Quartic Quasi-Topological-Born-Infeld Gravity
aa r X i v : . [ g r- q c ] S e p Quartic Quasi-Topological-Born-Infeld Gravity
Mohammad Ghanaatian
Department of Physics, Payame Noor University, Iran
Abstract
In this paper, quartic quasi-topological black holes in the presence of a nonlinear electromagneticBorn-Infeld field is presented. By using the metric parameters, the charged black hole solutionsof quasi-topological Born-Infeld gravity is considered. The thermodynamics of these black holesare investigated and I show that the thermodynamics and conserved quantities verify the first lawof thermodynamics. I also introduce the thermodynamics of asymptotically AdS rotating blackbranes with flat horizon of these class of solutions and I calculate the finite action by use of thecounterterm method inspired by AdS/CFT correspondence. . INTRODUCTION The paradigm of extra dimensions does much more that excite our imagination. It solvesthe so called ”hierarchy of scales” problem. Several extra-dimensional models have beenintroduced in the past few years. It is well known that the natural generalization of theEinstein-Hilbert action to higher dimensional spacetime, and higher order gravity with sec-ond order equation of motion, is the Lovelock action [1]. Because of the topological origin ofthe Lovelock terms, the second term of the Lovelock action (the Gauss-Bonnet term) doesnot have any dynamical effect in four dimensions. Similarly, the cubic term just contributesto the equations of motion in seven dimentions or greater. A modification of higher orderLovelock gravity which contains cubic and quartic terms of Riemann tensor and contributesto the equation of motions in five dimensions is quasi-topological gravity [2–4]. In second,third- [2, 3], and fourth-order [4] quasi-topological gravity, we deal with the field equationsof higher-derivative gravity in five and higher dimensions. The equations of motion of thesetheories are second-order differential equations for the spherically symmetric metric. Inorder to the fact that the exact black hole solutions have been constructed and also theequation of motion are only second-order in derivatives in spherically symmetric setting,contrary to the fact the quartic quasi-topological gravity is a higher carvature theory ofgravity, the holographic computations in this model are still under control, at least for thespherical symmetry. Furthermore, Since the quasi-topological theory contains derivativesof metrics of order not higher than two (for the spherical symmetry), the quantization oflinearized quasi-topological theory is free of ghosts. Thus, it is natural to study the effectsof these higher curvature terms on the properties and thermodynamics of black holes andblack branes [5–12].It is well known that the rsh of gravitational equations (like Einstein equation) isenergy-momentum tensor, which relate to matter and various fields. Among these fieldsthe electromagnetic field is so important. Without exaggeration the linear electromagneticfield is one of the most successful theories of electromagnetic fields. In order to solve theproblem of self-energy of electron, the theory of the non-linear electromagnetic field wasintroduced by Born and Infeld (BI) in 1934 [13]. In the limits of weak fields, the Born-Infeldlagrangian reduces to Maxwell lagrangian plus some small corrections. If one is to considerthe Maxwell fields coupled to a gravitational action, which also includes string generated2orrections at higher orders, then, it is natural to consider string generated corrections tothe electromagnetic field action as well. It is known that there are Born-Infeld terms whichappear as higher order corrections to the Maxwell action. In fact, considering the analogybetween the quasi-topological and the Born-Infeld terms, it is worth to include both thesecorrections simultaneously. In this letter, I will consider the quartic quasi-topological gravityin the presence of nonlinear electrmagnetic field and consider the black hole and black branesolutions of this theory.The outline of this paper is as follows: In Sec. II, a brief review of the quartic quasi-topological gravity in the presence of a nonlinear Born-Infield electromagnetic field is pre-sented. In section III, I consider the charged black holes of quasi-topological gravity in thepresence of Born-Infeld electromagnetic field. Section IV is devoted to the investigation ofthe thermodynamic properties of these solutions and the first law of thermodynamics. InSec. V, I endow the solutions with asymptotically AdS charged rotating black branes andstudy the thermodynamic properties of them with flat horizon. Then, in Sec. VI, the finiteaction and conserved quantities of the solutions are calculated. Finally, I finish my paperwith some concluding remarks in section VII.
