Axionic black branes in the k-essence sector of the Horndeski model
Adolfo Cisterna, Mokhtar Hassaine, Julio Oliva, Massimiliano Rinaldi
AAxionic black branes in the k -essence sector of the Horndeskimodel. Adolfo Cisterna,
1, 2, ∗ Mokhtar Hassaine, † Julio Oliva, ‡ and Massimiliano Rinaldi
2, 5, § Vicerrector´ıa Acad´emica, Toesca 1783,Universidad Central de Chile, Santiago, Chile Dipartimento di Fisica, Universit`a di Trento,Via Sommarive 14,38123 Povo (TN), Italy Instituto de Matem´atica y F´ısica, Universidad de Talca, Casilla 747, Talca, Chile Departamento de F´ısica, Universidad de Concepci´on, Casilla, 160-C,Concepci´on, Chile TIFPA - INFN,Via Sommarive 14, 38123 Trento, Italy
Abstract
We construct new black brane solutions in the context of Horndeski gravity, in particular in its K-essence sector. These models are supported by axion scalar fields that depend only on the horizoncoordinates. The dynamics of these fields is determined by a K-essence term that includes thestandard kinetic term X and a correction of the form X k . We find both neutral and charged exactand analytic solutions in D -dimensions, which are asymptotically anti de Sitter. Then, we describein detail the thermodynamical properties of the four-dimensional solutions and we compute thedual holographic DC conductivity. PACS numbers: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ h e p - t h ] D ec . INTRODUCTION The observed current Universe is not only expanding but also accelerating because of thepresence of a source to the Einstein equations that differs from the usual mixture of darkmatter, baryonic matter, and radiation. In fact, the simplest phenomenological explanationfor the acceleration is the presence of a cosmological constant Λ in the Einstein-Hilbertaction. At the quantum level, such a constant can be interpreted as a renormalized vacuumenergy. The standard model of cosmology assumes that the current Universe is dominatedby the vacuum energy together with a large amount of cold dark matter and a tiny fractionof baryonic matter and it is called ΛCMD model.From a fundamental point of view, however, the cosmological constant has a series offundamental and conceptual issues, which makes alternatives rather appealing [1]. In generalterms, one can replace the cosmological constant with a dynamical degree of freedom that isoften modeled as a fluid with special properties that goes under the name of “dark energy”.In this way, the matter sector of the theory is implemented by a fluid with unusual but stillreasonable properties, whose dynamics dominates at late times (for a comprehensive review,see e.g. [2]). Finally, the acceleration of the Universe could also be driven by the dynamicsof the classical counterparts of the standard model fields [3].There is a somewhat more radical approach to the problem of dark energy that reliesupon a modification of general relativity (GR) in the infrared. In other words, this meansthat the Einstein equations are different at cosmological scales. Several models of modifiedgravity have been explored during the last decade [4, 5]. One of the most popular is theso-called scalar-tensor theories of gravity (STT), first proposed in the late sixties by Brans-Dicke [6] (for a modern review see e.g. [7]). STT represent the simplest way to describe adiffeomorphism invariant theory in four dimensions that avoids Ostrogradski instabilities,which typically arise in higher order theories [8, 9]. The price to pay is to introduce newdegrees of freedom in the form of one or more dynamical scalar fields. Through a suitableWeyl rescaling of the fields, it is always possible to write STT in terms of modified gravityactions where the Ricci scalar R is replaced by some arbitrary function of it, f ( R ). At leastat the classical level, STT and f ( R ) gravity are perfectly equivalent [10].In STT, gravity is described by the graviton spin-two field and one or more spin-zeroparticles, represented by scalar fields. In order to avoid possible violations of the Einstein2quivalence principle the usual prescription is that the scalar fields are only coupled to themetric and not to matter particles. This means that, in the matter action, there is nocoupling between the new degrees of freedom and ordinary matter. However, due to thenonlinearity of Einstein equations, scalar fields produce a backreaction on the metric that,in