Hyperscaling-Violating Lifshitz hydrodynamics from black-holes: Part II

The derivation of Lifshitz-invariant hydrodynamics from holography, presented in [arXiv:1508.02494] is generalized to arbitrary hyperscaling violating Lifshitz scaling theories with an unbroken U(1) symmetry. The hydrodynamics emerging is non-relativistic with scalar"forcing". By a redefinition of the pressure it becomes standard non-relativistic hydrodynamics in the presence of specific chemical potential for the mass current. The hydrodynamics is compatible with the scaling theory of Lifshitz invariance with hyperscaling violation. The bulk viscosity vanishes while the shear viscosity to entropy ratio is the same as in the relativistic case. We also consider the dimensional reduction ansatz for the hydrodynamics and clarify the difference with previous results suggesting a non-vanishing bulk viscosity.


Introduction, results and outlook
Holography [2,3,4,5] is a correspondence between quantum field theory and gravity. It provides a powerful tool to analyze strong-coupling theories by using the dual gravity description. At finite temperature and in the long-wavelength regime, holography provides a correspondence between fluids and black holes. This fluid/gravity correspondence was first studied in the AdS/CFT correspondence by [6,7,8,9] in the linearized approximation and in [10] by studying the fluctuations of the bulk black-hole horizon. Holography has also been applied to geometries with Lifshitz or Schrödinger symmetry mainly for application to condensed matter systems [12,13,14,15,16,17,18,19] 1 . Many condensed matter systems have non-relativistic scale invariance [21,22,23,24,25,26], and some of them have Lifshitz or Schrödinger symmetry [27,28,29]. 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, [35], other strongly correlated systems, [36] and graphene, [37]. Recent experiments in various materials that are strongly coupled as well as graphene have indicated that the hydrodynamics of electrons is observable and it exhibits non-trivial shear viscosity, [38,39,40,41,42,43].
The fluid/gravity correspondences in Schrödinger spacetimes has been studied in [30,31,32], where the dual description for non-relativistic fluid mechanics was given. A proposal for hydrodynamics in Lifshitz invariant theories was considered in [33,34] from the effective field theory point of view.
In [17,18,19], all quantum critical holographic scaling geometries with a U(1) symmetry respecting translation invariance and spatial rotation invariance were classified in terms of three scaling exponents. Two of them (z, θ) appear in the metric while another exponent, which is referred to as ζ in [17,18,19], appears in the profile of the U(1) gauge field. 2 The exponent z is the Lifshitz (dynamical) scaling exponent, and θ is the hyperscaling violation exponent, [18,47]. Even though such theories have been studied intensively, many of their aspects are still unclear. In particular, hydrodynamics with Lifshitz scaling symmetry is not fully understood.
In non-relativist quantum field theories, the geometry that captures the symmetry and dynamics is the Newton Cartan geometry, [48]. For Lifshitz space-times, the dual field theory is non-relativistic and the source terms at the boundary are related to the torsional Newton-Cartan theory [49,50,51,52,53,54]. In particular, the role of torsional Newton-Cartan geometry in the boundary structure of bulk non-relativistic solutions as well as on the boundary symmetries was investigated in [52,53,54]. Alternatively the boundary structure can be analyzed using the Hamilton-Jacobi method, [55,56].
In [1], the correspondence between fluids in the torsional Newton-Cartan theory and black holes in Lifshitz space-times has been analyzed. Lifshitz space-times with unbroken U(1) gauge symmetry were considered, that were solutions of the Einstein-Maxwell-dilaton (EMD) theories with constant potential. Although the geometry has Lifshitz scaling symmetry with dynamical exponent z, there was a non-trivial gauge field in the bulk that introduced non-trivial scaling exponents in the charged sector. The black-hole solution of that theory was considered, boosted using Galilean boosts and then all parameters of the solution including the velocities, were made xdependent. Using the technique introduced in [10], the bulk equations of motion were solved order by order in boundary derivatives and the (fluid) stress-energy tensor was computed and renormalized. The results found were as follows: • The stress-energy tensor was expressed in terms of the fluid variables: velocity field v i , energy density E and pressure P , but it also contained the (particle number) density n and external source A i associated to the U(1) symmetry current. It satisfied the scaling condition for Lifshitz invariant theories zE = (d − 1)P .
• It was found that the stress tensor satisfied the conservation equations of the Newton-Cartan theory.
• The role of the (unbroken) U(1) symmetry in this class of theories is important. It was found that it behaves very closely to the U(1) mass conservation symmetry in non-relativistic hydrodynamics.
• The fluid equations found were non-relativistic in contrast to the relativistic fluid analysis in [33,34]. Even though the continuity equation and energy conservation equation agree with those in the ordinary non-relativistic fluids, the Navier-Stokes equation was different from that in the ordinary non-relativistic fluids.
• By redefining variables and allowing a (Milne-invariant) Newton potential in the sources, [52]- [54] the fluid equations can be mapped to the standard nonrelativistic fluid equations coupled to the torsional Newton-Cartan geometry in the presence of a Newtonian potential.
• It was found that the form of the fluid 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 satisfies the local thermodynamic relation with the energy density and pressure. The divergence of the entropy current is non-negative, compatible with the second law.
The hydrodynamics analysis of [1] was generalized beyond the hydrodynamic limit in [57] using numerical techniques.
In this paper, we generalize the studies in [1] to those with hyperscaling-violation in the associated holographic geometries. In [1], we have considered only the θ = 0 cases in which hyperscaling-violation comes only from the gauge field. Here, we introduce a non-trivial coupling between the dilaton and cosmological constant in order to consider θ = 0. Fluids now, have hyperscaling-violation. We find the following results: • As in [1], the fluid equations and the stress-energy tensor reproduce those in the Newton-Cartan theory if the holographic gauge field is identified to that that enters the Newton-Cartan theory.
• The fluid equations are similar to those in [1] but for additional external forcing terms which come from the coupling to the dilaton and its external source term. Note however that in the Lifshitz case with θ = 0, although the dilaton was non-trivial, no such terms appeared in the hydrodynamics. Their appearance is therefore tied to hyperscaling violation in the bulk geometry.
• With a judicious choice of the pressure, these forcing terms can be redefined away. The hydrodynamic equations are now equivalent to the non-relativistic hydrodynamics equations with a conserved mass current, but with an additional chemical potential for the mass current, that is given in (5.8). This chemical potential is not related to external sources like the Newtonian potential but is directly related to the fact that the theory violates hyperscaling.
• The bulk viscosity is zero even in the presence of hyperscaling-violation.
• The hydrodynamic equations respect the full scaling symmetry of Lifshitz invariance with hyperscaling violation as detailed in appendix D.
The Lifshitz space-time with hyperscaling-violation can be also obtained by dimensional reduction from higher-dimensional Lifshitz space-times without hyperscaling violation, [18]. In this case, the model has two dilatons. We derive also the associated hydrodynamics from the black holes. We find the following: • The bulk viscosity is non-zero if we consider the naive dimensional reduction, i.e., with trivial background in the extra dimensions and fixed volume.
• When the internal metric on the extra dimensions, or equivalently, the volume of the extra dimensions satisfies a specific relation to the U(1) charge, the bulk viscosity becomes zero. This reduction gives the lower dimensional hydrodynamics we derived earlier.
• We also present the hydrodynamic ansatz which describes a Lifshitz-invariant fluid with hyperscaling violation on a non-trivial (conformally-flat ) boundary metric. The hydrodynamic equations in such a metric turn out to be simpler than on flat space.
The hydrodynamic equations we derive are general and are given by (5.52)-(5.54) and supplemented by (5.56), with independent variables the temperature T , the (mass) charge density n and velocities v i . All other variables like the energy E, the pressure P and the chemical potential as well as the transport coefficients are determined in terms of T, n by constitutive relations that are dynamics/theory specific. 3 Finally there is the Lifshitz invariance Ward identity, namely (5.55).
The z, θ dependence does not appear explicitly in the hydrodynamic equations. To leading order, it appears only through the constituent relations that express the energy, pressure and (mass) chemical potential as functions of temperature and mass density. To next order the transport coefficients may also depend on z and θ. For this reason we expect that although the hydrodynamic equations we derived strictly apply to the duals of EMD theories we considered, their validity is universal.
This analysis completes the derivation of fluid dynamics for non-relativistic scaleinvariant and U(1)-invariant theories with Lifshitz scaling and a violation of hyperscaling. Remains open the case of such theories without a U(1) symmetry (ie. a broken U(1) symmetry). The case of a perfect fluid was studied recently in [58]. It is highly probable that the hydrodynamics in this case will turn out to be nonrelativistic hydrodynamics in the absence of a conserved "mass" current.
A further interesting question concerns the appropriate hydrodynamics for QFTs that are described by RG flows that interpolate between relativistic and non-relativistic theories. To motivate the answer to this question, we consider first non-Lorentz invariant (but rotationally invariant) flows between Lorentz invariant fixed points, 4 [60,61], but where the velocity of light in the IR is different for that in the UV. In such a case, the hydrodynamics of this theory, is quasi-relativistic, but with a speed of light that is temperature dependent. This context resembles more the proposal of [33,34]. 3 Here, n plays the role of the density of the non-relativistic fluid. We use this (and not the associated chemical potential) as a fluid variable because of the important role it plays in the realization of the non-relativistic momenta. It is the same reason that the mass density is usually used in non-relativistic fluid mechanics. 4 The fact that the speed off light can vary on branes was pointed out first in [59].
The example above suggests that in a (Lorentz-violating) RG flow from a CFT (with a un unbroken U(1) symmetry that is used to drive the breaking of Lorentz invariance) to an IR non-relativistic scaling (rotational invariant) geometry at an arbitrary temperature, the hydrodynamics will be again of the relativistic form (but with general equation of state) and with a speed of light c(T ) that is again temperature dependent. In the IR, c(T → 0) = ∞ and the hydrodynamics reduces to the one found here with the U(1) symmetry becoming the mass-related symmetry. This is the standard non-relativistic limit of the relativistic hydrodynamics while all thermodynamic functions and transport coefficients are smooth functions of T (if no phase transition exists at finite T ). Otherwise they follow the standard behavior at phase transitions.
More general breaking of Lorentz invariance during RG flow must involve higher form fields of tensors in the bulk, and the details of the RG flow become complicated. It is plausible that a RG-covariant definition of velocities and other state functions is necessary in order to define hydrodynamics globally in the full energy landscape. 5 This paper is organized as follows. In Section 2, we introduce the model and its Lifshitz solution with hyperscaling-violation. In Section 3, we introduce the hydrodynamic ansatz. We consider the derivative expansion and solve the equations of motion to first order. We calculate the stress-energy tensor in Section 4, and discuss the relation to non-relativistic fluids in Section 5. In Section 6, we consider the relation to the dimensional reduction from higher-dimensional Lifshitz space-times without hyperscaling-violation. In Section 7, we show that a simpler hydrodynamic ansatz gives a fluid moving in a non-trivial but conformally flat metric. In Appendix B, we review the Newton-Cartan theory and its application to fluids. In Appendix D, we discuss the scaling dimensions of the fluid variables and other relevant coefficients.

