Aplicações do Setor John da Gravidade de Horndeski nos Cenários de Brana Negra e Relação de viscosidade/entropia, Mundo Brana e Cosmologia (In Portuguese)
aa r X i v : . [ h e p - t h ] J u l Constraints of Horndeski parameters in AdS/BCFT
F. F. Santos ∗ Departamento de Física, Universidade Federal da Paraíba,Caixa Postal 5008, 58051-970 João Pessoa PB, Brazil (Dated: July 14, 2020)In this article, we propose a study of the implications of the AdS/BCFT corre-spondence for the parameters of Horndeski’s theory. To carry out this investigation,we introduced a Gibbons-Hawking surface term with γ dependent terms associatedwith the term Horndeski. On the gravity side, Horndeksi’s gravity has a solutionfor the black hole BTZ and we investigate the restrictions that these parameters aregiven by α and γ suffer for the thermodynamics of this black hole, as well as theireffects on the fluid/gravity correspondence. ∗ Electronic address: fabiano.ff[email protected]
I. INTRODUCTION
In the last few years, investigations involving AdS/CFT correspondence have given greatsupport to the dynamics of tightly coupled condensed matter systems, especially in thestudy of the universal limits of the transport coefficients [1–8]. One of these universal limitsis the known shear viscosity [5, 6] which is conjectured based on holographic bottom-upmodels. We recently discussed the impact of Horndesdi’s theory parameters to study thebehavior of the viscosity-entropy ratio, which provided us that for some common substances:helium, nitrogen and water, the ratio is always substantially greater than its value in dualgravity theories [9], that is, η/s > / (4 π ) and that for unconventional superconductingsystems, the entropy viscosity/density ratio is η/s < / (4 π ) which violates the KSS limit[5]. In addition to this, the violation of this limit made it possible to probe the violation ofthermal conductivity in Horndeski’s gravity as discussed by [10]. In this way, Horndeski’sgravity has given remarkable support in the violation of the transport coefficients [9–11].Furthermore, new considerations for Hondeski’s gravity involving a quartic model have beenelegantly presented by [12] where this model is considered to be shift-invariant. Within thismodel, it is possible to build planar black holes with a non-trivial axis profile, which allowsexploring thermodynamic properties. However, the profile of the scalar field dissipates themoment in the contour theory. Thus, within the context presented by the authors, it ispossible to derive a transport matrix allowing a description of the holographic dual linearthermoelectric response to an external electric field with a thermal gradient. For holographicscenarios, the properties of Horndeski’s theories for a general coupling constant, using the a -theorem in Horndeski’s gravity, showed that there are critical points and the a -theorem isestablished [13]. In such a prescription, we have that the Horndeski scalar can be performedas a holographic flow of RG.In addition, from the AdS/CFT correspondence, another boundary theory has drawnattention in the study of holographic transport coefficients where these transport coefficientswere presented by [14–16]. This theory was proposed by Takayanagi [17]. Where the ideabehind this theory is an extension of the AdS/CFT correspondence [18–21], to which we adda Gibbons-Hawking boundary term [22] and this prescription became known as BoundaryConformal Field Theory (BCFT) and correspondence became known as AdS/BCFT [14–17, 22]. Motivated by the recent applications of AdS/CFT correspondence in the Horndeskiscenario, we propose a study of the impact of the parameters of this theory in the AdS/BCFTscenario. Thus, a Gibbons-Hawking surface term for Horndeski’s gravity is necessary, whichwas recently introduced by [23]. However, despite the restrictions imposed for