Black branes in asymptotically Lifshitz spacetime and viscosity/entropy ratios in Horndeski gravity

# Black branes in asymptotically Lifshitz spacetime and viscosity/entropy ratios in Horndeski gravity

F. A. Brito and F. F. Santos Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, 58109-970, Campina Grande PB, Brazil Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa PB, Brazil
July 31, 2019
###### Abstract

We investigate black brane solutions in asymptotically Lifshitz spacetime in 3+1-dimensional Horndeski gravity, which parameters are related to the cosmological constant as and depend on arbitrary values of the dynamical critical exponent since . For the case we recover black brane solutions in asymptotically AdS spacetime. We also investigate the shear viscosity in the 2+1-dimensional dual boundary field theory via holographic correspondence. We show that for arbitrary values of AdS radius , only two specific critical exponents are allowed: or . At the former value, we find that the bound for viscosity to entropy density ratio is violated.

## I Introduction

The celebrated Einstein gavity has been supported by strong observational evidence in many astrophysical scenarios. Despite of this, there are still fundamental problems such as dark matter, dark energy and inflationary phase of the early Universe to be well-understood in this framework. One of the principal attempts to deal with such problems in Einstein gravity concerns to coupling the theory to scalar fields. Such efforts have led to the development of the now well-nown Galileon theories which are scalar-tensor theories Nicolis:2008in (). Particularly, these studies have led to the rediscovery of the Horndeski gravity. Horndeski gravity is the most general scalar-tensor theory that was originally discovered in 1974 Horndeski:1974wa (); Anabalon:2013oea (); Cisterna:2014nua (); Heisenberg:2018vsk () and is characterized by a single scalar-tensor theory with second-order field equations and second order energy-momentum tensor. The Lagrangian producing second order equations of motion as discussed in Deffayet:2011gz (); Anabalon:2013oea (); VanAcoleyen:2011mj (); Gomes:2015dhl (); Rinaldi:2016oqp (); Cisterna:2017jmv (); Cisterna:2014nua () includes four arbitrary functions of the scalar field and its kinetic term Gleyzes:2013ooa (); Zumalacarregui:2013pma (). The term we shall focus our attention includes a nonminimal coupling between the standard scalar kinetic term and the Einstein tensor. Besides the cosmological interest this theory has also attracted attention in astrophysics such as in the searching for black hole solutions. In a more recent investigations of this theory it was shown to admit construction of black holes in Horndeski gravity that develop Hawking-Page phase transitions at a critical temperature Anabalon:2013oea (); Rinaldi:2012vy ().

The AdS/CFT correspondence (holographic description) Maldacena:1997re (); Gubser:1998bc (); Witten:1998qj (); Aharony:1999ti () has become an important tool to explore strongly coupled field theories. Black branes are fundamental as gravity duals in this correspondence. They, for instance, can be dual to strongly coupled plasma and allow to determine hydrodynamic and thermodynamic properties. Black branes in the context of the Horndeski gravity have been proposed recently Cisterna:2017jmv () in a particular sector of the theory, well known as K-essence sector, which are supported by axion scalar fields that depend on the horizon coordinates. Further scenarios involving descriptions of static black branes supported by axionic scalars/two 3-form fields were proposed in Bardoux:2012aw (). These solutions have planar (or toroidal) horizons.

In quantum critical systems develops the Lifshitz scaling , , where stands for a critical exponent. This scaling is similar to the scaling invariance of the pure AdS spacetime () in Poincaré coordinates. The holographic point of view suggests that such scaling realizes an isometry in the spacetime metric as long as the radial coordinate scales as Bertoldi:2009dt (); Bertoldi:2010ca (); Bertoldi:2009vn (). Holographic description of models involving Lifshitz scaling have attracted strong interest in recent years, mainly due to applications in condensed matter systems Kachru:2008yh (). Specially, planar black holes involving Lifshitz superconductors with an axion field as proposed in Tallarita:2014bga () are of special interest in the context of the AdS/CFT correspondence due to the application in non conventional superconductor systems.