Hyperscaling-violating solutions and black holes
We consider a holographic model with Lifshitz scaling symmetry and hyperscalingviolation. The gravity dual is (d + 1)-dimensional Einstein gravity with a Maxwell field A µ and a dilaton φ. The action is given by where F = dA is the field strength of the gauge field, Λ is a negative cosmological constant with a coupling to dilaton, and λ and ν are dimensionless coupling constants.
The equations of motion are given by This model has the Lifshitz geometry with hyperscaling-violation as a solution; The solution for the gauge field and dilaton are given by Here, the cosmological constant is related to a length scale of the solution, which we fixed to be 1. Λ is then given by (2.10) The exponents z, θ and constants φ 0 , χ 0 , µ are determined by the coupling constants in the action λ, ν, and the free parameter a of the solution by the following relations; log a , (2.13) log a , (2.14) To simplify further calculations, we rescale the gauge field such that the solution becomes A t = ar z+d−1−θ .
(2.16) the original action (2.1) becomes The following black hole geometry is also a solution of this theory; ds 2 = e 2χ ds 2 (2.18) The radius of the horizon is given by r 0 and the Hawking temperature of the black hole is The gauge field and dilaton take almost the same form as in the the zero temperature solution but the constant mode of A t is chosen such that A t vanishes at the horizon for regularity.