theoriesinvolving kinetic couplings like the Jhon sector of Horndeski’s gravity by the recent eventGW170817, this sector can be revived to accommodate a description in which there is nophysical process of particles for the inflationary and post-inflationary era that changes thegravitational mass [24].As Horndeski’s theory involves additional degrees of freedom, which are scalar fields [25],we have to contour the conditions for the scalar field, telling us that it must fall fast enoughnear the limit (i.e., reaches zero or a constant), close to the limit [4, 25]. In this sense,Neumann boundary conditions or Dirichlet boundary conditions can be imposed for freescalars [26–29]. In this work, we will investigate the implications of Horndeski’s parametersfor the entropy of a BTZ black hole, where we have a high-temperature phase in the "bulk"that is described by the BTZ black hole. Another investigation into holographic transportcoefficients using a BTZ black hole in the context of Horndeski was presented elegantly by[2]. Furthermore, as we know it is an extremely important state function, and in informationtheory. Regarding the information process, we have that the entanglement entropy offersus an important observation when the spacetime M has an additional contour crossing itscontour ∂A [30–32]. For this case, we have that the theory is non-gravitational and lives in avariety with an outline. On the other hand, for the AdS/CFT case, we can observe that thissituation occurs when the theory of the conforming field (CFT) is defined in a variety witha contour, called field theory according to the contour (BCFT) [17]. Recent investigationshave shown that interlacing entropy has been computed for the AdS/BCFT configurationand has been shown to characterize BCFT.Through limited resources, for which there are limits to the storage and processing ofinformation [33]. Due to the fact that the limits of information storage are much more stud-ied and well understood, these in turn impose restrictions on the parameters of Horndeski’stheory. Furthermore, as the entropy of the black hole involves the contributions of "bulk"and the outline we will discuss the holographic g -theorem where we have g UV > g IR . Thiscondition is established based on Horndeski’s parameters. In the fluid/gravity correspon-dence we will investigate the implications of Horndeski’s parameters for fluid regimes. In thisduality, there is an equivalence between the gravitational dynamics in bulk and equationsof motion of the theory of double fields in the hydrodynamic regime. In addition, we havethat the connection resides in the definition of gravity of the limit voltage energy tensor,which is used in fluid/gravity correspondence and is therefore equivalent to the Neumannlimit condition for the metric [16]. Thus, we can adopt a fluid/gravity structure to probe theeffects of Horndeski’s parameters on the AdS /BCFT problem. This work is summarizedas follows. In Section II, we present the total action with the boundary term [23] for thescenario of Horndeski [9, 25]. In the Section III, we consider a BTZ black hole and find theQ contour profile. In the Section IV,although the scalar field is constant at the boundary, itis still possible to evaluate the restrictions imposed by thermodynamics and the g -theoremfor the parameters of Horndeski’s theory. In Sec. V motivated by the works [15, 16] wepresent a boundary fluid from AdS/BCFT and discuss Horndeski parameters for that fluid.Finally, in the Section VI, we present our conclusions. II. THE SETUP
In this section, we will present the outline systems for Horndeski’s gravity. Thus, asdiscussed by [17, 22] for the construction of boundary systems we need to add a Gibbons-Hawking surface term. In addition, this Gibbons-Hawking surface term for the Horndeski γ -dependent gravity scenario was proposed by [23]. Motivated by these works, we proposea total action as follows: S = S N + S Nm + S Q + S Qmat + S ct = Z N d x √− g L H + S m + 2 κ Z bdry d x √− h L bdry + 2 κ Z Q d x L mat + S ct (1) L H = κ ( R − −