More recently, several considerations of the AdS/CFT correspondence in the context of the Horndeski gravity have been put forward in Feng:2015oea (). By using AdS/CFT correspondence, among several other quantities, an important relation well known as the shear viscosity to entropy density ratio Feng:2015oea (); Policastro:2001yc (); Policastro:2002se (); Kovtun:2004de (); Sadeghi:2018vrf (); Kovtun:2003wp () can be computed in the dual conformal field theory. This viscosity/entropy density ratio, which is given by the bound , can be violated by the addition of higher-order curvature terms Kats:2007mq (); Brigante:2007nu (). However, as shown in Feng:2015oea () by using the Horndeski gravity this ratio can be simply violated by adjusting the Horndeski parameters, without the presence of any higher-order curvature term in the bulk. For further investigations on these issues see Feng:2015wvb () for Horndeski gravity and Sadeghi:2018vrf () for Rastall AdS massive gravity. As we shall show, in our black brane solution, the violation of the viscosity/entropy density ratio is developed by a specific critical exponent consistent with the holographic condition on the boundary. In our analyzes the violation of the ‘universal’ Kovtun-Son-Starinet (KSS) Kovtun:2003wp (); Policastro:2001yc () bound is due to the critical exponent which implies and confirms the aforementioned violations of such bound.

In the present study we find black brane solutions in asymptotically Lifshitz spacetime in Horndeski gravity. They are black hole solutions for the Horndeski gravity with flat horizon topology Tallarita:2014bga (); Feng:2015oea (). Investigations of black branes in asymptotically Lifshitz spacetime have been considered previously in Bertoldi:2009dt (); Bertoldi:2010ca (); Bertoldi:2009vn (); Tallarita:2014bga (). Lifshitz black holes with a time-dependent scalar field in Horndeski theory were studied in Bravo-Gaete:2013dca (). In our case, we find static Lifshitz black holes for Horndeski parameters related as with cosmological constant for arbitrary dynamical critical exponent . For Einstein gravity, Lifshitz solutions to Einstein equations are obtained due to massive vector fields, scalar and/or massless vector fields Hoyos:2013cba ().

Finally, to compute the shear viscosity to entropy density living in a 2+1-dimensional dual boundary field theory we consider the black brane background as the gravity dual assuming that the black brane is extended along of the two spatial dimensions where the linearization of the field equations is performed. The graviton fluctuations decouple from all modes DeWolfe:1999cp (); Brito:2018pwe () and can be treated separately Kovtun:2004de (). The shear viscosity is then computed by finding the retarded Green’s function through the holographic correspondence Maldacena:1997re (); Gubser:1998bc (); Witten:1998qj (); Aharony:1999ti (); Policastro:2001yc (); Hartnoll:2016tri (); Lucas:2015vna ().

This work is summarized as follows. In Sec. II we address the issue of finding black brane solutions in asymptotically Lifshitz spacetime. In Sec. III we compute the shear viscosity/entropy density ratio Policastro:2001yc (); Policastro:2002se (); Hartnoll:2016tri (); Eling:2011ms (); Eling:2011ct (); Lucas:2015vna (); Son:2002sd (); Jain:2015txa () and we show that the Horndeski parameters that depend on the critical exponent can violate the ‘universal’ bound . Finally, in Sec. IV we present our conclusions.

## Ii Black brane solutions in asymptotically Lifshitz spacetime in Horndeski gravity

In this section we address the issue of finding black brane solutions in asymptotically Lifshitz spacetime Bertoldi:2009dt (); Bertoldi:2010ca () in Horndeski gravity Horndeski:1974wa (); Cisterna:2014nua (). Lifshitz black holes with a time-dependent scalar field in Horndeski theory have been previously studied in Bravo-Gaete:2013dca () and black brane solutions in Horndeski gravity have been considered in Cisterna:2017jmv (). Let us now focus on the following action

 I[gμν,ϕ]=∫√−gd4x[κ(R−2Λ)−12(αgμν−γGμν)∇μϕ∇νϕ]. (1)