Solving the hydrodynamics ansatz
In this paper, we focus on the case of d = 4. In some parts however, we may give results in arbitrary dimension d. Extensions to other dimensions (namely d = 3 and d > 4) are expected to be straightforward. We introduce the hydrodynamic ansatz, which describes the fluid mechanics on the field theory side. For the regularity at the future horizon, we transform the black-hole solution to Eddington-Finkelstein coordinates; Hereafter, we will always work in Eddington-Finkelstein coordinates and t will henceforth stand for the null coordinate t + . In these coordinates, the black hole solution for the metric and gauge field are expressed as

3)
where we have taken the gauge such that A r vanishes in the original Fefferman-Graham coordinates. Then, we perform the Galilean boost where v i is a set of constant parameters of the Galilean boost (that we call velocity). Note that this is a general coordinate transformation and for constant v i it provides a new class of solutions parametrized by v. Moreover, we assume in this paper 6 that z = 1 and therefore such a coordinate transformation keeps the boundary sources 7 invariant, as it should. It therefore provides a different state (more precisely ensemble) of the same boundary theory. The black hole geometry becomes

7)
e χ = e χ 0 r −θ/3 (3.8) and the solution for the gauge field and dilaton remains invariant; In order to describe the dynamics of the fluid, we replace the free parameters r 0 , v i and a by slowly varying functions of the boundary coordinates ( x, t). However, this procedure generates a non-trivial background metric at the boundary since a appears as an overall factor of the metric. In order to obtain a flat space background at the boundary, we absorb this overall factor by introducing the additional coordinate transformation with χ 0 given in (2.14), before replacing the parameters by functions. Then, the metric becomes 11) and the gauge field is also rescaled as Since we have rescaled the time coordinate, the Hawking temperature is also rescaled as (3.13) 6 The hydrodynamics of z = 1 with hyperscaling violation was considered in [67] obtaining such solutions by dimensional reduction from higher dimensional AdS geometries. 7 They are defined as the most divergent parts of the metric and gauge field near the boundary. Now, we replace the parameters r 0 , v i and a by slowly varying function of the boundary coordinates x µ . Moreover, we promote the constant part of A i , which is usually gauged away to x µ -dependent functions as was also done in [1]. Then, the metric, gauge field and dilaton become where x µ -dependence of φ 0 (x) and χ 0 (x) comes from that of a(x). We have also introduced A t (x) and A i (x), that originate in the constant parts of A t and A i , respectively, and which have been now replaced by functions of the boundary coordinates. The relations (2.13)-(2.15) are however preserved. The above is no longer a solution of the equations of motion, and we must introduce additional correction terms. We consider the derivative expansion in t, x and calculate the first order solution for the hydrodynamic ansatz. We first expand (3.14)-(3.17) at a point, which we can take x µ = 0 without loss of generality. Then, we assume that the derivatives are small since x µ dependence appears only through the slowly varying functions r 0 (x), v i (x), a(x), etc., and expand with respect to the derivatives ∂ µ ;

18)
where g (n) µν , etc. stands for n-th order terms in the derivative expansion. At the leading order of the derivative expansion, the equations of motion contain only the leading order terms g µ and φ (0) which are equivalent to the solutions (3.6)-(3.9) before replacing the parameters by slowly varying functions, and hence, are satisfied. At the next-to-leading order, only the linear order terms of the derivatives ∂ µ appear, and the equations of motion are not satisfied due to these linear order terms. Now, we introduce the correction terms to (3.14)-(3.17) as

21)
where h µ and ϕ (n) are the correction terms at the n-th order of the derivative expansion, which start from n = 1 since the equations of motion satisfied without the correction terms at leading order. At the next-to-leading order, or equivalently, at the linear order of the derivative expansion, the equations of motion give the inhomogeneous linear differential equations for the correction terms, h (1) µν , a (1) µ and ϕ (1) . The inhomogeneity comes from the first order terms of (3.14)-(3.17), namely, g (1) µν , A (1) µ and φ (1) . By solving the differential equations for the correction terms, we obtain the following first order solution of the derivative expansion; where the first line is the original solution and the the rest are the corrections terms. σ ij is the shear tensor 25) and the functions F i (r) are given by (3.33) The first order solution for the gauge field is 34) and the dilaton has no correction term, We find that, in order for (3.24), (3.34) and (3.35) to be a solution of the equations of motion, the functions r 0 (x), v i (x), a(x) and A µ (x) must satisfy the following constrains; where F = dA.

Calculation of the stress tensor
We will calculate now the boundary stress-energy tensor. The asymptotically Lifshitz space-times have anisotropic behavior in temporal and spatial directions, and hence it is convenient to introduce the vielbeins in order to consider their general asymptotic behavior.
The metric is expressed in terms of the vielbein E A as For the asymptotic Lifshitz space-time in Fefferman-Graham coordinates, the vielbein can be expressed as [63] E r = e χ dr r , where the one forms τ (r, x µ ) andê a (r, x µ ) have a finite and non-degenerate limit near the boundary, r → ∞, and provide the characteristic quantities of Newton Cartan geometry. Eq. (4.2) does not however fix the r → ∞ limit of τ (r, x µ ) andê a (r, x µ ), uniquely. For our solution (3.24), we take the vielbein E A such that τ (r, x µ ) and e a (r, x µ ) behave in r → ∞ as Then, the induced metric at transverse surfaces, dr = 0, is expressed for large r as wherev µ and e µ a are the inverse vielbeins, which take the following form 8 as r → ∞ for (3.24);v Relations (4.4) and (4.5) do not completely fix the r-dependent vielbeins τ µê a µ andv µ . For example, the one form τ µ may also have subleading terms of order O(r 2−2z ), which appear at the same order as the leading terms ofê a µ . This would changeê a µ andv µ to leading order. This ambiguity is equivalent to a Milne boost at the boundary. We fix this ambiguity in the sequel in order to proceed. We will discuss the action of Milne boosts at the boundary data later.
To summarize, for our solution we take that describe the asymptotic Newton Cartan geometry in what we called in [1] "the holographic frame". For the gauge field with boundary Lorentz indices we definê The variation of the action with respect to these boundary variables is given by where S r is the renormalized action with appropriate (boundary) counter terms.
Since the volume form behaves as The definition above does not give the ordinary stress-energy tensor but the one with contributions from the gauge field and current. It is related to the ordinary Brown-York tensor asŜ where T µν (BY) is the Brown-York tensor which is defined in terms of the extrinsic curvature K µν as and T µν (ct) is the terms from the counter terms. By using appropriate counter terms, the stress-energy tensor becomes finite at the boundary r → ∞ and we define 13) and the current is given by (4.14) As we will see later, the stress tensor in (4.13) is also different from the ordinary stress-energy tensor on the boundary as it contains the contributions from the external gauge field. Now, we calculate the renormalized stress-energy tensor from the first order solution (3.24). In order to regularize the stress-energy tensor and other expectation values, we introduce the following counter terms; (4.15) Although the last counterterm is apparently not gauge invariant, the effective action is still invariant under boundary gauge transformations (up to boundary terms on the boundary). To see this, the second term in (4.15) is expanded near the boundary as Near the boundary, r → ∞, the first term is leading and will cancel with the divergent part in the original action. The second and third terms give finite contributions to the effective action. The ellipsis denotes terms that are vanishing as we take the cutoff surface to the boundary. We therefore observe that indeed the finite terms are not gauge invariant but are linear in the gauge field. Under the boundary gauge transformation δA = dΛ, the finite term in (4.15) transforms as where we used (3.36). Therefore the transformed terms are surface terms and should vanish at infinity on the boundary. The coefficient of these terms is indeed the conserved current (see equation (4.23)) and (4.23) can be written as Therefore if no charge is leaking to infinity on the boundary the effective action is gauge invariant. By using the counter terms (4.15), the renormalized stress-energy tensor is calculated as The current J µ is obtained as The expectation value of the dual operator of the dilaton φ is also calculated as We also calculate the entropy current that is given by, [11], where √ h is the volume form on the time-slice at the horizon and n µ is the normal vector to the horizon. By using the first order solution (3.24), the entropy current becomes