turn, affects the motion of test particles. Therefore the dynamics of matter fluids isinfluenced by scalar fields even in the case of minimal coupling [7, 11].The most general STT constructed in four dimensions and yielding at most second orderequations of motion is known as the Horndeski model [12], which is better known in its mod-ern version as the Galileon theory [13]. The latter is a STT coming from the generalizationof the decoupling limit of the brane-inspired Dvali-Gabadadze-Porrati model [14]. Galileontheory, which exhibits Galilean symmetry in Minkowski spacetime, was further covariantizedin [15] and it was finally shown to be equivalent to the original Horndeski action; see [16].The Galileon action exhibits shift symmetry and, in its covariant form, is given by L = K ( X ) − G ( X ) (cid:3) φ + G ( X ) R + G ,X ( X )[( (cid:3) φ ) − ( ∇ µ ∇ ν φ ) ] + G ( X ) G µν ∇ µ ∇ ν φ − G ,X (cid:3) φ ) + 2( ∇ µ ∇ ν φ ) − (cid:3) φ ( ∇ µ ∇ ν φ ) ] . (1)Here, X represents the canonical kinetic term for the scalar field φ , while K and G i arearbitrary functions of X . Each function can be generalized to the case in which it alsoexplicitly depends on the scalar field itself. Nevertheless, in such a case, the shift invarianceof theory is lost and this considerably complicates the integration of the field equations.Many sectors of the theory (1) have been investigated in cosmology. For instance, itwas shown that (1) contains a subset which possesses a self-tuning mechanism that allowsto circumvent Weinberg’s theorem on the cosmological constant [17]. Moreover, the sectordefined by the nonminimal kinetic coupling controlled by the Einstein tensor exhibits inter-esting inflationary properties without the need of ad hoc potential terms [18–22]. Likewise,the nonminimal coupling between the Einstein tensor and the scalar field kinetic term can,on large scales, mimic cold dark matter and also flatten the rotational curves of galaxies [23].Finally, several works have been devoted to the study of cosmological perturbations with theaim of finding observable deviations from GR in large-scale structures and the conditionson the parameter space that avoid too large gravitational instabilities [24].Technically speaking, the so-called k -essence models of dark energy belong to the classof gravitational theories represented by (1). In k -essence, the acceleration of the Universe3both at early and late times) can be driven by the kinetic energy instead of the potentialenergy of the scalar field [25]. The model was first introduced in [26] and then specificallyused as dark energy models in [27–32]. These models are characterized by a nonlinear kineticterm for the scalar field and are expressed in (1) by the arbitrary function K ( X ) (togetherwith with G = 1 and G = G = 0). Quantum and classical stability of k -essence have beeninvestigated [33]. In particular, the classical stability and their perturbations are crucial todiscriminate the model from standard GR in view of the forthcoming Euclid mission [34].One fundamental step that may put the theory on a solid theoretical foot is the con-struction of black hole solutions. In principle, there is a no-hair theorem that prevents theexistence of nontrivial black hole solutions in Galileon gravity [35]. However, there are waysto get around this theorem and several black hole solutions have been found for particularsectors of (1), in particular, the one containing the nonminimal coupling between the Ein-stein tensor and the kinetic term. Spherically symmetric solutions were found in [36–39]where their thermodynamical properties were also studied. Moreover, anti-de Sitter (AdS),asymptotically flat stealth and Lifshitz solutions with a self-tuned effective cosmologicalconstant were found making use of a time-dependent scalar field in [44, 45]. Charged solu-tions were found in [46, 47]. Recently also, for the sectors of (1) controlled by G and G ,analytical and numerical solutions have been found [48, 49]. There is still one sector of (1), where black holes solutions are little known, that isthe k -essence sector governed by K ( X ). This work aims to fill this gap, at least partially,by exploring black hole configurations in the sector of (1) that contains terms like X k ,in addition to the usual kinetic term. To construct our solutions, instead of considering aspherically symmetric scalar