II. QUASI-TOPOLOGICAL-BORN-INFELD ACTION
The action of quartic quasi-topological gravity in ( n + 1) dimensions in the presence of anonlinear Born-Infeld electromagnetic field can be written as I G = 116 π Z d n +1 x √− g [ −
2Λ + L + µ L + µ X + µ X + L ( F )] . (1)where Λ = − n ( n − / l is the cosmological constant, L = R is just the Einstein-HilbertLagrangian, L = R abcd R abcd − R ab R ab + R is the second order Lovelock (Gauss-Bonnet)Lagrangian, X is the curvature-cubed Lagrangian [3] X = R cdab R e fcd R a be f + 1(2 n − n − (cid:18) n − R abcd R abcd R − n − R abcd R abce R de + 3( n + 1) R abcd R ac R bd + 6( n − R ab R bc R ca − n − R ba R ab R + 3( n + 1)8 R (cid:19) . (2)3nd X is the fourth order term of quasi-topological gravity [4] X = c R abcd R cdef R hgef R hgab + c R abcd R abcd R ef R ef + c RR ab R ac R cb + c ( R abcd R abcd ) + c R ab R ac R cd R db + c RR abcd R ac R db + c R abcd R ac R be R de + c R abcd R acef R be R df + c R abcd R ac R ef R bedf + c R + c R R abcd R abcd + c R R ab R ab + c R abcd R abef R ef cg R dg + c R abcd R aecf R gehf R gbhd . (3)with c = − ( n − (cid:0) n − n − n + 170 n − n + 348 n − n + 36 (cid:1) ,c = − n − (cid:0) n − n + 65 n − n + 13 n + 45 n − (cid:1) ,c = −
64 ( n − (cid:0) n − n + 3 (cid:1) (cid:0) n − n + 3 (cid:1) ,c = − ( n − n + 12 n − n + 114 n − n + 468 n − n + 54) ,c = 16 ( n − (cid:0) n − n + 93 n − n + 18 (cid:1) ,c = − −
32 ( n − ( n − (cid:0) n − n + 3 (cid:1) ,c = 64 ( n −
2) ( n − (cid:0) n − n + 27 n − (cid:1) ,c = −
96 ( n −
1) ( n − (cid:0) n − n + 4 n + 6 n − (cid:1) ,c = 16 ( n − (cid:0) n − n + 93 n − n + 36 (cid:1) ,c = n − n + 168 n − n + 330 n − ,c = 2 (6 n − n + 311 n − n + 936 n − n + 126) ,c = 8 (7 n − n + 121 n − n + 63 n − ,c = 16 n ( n −
1) ( n −
2) ( n − (cid:0) n − n + 3 (cid:1) ,c = 8 ( n − (cid:0) n − n − n + 122 n − n + 297 n − n + 18 (cid:1) . In the action (1), L ( F ) is the Born-Infeld Lagrangian given as [14–19] L ( F ) = 4 β − s F β ! . (4)where F = F µν F µν , F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor and A µ is thevector potential. One may note that in the limit β −→ ∞ reduces to the standard Maxwellform L ( F ) = − F . 4 II. CHARGED QUASI-TOPOLOGICAL-BORN-INFELD BLACK HOLE SOLU-TIONS
Now, I introduce the charged black hole solutions of quasi-topological gravity in thepresence of nonlinear Born-Infeld electromagnetic field with Lagrangian (4). The metric hasthe following form: ds = − f ( ρ ) dt + dρ f ( ρ ) + ρ d Ω . (5)where d Ω = dθ + n − P i =2 i − Q j =1 sin θ j dθ i k = 1 dθ + sinh θ dθ + sinh θ n − P i =3 i − Q j =2 sin θ j dθ i k = − n − P i =1 dφ i k = 0 represents the line element of an ( n − n − n − k and volume V n − . Using the metric (5) and A µ = h ( ρ ) δ µ . (6)for the vector potential, one can calculate the one dimensional action after integration byparts. One obtains the action per unit volume as I G = ( n − πl Z dtdρ [ (cid:2) ρ n (1 + ψ + ˆ µ ψ + ˆ µ ψ + ˆ µ ψ ) (cid:3) ′ + 4 l β ρ ( n − (1 − q − h ′ β )( n −
1) ] . (7)where ψ = l ρ − ( k − f ) and the dimensionless parameters ˆ µ , ˆ µ and ˆ µ are defined as:ˆ µ ≡ ( n − n − l µ , ˆ µ ≡ ( n − n − n − n + 4)8(2 n − l µ , ˆ µ ≡ n ( n −
1) ( n − ( n −
3) ( n −