Thermodynamics
In order to consider the relation between the form of the stress-energy tensor (4.13) and that for fluids, we first calculate the thermodynamic functions. The energy E, entropy S and charge N in volume V (in the x i directions) of the Lifshitz black hole geometry are given by 10 where we have also defined E, s and n, which are densities of energy, entropy and charge. Here we have also taken into account the effect of the coordinate redefinition (3.10), and hence the energy has an additional factor of a . Now, we consider the first law of thermodynamics; where the variations with respect to entropy S, volume V and charge N give the temperature T , pressure P and chemical potential µ, which are calculated as For d = 4, the energy density, pressure, entropy density, charge density and chemical potential are given by Note that at θ = 0, the chemical potential µ vanishes. 10 In this subsection we return temporarily to arbitrary dimension, d.

Relation to hydrodynamics in Newton-Cartan theory
In this section we will rewrite the boundary stress tensor in terms of the Newton-Cartan geometry data along the lines of [52]- [54] following the detailed formalism of [1].
In the Newton-Cartan theory, (that is explained in more detail in appendix B), the spacelike vielbein and inverse timelike vielbeinē a µ andv µ , transform under the Milne boost. In this paper,v µ andē a µ indicate the vielbeins in an arbitrary (Milne) frame.
There is a special frame, the "holographic frame," introduced in [1], where the vielbeins are given byv wherev µ andê a µ are defined on the gravity side, in (4.6), andv µ always satisfies in any framev where u µ is the fluid velocity. The timelike vielbein and inverse spacelike vielbein do not transform under the Milne boost and hence are simply referred to as τ µ and e µ a , respectively.
The renormalized boundary stress-energy tensor, which we have calculated in the previous section, can be expressed in the following form; where the spatial metric is defined in terms of the vielbeins e µ a andê a µ ; The energy density E, pressure P , particle number density n and chemical potential µ are given by (5.8). The transport coefficients like heat conductivity κ, shear viscosity η and bulk viscosity ζ are also read off (in d = 4) as where the bulk viscosity ζ is the coefficient of the expansion ∂ i v i in the stress-energy tensor (5.11). Hence, ζ = 0 can be deduced from the absence of ∂ i v i in (5.11). The stress-energy tensor can also be expressed in terms of the energy flow E µ , momentum density P µ , stress tensor T µ ν and current J µ , which are defined by Then, T µ ν is written as Comparing (5.18) with (5.11), we obtain The above stress-energy tensor can be identified with that in the Newton-Cartan theory 11 . In the Newton-Cartan theory, the geometry is described by the timelike vielbein τ µ , spatial (inverse) metric h µν , timelike inverse vielbeinv µ and gauge field B µ . Here, the spacelike vielbein, its inverse and spatial metric with lower indices are denoted asē a µ , e µ a andh µν . The energy current E µ , stress tensor T µν , momentum density P µ and mass current J µ are the conserved quantities associated to τ µ , h µν ,v µ and B µ , respectively. For a generic fluid in Eckart frame, they are given by [65] P µ ν is the projector to the spatial directions and ϑ is the expansion of the fluid. The fluid velocity u µ satisfies the normalization condition τ µ u µ = 1 and is given by In the Newton-Cartan theory, the generic stress-energy tensor (not only for the fluid above) is defined from the energy current E µ , stress tensor T µν , and momentum density P µ as 12T (5.28) 11 There can be ambiguities in the definition of the stress tensor. In relativistic CFTs they are known to affect higher-order transport coefficients, [70]. In non-relativistic scaling theories the situation is even more sensitive as they affect even the ideal hydrodynamics equations as we will see further. Moreover, as we discuss in section 6, they also affect the bulk viscosity. 12 We always denote stress-energy tensors by the letter T and their spatial projection (the stress tensor) by the letter T .
The Newton-Cartan theory has a symmetry which is known as Milne boost; 13 and we can introduce the Milne boost invariant combination for the gauge field 1form as [65,54] where We also define a Milne boost invariant stress-energy tensor as We choosev µ = u µ and we have and then, (5.31) and (5.33) give Therefore, the Milne boost invariants B and T µ ν are the same as B andT µ ν in thē v µ = u µ frame.
We further define the Milne boost invariants for the energy flow E µ , momentum density P µ , stress tensor T µ ν from the Milne boost-invariant stress-energy tensor T µ ν as and then, T µ ν is written as and hereh µν andP µ ν are given by those inv µ = u µ frame. Then, (5.23)-(5.25) take the same form to (5.41)-(5.43), respectively, for v i = u i and ζ = 0. Therefore, we identify the energy flow E µ , momentum density P µ and stress tensor T µ ν , which are calculated from the black hole geometry in the previous section, with the Milne boost-invariants E µ , P µ and T µ ν , respectively. We also identify the gauge field A µ , which originates in the constant mode of the bulk gauge field A µ , with the Milne boost-invariant combination B µ as where m is the coupling constant for the gauge field B µ , which is equivalently in the non-relativistic language the mass per particle. Then, the mass current J µ is related to the particle number current J µ as It is then straightforward to verify that the stress-energy tensor T µ ν , which we calculated from the black hole geometry, is related to the Milne boost-invariant stressenergy tensor T µ ν in the Newton-Cartan theory as The conservation law for these Milne boost-invariants is given by [66] where H = d B. The covariant derivative D µ is defined as usual in the Newton-Cartan geometry (see appendix B).
In terms of the Milne boost invariant stress-energy tensor, the conservation law can be expressed as since P µ = 0. More details are given in Appendix B.
The renormalized stress-energy tensor T µ ν satisfies the conservation equations which are equivalent to (5.47) and (5.48). These equations come from the following components of the bulk equations of motion; where n µ and γ µν are the normal vector and the induced metric on the boundary, respectively. In order to derive the conservation equations for the first order fluids, we have to calculate the constraint equations (5.50), (5.51) to second order in the derivative expansion. The correction terms at each order do not contribute to the constraint equations at that same order, and hence, we do not need to solve the differential equation for the correction terms to second order. The constraint equations (5.50), (5.51) can be expressed in terms of the fluid variables E, P , n and v i defined in (5.23)-(5.26) as Note that the main difference in these equations for the hyperscaling violating case studied here and the case with θ = 0 studied in [1] is the appearance of the term n∂ i µ in the Navier-Stokes-like equation (5.53). Here we have a chemical potential in the absence of an external Newtonian potential and this is due to hyperscaling violation. The fluid equations (5.52)-(5.54) are equations for energy density E, pressure P , velocity field v i , particle number density n and temperature T . There are also constituent relations between these variables. In particular, the temperature and energy density are related to each other, and hence are not independent. For the fluid dual to the Lifshitz geometry, the energy density is constrained first by the Lifshitz Ward identity with hyperscaling violation. This is given by The transport coefficients η and κ are not constant but depend on the temperature, and also on the particle number density in this case, as in (5.13). By using (5.31) and (5.44), the gauge field A is rewritten in terms of the velocity field v i and the external gauge field of B; Therefore, we have 5 equations for 5 independent variables E, v i and n.
We also have external sources in the hydrodynamic equations: one is of course the space metric that is flat here (but in subsequent sections this will change). The other is the external gauge field B µ that couples to the U(1) current. In ordinary non-relativistic hydrodynamics, B t would be the ordinary Newtonian potential that couples to mass [52,53,54,1].