Note that we have a non-minimal scalar-tensor coupling where we can define a new field . This field has dimension of and the parameters and control the strength of the kinetic couplings, is dimensionless and has dimension of . Thus, the Einstein-Horndeski field equations can be formally written as in the usual way

 Gμν+Λgμν=12κTμν, (2)

where with and the scalar field equation is given by

 ∇μ[(αgμν−γGμν)∇νϕ]=0. (3)

The energy-momentum tensors and take the following form

 T(1)μν=∇μϕ∇νϕ−12gμν∇λϕ∇λϕT(2)μν=12∇μϕ∇νϕR−2∇λϕ∇(μϕRλν)−∇λϕ∇ρϕRμλνρ−(∇μ∇λϕ)(∇ν∇λϕ)+(∇μ∇νϕ)□ϕ+12Gμν(∇ϕ)2−gμν[−12(∇λ∇ρϕ)(∇λ∇ρϕ)+12(□ϕ)2−(∇λϕ∇ρϕ)Rλρ]. (4)

In our case for Einstein-Horndeski gravity we consider the following Ansatz for a general four-dimensional Lifshitz black hole with flat horizon topology Tallarita:2014bga ()

 ds2=L2(−r2zf(r)dt2+r2(dx2+dy2)+dr2r2f(r)). (5)

Now computing the equations of motion with and for the component we have the equation

 4L2κ(L2Λ−rf′(r))−f(r)[12L2κ−r2ψ2(r)(α−3γrΛf′(r))]+γr2Λf2(r)ψ(r)(7ψ(r)+4rψ′(r))=0 (6)

and for or components we find

 −8L4Λκ2+2αL2r2f(r)ψ2(r)+4κL2[2(1+z+z2)f(r)+r(3(1+z)f′(r)+rf′′(r))]+ −γr2ψ2(r)[2f2(r)(3+z(3+z))+rf′2(r)+rf(r)((7+5z)f′(r)+rf′′(r))]+ −γr2ψ(r)[2rf(r)ψ′(r)(2(1+z)f(r)+rf′(r))]=0. (7)

Finally the component can be recast in the form

 ψ(r)=  ⎷−L2Λ+f(r)+2zf(r)+rf′(r)γr2f(r)[α4κ+Λ4κ(f(r)+2zf(r)+rf′(r))]. (8)

 ϕ′(r)[rf′(r)(αL2+γrf′(r))+f(r)(αL2(3+z)+γr(rf′′(r)+(6+5z)f′(r)))]+ (9) +ϕ′(r)[(3+z)(1+2z)γf2(r)]+rf(r)ϕ′′(r)(αL2+γrf′(r)+γ(1+2z)f(r))=0.

In Hui:2012qt () static spherically symmetric configurations of certain Galileons with shift invariance was first argued to admit a no-hair theorem. The no-hair theorem for Galileons requires that the square of the conserved current , defined in (3), should not diverge at horizon. Thus, to scape from the no-hair theorem Hui:2012qt (); Bravo-Gaete:2013dca (); Rinaldi:2012vy (); Babichev:2013cya (), we need to impose that the radial component of the conserved current vanishes identically without restricting the radial dependence of the scalar field Bravo-Gaete:2013dca ():

 αgrr−γGrr=0. (10)

Recalling that we can easily note that this condition annihilates regardless its behavior at horizon.

One can show that the equation (9) is satisfied by the following solution

 f(r) = −αL2γ(2z+1)−(r0r)2z+1, (11) ψ(r) = ±√2κL2(1+α/γΛ)αr2f(r). (12)

The Einstein-Horndeski field equations (6) and (7) are satisfied by these equations as long as . This allows us to achieve a black hole solution with flat horizon topology Tallarita:2014bga (), i.e., the topology corresponding to a toroidal horizon Feng:2015oea (). Note that when , that is in accord with Bertoldi:2009dt (); Bertoldi:2010ca (); Bertoldi:2009vn () we have the black brane solution in asymptotically Lifshitz spacetime

 f(r) = 1−(r0r)2z+1, (13) ψ(r) = ±√4κL2γΛr2f(r). (14)

In Fig. (1) is depicted the behavior of and positive given above, for several critical exponents. The values and will be justified in the next section. For we find , which is the limit where we recover a solution that corresponds to black brane solutions for asymptotically AdS spacetime Bertoldi:2009dt (), and the effective AdS radius obeys the relationship . In addition, the fact that into the action (1) ensures that this is a genuine vacuum solution.