7) ( n − n + 72 n − n + 150 n − l µ . Variation with respect to h ( ρ ) gives( n − h ′ ( β − h ′ ) + ρh ′′ β = 0 , (8)and therefore one can show that the vector potential can be written as h ( ρ ) = − r ( n − n − qρ n − Γ( η ) , (9)5here q is an integration constant which is related to the charge parameter and η = ( n − n − q β ρ n − . In Eq. (9) and throughout the paper, the following abbreviation for the hypergeometricfunction is used, F (cid:18)(cid:20) , n − n − (cid:21) , (cid:20) n − n − (cid:21) , − z (cid:19) = Γ( z ) . (10)The hypergeometric function Γ( η ) → η → β → ∞ ) and therefore h ( ρ ) of Eq. (9)reduces to the gauge potential of Maxwell field.Varying the action (7) with respect to ψ ( ρ ) yields (cid:0) µ ψ + 3ˆ µ ψ + 4ˆ µ ψ (cid:1) dN ( ρ ) dρ = 0 , (11)which shows that N ( ρ ) should be a constant. Variation with respect to N ( ρ ) and substituting N ( ρ ) = 1 gives ˆ µ ψ + ˆ µ ψ + ˆ µ ψ + ψ + κ = 0 , (12)where κ = ˆ µ − mρ n + 4 l β n ( n −
1) [1 − p η − ηn − F ( η )] (13)and m is an integration constant which is related to the mass of the spacetime. In order toobtain the black hole solutions, I choose two solutions of f ( ρ ) as f ( ρ ) = k + ρ l (cid:18) ˆ µ µ + 12 R − E (cid:19) . (14) f ( ρ ) = k + ρ l (cid:18) ˆ µ µ − R + 12 K (cid:19) . (15)where R = (cid:18) ˆ µ µ − ˆ µ ˆ µ + y (cid:19) / , (16) E = (cid:18) µ µ − µ ˆ µ − R − R (cid:20) µ ˆ µ ˆ µ − µ − ˆ µ ˆ µ (cid:21)(cid:19) / , (17) K = (cid:18) µ µ − µ ˆ µ − R + 14 R (cid:20) µ ˆ µ ˆ µ − µ − ˆ µ ˆ µ (cid:21)(cid:19) / (18)∆ = H
27 + D , H = 3ˆ µ − ˆ µ µ − κ ˆ µ , (19) D = 227 ˆ µ ˆ µ − (cid:18) ˆ µ ˆ µ + 8 κ ˆ µ (cid:19) ˆ µ ˆ µ + ˆ µ κ ˆ µ + 1ˆ µ . (20)6nd y is the real root of following equation: y − µ y µ + (cid:18) µ µ − κµ (cid:19) y − µ κµ + 4 µ κµ − µ = 0 (21)The metric function f ( ρ ) for the uncharged solution ( q = 0) is real in the whole range0 ≤ ρ < ∞ . But for charged solutions, one should restrict the spacetime to the region ρ ≥ r ,where r is the largest real root of ∆ = ∆( κ = κ ), R = R ( κ = κ ), E = E ( κ = κ ) and K = K ( κ = κ ), and κ is κ = ˆ µ − mr n + 4 l β n ( n −
1) [1 − p η − η n − F ( η )] (22)where η = ( n − n − q β r n − . Performing the transformation r = p ρ − r ⇒ dρ = r r + r dr (23)the metric becomes ds = − f ( r ) dt + r dr ( r + r ) f ( r ) + ( r + r ) n − X i =1 dφ i . (24)where now the functions η , h ( r ) and κ are η = ( n − n − q β ( r + r ) n − / .h ( r ) = − r ( n − n − q ( r + r ) ( n − / Γ( η ) , (25) κ = ˆ µ − m ( r + r ) n/ + 4 l β n ( n −
1) [1 − p η − ηn − F ( η )] (26)7 V. THERMODYNAMICS OF QUASI-TOPOLOGICAL-BORN-INFELD BLACKHOLES
One can obtain the Hawking temperature of the black hole solutions that are consideredin the previous section as: T + = f ′ ( r + )4 π s r r = ( n − n ˆ µ Υ + ( n − kl Υ + ( n − k ˆ µ l Υ + ( n − k ˆ µ l Υ + ( n − k ˆ µ l ]( Υ + 2 k ˆ µ l Υ + 3 k ˆ µ l Υ + 4ˆ µ k l ) 4 π ( n − l Υ + + 4Υ β (cid:0) − √ + (cid:1) ( Υ + 2 k ˆ µ l Υ + 3 k ˆ µ l Υ + 4ˆ µ k l ) 4 π ( n − l Υ + (27)where Υ + = p r + r and r + is the largest real root of f ( r ).The entropy density for black hole in quartic quasi-topological gravity becomes, S = r n − (cid:18) k ( n −
1) ˆ µ l ( n − r + 3 k ( n −
