aa r X i v : . [ phy s i c s . g e n - ph ] O c t Lovelock gravity from entropic force
A. Sheykhi , , ∗ H. Moradpour and N. Riazi † hysics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
In this paper, we first generalize the formulation of entropic gravity to ( n + 1)-dimensional space-time. Then, we propose an entropic origin for Gauss-Bonnet gravity and more general Lovelockgravity in arbitrary dimensions. As a result, we are able to derive Newton’s law of gravitation aswell as the corresponding Friedmann equations in these gravity theories. This procedure naturallyleads to a derivation of the higher dimensional gravitational coupling constant of Friedmann/Einsteinequation which is in complete agreement with the results obtained by comparing the weak field limitof Einstein equation with Poisson equation in higher dimensions. Our study shows that the approachpresented here is powerful enough to derive the gravitational field equations in any gravity theory.keywords: entropic; gravity; thermodynamics. I. INTRODUCTION
Nowadays, it is a general belief that there should be some deep connection between gravity and thermodynamics.Indeed, this connection has a long history since the discovery of black holes thermodynamics in 1970’s by Bekensteinand Hawking [1]. The studies on the profound connection between gravity and thermodynamics have been continued[2, 3] until in 1995 Jacobson [4] disclosed that the Einstein field equation is just an equation of state for spacetime andin particular it can be derived from the the first law of thermodynamics together with the relation between the horizonarea and entropy. Inspired by Jacobson’s arguments, an overwhelming flood of papers has appeared which attempt toshow that there is indeed a deeper connection between gravitational field equations and horizon thermodynamics. Ithas been shown that the gravitational field equations in a wide variety of theories, when evaluated on a horizon, reduceto the first law of thermodynamics and vice versa. This result, first pointed out in [5], has now been demonstratedin various theories including f(R) gravity [6], cosmological setups [7–12], and in braneworld scenarios [13, 14]. For arecent review on the thermodynamical aspects of gravity and complete list of references see [15]. The deep connectionbetween horizon thermodynamics and gravitational field equations, help to understand why the field equations shouldencode information about horizon thermodynamics. These results prompt people to take a statistical physics pointof view on gravity.A remarkable new perspective was recently suggested by Verlinde [16] who claimed that the laws of gravitation areno longer fundamental, but rather emerge naturally from the second law of thermodynamics as an “entropic force”.Similar discoveries are also made by Padmanabhan [17] who observed that the equipartition law for horizon degreesof freedom combined with the Smarr formula leads to the Newton’s law of gravity. This may imply that the entropylinks general relativity with the statistical description of unknown spacetime microscopic structure when a horizon ispresent. The investigations on the entropic gravity has attracted a lot of interest recently [18–29].On the other hand, the effect of string theory on classical gravitational physics is usually investigated by meansof a low energy effective action which describes gravity at the classical level. This effective action consists of theEinstein-Hilbert action plus curvature-squared (Gauss-Bonnet) term and also higher order derivatives curvature terms.Lovelock gravity [30, 31] which is a natural generalization of Einstein gravity in higher dimensional spacetimes containshigher order derivatives curvature