12 ( αg µν − γG µν ) ∇ µ φ ∇ ν φ (2) L bdry = ( K − Σ) + γ ∇ µ φ ∇ ν φn µ n ν − ( ∇ φ ) ) K + γ ∇ µ φ ∇ ν φK µν (3) L ct = c + c R + c R ij R ij + c R + b ( ∂ i φ∂ i φ ) + ... (4)Where φ = φ ( r ) and we define a new field φ ′ ≡ ψ and κ = (16 πG ) − . In the action N the field has dimension of ( mass ) and the parameters α and γ control the strength of thekinetic couplings, α is dimensionless and γ has dimension of ( mass ) − [9, 25]. L mat is aLagrangian of possible matter fields on Q and L bdry term corresponds with γ -dependentterms are associated with the Horndeski term [23]. In the bulk action where K µν = h βµ ∇ β n ν is the extrinsic curvature and the traceless is given by contraction K = h µν K µν and h isthe induced metrics on the hypersurface Q [15–17, 22]. Furthermore, S ct is the boundarycounterterm that is necessary for asymptotic AdS spacetime. For the action (1), we haveit is invariable under the displacement symmetry φ → φ + constant and under the discretetransformation φ → − φ . In this way, by imposing the Neumann boundary condition insteadof the Dirichlet one, we obtain the boundary condition K αβ − h αβ ( K − Σ) + γ H αβ = κS Qαβ (5)With H αβ = − √− h δS K,φ δh αβ (6) S Qαβ = − √− h δI Q δh αβ ; I Q = 1 κ Z √− h L mat (7)In our case we impose the Neumann boundary condition considering I Q [ matter ] = constant [17] for the second term in the total action (1), which imply that the S Qαβ = 0 [15–17, 22],so we can write K αβ − h αβ ( K − Σ) + γ H αβ = 0 (8)On the gravitational side for Einstein-Horndeski gravity we have E µν [ g µν , φ ] = S Qαβ h αµ h βν δ ( r ) (9) E φ [ g µν , φ ] = 1 √− h δ L bdry δφ (10)With E µν [ g µν , φ ] = G µν + Λ g µν − α κ (cid:18) ∇ µ φ ∇ ν φ − g µν ∇ λ φ ∇ λ φ (cid:19) (11) − γ κ (cid:18) ∇ µ φ ∇ ν φR − ∇ λ φ ∇ ( µ φR λν ) − ∇ λ φ ∇ ρ φR µλνρ (cid:19) − γ κ (cid:18) − ( ∇ µ ∇ λ φ )( ∇ ν ∇ λ φ ) + ( ∇ µ ∇ ν φ ) (cid:3) φ + 12 G µν ( ∇ φ ) (cid:19) − γ κ (cid:20) − g µν (cid:18) −
12 ( ∇ λ ∇ ρ φ )( ∇ λ ∇ ρ φ ) + 12 ( (cid:3) φ ) − ( ∇ λ φ ∇ ρ φ ) R λρ (cid:19)(cid:21) ,E φ [ g µν , φ ] = ∇ µ [( αg µν − γG µν ) ∇ ν φ ] . (12)We can see that as S Qαβ = 0 , we have to E µν [ g µν , φ ] = 0 . III. BTZ BLACK HOLE
We will now present some necessary boundary conditions to work with the equation (8)to investigate the incorporation function represented by y ( r ) [15–17, 22]. In that way, let’sconsider the black hole BTZ in three-dimensional shape ds = L r (cid:18) − f ( r ) dt + dy + dr f ( r ) (cid:19) (13)A condition that deals with static configurations of black holes, which can be sphericallysymmetric for certain Galileons, which was presented by [34] to discuss the no-hair theorem.However, in order to escape this no-hair theorem, we have to keep the radial componentof the conserved current disappearing in an identical way without restricting the radialdependence of the scalar field: αg rr − γG rr = 0 . (14)Thus, for this condition we have E φ [ g µν , φ ] = 0 . Thus, we have to φ ′ ( r ) ≡ ψ ( r ) , providingthe annihilation of ψ ( r ) , regardless of its behavior on the horizon. Thus, we have thatthe metric function f ( r ) can be found using the equation (14). It can be shown that theequation E φ [ g µν , φ ] = 0 is satisfied by the following solution f ( r ) = αL γ − (cid:18) rr h (cid:19) , (15) ψ ( r ) = − L κ ( α + γ Λ) αγr f ( r ) . (16)these equations satisfy both E φ [ g µν , φ ] = 0 and E µν [ g µν , φ ] = 0 e this fact confirms that wecan assume fields of generic matter on the sides of the boundary Q and on the gravitationalside. In addition, looking at the equation (15), we have α/ (3 γ ) = L − which is definedas an effective radius of AdS [35] where the solutions can be asymptotically dS or AdS forthe following conditions α/γ < and α/γ > , respectively. The scalar field given by theequation (16) is real for α > and γ < . In this way, we can write that γ = − α/ Λ and thatpoint gives us a constant scalar field φ bdry = constant, this fact is in full agreement with thefact that the scalar field must fall sufficiently fast near the boundary (i.e it reaches zero ora constant) [4, 25]. The dual BCFT temperature is given by T BCF T = 1 / πr h . We can seethat this condition for a constant scalar field, we have a reduction of Horndeski’s gravity toEinstein’s gravity, and the equation (8) has been reduced to the usual case. Furthermore,the conditions E φ [ g µν , φ ] = 0 and E µν [ g µν , φ ] = 0 implies that we have a full agreementwith the total variation of the stock, that is, the variation of δS , will imply the individualvariation of each term. Thus, we can analyze the Q limit for this condition, as we know thatthe normal vectors and the induced metric can be presented as n µ = (cid:18) , rLg ( r ) , − rf ( r ) y ′ ( r ) Lg ( r ) (cid:19) (17)We can see that g ( r ) = 1 + y ′ ( r ) f ( r ) and y ′ ( r ) = dy/dr . So, solving the equation K αβ − h αβ ( K − Σ) = 0 (18)For, the equations (11,12), we have y ′ ( r ) = (Σ L ) s − (Σ L ) (cid:18) − (cid:16) rr h (cid:17) (cid:19) (19)as L = p γ/α for the case where γ = 0 , we have a non-trivial gravity solution with anon-zero stress where the existence of such gravity solutions for RS branes was recentlyaddressed in [36]. The equation (19) can be solved and we obtain as a solution: y ( r ) − y = r h sinh − r (Σ L ) r h p − (Σ L ) ! (20)where we can introduce Σ L = cos( θ ′ ) with θ ′ where the angle between the positive directionof the y axis and the hypersurface Q [15, 16]. Performing an expansion around r → andassuming y = 0 , we can write that y ( r ) = r cot( θ ′ ) . FIG. 1: Q boundary profile for the BTZ black hole. The red regions show the "shadows" of the Qboundaries on the horizon, which contribute to the boundary entropy.
IV. BLACK HOLE THERMODYNAMICS
In this section we will present the implications of the truncation γ = − α/ Λ for the blackhole thermodynamics and the g -theorem. So, let’s start by first investigating BCFT’s equi-librium thermodynamics for the BTZ black hole. So, let’s consider the entropy calculationfor the BT Z black hole, so we start with the Euclidean action given by I E = I bulk + 2 I bdry where truncation takes us to the usual case of the spacetime of [16, 17], i.e., I bulk = − πG N Z N ′ d x √ g ( R − − πG N Z M d x √ γ ( K ( γ ) − Σ ( γ ) ) , (21)where g µν is the metric on the bulk N ′ . γ and Σ ( γ ) are the induced metric and the surfacetension on M , respectively. K ( γ ) is the trace of the extrinsic curvature on the surface M .On the other hand, for the boundary we have I bdry = − πG N Z N d x √ g ( R − − πG N Z Q d x √ h ( K − Σ) , (22)Computing the Euclidean action and for r = r h we can have that y ( z h ) − y ( z ) = z h arc sinh cot( θ ′ ) , that is, ∆ y ′ = z h arc sinh cot( θ ′ ) , we can write that I E = − L ∆ y r h G N − L ∆ y ′ r h G N (23)With the entropy given by the equation S = − ∂F∂T H = − ∂ ( T H I E ) ∂T H , (24) S = L ∆ y + 4 L ∆ y ′ r h G N = A G N . (25) F = T H I E = − L ∆ y + 4 L ∆ y ′ πr h G N (26) E = L ∆ y + 4 L ∆ y ′ πr h G N (27)The equation (25,27) satisfy the thermodynamical relation F = E − T H S where A = L ∆ y +4 L ∆ y ′ r h is the total area of the AdS, in this sense the equation (25) satisfies the en-tropy of Bekenstein-Hawking. However, we can also note that the mass term provides thestandard entropy, which is proportional to the size of the boundary system through theBekenstein-Hawking entropy density, in this sense the contribution of the boundary doesnot have a "size" however, its entropy has a geometric interpretation, in