1) ˆ µ l ( n − r + 4ˆ µ k l ( n − r ( n − (cid:19) (28)Calculating the flux of the electric field at infinity, one can find the charge of the blackhole as Q = V n − π r ( n − n − q (29)The electric potential Φ at infinity with respect to the horizon can be defined by [20, 21],Φ = A µ χ µ | r →∞ − A µ χ µ | r = r + (30)where χ = ∂/∂t is the null generator of the horizon. The electric potential Φ can be foundas follow: Φ = s ( n − n − q ( r + r ) ( n − / Γ( η + ) . (31)By using the behavior of the metric at large r , the ADM (Arnowitt-Deser-Misner) massof black hole can be arrived. One can easily show that the mass of the black hole is M = V n − π ( n − m. (32)In order to investigate the first law of thermodynamics, I use the expression for theentropy, the charge, and the mass that are given in Eqs. (28), (29) and (32), and keep f ( r + ) = 0 in the mind, I introduce as 8 ( S, Q ) = ( n −
1) ( r + r ) n/ π (cid:26) β n ( n − (cid:20) − √ ℑ + ( n − ℑ ( n −
2) Γ( ℑ ) (cid:21) +ˆ µ + k l ( r + r ) + ˆ µ k l ( r + r ) + ˆ µ k l ( r + r ) + ˆ µ k l ( r + r ) ) , (33)where ℑ = 16 π Q β ( r + r ) n − . In Eq. (33), r + is the real root of Eq. (28) which is a function of S . I can regard the param-eters S and Q as a complete set of extensive parameters for the mass M ( S, Q ) and definethe intensive parameters T and Φ conjugate to them. These quantities are the temperatureand the electric potential T = (cid:18) ∂M∂S (cid:19) Q , Φ = (cid:18) ∂M∂Q (cid:19) S . (34)It is easy to show that the intensive quantities calculated by Eq. (34) that are obtained bycomputing ∂M/∂r + and ∂S/∂r + and using the chain rule, coincide with Eqs. (27) and (31),respectively. So, the thermodynamic quantities calculated in Eqs. (27) and (31) lead to thefirst law of thermodynamics, dM = T dS + Φ dQ. (35)
V. THERMODYNAMICS OF CHARGED ROTATING QUASI-TOPOLOGICAL-BORN-INFELD BLACK BRANES
Now, I apply the spacetime solution (5) for k = 0 with a global rotation. One mayperform the following rotation boost in the t − φ i planes to add angular momentum to thespacetime t Ξ t − a i φ i , φ i Ξ φ i − a i l t (36)where [ x ] is the integer part of x for i = 1 ... [ n/ SO ( n ) rotation group in n + 1 dimen-sions shows the maximum number of rotation parameters. So, the number of independentrotation parameters is [ n/ p ≤ [ n/ ds = − f ( r ) Ξ dt − p X i =1 a i dφ i ! + ( r + r ) l p X i =1 (cid:0) a i dt − Ξ l dφ i (cid:1) + r dr ( r + r ) f ( r ) − ( r + r ) l p X i The AdS/CFT correspondence mostly apply for infinite boundaries, but sometimes it isused for finite boundaries in order to drive the conserved and thermodynamic quantities[23, 24]. Considering the thermodynamics of AdS black holes by using the AdS/CFT corre-spondence, the deep insights in the characteristics and phase structures of strong ’t Hooftcoupling CFTs can be obtained. In this section, I drive the action and conserved quantitiesof the solutions. In general, the action and conserved quantities of the spacetime are diver-gent when evaluated on the solutions. By using the counterterms method and AdS/CFT10orrespondence, one can consider this divergence for asymptotically AdS solutions of Ein-stein gravity [25]. The finite action for asymptotically AdS solutions with flat boundary, b R abcd ( γ ) = 0 may be written as follow: I = I G + I b + I ct (43)In this finite action, boundary term is I