terms, however there are no terms with more than second order derivatives of metricin equations of motion just as in Gauss-Bonnet gravity. Since the Lovelock tensor contains metric derivatives no higherthan second order, the quantization of the linearized Lovelock theory is ghost-free [32].Since the entropic gravity is fundamentally based on the holographic principle, one expects that entropic gravitycan be generalized to any arbitrary dimension [16]. The motivation for studying higher dimensional gravity originatesfrom string theory, which is a promising approach to quantum gravity. String theory predicts that spacetime has morethan four dimensions. Another striking motivation for studying higher dimensional gravity comes from AdS/CFTcorrespondence conjecture [33], which associates an n -dimensional conformal field theory with a gravitational theoryin ( n + 1) dimension. The generalization of this duality is embodied by the holographic principle [34], which positsthat the entropy content of any region of space is defined by the bounding area of the region. These considerationshave provided us enough motivation to study the formulation of the entropic gravity in ( n + 1)-dimensional spacetime. ∗ [email protected] † [email protected] In this paper, we consider the problem of formulating entropic gravity in all higher dimensions. We also show thatin an string inspired model of gravity the formalism of entropic force works well and can be employed to derive theNewton’s law of gravity as well as the ( n + 1)-dimensional Friedmann equation in Gauss-Bonnet theory and moregeneral Lovelock gravity.This paper is organized as follow. In the next section we generalize the entropic gravity to arbitrary dimensions andwill derive successfully Newton’s law of gravitation as well as Friedmann equation in ( n + 1)-dimensions. In sectionIII, we derive Newton’s law of gravity and the ( n + 1)-dimensional Friedmann equation in Gauss-Bonnet theory fromthe entropic gravity perspective. In section IV, we generalize our study to the more general Lovelock gravity. Thelast section is devoted to conclusions and discussions. II. ENTROPIC GRAVITY IN ( n + 1 )-DIMENSIONS According to Verlinde, when a particle is on one side of screen and the screen carries a temperature, it will experiencean entropic force equal to F = − T △ S △ x . (1)By definition, F is a force resulting from the tendency of a system to increase its entropy. Note that △ S > △ x as it relates to the proposed system.Here △ x is the displacement of the particle from the holographic screen, while T and △ S are the temperature and theentropy change on the screen, respectively. Suppose we have a mass distribution M which is distributed uniformlyinside an screen Σ. We have also a test mass m which is located outside the screen. The surface Σ surrounds themass distribution M has a spherically symmetric property, while the test mass m is assumed to be very close to Σcomparing to its reduced Compton wavelength λ m = ~ mc . Now, consider an ( n + 1)-dimensional spacetime with n spacial dimensions. The mass M induces a holographic screen Σ n at distance R that has encoded on it gravitationalinformation. The volume and area of this n -sphere are V n = Ω n R n , Σ n = n Ω n R n − , (2)where Ω n = π n/ Γ( n + 1) , Γ( n n n − . (3)According to the holographic principle, the screen encodes all physical information contained within its volume inbits on the screen. The maximal storage space, or total number of bits, is proportional to the area Σ n . Let us denotethe number of used bits by N . It is natural to assume that this number is proportional to the area Σ n , namelyΣ n = N Q, (4)where Q is a constant which should be specified later. Since N