In the following we address the issue of singularity through the curvature invariant given by the Ricci scalar

 R = −r2f′′(r)+rf′(r)(5+3z)+f(r)(2z2+4z+6)L2 (15) = 2α(z2+2z+6)γ(2z+1)+(r0r)2z+1(−6z2+3z+3)L2(2z+1). (16)

We can see that for , we find a curvature singularity since the Ricci scalar diverges, which implies that we found a solution that is in fact a black brane solution. This singularity, however, can be removed for the critical exponent and using with the fact that , we have , that is, which characterizes the AdS spacetime. Note also that from equations (11)-(12) the black hole geometry is regular everywhere (except at the central singularity), the scalar field derivative diverges at horizon Anabalon:2013oea (); Feng:2015oea (); Babichev:2013cya () — see Fig. 1, but the scalar field does not explodes at horizon since it approaches to a constant near the horizon as These facts are in complete agreement with the no-hair theorem as previously discussed and evade the issues raised in Babichev:2013cya (). The scalar field equation (12) is a real function outside the horizon since for we have , and because for . We can see that at infinity the scalar field itself diverges as , but not its derivatives that are the ones present in the action, which are finite at asymptotic infinity Babichev:2013cya () — see Fig. 1.

### ii.1 Thermodynamics

Before finishing this section, for later use, we anticipate the definitions of temperature and entropy density Jain:2015txa () as given by

 T = −g′tt(r→r0)4π(√−gtt(r→r0)grr(r→r0)), (17) s = 14G√−g(r→r0)√−gtt(r→r0)grr(r→r0), (18)

where and . Using these equations we can compute the Smarr relation HosseiniMansoori:2018gdu () through the thermodynamics first law or in terms of energy density and entropy density , . Here is the volume of the unit horizon 2-manifold. In our case, that concerns a two-dimensional toroidal horizon, we assume , as in Ref. Feng:2015oea (), such that and coincide with densities. Now computing the energy density, we have

 E(r0)=∫Tds=∫r00T(r0)ds(r0)dr0dr0=(2z+1)L2rz+208πG(z+2)=2Ts(z+2). (19)

This result is in accord with Feng:2015oea (); Bertoldi:2009dt (); Bertoldi:2010ca (); Bertoldi:2009vn ().

## Iii Viscosity/entropy density ratio

In this section we present the computation of the shear viscosity in the boundary field theory through holographic correspondence Feng:2015oea (); Sadeghi:2018vrf (); Kovtun:2003wp (). We do this in the Horndeski gravity context Feng:2015oea (); Liu:2016njg (), where is found a black brane solution in the presence of asymptotically Lifshitz spacetime. In the gravity side, this planar black brane plays the role of the gravitational dual of a certain fluid. To compute the shear viscosity through holographic correspondence, we need to linearize the field equations Kovtun:2004de (); Sadeghi:2018vrf (); Kovtun:2003wp (). Thus, the effective hydrodynamics in the boundary field theory are constructed in terms of conserved currents and energy-momentum tensor by considering small fluctuations around the black brane background , where Sadeghi:2018vrf (); Kovtun:2003wp (); Liu:2016njg (); Baier:2007ix () is a small perturbation — related issues in braneworlds at Einstein and Horndeski gravity can be seen, e.g., in DeWolfe:1999cp (); Brito:2018pwe (). For the metric (5) we find the fluctuations of Ricci tensor in the form

 δ(1)Rxy=−r2f(r)2L2h′′xy+¨hxy2r2zf(r)L2−(rf(r)(z+3)+r2f′(r))2L2h′xy. (20)