field, we use axion fields which depend linearly on the Cartesiancoordinates along the flat horizon. We see later that these kinds of configurations areused in the context of dual condensed matter systems due to the fact that they break thetranslational invariance of the dual field theory [52]. This is an easy way to circumvent theno-hair theorem [35], which is mostly based on the fact that the equation of motion for the This model has been also thoroughly investigated in the context of astrophysical configurations such asneutron and boson stars [40–43]. In [50] solutions have been obtained considering a kinetic coupling controlled by the Gauss-Bonet invariant. These kind of Lagrangians are used to obtain cosmological models with equation of state parametersatisfying ω < ∇ µ J µ = 0.For the case of spherically symmetric scalar fields, this current is given by the component J r whose modulus diverges at the horizon. In the case studied here, our axion fields yielda finite current on the black hole horizon while simultaneously satisfying the Klein-Gordonequation. Moreover, the contribution to the equations of motion coming from the K-essenceterm is still spherically symmetric and the energy of the solutions remains finite. In Ref. [53], a static black brane with axionic charge generated by the presence of two3-form fields was presented. The symmetry of the solution is endowed with a planar hori-zon with a lapse function mimicking that of a hyperbolic black hole in AdS. This apparentdiscrepancy between the horizon topology and the metric behavior was shown to be dueto the presence of the axionic charges, which play the role of an effective curvature term.The thermodynamical properties and the possibility of phase transitions were also reportedin [53]. These ideas were also applied to construct black branes with a source given by ascalar field nonminimally coupled to gravity [54, 55]. Planar/toroidal black holes with ascalar field are of special interest in the context of the AdS/CFT correspondence [57] due, inparticular, to their applications in nonconventional superconductor systems [58, 59]. Withinthis approach, the nonzero condensate behavior of the unconventional superconductors canbe reproduced by means of a hairy black hole at low temperature with a hair that shoulddisappear as the temperature increases. Usually, planar/toroidal solutions suffer from singu-lar behaviors due to the lack of a curvature scale on the horizon. Nevertheless, this situationis successfully circumvented using axion fields which are homogeneously distributed alongthe horizon coordinates, providing in this way an effective curvature scale which makes thespacetime nonsingular. Several solutions with these ingredients have been reported in orderto study different aspects of their holographic dual systems [52, 61–67]. A very interestingapplication is the construction of homogeneous black string and black p-branes with negativecosmological constant, with no more ingredients that minimally coupled scalar fields [68].Moreover, recently these ideas have been applied to the case of Horndeski theory, specificallyto the nonminimal kinetic coupled sector [69–71]. This is similar to what happens in the case of the linearly time-dependent scalar fields considered in [44]. A new planar solution with a conformally coupled scalar field will be presented in [56] This solutionrepresents a novel generalization of the Bekenstein black hole plus cosmological constant, without selfinteraction and free of self-tuned parameters.
II. THE MODEL
We consider the following K-essence Lagrangian in four dimensions L = K ( X , X ) (2)where the two scalar fields, with their kinetic terms X and X , correspond to the twoaxion fields. As considered below, the axion fields are homogenously distributed along thecoordinates of the planar horizon. This explains the reasons for considering two axion fieldsin four dimensions. As mentioned before, we study the case in which the dynamics ofeach scalar field is governed by a standard kinetic term plus a nonlinear contribution givenby an arbitrary power of X . More precisely, we consider a K -term of the form K ( X i ) = − (cid:80) i =1 ( X i + γX ki ), and hence our four-dimensional action reads I [ g µν , φ i ] = (cid:90) (cid:34) κ ( R − − (cid:88) i =1 (cid:32) ∇ µ φ i ∇ µ φ i + γ (cid:18) ∇ µ φ i ∇ µ φ i (cid:19) k (cid:33)(cid:35) d x √− g , (3)where we have defined X i = ∇ µ φ i ∇ µ φ i with i = 1 ,