denotes the number of bits, thus for one unit changewe find △ N = 1. Therefore, from relation (4) one gets △ Σ n = Q . Motivated by Bekenstein’s area law of black holeentropy, we assume the entropy of the ( n − S = k B c Σ n ~ G n +1 , (5)where G n +1 = 2 π − n/ Γ( n c ℓ n − p ~ , (6)is the ( n + 1)-dimensional gravitational constant [29]. We also assume the entropy change △ S = k B c △ Σ n ~ G n +1 = k B c Q ~ G n +1 . (7)is one fundamental unit of entropy when △ x = ~ mc , and the entropy gradient points radially from the outside of thesurface to inside. Assuming that the total energy of the system, E = M c , (8)is evenly distributed over the bits. Then according to the equipartition law of energy [35], the total energy on thescreen is E = 12 N k B T. (9)Combining Eqs. (4), (8) and (9), we find T = 2 M c Q Σ n k B . (10)Finally, inserting Eqs. (7) and (10) as well as relation △ x = ~ mc in Eq. (1), after using relation Σ n = n Ω n R n − , it isstraightforward to show that the entropic force yields the ( n + 1)-dimensional Newton’s law of gravitation F = − M mR n − (cid:20) Q c n Ω n ~ G n +1 (cid:21) , (11)This is nothing but the Newton’s law of gravitation in arbitrary dimensions provided we define Q ≡ ~ c n Ω n G n +1 . (12)For n = 3 we have G = G = ℓ p c / ~ and the above expression reduces to Q = 8 πℓ p [24]. It is important to note thatthe relations N = Ac G ~ and △ S = 2 πk B postulated by Verlinde [16]. Our assumption here differs a bit from Verlindepostulates. For example, we have taken △ S = k B c Q ~ G instead of △ S = 2 πk B . Indeed, △ S in our paper is √ π × △ S Verlinde. So we do not expect to have exactly Q = G ~ c . Although our main assumptions Σ n ∝ N and △ S ∝ k B aresimilar but the constants of proportionality are just assumption for later convenience [16]. Combining Eq. (12) with(11) we reach F = − G n +1 M mR n − . (13)As the next step, we generalize the study to the cosmological setup. Assuming a homogeneous and isotropic Friedmann-Robertson-Walker (FRW) spacetime which is described by the line element ds = h µν dx µ dx ν + R d Ω n − . (14)Here R = a ( t ) r , x = t, x = r , and h µν =diag ( − , a / (1 − kr )) is the two dimensional metric, while d Ω n − isthe metric of ( n − h µν ∂ µ R∂ ν R = 0. It is a straightforward calculation to show that the radius of the apparent horizon for the FRWuniverse becomes R = ar = 1 p H + k/a . (15)We also assume the matter source in the FRW universe is a perfect fluid with stress-energy tensor T µν = ( ρ + p ) u µ u ν + pg µν . (16)Conservation of energy-momentum in ( n + 1)-dimensions leads to the following continuity equation˙ ρ + nH ( ρ + p ) = 0 , (17)where H = ˙ a/a is the Hubble parameter. First of all, we derive the dynamical equation for Newtonian cosmology.Consider a compact spatial region V n with a compact boundary Σ n , which is a sphere with physical radius R = a ( t ) r .If we combine the gravitational force (13) with the second law of Newton for the test particle m near the screen Σ n ,then we obtain F = m ¨ R = m ¨ ar = − G n +1 M mR n − . (18)The total physical mass M in the spatial region V n is defined as [18] M = Z dV ( T µν u µ u ν ) = Ω n R n ρ, (19)where ρ = M/V n is the energy density of the matter inside the the volume V n = Ω n R n . Combining Eqs. (18) and(19) we reach ¨ aa = − G n +1 Ω n ρ = − G n +1 π n/ n ( n − ρ. (20)This is the dynamical equation for ( n + 1)-dimensional Newtonian cosmology. In four dimensional spacetime where n = 3, we recover the well-known formula, ¨ aa = − πG ρ. (21)In order to derive the ( n + 1)-dimensional Friedmann equations of the FRW universe, let us notice that the quantitywhich produces the acceleration in a dynamical background is the active gravitational