b = I (1) b + I (2) b + I (3) b (44)where, the following boundary term makes the Einstein-Hilbert action well-defined [26], I (1) b = 18 π Z ∂ M d n x √− γK. (45)and the proper surface term for the Gauss-Bonnet term is [27–30], I (2) b = 18 π Z ∂ M d n x √− γ (cid:26) µ l ( n − n − J (cid:27) . (46)where J is the trace of J ab = 13 (2 KK ac K cb + K cd K cd K ab − K ac K cd K db − K K ab ) . (47)and the surface terms for the curvature-cubed term of quasi-topological gravity have beendriven in Ref. [6] as I (3) b = 18 π Z ∂ M d n x √− γ n µ l n ( n − n − ( n − 5) ( nK − K K ab K ab +4( n − K ab K ab K cd K de K ec − (5 n − KK ab [ nK ab K cd K cd − ( n − K ac K bd K cd ]) o . (48)and I (4) b = 18 π Z ∂ M d n x √− γ ˆ2 µ l n ( n − 1) ( n − 2) ( n − 7) ( n − n + 3) n α K K ab K ac K bd K cd + α K K ab K ab K cd K ec K de + α K K ab K ac K bd K ce K de + α KK ab K ab K cd K ec K fd K ef + α KK ab K ca K bc K de K fd K ef + α KK ab K ac K bd K ce K df K ef + α K ab K ca K bc K de K df K eg K fg o . (49)presents a surface term which makes the action of quartic quasi-topological gravity well-defined [8]. And the last term, I ct , which is a functional of the boundary curvature invariants,is counterterm and may be written as follow [31–34]: I ct = − π Z ∂ M d n x √− γ ( n − l eff . (50)11here l eff is a scale length factor which depends on l and coupling constants of gravityand should reduce to l in the absence of higher curvature term. By using this countertermmethod, the action (43) may be finite and can be used it to calculate the conserved andthermodynamics quantities.The conserved quantities associated with the Killing vectors ∂/∂t and ∂/∂φ i can beobtained as M = 116 π m (cid:0) n Ξ − (cid:1) . (51) J i = 116 π n Ξ ma i . (52)which are the mass and angular momentum of the solution.By using Gibbs-Duhem relation S = 1 T ( M − Q Φ − k X i =1 Ω i J i ) − I, (53)one can introduce the entropy per unit volume V n − as follow: S = Ξ4 r n − (54)This shows that the entropy obeys the area law for our case where the horizon is flat. VII. CONCLUDING REMARKS In this paper, I introduced the quartic quasi-topological gravity in the presence of Born-Infeld field which is a nonlinear electromagnetic field. I computed the charged black holesolutions of this theory. These solutions presented a black holes with one or two horizons or anaked singularity depending on the values of charge and mass parameters. I investigated thatthe solutions reduce to the solutions of Einstein-Born-Infeld when the the Gauss-Bonnet andquasi-topological coefficients ( µ , µ and µ ) vanish, and reduce to the solutions of quarticquasi-topological gravity in the presence of Maxwell field as β goes to infinity. In orderto verify the first law of thermodynamics of these black hole solutions, I calculated thethermodynamic quantities S , Q and M where the mass was driven as a function of theextensive parameters S and Q .Then, I considered the thermodynamics of asymptotically AdS black branes with a flathorizon. I used the boundary term to find the on-shell action of the quartic quasi-topological12ravity. However, one can use this method in the Hamiltonian formalism of quasi-topologicalgravity. 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