Here we have disregarded the dependence of on . Recalling that and using Einstein-Horndeski equation in the Ricci form then , such that we find a Klein-Gordon-like equation with a position dependent mass as follows

 1√−g∂α(√−ggαβ∂βhxy)=2m2(r)hxy, (21)

where

 m2(r)=[2(z2−3z−1)3L2+2(z−1)3L2(r0r)2z+1]. (22)

Now considering the following Ansatz

 hxy(x,r)=∫d3k(2π)3eikxχ(r,k), (23)

with and . The equation of motion for the fluctuations assumes the following form

 1√−g∂r(√−ggrr∂rχ(r,k))=2m2(r)χ(r,k). (24)

In addition to the mass term, in general this equation also contains the contribution , but we have considered the limit and spatial momentum . For , this equation has the asymptotic solution , for , as one can be seen through the use of the metric (5) with function (13). This means that the solution is regular near the boundary for .

In our present scenario, however, from (22) we realize that , but it can be made zero. This happens as the critical exponent assumes , or approximately, or . Non integer critical exponents have appeared in previous studies, e.g., Bravo-Gaete:2013dca (); MohammadiMozaffar:2017nri (); MohammadiMozaffar:2017chk (); Passos:2016bbc (); Anacleto:2018wlj () — and references therein. Specially in MohammadiMozaffar:2017nri (); MohammadiMozaffar:2017chk () the authors argue that although the Lifshitz critical exponents in the action of a quantum field theory developing Lifshitz symmetry are assumed to be integer, there is no such limitation indeed since the obtained dispersion relation in the Hamiltonian density associated with the quantized theory shows the exact analytic continuation to non integer values of .

Furthermore, as has been shown in Foster:2016abe (), negative critical exponents although do not violate the null energy condition, they drive the holographic flow from IR to UV Kachru:2008yh () in an opposite way, that is, they correspond to directions opening up in the IR, as opposed to the UV. In the presence of one type of dynamical critical exponent, as in the present study, for the negative exponent , this reversed flow can be seen as a simply interchange of what its called the UV and the IR in the holographic RG flow.

The general explicit solution for the differential equation (24) can be given by Liu:2016njg ()

 χ(r) = C1(r/r0)β+F1(a−,b−,c−;(r0r)2z+1)+C2F1(a+,b+,c+;(r0r)2z+1)(r/r0)β−, (25) β± = −1−z2±√57(361z+38)√z−2144, (26) a∓ = ∓√57(361z+38)√z−2±19√9z−57√z−1∓114z∓57114(2z+1), (27) b∓ = ∓√57(361z+38)√z−2∓19√9z−57√z−1∓114z∓57114(2z+1), (28) c∓ = ∓√57(361z+38)√z−2∓144z∓5757(2z+1). (29)

On the horizon, this solution is

 χ(r0)=C1Γ(c−)Γ(a−+b−−c−)Γ(c−−a−)Γ(c−−b−)+C2Γ(c+)Γ(a++b+−c+)Γ(c+−a+)Γ(c+−b+), (30)

which diverges. We can remove the divergence by considering the choice

 C2C1=Γ(c−)Γ(c+−a+)Γ(c+−b+)Γ(c+)Γ(c−−a−)Γ(c−−b−). (31)

This allows us to find the following regularized solution

 χ(r0)=2πC1Γ(c−)Γ(c−−a−)Γ(c−−b−), (32)

that will be crucial to our following considerations. The integration constant can be set to unity without loss of generality.

Let us now focus on the computation of the shear viscosity of the fluid in 2+1-dimensional dual boundary field theory by proceeding as follows. We can see that we can propose an effective action for the equation (24) written as

 S=−116πG∫d3kdr(2π)3(N(r)2dχ(r,k)drdχ(r,−k)dr+M(r)2χ(r,k)χ(r,−k)), (33)

where and . This action on-shell reduces to the surface term

 S=−116πG∫d3k(2π)3(12N(r)χ(r,k)∂rχ(r,−k))∣∣∣∞r0. (34)