The entropy current
The entropy current (4.25) can be expressed as where the entropy density s is given by The entropy current satisfies the following relation with the (internal) energy density E and pressure P ; It also satisfies the second law. By using (5.52), the divergence of J µ S is expressed as which is manifestly non-negative. Therefore, the entropy current satisfies the second law. The entropy density s = J 0 S is such that the KSS bound, [64], is saturated;

The relation to dimensional reduction
In the previous section, we studied the fluids in the field theory dual of the Lifshitz space-time with hyperscaling-violation. We found that the fluid has zero bulk viscosity ζ = 0.
It is known that the Lifshitz space-time with hyperscaling-violation can be obtained by the dimensional reduction from the higher-dimensional Lifshitz space-time without hyperscaling-violation, [18]. As discussed in [1], the fluid for the Lifshitz space-time without hyperscaling-violation also has zero bulk viscosity. Upon compactification, this Lifshitz geometry with flat internal space and constant radius, becomes the lower dimensional geometry with hyperscaling violation. In such a case it was shown, [67,68], that the bulk viscosity becomes non-zero after dimensional reduction.
In this section, we will first derive the general fluid equations in the higherdimensional theory by allowing also the internal volume to be an additional thermodynamic variable. Then we will reduce to the lower dimension. We will show that the appropriate reduction that corresponds to the thermodynamic ansatz in the lower dimension is compatible with our four-dimensional results of the previous section.
We first consider the following theory, which is the Einstein gravity with the Maxwell field and a single scalar field; where D = d − θ, and we assumed that θ is a negative integer at this moment. This model has the Lifshitz space-time without hyperscaling-violation as a solution; with the following gauge field and dilaton; where the parameters z, a and µ are related to the parameters of the action (coupling constants) asλ We compactify the (−θ)-dimensional extra dimensions and consider the dimensional reduction to (d + 1)-dimensional spacetime. The metric is decomposed as where ds 2 is the metric after the dimensional reduction and we compactified x idirections with i = d, · · · , d − θ − 1 to circles with period x i ∼ x i + 1. The constant ν is given byν .
Then, after the dimensional reduction, the action becomes 9) Now, the geometry has hyperscaling-violation because of the redefinition of the metric, and is given by while the additional dilaton field φ 2 is More generally, φ 2 also has the constant mode as is φ 1 but it is independent from the gauge field. Then, the solution becomes ds 2 = e 2χ ds 2 (6.13) 14) where χ 0 is an arbitrary constant. We define a new parameter b as In a sense b parametrizes the internal volume. Now, the model has 2 dilatons instead of one. This solution has an additional parameter b which is the constant mode of the additional dilaton φ 2 . The solution becomes equivalent to that in the previous sections if the dilatons satisfy the following condition; The solution which describes the hydrodynamics can be calculated in a similar fashion to the previous sections. We now specialize to d = 4. The first order solution of the derivative expansion is obtained as 20) where θ must be negative integer for the metric before the dimensional reduction ds 2 D . The non-zero components of v i and A i are introduced only for i = 1, 2, 3 and the parameters v i , r 0 , a, b and A µ are replaced by functions which depend on x µ with µ = 0, · · · , 3. We have redefined the coordinates x µ as (3.10) in the previous section and hence the Hawking temperature is also rescaled as where we have written this relation in arbitrary dimension. The function F (r) is given by dr r 6−z+θ F 2 (r) (6.27) F 4 (r) = 2(z − 1)r z+3−θ − (z − 5 + θ)r z+3−θ 0 dr F 2 (r) (6.28) (6.32) The first order solution of the gauge field is and φ 1 has no correction term, while φ 2 receives the correction term ϕ 2 ; The solution above must satisfy the following constraints; The hydrodynamic solution from the dimensional reduction also reproduce the result in the previous sections, if b = a.

Higher-dimensional thermodynamics and hydrodynamics
We first consider thermodynamics before the dimensional reduction. For D-dimensional space-time with arbitrary D = d − θ, the energy, entropy and charge are given by where V D−1 is the volume of the (D − 1)-dimensional space. The 1st law of thermodynamics is expressed as and the temperature, pressure and chemical potential are calculated as Here, the temperature T D agrees with that for the local observer where T is the Hawking temperature of the black hole. Next, we consider the fluid in D-dimensional space-time with D − 1 = 3 − θ. The vielbein behaves near the boundary r → ∞ as where τ D = e −χ 0 dt , (6.47) This implies that the Newton-Cartan data on the boundary is given by h µν D = diag(0, e 2χ 0 , · · · , e 2χ 0 , e 6 θ χ 0 , · · · , e 6 θ χ 0 ) , (6.52) The stress-energy tensor on the boundary is calculated from the first order solution for the metric before the dimensional reduction (6.20), in a similar fashion to Section 4 as where the shear tensor is given by This stress-energy tensor can be expressed in the context of the Newton-Cartan theory as The shear tensor can be expressed in terms of the Newton-Cartan geometry as where D µ is the covariant derivative in the Newton-Cartan geometry in the holographic framev µ =v µ D , (whose Christoffel symbol is given by (B.6) withv µ =v µ D , see Appendix B for more details), and £ is the Lie derivative.
The energy density, pressure, particle number density and the temperature are given by The transport coefficients are read off as The stress-energy tensor can also be expressed in terms of the energy current, momentum density, stress tensor and particle number current as The conservation equations in the Newton-Cartan theory is given by Then, they can be expressed in terms of the fluid variables as It is straightforward to verify that these equations are equivalent to the constraint equations in the bulk equations of motion.