the sense that itis the Bekenstein-Hawking coefficient times the horizon area of the black hole immediatelybelow the whitener Q (see figure 1). We can see that as L = p γ/α , we have A = r γα ∆ y + 4∆ y ′ r h (28)where information storage adds limitations to Horndeski’s parameters, that informationis bounded by the BTZ’s black hole area. Thus, when an object such as a black holecaptures mass it can be forced to undergo a gravitational collapse and the second law ofthermodynamics insists that it must have less entropy than the resulting black hole, this0fact implies that γ is very small, we have the entropy decrease. So, for boundary entropy: S bdry = 1 G N r γα arc sinh Σ r γα ! = C arc sinh Σ r γα ! ; C = 1 G N r γα (29)where we can see that if the entropy of the limit is S bdry = 0 it implies that γ → and, inthat sense, the conditions of the limit can be preserved [17]. An interesting aspect of theAdS/CFT conjecture is that for this duality infrared-IR divergences in the AdS correspondto ultraviolet-UV regimes on the CFT side. Thus, we have that this relationship is called theIR-UV connection. Thus, we have that γ → ∞ represents a UV divergence for the boundary.And in this way we can establish the C -theorem, which establishes that the central chargesdecrease the flow of RG [37, 38], however, for the BCFT case, we have that the analogousquantity is the g holographic function [17, 22]. Now let’s address this holographic g theorem,for which we have S bdry = ln g ( r ) = 12 G N r γα arc sinh (cid:18) y ( r ) r (cid:19) (30)Taking the derivative, we have ∂ ln g ( r ) ∂r = y ′ ( r ) r − y ( r ) p r + y . (31)We can see that for y ′ ( r ) r − y ( r ) negative, it disappears at r = 0 and y ′′ ( r ) ≤ leads to ( y ′ ( r ) r − y ( r )) ′ = y ′′ ( r ) ≤ . Thus, we have that the g -theorem is established in our scenario.Note that we can choose y ( r ) so that g ( r ) can flow from g UV to g IR and in that sense wecan have g UV > g IR . Furthermore, as g = e S bdry for ( √ γ → ∞ ) UV the g function grow up,we have g UV and for ( √ γ → −∞ ) IR the g function decreases [38]. V. PERFECT FLUID IN BTZ BLACK HOLE
In this section, we present the hydrodynamical quantities where for the boundary fluidfrom AdS/BCFT correspondence, we have that on the hypersurface Q , which the stress-energy tensor residing on it is defined through the variation of the action with respect toinduced metric on Q [15–17, 22]. Furthermore, due to the truncation γ = − α/ Λ we can1eliminate possible dissipation’s that the scalar field could generate in the fluid. However,we have that the renormalization procedure [39] leads to the following form of stress-energytensor T ab as T αβ = − Lrκ (cid:20) K αβ − h αβ ( K − Σ) + 2 √− h δS ct δh αβ (cid:21) (32)Here S ct is the counter term action, which we will add in order to obtain a finite stresstensor. However, neglecting S ct , we have T αβ = − Lrκ [ K αβ − h αβ ( K − Σ)] (33)Thus, we can write the pressure and energy density by mean the equations: ǫ = 12 κr " r γα + ( rf ′ ( r ) − f ( r )) y ′ ( r ) − f ( r ) y ′ ( r ) + 2 rf ( r ) y ′′ ( r )(1 + f ( r ) y ′ ( r )) / (34) p = 12 κr " − r γα + (4 f ( r ) − rf ′ ( r )) y ′ ( r )(1 + f ( r ) y ′ ( r )) / (35)Here the stress-energy tensor can at all describe hydrodynamical quantities, where f ( r ) = 1 is in according to [16] for empty AdS-space, providing ǫ = − p , in this case as we have that T ab describes an energy dominated universe. however, as expected these hydrodynamicalquantities diverge as r → . Thus, for a finite temperature, we have for a general Σ that r → is an asymptotic regime. Apparently, this is a restriction on the profile y ( r ) . Inaddition, choosing Σ = p α/ γ removes divergences from the UV, that is, we have the so-called holographic renormalization. Although empty AdS-space imply an energy dominateduniverse, we