mass M [36] rather than thetotal mass M . To determine the active gravitational mass, we should express M in terms of energy-momentum tensor T µν . The key point here is to connect the energy momentum T µν with the spacetime curvature with the use of theTolman-Komar’s definition of active gravitational mass. The active gravitational mass in ( n + 1)-dimension is definedas [18] M = n − n − Z V dV (cid:18) T µν − n − T g µν (cid:19) u µ u ν . (22)It is a matter of calculation to show that M = Ω n R n n − n − ρ + np ] = 2 π n/ R n n ( n − n − n − ρ + np ] . (23)Now, we can combine Eq. (23) with (18) provided we replace M in Eq. (18) with induced active gravitational mass M . This can be done because according to the weak equivalence principle of general relativity, the active gravitationalmass of a system (here the universe) in general relativity is equal to its total mass in Newtonian gravity. We find¨ aa = − G n +1 n − n [( n − ρ + np ] = − G n +1 π n/ n ( n − n − n − ρ + np ] . (24)This is the acceleration equation for the dynamical evolution of the FRW universe in ( n + 1)-dimensional spacetime.Multiplying ˙ aa on both sides of Eq. (24), and using the continuity equation (17), after integrating we find H + ka = 2 G n +1 n − n ρ = 4 G n +1 π n/ n ( n − n − ρ, (25)where k is an integration constant. When n = 3, we have Ω = 4 π/ H + ka = 8 πG ρ. (26)Is it worth noting that in the literature the ( n + 1)-dimensional Friedmann equation, in Einstein gravity, usually iswritten as H + ka = 2 κ n n ( n − ρ, (27)with κ n = 8 πG n +1 (sometimes it is also written κ n = 8 πG ). Let us note that the coupling in the r.h.s of thisequation differs from that we derived in Eq. (25) for n ≥
4. A question then arises, which one is the correct Einsteingravitational constant? We believe that the coupling constant we derived from the entropic force approach is thecorrect one. To show this, let us note that the root of the factor 8 π in Eq. (26) and also (27) is the relation R = ∇ φ, (28)where φ is the Newtonian gravitational potential and R is the (00) component of the Ricci tensor. Now, thecoefficient in the Poisson equation, i.e. 4 π has been obtained using the Gauss law for 3-dimensional space. Thus weshould first derive the correct coefficient for n -dimensional space. Applying Guass’s law for an n -dimensional volume,one finds the Poisson equation for arbitrary fixed dimension [37] ∇ φ = 2 G n +1 π n/ ( n − ρ, (29)On the other hand for n ≥
3, one finds [37] R = (cid:18) n − n − (cid:19) κ n ρ, (30)Comparing Eqs. (28), (29) and (30) gives us the following modified Einstein gravitational constant for arbitrary n ≥ κ n = 2( n − π n/ G n +1 ( n − n − . (31)Substituting relation (31) in (27), immediately shows that the correct form of the Friedmann equation in n ≥ III. GAUSS-BONNET ENTROPIC GRAVITY
Next we study the entropic force idea in Gauss-Bonnet gravity. This theory contains a special combination ofcurvature-squared term, added to the Einstein-Hilbert action. The key point which should be noticed here is thatin Gauss-Bonnet gravity the entropy of the holographic screen does not obey the area law. The lagrangian of theGauss-Bonnet correction term is given by L GB = R − R ab R ab + R abcd R abcd . (32)The low energy effective action of heterotic string theory naturally produces the Gauss-Bonnet correction term. TheGauss-Bonnet term does not have any dynamical effect in four dimensions since it is just a topological term in fourdimensions. Static black hole solutions of Gauss-Bonnet gravity have been found and their thermodynamics have beeninvestigated in ample details [38, 39]. The entropy of the static spherically symmetric black hole in Gauss-Bonnettheory has the following expression [39] S = k B c Σ n ~ G n +1 (cid:20) n − n − αr (cid:21) , (33)where Σ n is the horizon area and r + is the horizon radius. In the above expression ˜ α = ( n − n − α , where α isthe Gauss-Bonnet coefficient which is positive [38], namely α >
