in the imaginary (euclidean) time, it becomes two surfaces, future and past horizon in real time. In this section, we show that the singularity appears only in the past horizon even for At 6= 0, if we take the Eddington-Finkelstein coordinates. Here, we focus on the near horizon region and discuss about the regularity of the gauge eld at the horizon. In the near horizon region, the metric of the non-extremal black holes is universally given by the Rindler space; ds2 = rdt2 + + (dxi)2 ; where r = 0 is the horizon of the black hole. Only outside of the horizon is covered by this coordinates. In order to move to the Eddington-Finkelstein coordinates, we de ne null and then, the metric is expressed as t = t log r ds2 = rdt2 2drdt + (dxi)2 : The ingoing (outgoing) Eddington-Finkelstein coordinates (with t+ (t )) also cover inside of the future (past) horizon. The Kruskal coordinate is de ned by x = e t =2 ds2 = dx+dx + (dxi)2 : and then, the metric becomes This covers all region, and x+ = 0 and x = 0 are the past and future horizon, respectively. If the gauge eld has non-vanishing At and regular Ar at the horizon, it is singular there. It can be seen as follows. In the Kruskal coordinates, the gauge eld becomes A = Atdt = At : Therefore, the gauge eld is singular at the future and past horizon. In the EddingtonFinkelstein coordinates, it is expressed as A = At dt dx x : (E.1) (E.2) (E.3) (E.4) (E.5) (E.6) (E.7) (E.8) However, if we take the ingoing Eddington-Finkelstein coordinates, and if Ar is not singular at the horizon, the gauge eld is singular only at the past horizon and is regular at the future horizon. In the Kruskal coordinates, the gauge eld is expressed as A = A+dt+ + Ardr = A+ x+ dx+ + Ar (x+dx + x dx+) : Therefore the gauge eld is singular at the past horizon x+ = 0 but regular at the future horizon x = 0. First order solution for general z The correction terms can be calculated in a similar fashion to the z = 2 case. We de ne h , a and ' as gtt = r2z( f + htt) ; rz 1 2 gtr = gti = r2hti ; htr ; gij = r2( ij + hxx ij + hij) ; ' = log( r 6) : A = a(x) r5 Ai(x)vi(x) dt a(x)r2dr + Ai(x)dxi + atdt + aidxi ; and In the vi(0) = 0 gauge, the constrains are expressed as The rst order solution for the sound modes is p(x) = a(x); 1 2(z 1) 0 1)r z 3h(0) 1 r z 3 rh3(r) 2(z r 3 (z 2 5)pz (z + 3)2 1 at(1) 1) Z dr h2(r) (F.1) (F.2) (F.3) (F.4) (F.6) (F.7) (F.8) (F.9) (F.10) (F.11) (F.12) (F.13) (F.14) z rz+3 0 rz+3 a(1) t h2(r) r z 3 2z (z rz+3 3) r0z+3 h(0) 1 3) r0z+3 rz+3 Z dr h2(r) htt = 2 6 p p z z + 3 + r z 2 1 2 3 r 2 pz 3 z + 3 h1 = at = at(0) 3 ' = (0) r 3 1 a(1) t + r z 3h(10) + r z 3 Z dr h2(r) 3 z z + 3 ah(10) 1 1 z + 3 rz+3at(1) + z + 3 Z z dr h2(r) (10), a(0) and at(1) are integration constants. The function h2 is the solution t of the following di erential equation; 0 = r2 rz+3 rz+3 h020(r) r zrz+3 + (2z + 3)r0z+3 h02(r) 0 (z + 2) (2z + 7)rz+3 + (z + 2)r0z+3 h2(r) 2F1 + C2r 21 z+1 p(z+3)(9z+19) 2F1 ; 2 ; r0z+3 rz+3 +; +; 2 +; r0z+3 rz+3 (F.15) (F.16) (F.17) (F.18) (F.19) (F.20) (F.21) (F.22) (F.23) The rst order solution for the vector modes is = 2 1 pz + 3 p9z + 19 z + 3 : hti = Z r6 z ai = 2(z ht(i0) 1)rz+3 2(z (z 1) ai 5)r0z+3 Z where ht(i0) and ai(0) are integration constants. The function a1 is given by r7a2(r) a2(r) = C3 1)rz+3 (z 5)r0z+3]2 (z + 3)arz 5 rz+3 rz+3 ht(i0) 0 z + 3 rz+2 0 2(z 1) r5 a 10(z 1)rz+3 (z 5)(z and a2(r) is expanded around r = r0 as In order for the solution to be non-singular at r = r0, we have to take a2(r) = C3 + 2(z 1) z(z + 3)2ar02z@ir0 + O(r r0): C3 = z(z + 3)2ar02z@ir0 2(z 1) and then, a2(r) becomes a2(r) = (z + 3)arz 5 rz+3 rz+3 ht(i0) 0 z +3 rz+2 2(z 1) r5 a 10(z 1)rz+3r02 +z(z +3)r5r0z (z j!5)(z 2)rz+3 @ir0 : (F.24) 0 hij = Z r2dr rz+3 0 rz+3 + C4 Z r(rz+3 The regularity at r0 implies C4 = r03. Then, we obtain hij = Z Counter terms for general z In order to obtain regular stress-energy tensor, we introduce the counter terms; Sct = 1 Z d x 4 p (4 + 2z) + C + A A : z + d 2(z 1 1) C e Since for general z, the volume form on the boundary behaves as the regular contribution to the stress-energy tensor is given by O(r z 3) terms of Tr . Then, the renormalized stress-energy tensor is obtained as p rz+3 ; a 1 viAi ; 1 3 a a 1 vivj Aj + 1 3 ; 1 1 1 z 1 2 C 2 1 z + 2 2 0 rz+3 0 2 Ai ; (1 + C)r0z+3 ij (F.25) (F.26) (G.2) (G.3) (G.5) (G.6) (G.7) (G.8) (G.9) (G.10) and they satis es becomes O Or = = lim rz+3 1 Or For C = z 1, the energy density and the pressure becomes E = rz+3 ; P = 16 G 0 rz+3 ; The constant C in the counter terms can be xed by the regularity of the dual operator to the dilaton . By introducing the counter terms, the expectation value of the operator r 1) C C r r 3(z 2 1) ! rz+3 # 0 rz+3 In order for the above expression to be regular, we have to take C = z 1, and then, we obtain and 1 1 1 z 2 0 1 viAi ; Ai ; z rz+3 2 0 1 vivj Aj + 1 3 j ; 1 3 1 viAj ; O 1 16 G r 3(z 2 1) rz+3 : 0 (G.11) (G.13) (G.14) (G.15) 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. [1] J.M. 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Elias Kiritsis, Yoshinori Matsuo. Charge-hyperscaling violating Lifshitz hydrodynamics from black-holes, Journal of High Energy Physics, 2015, 1-51, DOI: 10.1007/JHEP12(2015)076