Thermodynamics after the dimensional reduction
In this subsection we take the lower dimensional boundary theory to have space dimension d − 1. For general b, we consider in the following the first law, where the thermodynamic variables are the entropy S, the (d − 1)-dimensional volume 14 V and the charge N, which is related to the variable a. We have already found that (in general d) In general, b is not related to the particle number density n and hence the energy does not depend on the charge. Then the first law can be written as, The black hole solution also has another variable b, which is related to the scalar source asφ Then, by taking into account this scalar source as an additional thermodynamic variable, the first law of thermodynamics may be expressed as where O 2 is the vev of the dual operator to the scalar φ 2 , which is expressed as We will discuss this operator later.
In the previous section, we obtained the thermodynamics in the lower dimension, with one less variable. To recover it from the one here we must take a codimension one section of the thermodynamic variables. As argued already, the correct section involves taking b = a above. With this constraint, we obtain (6.87) and the first law takes the form The temperature, pressure and chemical potential are now given by Note that these are exactly the thermodynamic quantities and the first law we obtained in the previous section (for d = 4).
The Ward identity for the scaling symmetry is simply expressed as (6.92)

Hydrodynamics after the dimensional reduction
We now set back d = 4. The stress-energy tensor and the fluid equations in lower dimensional theory can be calculated straightforwardly. The stress-energy tensor is obtained as For b = a, the results above agree with those in Section 4. The stress-energy tensor after the dimensional reduction T µ ν can be expressed in terms of the fluid variables as whereθ is defined byθ If b = 1, this term gives the expansion term ∂ i v i . The energy density E, pressure P , particle number density n are the same as those in Section 6.2; The transport coefficients, heat conductivity κ and shear viscosity η are also read off as For b = a,θ vanishes but it does not mean that the fluid is incompressible, sincē ϑ is not only the expansion but has extra terms. The expansion is non-zero but cancels with the extra terms, and hence, this implies that the bulk viscosity vanishes in the lower dimension. For the dimensional reduction with b = 1 as in (6.2),θ simply gives the expansion and hence the bulk viscosity ζ in that case is It should be noted that θ is negative for the dimensional reduction and hence ζ is positive.
The constraint equations can be written in terms of the fluid variables as where O 2 is the dual of the dilaton φ 2 but without the contribution from the counter term A 2 ; (6.107) This expression is consistent to the dimensional reduction from the higher dimensional fluid; where T D µ ν is the stress-energy tensor of the fluid before the dimensional reduction. Thus, O 2 is related to the other fluid variables if the fluid is obtained by the dimensional reduction. It should be noted that the constant mode of φ 2 is given byν log b, and hence,the contribution from O 2 is interpreted as the coupling to the external source of φ 2 . Contrary to the single dilaton case in the previous sections, the variable b is independent from the fluid variables and is interpreted as an external field. The transport coefficients also depend on temperature and external field b, but are independent of the particle number density.
The energy E, stress tensor T i j and scalar operator O 2 satisfy the following condition; where the trace of the Milne invariant stress tensor is given by The above condition is nothing but the Ward identity of the Lifshitz scaling symmetry with the hyperscaling-violation. The coefficients of E, P andν O 2 equal to the scaling dimensions of t, x i and b with appropriate signs, respectively. If the fluid satisfies the conditionθ = 0, the fluid equations (6.104) and (6.105) can be rewritten as where The Ward identity of the Lifshitz scaling symmetry can also be expressed in terms of P as For b = a, these equations agree with those in the previous section. The above effective pressure (6.113) and chemical potential (6.115) agree with those in the thermodynamic relation (6.90) and (6.91), respectively. Then, the thermodynamic relations, fluid equations and Ward identity, as well as the stress-energy tensor reproduce the result in the previous section.
The result in this section is a generalization of [67] to non-relativistic and z = 1 cases. The contributions from the gauge field and φ 1 vanish for z = 1 limit in which the Lifshitz black hole geometry becomes the Schwarzschild-AdS. The hydrodynamic ansatz should be given by using the Lorentz boost for z = 1, and hence the fluid will be relativistic. The results in this section are not well defined in z = 1 limit, since the ansatz is obtained by using the the Galilean boost. The non-relativistic fluid which is obtained in this section agrees with the non-relativistic limit of [67] for z = 1.

Lifshitz hydrodynamics on a conformally flat background
We have introduced the redefinition of the boundary coordinate before replacing the parameters by slowly varying functions. This coordinate redefinition is introduced to make a flat space background on the boundary. Here, we show that the naive hydrodynamic ansatz without such a coordinate redefinition gives fluids on a nontrivial but conformally flat background.
It can be calculated straightforwardly in a similar fashion to Section 3 but without introducing the coordinate redefinition (3.10). Then, the first order constraint equations, which are equivalent to the fluid equations in the perfect fluid limit, are obtained as These equations are different from (3.36)-(3.38), and as we will explain below, the difference can be interpreted as the effect of the non-trivial background geometry at the boundary. It is natural to expect that fluid variables as energy density, pressure and particle number density are not affected by the background geometry. In fact, we can calculate the stress-energy tensor straightforwardly and they are read off as which are the same as in (5.8).
In order to consider the fluid mechanics in the non-trivial background, we first introduce the fluid velocity field u µ which is normalized as where the timelike vielbein is given by τ = e χ 0 dt , (7.6) and hence the normalized velocity field u µ is The constraint equations (7.1)-(7.3) can be expressed as where the particle number current J µ is defined by Eq. (7.9)-(7.11) are nothing but the fluid equations in the perfect fluid limit in Newton-Cartan theory, and the generalization to the first order fluid is straightforward.
The first order stress-energy tensor is obtained in the following form; where the transport coefficients are given bỹ 14) The bulk viscosity is zero as for the flat background (5.13). The difference of heat conductivity would come from the difference of the temperature. In this case, the Hawking temperature is simply given by (2.22) on the contrary to that in Section 3 where the temperature is rescaled due to the coordinate transformation. In the curved background, this should be expressed in terms of the local temperature T T = e −χ 0 T asκ 15) and then, the heat conductivity is same as (5.13); The difference of the shear viscosity implies that the shear tensor must be written in terms of the normalized velocity field u µ ; Then, the shear can be written as Then, the shear viscosity η equals to (5.13);

Dimensional reduction for conformally flat background
We can also consider the naive hydrodynamic ansatz for the dimensional reduction from the higher dimensional Lifshitz geometry. In this case, we can see the effects of the conformal factor in the metric more explicitly. The perfect fluid limit of the fluid equations are obtained as where b comes from the effects of the non-trivial background; The fluid variables are the same as in (6.101) 24) and the fluid equations can be written in terms of the normalized velocity field u µ but now χ 0 is independent from the particle number density n ∼ 1/a; The first order stress-energy tensor is given by (7.28) In this case, the expansion appears in the combination of and it vanishes for b = a by substituting the constraint equation. For b = 1, the background becomes flat and hence this agrees with (6.99).