0. We assume the entropy expression (33) also holdsfor the apparent horizon of the FRW universe in Gauss-Bonnet gravity [7]. The only change we need to apply is thereplacement of the horizon radius r + with the apparent horizon radius R , namely S = k B c Σ n ~ G n +1 (cid:20) n − n − αR (cid:21) . (34)For n = 3 we have ˜ α = 0, thus the Gauss-Bonnet correction term contributes only for n ≥ △ S = k B c △ Σ n ~ G n +1 + n − n − k B c ˜ α ~ G n +1 △ (cid:18) Σ n R (cid:19) . (35)Using the relation Σ n = n Ω n R n − we have △ Σ n R = ( n − n △ R . Combining this expression with Eq. (35) afterusing relation △ Σ n = Q , we obtain △ S = Qk B c G n +1 ~ (cid:20) αR (cid:21) . (36)Inserting Eqs. (10), (12) and (36) in Eq. (1) we find F = − G n +1 M mR n − (cid:20) αR (cid:21) . (37)This is the Newton’s law of gravitation in Gauss-Bonnet gravity resulting from the entropic force approach. In theabsence of Gauss-Bonnet term (˜ α = 0) one recovers Eq. (13). It is worth mentioning that the correction term in Eq.(37) can be comparable to the first term only when R is very small, namely for strong gravity. This implies that thecorrection make sense only at the very small distances. When R becomes large, i.e. for weak gravity, the modifiedNewton’s law reduces to the usual Newton’s law of gravitation.Finally, we derive the ( n + 1)-dimensional Friedmann equation of FRW universe in Gauss-Bonnet gravity using theapproach we developed in the previous section. In the presence of Gauss-Bonnet term Eq. (20) is modified as¨ aa = − G n +1 π n/ n ( n − ρ (cid:20) αR (cid:21) . (38)Note that R = a ( t ) r is a function of time. Eq. (38) is the dynamical equation for ( n + 1)-dimensional Newtoniancosmology in Gauss-Bonnet gravity. The main difference between this equation and Eq. (20) is that the correctionterm depends explicitly on the radius R . In order to remove this confusion, we suppose that for Newtonian cosmologythe spacetime is Minkowskian with k = 0. In this case we have R = 1 /H , and thus we can rewrite Eq. (38) as¨ aa = − G n +1 π n/ n ( n − ρ " α (cid:18) ˙ aa (cid:19) . (39)Combining Eq. (38) with (23), after replacing M by M , we get¨ aa = − G n +1 π n/ n ( n − n − n − ρ + np ] (cid:20) αR (cid:21) . (40)Thus we have derived the acceleration equation for the dynamical evolution of the FRW universe in Gauss-Bonnettheory. Multiplying ˙ aa on both sides of Eq. (40), and using the continuity equation (17), we get d ( ˙ a ) = 4 G n +1 π n/ n ( n − n − (cid:20) d ( ρa ) + 2 ˜ αr d ( ρa ) a (cid:21) . (41)Integrating yields H + ka = 4 G n +1 π n/ n ( n − n − ρ (cid:20) αρR Z d ( ρa ) a (cid:21) . (42)Now, in order to calculate the correction term we need to find ρ = ρ ( a ). Suppose a constant equation of stateparameter w = p/ρ , integrating the continuity equation (17) immediately yields ρ = ρ a − n (1+ w ) , (43)where ρ , an integration constant, is the present value of the energy density. Inserting relation (43) in Eq. (42), afterintegration, we obtain H + ka = 4 G n +1 π n/ n ( n − n − ρ (cid:20) αR (cid:18) n (1 + w ) − n (1 + w ) (cid:19)(cid:21) . (44)Using Eq. (15) we can further rewrite the above equation as (cid:18) H + ka (cid:19) (cid:20) α (cid:18) H + ka (cid:19) n (1 + w ) − n (1 + w ) (cid:21) − = 4 G n +1 π n/ n ( n − n − ρ (45)Next, we expand the above equation up to the linear order of ˜ α . We find (cid:18) H + ka (cid:19) + α ′ (cid:18) H + ka (cid:19) = 4 G n +1 π n/ n ( n − n − ρ, (46)where we have defined α ′ ≡ α [2 − n (1 + w )] n (1 + w ) , (47)and we have neglected O (˜ α ) terms and higher powers of ˜ α . This is due to the fact that at the present time R ≫ H + k/a ≪