2. The coupling γ (with mass dimension4 − k )is supposed to be positive in order to avoid phantom contributions. For γ = 0 werecover the case of two minimally coupled scalar fields studied in [52, 53]. Even if thesolutions can be constructed in arbitrary dimensions we focus our attention on the four-dimensional case, leaving the D-dimensional extension to Appendix A. The variation of theaction with respect to the metric yields the following Einstein equations, κ ( G µν + Λ g µν ) = 12 (cid:88) i (cid:2) ∂ µ φ i ∂ ν φ i − g µν X i + γ ( kX k − i ∂ µ φ i ∂ ν φ i − g µν X ki ) (cid:3) , (4) Recently, solutions for the minimally coupled case with phantom axion fields where studied in [72]. (cid:2) (1 + γkX k − i ) g µν + γk ( k − X k − i ∇ µ φ i ∇ ν φ i (cid:3) ∇ µ ∇ ν φ i = 0 . (5)We now impose the planar metric ansatz ds = − F ( r ) dt + dr G ( r ) + r ( dx + dx ) , (6)and we assume that the axion fields depend on the coordinates ( x , x ) only. In the case F = G , the Klein-Gordon equations are easily solved by φ = λx , φ = λx . (7)Note that these scalar fields can be dualized to construct solutions with backreacting 2-forms, B (2) , by setting H ( i )(3) = dB ( i )(2) = (cid:63)dφ i , i = 1 , , (8)where (cid:63) denotes the Hodge dual.In order to ensure that the solutions of the previous equations do not generate ghosts, itis important to check whether or not they satisfy the null energy condition given by T µν n µ n ν ≥ , i = 1 , , . (9)Classical stability is instead guaranteed by a positive sound speed, namely c s = K , X i K , X i +2 X i K , X i ,X i > . (10)In our model a sufficient condition to satisfy simultaneously both requirements is k > / k > / In [53] the authors considered Einstein gravity with a source given by two 3-form fields (whose Hodgeduals can be identified with the exterior derivatives of two scalar fields). A Birkoff’s like theorem wasestablished where it is shown that each of the two 3-form fields must depend on one for the transversespatial coordinate. II. K-ESSENCE BLACK HOLES WITH AXIONS
Under the conditions described above Eqs. (4) and (5) have the following exact black branesolution: F ( r ) = G ( r ) = r l − Mr − λ κ + γλ k k (2 k − κ r − k ) (11) φ = λx , φ = λx . The case k = 3 / / < k < /
2, the asymptotic behavior of our AdS solutions differs from the standard onesdefined in [73], and, as a consequence, configurations with infinite mass could be obtained.Thus, from now on we consider only the case k > /
2, which allows the use of standardmethods to compute the mass of our solutions.It is evident that the effect of the axion fields is to include an effective hyperbolic curvaturescale on the metric proportional to the axion parameter λ . By setting γ = 0, we find thesolution first described in [53].These solutions can be easily generalized to include electric and magnetic monopolecharges. In order to do this it is sufficient to include in the action (3) the standard Maxwellterm S = − (cid:90) F µν F µν d x √− g. (12)Then, the Maxwell equation ∇ µ F µν = 0 , (13)is easily solved by A = − Q e r dt + Q m x dx − x dx ) , (14)where Q e and Q m are the electric and magnetic monopole charges. Finally, the generalcharged solution of the Einstein equations reads F ( r ) = G ( r ) = r l − Mr − λ κ + γ λ k k (2 k − κ r − k ) + 14 κr ( Q e + Q m ) . (15)Solutions (11) and (15) are the neutral and charged K-essence generalization of the solutionsfound in [52, 53], which are known to possess interesting holographic properties. We willdiscuss a particular application in Sec. V. 8 very interesting case is the one corresponding to Λ = 0. In particular, the unchargedsolution takes the form F ( r ) = G ( r ) = − Mr − λ κ + γ λ k k (2 k − κ r − k ) . (16)This solution can have two horizons. To show this in a simple way let us consider the case k = 2. Then, the horizon location can be found algebraically by solving the equation F ( r ) = G ( r ) = − Mr − λ κ + γ λ κr . (17)The two distinct solutions are r = − M κ + (cid:112) M κ + 2 λ γ λ , (18) r = − M κ + (cid:112) M κ + 2 λ γ λ . (19)For γ > r = r c = r ) which surrounds a curvature singularity located at the horizon. However, if both γ and M are negative, it is possible to find two horizons. This is evident upon the substitutions M → −| M | and γ → −| γ | , which gives the location of an event and a cosmological horizonlocated respectively at r = r h and r = r c , with r c = 2 | M | κ + (cid:112) | M | κ − λ | γ | λ (20) r h = 2 | M | κ − (cid:112) | M | κ − λ | γ | λ . (21)As we mentioned above, negative values of γ could induce violations of the null energycondition or nonhyperbolicity of the Klein-Gordon equation. However, this violation maybe hidden behind the event horizon provided the condition | M | > √ λ κ (cid:112) | γ | (22)is satisfied. IV. THERMODYNAMICAL PROPERTIES OF ADS K-ESSENCE BLACKBRANES