A. Notations
Variables defined on the bulk (gravity) side • v i : Boost parameter introduced into the (static) black hole geometry in (3.11). It becomes the velocity of the fluid.
• A µ : the constant part of the gauge field, which is defined in (3.16). This corresponds to the Milne boost-invariant gauge field in the Newton-Cartan theory B, or equivalently, B in the holographic framev µ =v µ . In this paper, it is sometimes expressed as the 1-form A = A µ dx µ .
• τ µ : the timelike vielbein on the boundary which is defined up to the factor of e χ r z , and given by (4.3) in r → ∞. This corresponds to the timelike unit normal which defines the time direction in the Newton-Cartan theory. It is automatically invariant under the Milne boost.
•ê a µ : the spacelike vielbein on the boundary which is defined up to the factor of e χ r, and given by (4.3) in r → ∞. This corresponds to the spacelike vielbein in the Newton-Cartan theory if we take the holographic framev µ =v µ .
•v µ : the timelike inverse vielbein on the boundary which is defined up to the factor e −χ r −z , and given by (4.6) in r → ∞. This corresponds to the inverse timelike vielbein in the Newton-Cartan theory if we take the holographic framē v µ =v µ , since the holographic frame is defined byv µ =v µ .
• e µ a : the spacelike inverse vielbein on the boundary which is defined up to the factor of e −χ r −1 , and given by (4.6) in r → ∞. This corresponds to the spacelike inverse vielbein in the Newton-Cartan theory. It is automatically invariant under the Milne boost.
• T µ ν : the energy-momentum tensor calculated from the black hole geometry (3.24), which is defined by (4.13) • E µ : the energy current which is calculated from the black hole geometry (3.24).
It is defined by (5.18).
• P µ : the momentum density which is calculated from the black hole geometry (3.24). It is defined by (5.18).
• T µ ν : the stress tensor which is calculated from the black hole geometry (3.24). It is defined by (5.18).
• J µ : the particle number current which is calculated from the black hole geometry (3.24). It is given by (5.22).

Variables defined in the (boundary) Newton-Cartan theory
The following variables in the Newton-Cartan theory are appears in Section 5. See Appendix B for more details.
•v µ : the timelike inverse vielbein in Newton-Cartan theory. The timelike inverse vielbein is not invariant under the Milne boost. In the literature it is sometimes called "velocity" but must be distinguished from the velocity of the fluid.
• h µν : the induced contravariant metric on the time-slice. It is invariant under the Milne boost.
•h µν : the induced covariant metric on the time-slice. It is not invariant under the Milne boost.
• B µ : the gauge field in the Newton-Cartan theory. It is not invariant under the Milne boost.
• E µ : the energy current in the Newton-Cartan theory. For fluids, it is given by (5.23).
• P µ : the momentum density in the Newton-Cartan theory. For fluids, it is given by (5.24).
• T µ ν : the generic stress tensor in the Newton-Cartan theory. For fluids, it is given by (5.25).
• J µ : the mass current in the Newton-Cartan theory. For fluids, it is given by (5.26).
• B: a Milne boost invariant combination for the gauge field, or equivalently, the gauge field in the holographic framev µ =v µ . It is defined by (5.31).
• E µ : a Milne boost invariant combination for the energy current. or equivalently, the energy current in the holographic framev µ =v µ . It is defined by (5.40).
• P µ : a Milne boost invariant combination for the stress tensor, or equivalently, the momentum density in the holographic framev µ =v µ . It is defined by (5.40).
• T µ ν : a Milne boost invariant combination for the stress tensor, or equivalently, the stress tensor in the holographic framev µ =v µ . It is defined by (5.40).
• T µ ν : a Milne boost invariant combination of the stress-energy tensor, or equivalently the stress-energy tensor in the holographic framev µ =v µ . It is defined by (5.33) and can be decomposed into E µ , P µ and T µ ν .