1. Indeed for the present time where the apparent horizon radius becomes large, the correctionterm is relatively small and the usual Friedman equation is recovered. Thus, the correction make sense only at theearly stage of the universe where a →
0. When a →
0, even the higher powers of ˜ α should be considered. Withexpansion of the universe, the modified Friedmann equation reduces to the usual Friedman equation. From Eq. (46)we see that the correction term explicitly depends on the matter content through the equation of state parameter, w ,where we have assumed to be a constant.Eq. (46) is the ( n +1)-dimensional Friedmann equation in Gauss-Bonnet Gravity. The Friedmann equation obtainedhere from entropic force approach is in good agreement with that obtained from the gravitational field equation inGauss-Bonnet gravity [40]. This fact further supports the viability of Verlinde formalism. IV. LOVELOCK ENTROPIC GRAVITY
Finally we generalize our discussion to a more general case, the so-called Lovelock gravity, which is a generalizationof the Gauss-Bonnet gravity. The most general lagrangian which keeps the field equations of motion for the metricof second order, as the pure Einstein-Hilbert action, is Lovelock lagrangian [30]. This lagrangian is constructed fromthe dimensionally extended Euler densities and can be written as L = m X p =0 c p L p , (48)where c p and L p are arbitrary constant and Euler density, respectively. L set to be one, so c plays the role ofthe cosmological constant, L and L are, respectively, the usual curvature scalar and Gauss-Bonnet term. In an( n + 1)-dimensional spacetime m = [ n/ S = k B c Σ n ~ G n +1 m X i =1 i ( n − n − i + 1 ˆ c i r +2 − i . (49)where Σ n = n Ω n r n − is the horizon area. In the above expression the coefficients ˆ c i are given byˆ c = c n ( n − , ˆ c = 1 , ˆ c i = c i m Y j =3 ( n + 1 − j ) i > . (50)Note that in expression (49) for entropy, the cosmological constant term ˆ c doesn’t appear. This is a reasonable result,and due to the fact that the black hole entropy depends only on its horizon geometry. We further assume the entropyexpression (49) are valid for a FRW universe bounded by the apparent horizon in the Lovelock gravity provided wereplace the horizon radius r + with the apparent horizon radius R , namely S = k B c Σ n ~ G n +1 m X i =1 i ( n − n − i + 1 ˆ c i R − i . (51)It is easy to show that, the first term in the above expression leads to the well-known area law. The second termyields the apparent horizon entropy in Gauss-Bonnet gravity. The change in the general entropy expression of Lovelockgravity is obtained as △ S = k B c Q ~ G n +1 m X i =1 i ˆ c i R − i . (52)where we have used Eq. (4). Inserting Eq. (10), (12) and (52) in Eq. (1) one finds F = − G n +1 M mR n − m X i =1 i ˆ c i R − i . (53)Thus we have derived the Newton’s law of gravitation in Lovelock gravity resulting from the entropic force. It is obviousthat the first term of the above expression yields the famous Newton’s law of gravity, and the others terms will beimportant only for strong gravity or small distances. In this manner, the dynamical equation for ( n + 1)-dimensionalNewtonian cosmology takes the following form¨ aa = − G n +1 π n/ n ( n − ρ m X i =1 i ˆ c i (cid:18) ˙ aa (cid:19) i − . (54)The acceleration equation for the dynamical evolution of the FRW universe in ( n + 1)-dimensional Lovelock gravityis obtained following the method developed in the previous section. The result is¨ aa = − G n +1 π n/ n ( n − n − n − ρ + np ] m X i =1 i ˆ c i R − i . (55)Multiplying ˙ aa on both sides of Eq. (55), and using the continuity equation (17), after integrating, we get H + ka = 4 G n +1 π n/ n ( n − n − ρ " m X i =2 i ˆ c i ρa r i − Z d ( ρa ) a i − . (56)Using Eq. (43), we can perform the integration. We obtain H + ka = 4 G n +1 π n/ n ( n − n − ρ " m X i =2 [2 − n (1 + w )] i ˆ c i [2(2 − i ) − n (1 + w )] × R i − . (57)Eq. (57) can be rewritten in the following form (cid:18) H + ka (cid:19) " m X i =2 [2 − n (1 + w )] i ˆ c i [2(2 − i ) − n (1 + w )] × R i − − = 4 G n +1 π n/ n ( n − n − ρ. (58)At the present time where R ≫