From the holographic correspondence Maldacena:1997re (); Gubser:1998bc (); Witten:1998qj (); Aharony:1999ti (); Hartnoll:2016tri (); Lucas:2015vna (); Son:2002sd (); Jain:2015txa (); Pang:2009wa () the Green’s retarded function can be computed via two-point function from the generator of connected correlation functions on the boundary which is given in terms of the classical action (34). Thus as is well known we have that the retarded Green’s function reads

 GRxy,xy(ω,0)=−216πG(√−ggrr|r0)[χ(r,−ω)∂rχ(r,ω)|r0], (35)

where we have admitted spatial momentum . Since the imaginary part of the Green’s function does not dependent of the radial coordinate, we have conveniently chosen to compute it at horizon .

Now, expanding the derivative of around and taking the leading term Chakrabarti:2010xy () we can write this equation as in the following

 GRxy,xy(ω,0)=−2iωL2r2016πGχ2(r0). (36)

The shear viscosity Policastro:2001yc (); Policastro:2002se (); Hartnoll:2016tri (); Jain:2015txa (); Son:2002sd () is then given by

 η = −limω→012ωImGRxy,xy (37) = L2r2016πGχ2(r0), (38)

where is the entropy density Tallarita:2014bga (). Thus, we can now write the shear viscosity/entropy density ratio as in the form

 4πηs=[2πΓ(c−)Γ(c−−a−)Γ(c−−b−)]2. (39)

The table 1 shows the obtained values for the aforementioned critical exponents. The bound for viscosity to entropy density ratio is violated as .

One should notice that from equation (38), that is

 ηs=14πχ2(r0), (40)

the ‘universal’ bound is never achieved. This seems to be related to the fact that the mass term is zero at other critical exponents. More explicitly, the mass squared (22) would be zero at horizon only at . This would render the solution in (24), and the ‘universal’ bound would be satisfied. But these exponents are out of the values where the mass is asymptotically zero — see table 1.

We can also see that even for , the AdS regime, , which can be given in terms of the cosmological constant as , is a nonzero constant everywhere. Thus, the ‘universal’ result Policastro:2001yc (); Policastro:2002se (); Hartnoll:2016tri (); Lucas:2015vna (); Son:2002sd (); Jain:2015txa () cannot be recovered even in AdS spacetime because the mass depends of the Horndeski parameters in a nontrivial way as . The exception occurs in the limit , but this would imply a cosmological constant , and since , this would simply reduce the bulk background to an asymptotically Minkowski spacetime.

## Iv Conclusions

In this paper we show that black branes with asymptotically Lifshitz spacetime are solutions of Horndeski theory for the parameters of Horndeski gravity given in terms of the critical exponents as with cosmological constant . The solution is a Lifshitz black hole with flat or toroidal horizon topology. From the holographic point of view, the planar black brane plays the role of the gravitational dual of a boundary quantum field theory that describes a certain viscous fluid. The holographic description has become an important tool to explore strongly coupled field theories, more specifically strongly coupled plasmas. In this sense, the holographic description of hydrodynamics has attracted great interest in the literature for reasons similar to those that have attracted special interest in the application of this correspondence, for example, to non conventional superconductor systems. Specially, in our present system the holography imposes specific conditions on the critical exponent . Furthermore, from the gravity side point of view the negative critical exponent interchanges the roles of the UV and IR scales in the holographic RG flow Kachru:2008yh (). On the boundary quantum field side, the viscosity/entropy density ratio is for . In other words this means that the KSS bound is violated Foster:2016abe () for negative critical exponents in our setup in the context of the Horndeski gravity. For , we have . In both cases the system does not develop an ideal fluid since either increases or decreases with the critical exponent , but never vanishes. This fact clearly characterizes a viscous fluid with entropy production Rangamani:2009xk (). Finally, we have confirmed previous conclusions that the ‘universal’ KSS bound can be violated in a wide class of conventional theories with no higher-derivative terms in the Lagrangian.

###### Acknowledgements.
We would like to thank CNPq and CAPES for partial financial support. FAB acknowledges support from CNPq (Grant no. 309258/2014-6).

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