can see that through the regime γ → ∞ , we have ǫ → Σ κr r γα , (36) p → − Σ κr r γα . (37)The scenario described by the equations (36,37) represent an energy dominated universewith ǫ = − p where T ab describes an energy dominated universe, which has the followingequation of state as ω = p/ǫ = − . However, the regime of γ → in the equation (15)2imply f ( r ) ∼ αL / γ = 1 , that provides y ′′ (1 + y ′ ( r )) / = ǫ (1 + 4 ω ) . (38)For a region of positive entropy, we have ω > − . So for ω = − / , we have y ( r ) = cr this profile for the fluid in the equations (34,35), provides ω = − . However, currentobservational data shows that it is impossible to distinguish between phantom ω < − andnon-phantom ω ≥ − . However, the parameter α provides us with a ghost free solutions,since as α = − γ Λ with α > and γ < , we have to ω ≥ − . VI. CONCLUSION
In this work, we show the implications of Horndeski’s gravity parameters for theAdS/BCFT correspondence. An interesting fact is that the no-hair-theorem helps us withconsistent conditions for the boundary condition of Dirichlet and Neumann. Where the real-ity condition delimits that −∞ < γ ≤ α/ ( − Λ) , where the case Λ = − α/γ reduces the BTZsolution with an escaped scalar field [25, 40]. This condition provides us that Horndeski’sgravity is reduced to Einstein’s gravity in the construction of the AdS/BCFT, we have thatthese parameters provide us with the central frontier load. Furthermore, for AdS /BCFT ,we have that the partition function, which is interpreted as the g function, shows us a fullagreement for the g -theorem in which the holographic flow is, in fact, an IR to UV connec-tion, providing g UV > g IR . Thus, as we show the function that interpolates the two centralcharges in two CFTs, which are connected by an RG flow, it is the g function that is ourboundary entropy.Note that when calculating the free energy of BCFT, its entropy, and internal energy,they satisfy the thermodynamic relationship F = E − T H S . However, as the free energy F < implies global stability, that is, we have a positive specific heat c > , where we havea system with local stability [25]. On the other hand, c > suggests a higher value of themass of the black hole and is therefore at least locally stable. In this sense, if the blackhole captures mass it can be forced to undergo gravitational collapse and the second law ofthermodynamics insists that it must have less entropy than the resulting black hole. Thisfact fully agrees with the fact that the entropy of the black hole increases to a very large γ γ value, in this sense we can see that black holes saturatethe holographic bound.In the fluid/gravity duality, we show that for γ → ∞ or γ → , provided ǫ = − p ,in this case as we have that T ab describes an energy dominated universe, which has thefollowing equation of state as ω = p/ǫ = − . This energy that dominated the universe is theHolographic Dark Energy [41] with a perfect fluid, which has energy density ǫ and pressure p .Such fluid can be considered as a generalization of Chaplygin Gas, such a result is predictedfor a Horndeski subclass [42]. However, for sectors other than what is being treated herein this work, which is models of Kinetic Gravity Braid with symmetrical displacement, thefluids are imperfect, with zero vorticity and without dissipation. However, having someencoded diffusivity [43].We would like to thank CNPq and CAPES for partial financial support. I would also liketo thank Konstantinos Pallikaris, Vasilis Oikonomou and Adolfo Cisterna for their fruitfulcontributions at the end of this work. [1] W. J. Jiang, H. S. Liu, H. Lu and C. N. Pope, DC Conductivities with Momentum Dissipationin Horndeski Theories , JHEP , 084 (2017), [arXiv:1703.00922 [hep-th]].[2] A. Cisterna, M. Hassaine, J. Oliva and M. Rinaldi,
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