1, we can expand the l.h.s of the above equation. Using Eq. (15), we reach (cid:18) H + ka (cid:19) " − m X i =2 (2 − n (1 + w )) i ˆ c i [2(2 − i ) − n (1 + w )] (cid:18) H + ka (cid:19) i − = 4 G n +1 π n/ n ( n − n − ρ. (59)If we define β i ≡ i [2 − n (1 + w )]ˆ c i n (1 + w ) − − i ) , (60)then we can write Eq. (59) in the following form (cid:18) H + ka (cid:19) + m X i =2 β i (cid:18) H + ka (cid:19) i = 4 G n +1 π n/ n ( n − n − ρ. (61)In this way we derive the ( n +1)-dimensional Friedmann equations in Lovelock gravity from the entropic force approachwhich is consistent with the result obtained from different methods [9, 31]. When β i = 0 ( i ≥ c = ˜ α . In this case β is exactlythe α ′ of the previous section. Again, from Eqs. (60) and (61) we see that the correction terms explicitly depend onthe equation of state parameter, w . V. CONCLUSIONS AND DISCUSSIONS
According to Verlinde’s argument, the total number of bits on the holographic screen is proportional to the area, A ,and can be specified as N = Ac G ~ . Indeed, the derivation of Newton’s law of gravity as well as Friedmann equations,in Verlide formalism, depend on the entropy-area relationship S = Ac G ~ , where A = 4 πR represents the area of thehorizon [16]. However, it is well known that the area formula of black hole entropy no longer holds in higher derivativegravities. So it would be interesting to see whether one can derive Newton’s law of gravity as well as the correspondingFriedmann equations in these gravities in the framework of entropic force perspective developed by Verlinde [16].In this paper, starting from first principles and assuming the entropy associated with the holographicscreen/apparent horizon given by the expression previously known via black hole thermodynamics, we were ableto derive the Newton’s law of gravity as well as the cosmological equations (Friedmann equations) governing theevolution of the universe in any gravity theory including Einstein, Gauss-Bonnet and more general Lovelock gravityin arbitrary dimensions. We derived the Newton’s law of gravitation from entropic force directly. Then, we derivedthe Friedmann equation by equating the mass in Newtonian gravity with active gravitational mass. Therefore, theFriedmann equation derived here from entropic force too, but indirectly. In our derivation the assumption that theentropy of the apparent horizon of FRW universe in Gauss-Bonnet and Lovelock gravity have the same form as thespherically symmetric black hole entropy in these gravities, but replacing the black hole horizon radius by the apparenthorizon radius, plays a crucial role. Interestingly enough, we found that the higher dimensional gravitational couplingconstant of Friedmann/Einstein equation can be derived naturally from this approach which coincides with the resultobtained by comparing the weak field limit of Einstein equation with Poisson equation in higher dimension. Our studyshows that the approach here is powerful enough to derive the gravitational field equations in any gravity theory. Theresults obtained here in the framework of Gauss-Bonnet gravity and more general Lovelock gravity further supportthe viability of Verlinde’s formalism. Acknowledgments