B. The Newton-Cartan formalism
Here, we briefly review the Newton-Cartan formalism. The metric on the Galilei space-time is defined by a 1-form τ µ and a contravariant symmetric tensor h µν . The time direction of the Galilei space-time is defined by the 1-form τ µ . h µν is the spatial inverse metric on the time-slice. They satisfy the following orthogonality condition; Next, we introduce the covariant derivative D µ on the Galilei space-time. We impose the condition that the Galilei data (τ µ , h µν ) are constant under the covariant derivative Unlike Einstein gravity, the above conditions do not determine the Galilei connection uniquely. We further introduce a contravariant vectorv µ , which satisfies the following normalization condition; The vectorv µ is the contravariant timelike vielbein. It is sometimes referred to as the velocity field but is not to be confused with the velocity fluids. We also define the spatial covariant metrich µν by using the following conditions withv µ ; We impose the spatial torsion free condition, Then, the Newton-Cartan connection is determined up to a 2-form H µν as In general, the Newton-Cartan connection has the torsion; Without the spatial torsion free condition, (B.5), the connection can be more general [69]. We can also define the Milne boost invariant connection [66,69] and B is related to H in (B.6) by H = dB, which we will discuss, soon. The connection (B.8) has non-zero spatial torsion; or equivalently, (B.8) satisfies the modified spatial torsion free condition, h B σρ T ρ B µν = 0 instead of (B.5). Here, the spatial torsion free connection (B.6) is sufficient for our purposes and hence we do not consider other connections as (B.8).
The curvature is defined by using the commutator of the covariant derivative and given by For the torsion free case, dτ = 0, the Newtonian condition can be imposed where [· · · ] and (· · · ) in the indices stand for the antisymmetric part and symmetric part, respectively. The above condition implies that the 2-form H µν must be closed, Then, H is interpreted as the field strength of a gauge field; H = dB. For the torsional case, dτ = 0, no appropriate generalization of the Newtonian condition is known. Here, we simply impose the condition H = dB even for dτ = 0, as was proposed in [66].
To summarize, the Newton-Cartan geometry is described by the Newton-Cartan data, τ µ , h µν ,v µ and B µ .
In the Newton-Cartan data,v µ and B µ are introduced to define the Newton-Cartan connection. Two pairs (v µ , B µ ) and (v ′µ , B ′ µ ) are equivalent if they give the same Newton-Cartan connection. The transformation from (v µ , B µ ) to (v ′µ , B ′ µ ) gives an internal symmetry of the Newton-Cartan theory. The Newton-Cartan connection is invariant under the following transformation, [69]; This transformation is known as the Milne boost and V µ is a vector which parametrizes the Milne boost. It should be noted that the covariant spatial metrich µν is defined in terms ofv µ and is not invariant under the Milne boost. It transforms as Next, we consider the conservation laws in the Newton-Cartan theory. The energy current E µ , momentum density P µ , stress tensor T µν and mass current J µ are given by the variation of the action with respect to the Newton-Cartan data τ µ , h µν ,v µ and B µ ; where the variation δτ µ is arbitrary but δv µ and δh µν are only the variations which satisfy the orthogonality and normalization conditions, (B.1) and (B.3), namely, δv µ =P µ ν δv ν , etc. Invariance under coordinate transformations gives the following conservation equations Invariance under the U(1) gauge symmetry of the gauge field B µ gives the conservation equation of the mass current; Invariance under the Milne boost gives the following Ward identity; The energy flow, momentum density and stress tensor are not invariant under the Milne boost. The variations transform under the Milne boost as where we imposed V µv µ = 0 since V µ always appears together with h µν orP µ ν . It should also be noted that the variations in (B.17) are constrained by the orthogonality (B.1) and normalization (B.3) conditions and hence give additional contributions proportional to the variation of τ µ ; where the first terms in these equations are already included in (B.17) but their transformation under a Milne boost gives additional terms in the Milne transformation of the energy flow. Then, the transformation of the energy flow, momentum density and stress tensor are given by where the indices are raised or lowered by the spatial metric h µν orh µν , respectively, and the transformation of the stress tensor is given by (B We now introduce Milne boost invariant combinations. It is straightforward to see that the following combination is invariant under the Milne boost; which is an analogue of the stress-energy tensor, but constructed from the mass current, momentum density and stress tensor. We can also define the stress-energy tensor from the energy flow, momentum density and stress tensor; but this stress-energy tensor is not invariant under the Milne boost and transforms asT ′µ With this stress-energy tensor, a Milne boost invariant combination is given by It should be noted that the Milne boost combination T µ ν is not invariant (or covariant) under the U(1) gauge transformation associated to B µ . Now, we consider fluids in the Newton-Cartan theory. We introduce the fluid velocity field u µ , which is invariant under the Milne boost. For the first order fluid in the Eckart frame, the energy flow, momentum density, stress tensor and mass current are given by where the index of the velocity field is lowered by using the spatial metrich µν and u 2 is defined as u µ =h µν u ν , (B.40) The energy density E, pressure P and mass density ρ are invariant under the Milne boost. We construct a stress-energy tensor from the above quantities, and thenT µ ν is obtained asT µ ν = −Eu µ τ ν − 1 2 ρu 2 u µ + P P µ ν + ρu µ u ν − ησ ρσ h µρ P σ ν − ζθ P µ ν + κh µρ (∂ ρ T ) τ ν , (B.42) where P µ ν ≡ δ µ ν − u µ τ ν =P µ ν − h µρ u ρ τ ν . (B.43) We consider the Milne boost invariants in fluid mechanics. We introduce a Milne boost invariant combination for the gauge field by using u µ ; Then, the Milne boost invariant combination for the stress-energy tensor T µ ν is expressed in terms of the Milne boost invariant gauge field B µ as T µ ν =T µ ν + J µ B ν = −Eu µ τ ν + P P µ ν + J µ B ν − ησ ρσ h µρ P σ ν − ζθ P µ ν + κh µρ (∂ ρ T ) τ ν . (B.45) This stress-energy tensor is invariant under the Milne boost, but is not invariant under the U(1) gauge symmetry associated to B µ . We define the Milne boost invariant and U(1) gauge invariant stress-energy tensor, T µ ν from T µ ν and B µ such that Then, it is expressed as The Milne boost invariant stress-energy tensor (B.47) can be decomposed into the Milne boost invariant energy density E µ , momentum density P µ and stress tensor T µ ν as T µ ν = − E µ τ ν + u µ P ν + T µ ν (B.48) where E µ ≡ −T µ ν u ν = Eu µ − κh µν ∂ ν T (B.49) P µ ≡ T ρ ν τ ρ P ν µ = 0 , (B.50) T µ ν ≡ T ρ σ P µ ρ P σ ν = P P µ ν − ησ ρσ h µρ P σ ν − ζθ P µ ν . (B.51) By using the relation between T µ ν andT µ ν ; the Milne boost invariant energy flow, momentum density and stress tensor are expressed as E µ = E µ − T µ ν u ν −v µ P ν u ν + 1 2 u 2 J µ , (B.53) P µ = (P ν − τ ρ J ρ u ν ) P ν µ = 0 (B.54) T µ ν = h µρ (T ρσ − ρu ρ u σ ) P σ ν (B.55) where we have used the condition for the Eckart frame (B.39) in the last equality of (B.54) and in (B.55). The Milne boost invariant energy flow (B.53) is equivalent to that introduced in [65] In general, the Milne boost invariant quantities E µ , P µ , T µ ν and T µ ν are different from the physical quantities E µ , P µ , T µ ν and T µ ν , but they agree if we take a special frame of the Milne boost transformation by imposing the condition v µ = u µ , (B.56) and we can take this frame by using the Milne boost with This also implies that the conservation law for the Milne boost invariant quantities E µ , P µ and T µ ν is given by where D µ is the covariant derivative in this frame and G µ = −K µν u µ . (B.60) We call this frame the "holographic frame," since if we consider the holographic duality for the Lifshitz space-times, the hydrodynamic results are obtained in this frame.

C. Lifshitz fluid in scalar background
In section 5, we have seen that the Lifshitz black hole geometry corresponds to fluids with a non-zero chemical potential but without non-trivial scalar background. On the other hand, the dimensional reduction from the higher dimensional model in Section 6 gives fluids with zero chemical potential but with non-trivial scalar background. Here, we show that the results in Section 5 can also be interpreted as fluids with zero chemical potential but with non-trivial scalar background, even though the pressure is different from that in first law of thermodynamics. The stress-energy tensor T µ ν can be expressed in the following form; where the operator O φ and coupling constant c φ are defined by (C.8) and (C.7), respectively. The coefficients of E, P and c φ O φ are the scaling dimensions of t, x i and a with appropriate signs, respectively. (See Appendix D for more details on the scaling dimensions for the Lifshitz symmetry with hyperscaling-violation.) The thermodynamics relation given by This is consistent with µ = 0.

D. Scaling dimensions in fluids
If there is no hyperscaling-violation, the Lifshitz space-time is invariant under the following scaling transformation; t → c z t , However, in the presence of the hyperscaling-violation, the following metric and gauge field are not invariant under the above transformation; After the redefinition of the coordinate (3.10), the scaling dimension of the coordinates becomes [t] = −z + θ d − 1 , Then, the scaling dimension of the velocity field should be The constant modes of the gauge field have the following dimensions (D.11) From (5.8) and (3.13), or (6.101) and (6.24), the scaling dimensions of the energy density, pressure, the particle number density and temperature are given by .