In order to explore some holographic applications, we first provide a complete and detailedanalysis of the thermodynamic features of the electrically charged AdS solutions. Note that9uch studies have been done for Horndeski black holes with sources given by scalar fields,see e.g. [74].In our case, the thermodynamics analysis is carried out through the Euclidean approach.In this case, the partition function for a thermodynamical ensemble is identified with theEuclidean path integral in the saddle point approximation around the classical Euclideansolution [75]. Since we are interested in a static metric with a planar base manifold, it isenough to consider the following class of metric, ds = N ( r ) F ( r ) dτ + dr F ( r ) + r (cid:0) dx + dx (cid:1) , where τ is the periodic Euclidean time related to the Lorentzian time by τ = i t , and theradial coordinate r h ≤ r < ∞ . Now, in order to have a well-defined reduced action principlewith a Euclidean action depending only on the radial coordinate, some precautions mustbe taken. Indeed, in the present case, we are interested in configurations where the scalarfields φ i do not depend on the radial coordinate but rather on the planar coordinates.Nevertheless, since the scalar fields only appear in the action through their derivativesthat are constants, we can “artificially” introduce radial scalar fields and their associated“conjugate momentum” asΨ i ( r ) := (cid:90) r ∂ i φ i dr, Π ( i ) := − ∂ r Ψ i ( r ) , ˆΨ i ( r ) := (cid:90) r N ∂ r Ψ i dr, (23)Ψ i,k ( r ) := (cid:90) r ( ∂ i φ i ) k dr, Π ( i,r ) := − ∂ r Ψ i,k , ˆΨ i,k ( r ) := (cid:90) r N k − r − k ) ∂ r Ψ i,k dr. Under this prescription, the Euclidean action I E is given by I E = σβ (cid:90) Rr h (cid:88) i =1 (cid:110) N (cid:104) i ) + 2 γ k − r − k ) Π i,k ) + 12 r Π A + 2 κrF (cid:48) + 2 κF − κr l (cid:105) − i Π (cid:48) ( i ) − γ ˆΨ i,k Π (cid:48) ( i,k ) − A Π (cid:48) A (cid:111) dr + B E , (24)where β is the inverse of the temperature, σ stands for the volume of the two-dimensionalcompact flat space and Π A denotes the conjugate momentum to the vector potential A ,Π A = − r A (cid:48) N .
The Euclidean action is obtained in the limit R → ∞ and the boundary term B E is fixed byrequiring that the action has a well-defined extremum, i.e. δI E = 0. It is easy to check thatthe field equations obtained by varying the reduced action yield the electrically charged AdS10olution (15) with Q m = 0. In fact the variations with respect to F and A give respectively2 κrN (cid:48) = 0 and Π (cid:48) A = 0. The first equation implies that N is constant and without loss ofgenerality, can be taken to be N = 1. The second equation imposes the electric potentialto have the Coulomb form A t = Q e r . On the other hand, the variations with respect to theconjugate momenta Π ( i ) , Π ( i,k ) and Π A yield equations that are trivially satisfied while thoseobtained by variation with respect to ˆΨ i and ˆΨ i,k can be easily solved by choosingΠ ( i ) = − λ, Π ( i,k ) = − λ k ⇒ φ i = λx i . Finally, the equation obtained by varying N i ) + 2 γ k − r − k ) Π i,k ) + 12 r Π A + 2 κrF (cid:48) + 2 κF − κr l = 0 , gives rise to a differential equation for the metric function F whose integration yields (15).Now, in order to compute the boundary term, we consider the formalism of the grandcanonical ensemble where the temperature β − as well as the “potentials” at the horizon A ( r h ) , ˆΨ i ( r h ) and ˆΨ i,k ( r h ) are fixed. The extremal condition δI E = 0 implies that thecontribution of the boundary term must be given by δB E = (cid:104) (cid:88) i =1 (cid:16) − κσβN rδF + 2 σβ ˆΨ i δ Π ( i ) + 2 σβγ ˆΨ i,k δ Π ( i,k ) + σβAδ Π A (cid:17) (cid:105) r = Rr = r h . Without loss of generality, we can set again N = 1, and the contribution at infinity reducesto δB E ( R ) = 4 κσβδM = ⇒ B E ( R ) = 4 κσβM. At the horizon, in order to avoid the conical singularities, the variation of the metric functionat the horizon is given by δF | r h = − πβ δr h , and hence one gets B E ( r h ) = 4 πκσr h + σβA ( r h ) Q e − σβ (cid:88) i (cid:16) λ ˆΨ i ( r h ) + γλ k ˆΨ i,k ( r h ) (cid:17) . Finally, the boundary term becomes B E = 4 κσβM − πκσr h − σβA ( r h ) Q e + σβ (cid:88) i (cid:16) λ ˆΨ i ( r h ) + γλ k ˆΨ i,k ( r h ) (cid:17) . (25)The Euclidean action is related to the Gibbs free energy G by I E = β G = β M − S − βA ( r h ) Q e − β (cid:88) i (cid:16) ˆΨ i Q i + ˆΨ i,k Q i,k (cid:17) . M is given by M = (cid:18) ∂I E ∂β (cid:19) ˆΨ i , ˆΨ i,k − ˆΨ i β (cid:18) ∂I E ∂ ˆΨ i (cid:19) β − ˆΨ i,k β (cid:32) ∂I E ∂ ˆΨ i,k (cid:33) β = 4 κσM = 4 κσ (cid:18) r h l − λ r h κ + γλ k r − kh k +1 (2 k − κ + Q e κr h (cid:19) , (26)while the entropy S , the electric charge Q e and the axion charges Q i , Q i,k are defined by S = β (cid:18) ∂I E ∂β (cid:19) ˆΨ i , ˆΨ i,k − I E = 4 πκσr h , Q e = − β (cid:18) ∂I E ∂A ( r h ) (cid:19) β = σQ e , (27) Q i = − β (cid:18) ∂I E ∂ ˆΨ i (cid:19) β = − σλ, Q i,k = − β (cid:32) ∂I E ∂ ˆΨ i,k (cid:33) β = − σγλ k . With these results it is trivial to see that the first law holds, namely d M = T d S + A ( r h ) d Q e + (cid:88) i (cid:16) ˆΨ i ( r h ) d Q i + ˆΨ i,k ( r h ) d Q i,k (cid:17) . We conclude this section by comparing our results with those obtained recently for a similarmodel with phantom axion fields [72]. In this reference, the thermodynamics analysis of thephantom black hole solution is carried out without considering the axion parameter constant λ as an axion charge. Because of that, the thermal properties of the phantom solution arequite analogous to those of the Schwarzschild-AdS black hole. The importance of consideringaxion parameter constant λ as an axion charge is particularly important when holographicapplications and phase transitions are studied. V. HOLOGRAPHIC DC CONDUCTIVITY
Charged black brane solutions provide a perfect setup to compute holographic conduc-tivities [52, 69, 77–80]. This can be done by constructing a conserved current with radialdependence from which it is possible to obtain the holographic properties on the boundaryin terms of the black hole horizon data. Here, we are interested in the effects of the nonlinearkinetic term, controlled by the coupling constant γ , on the conductivity of the dual fieldtheory. Along the lines of [79], we introduce a perturbation of the fields of the form ds = − F ( r ) dt + dr G ( r ) + r (cid:0) dx + dx (cid:1) + 2 (cid:15)r h tx ( r ) dtdx + 2 (cid:15)r h rx ( r ) drdx (28) See also [76] for the computation of thermoelectric transport coefficients of systems which are dual tofive-dimensional, charged black holes with horizons modeled by Thurston geometries. A = µ (cid:16) − r r (cid:17) dt − (cid:15)Edt + (cid:15)a x ( r ) dx (29)for the gauge field, and φ = ˚ φ + (cid:15) Φ ( r ) λ (30)for one of the axion fields, with the background axion field fixed by ˚ φ = λx . Here µ = Q e /r is the chemical potential. Plugging this in the field equations and keeping the linear termsin (cid:15) , Maxwell equations allow one to construct the current density in terms of the horizonradius r h as J = λ r h + µ r h + 2 − k λ k r − kh γkλ r h + 2 − k λ k r − kh γk E, (31)which trivially leads to a DC conductivity of the form σ = ∂J∂E = λ r h + µ r h + 2 − k λ k r − kh γkλ r h + 2 − k λ k r − kh γk . (32)Note that when γ = 0, this expression coincides with the result obtained in Eq. (4.4)of reference [79] for minimally coupled axions with standard kinetic terms in D = 4 (seealso [52] and [69]). Note that in such a case, in terms of the chemical potential, the DCconductivity remains constant as the radius of the horizon changes. Figure 1 shows thebehavior of the DC conductivity for a quadratic derivative self-interaction ( k = 2) and fora cubic one ( k = 3). The plots show that in the limit T → γ = 0. As the strength of the nonlinearities of the axions (controlledby γ ) increases, the conductivity at low temperature decreases, and one can see that in sucha case larger temperatures are required to recover the result with minimally coupled, freeaxions. VI. CONCLUDING REMARKS