This work has been supported by Research Institute for Astronomy and Astrophysics of Maragha, Iran. [1] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973);S. W. Hawking, Commun Math. Phys. 43, 199 (1975);S. W. Hawking, Nature 248, 30 (1974).[2] J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).[3] P. C. W. Davies, J. Phys. A: Math. Gen. 8, 609 (1975);W. G. Unruh, Phys. Rev. D 14, 870 (1976);L. Susskind, J. Math. Phys. 36, 6377 (1995).[4] T. Jacobson, Phys. Rev. Lett. , 1260 (1995).[5] T. Padmanabhan, Class. Quantum. Grav. 19 (2002) 5387.[6] C. Eling, R. Guedens, and T. Jacobson, Phys. Rev. Lett. , 121301 (2006).[7] M. Akbar and R. G. Cai, Phys. Rev. D , 084003 (2007).[8] R. G. Cai and L. M. Cao, Phys.Rev. D , 064008 (2007).[9] R. G. Cai and S. P. Kim, JHEP , 050 (2005).[10] B. Wang, E. Abdalla and R. K. Su, Phys.Lett. B , 394 (2001);B. Wang, E. Abdalla and R. K. Su, Mod. Phys. Lett. A , 23 (2002).[11] R. G. Cai, L. M. Cao and Y. P. Hu, JHEP 0808 (2008) 090.[12] S. Nojiri and S. D. Odintsov, Gen. Relativ. Gravit. 38, 1285 (2006);A. Sheykhi, Class. Quantum Grav. 27 (2010) 025007;A. Sheykhi, Eur. Phys. J. C 69 (2010) 265.[13] A. Sheykhi, B. Wang and R. G. Cai, Nucl. Phys. B (2007)1;R. G. Cai and L. M. Cao, Nucl. Phys. B (2007) 135[14] A. Sheykhi, B. Wang and R. G. Cai, Phys. Rev. D (2007) 023515;A. Sheykhi, B. Wang, Phys. Lett. B 678 (2009) 434;A. Sheykhi, JCAP 05 (2009) 019[15] T. Padmanabhan, Rept. Prog. Phys. 73 (2010) 046901.[16] E. Verlinde, JHEP 1104, 029 (2011).[17] T. Padmanabhan, Mod. Phys. Lett. A 25 (2010) 1129.[18] R.G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81, (2010) 061501(R);R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81 (2010) 084012.[19] L. Smolin, arXiv:1001.3668.[20] M. Li and Y. Wang, Phys. Lett. B 687, 243 (2010).[21] Y. Tian and X. Wu, Phys. Rev. D , 104013 (2010);Y. S. Myung, arXiv:1002.0871. [22] I. V. Vancea and M. A. Santos, arXiv:1002.2454.[23] L. Modesto and A. Randono, arXiv:1003.1998.[24] A. Sheykhi, Phys. Rev. D , 104011 (2010).[25] B. Liu, Y. C. Dai, X. R. Hu and J. B. Deng, Mod. Phys. Lett. A , 489 (2011).[26] S. H. Hendi and A. Sheykhi, Phys. Rev. D (2011) 084012 ;A. Sheykhi and S. H. Hendi, Phys. Rev. D (2011) 044023;S. H. Hendi and A. Sheykhi, Int. J Theor. Phys. 51 (2012) 1125;A. Sheykhi, Z. Teimoori, Gen Relativ Gravit 44 (2012) 1129;A. Sheykhi, Int. J Theor. Phys. 51 (2012) 185.[27] W. Gu, M. Li and R. X. Miao, arXiv:1011.3419;R. X. Miao, J. Meng and M. Li, arXiv:1102.1166.[28] Y. X. Liu, Y. Q. Wang and S.W. Wei, Class. Quantum Grav. , 185002 (2010);V. V. Kiselev and S. A. Timofeev, Mod. Phys. Lett. A , 2223 (2010);R. A. Konoplya, Eur. Phys. J. C , 555 (2010);R. Banerjee and B. R. Majhi. Phys. Rev. D , 124006 (2010);P. Nicolini, Phys. Rev. D , 044030 (2010);C. Gao, Phys. Rev. D , 087306 (2010);Y. S. Myung and Y. W. Kim, Phys. Rev. D , 105012 (2010);H. Wei, Phys. Lett. B , 167 (2010);Y. Ling and J.P. Wu, JCAP , 017 (2010);D. A. Easson, P. H. Frampton and G. F. Smoot, Phys. Lett.B 696 (2011)273;D. A. Easson, P. H. Frampton and G. F. Smoot, arXiv:1003.1528;S. W. Wei, Y. X. Liu and Y. Q. Wang, Commun. Theor. Phys.56 (2011) 455.[29] R. B. Mann and J. R. Mureika, Phys. Lett. B 703 (2011) 167.[30] D. Lovelock, J. Math. Phys. (N.Y.) 12, 498 (1971).[31] N. Deruelle and L. Farina-Busto, Phys. Rev. D 41, 3696 (1990).[32] B. Zwiebach, Phys. Lett. B 156, 315 (1985);B. Zumino, Phys. Rep. 137, 109 (1986);D. J. Gross and J. H. Sloan, Nucl. Phys. B291, 41 (1987).[33] J.M. Maldacena, Adv. Theor.Math. Phys. 2, 231 (1998);Int. J. Theor. Phys. 38, 1113 (1999).[34] L. Susskind, J. Math. Phys. 36, 6377 (1995);R. Bousso, Rev. Mod. Phys. 74, 825 (2002).[35] T. Padmanabhan, Phys. Rev. D81