As we know, the Horndeski model [12] is the most general STT we can construct withsecond order equations of motion in four dimensions. In its shift invariant form, it is given13 = γ = γ = γ = σ DC ( k = ) σ DC ( k = ) γ = γ = γ = γ = FIG. 1: DC conductivities as a function of temperature for the cases k = 2 (left panel) and k = 3(right panel). Different curves on each panel correspond to different values of the derivative self-interaction coupling γ . We have set the chemical potential µ =1, as well as the AdS radius l = 1,the axions constant λ = 1 and κ = 1. by the covariant version of Galileon gravity [15] whose Lagrangian is given by (1). Black holesolutions with spherical symmetry and with flat asymptotic behavior are forbidden by theno-hair results of Hui and Nicolis [35]. Their argument relies on the shift invariance of (1)which forces the scalar field equation of motion to be written as a current conservation law.Then, by demanding that the norm of this current is finite on the horizon of the hypotheticalblack hole solution it is possible to show that, for spherically symmetric solutions with flatasymptotic geometry, the scalar field must be trivial. In spite of this, for the particularmodel of the nonminimal coupling between the Einstein tensor and the kinetic term of thescalar field, there are two ways to circumvent the no-hair conjecture. The first one is torelax the asymptotic flatness of the solutions allowing (A)dS behaviors [36–38]. The secondone is to consider scalar fields that do not share the same symmetries of the metric, butare nevertheless nontrivial. In the latter case, the simplest way is to consider scalar fieldslinearly dependent on time [44].In this work we have applied the second strategy in order to construct black brane so-lutions in the k -essence sector of Horndeski/Galileon gravity, specifically in the model inwhich, along with the standard kinetic term, a nonlinear contribution of the form X k is in-cluded. This sector represents scalar fields with nonlinear kinetic terms without need of anycoupling between the scalar field and the curvature. Specifically speaking, to construct our This could be interesting due to recent results which indicate that the inclusion of nonminimal couplings D − i = ( D −
2) scalar fields on the theory. It followsthat each scalar current norm | J i | does not diverge on the horizon and, at the same time,satisfies the continuity equation, ∇ µ J µi = 0, with a nontrivial profile for the scalar field.In order to satisfy the null energy condition and to ensure the hyperbolicity of the Klein-Gordon equation we have constrained the possible values of k to be greater than ( D − / k endows our solutions with the same asymptotic behavior of GR without affectingthe behavior of the mass term at infinity. Our solutions possess flat horizon; however theinclusion of the axion fields provides a new curvature scale including a noncanonical hy-perbolic term on the metric. The electrically and magnetically charged extension is shownto exist as well as the higher-dimensional extension. We observe that in the case in whichΛ = 0 solutions possessing a cosmological horizon are also possible provided both mass andcoupling γ are negative. We have analyzed the thermodynamical properties of the asymp-totically AdS solutions in order to study the possible holographic applications and we haveprovided an explicit computation for the DC conductivities in the holographic dual theoryof the electrically charged configurations. It would be also interesting to see whether ornot these solutions violate the reverse isoperimetric inequality (RII) as is the case of theHorndeski black brane solutions with axions recently constructed in [69] and studied in [70].Another possible extension of this work would be to see how the nonlinear contribution forthe scalar field affects the realization of the momentum dissipation phenomena extensivelystudied for minimally coupled scalar fields [52] and also recently in the Einstein coupledsector of Horndeski gravity [69]. VII. ACKNOWLEDGEMENTS
A.C and M.R acknowledge enlightening comments and discussions with Professors. L.Vanzo and S. Zerbini. A.C particularly expresses his gratitude to the Physics Departmentof the University of Trento for its kind hospitality during the development of this work. with the curvature might induce problems when defining a well-possed initial value problem in Horndeskitheory [81].
Appendix A: D-DIMENSIONAL SOLUTION
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