f(r) gravity and Maxwell equations from the holographic principle

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1 . Article. SCIENCE CHINA Physics, Mechanics & Astronomy March 2012 Vol. 55 No. 3: doi: /s f(r) gravity and Maxwell equations from the holographic principle MIAO RongXin 1*,MENGJun 2 &LIMiao 2 1 Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei , China; 2 Kavli Institute for Theoretical Physics, Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing , China Received December 27, 2011; accepted January 18, 2012; published online February 9, 2012 Extending the holographic program of our previous work, we derive f (R) gravity and the Maxwell equations from the holographic principle, using time-like holographic screens. We find that to derive the Einstein equations and f (R) gravity by a natural holographic approach, the quasi-static condition is necessary. We also find the surface stress tensor and the surface electric current, surface magnetic current on a holographic screen for f (R) gravity and Maxwell s theory, respectively. entropic gravity, f(r) gravity, Maxwell equations, the holographic principle PACS number(s): Cv, h Citation: Miao R X, Meng J, Li M. f (R) gravity and Maxwell equations from the holographic principle. Sci China-Phys Mech Astron, 2012, 55: , doi: /s Gravity may not be a fundamental force, this may be related to a deep principle underlying the working of our world: The holographic principle. The most recent concrete proposal for a macroscopic picture of gravity is due to Verlinde [1] (see also ref. [2]), which in turn is motivated by the gauge/gravity correspondence in string theory and an earlier proposal of Jacobson [3]. However, there are problems with Verlinde s proposal, as pointed out in ref. [4]. First, Verlinde s derivation of the Einstein equations always requires a positive temperature on a holographic screen, which is not attainable for an arbitrary screen. To avoid this problem, we propose to use a screen stress tensor to replace the assumption of equal partition of energy. Second, using Verlinde s proposal, we do not have a reasonable holographic thermodynamics. Instead, there is always an area term in the holographic entropy of a gravitational system. There is no such problem in our proposal. A prediction of the program of ref. [4] is that there is always a huge holographic entropy associated with a gravitational system, with a form similar to the Bekenstein bound. *Corresponding author ( mrx11@mail.ustc.edu.cn) Contributed by LI Miao (Editorial Board Member) In this paper, we would like to clarify the role of the adiabatic condition used in ref. [4] left unexplained in that paper. To see how far our program can get, we also use the same idea to derive f (R) gravity. We find that unlike Verlinde s proposal in which it is impossible to accommodate theories containing higher derivatives [5], there is a natural ansatz for the surface stress tensor to derive the f (R) theory. Last, using the same idea with a surface current replacing the surface stress tensor, we derive the Maxwell equations. Our success demonstrates that our program is more universal. This paper is organized as follows. In sect. 1, we review our holographic derivation of the Einstein equations on a time-like screen. In sects. 2 and 3, we derive the f (R) equations and the Maxwell equations by a similar holographic approach, respectively. We conclude in sect Einstein equations from the holographic principle In this section, we shall review our derivation of the Einstein equations in a holographic program [4] different from Verlinde s [1]. We will explain the physical reason to impose c Science China Press and Springer-Verlag Berlin Heidelberg 2012 phys.scichina.com

2 376 Miao R X, et al. Sci China-Phys Mech Astron March (2012) Vol. 55 No. 3 the quasi-static condition used in the previous paper [4] in order to derive the Einstein equations naturally on a time-like holographic screen. Let us start with some definitions. Our holographic screen is a (2+1) dimensional time-like hyper-surface, which can be open or closed, embedded in the (3+1) dimensional spacetime M. Weusex a, g ab, a, y i, γ ij and D i (here a, b run from 0to3,andi, j run from 0 to 2) to denote the coordinates, metric and covariant derivatives on M and, respectively. Unlike Jacobson s idea [3], we consider an energy flux δe passing through an open patch on a time-like holographic screen d =dadt (see Figure 1) δe = T ab ξ a N b dadt, (1) where T ab is the stress tensor of matter in the bulk M, ξ a is a time-like Killing vector, and N a is the unit vector normal to. To define N a we may assume to be specified by a function f (x a ) = c. (2) Thus, the normalized normal vector to is N a = gab b f b f b f. (3) As in ref. [4], we introduce the surface energy density σ and the surface energy flux j on the screen. These are quantities contained in the surface stress tensor τ ij and given by σ = u i τ ij ξ j, j = m i τ ij ξ j, (4) where u i, m i are the unit vectors normal to the screen s boundary (We will give the expressions of u i and m i shortly), and ξ i is a Killing vector on. In a quasi-static space-time u i is related to ξ i by u i = e φ ξ i,andφ is the Newton s potential defined by φ = 1 2 log( ξi ξ i ) on the screen. Note that σ and j are the energy density and energy flux on the screen measured by the observer at infinity. Apparently, central to our discussion is the choice of τ ij. Naturally, τ ij should depend on the extrinsic geometry of the screen, since extrinsic geometry contains both the information of the bulk M and the screen, thus it is a natural bridge relating both sides to realize the holographic principle. Thus we assume the following simplest and most general form τ ij = nk ij + qγ ij, (5) where n is a constant, q is a function to be determined and K ij is the extrinsic curvature on defined by K ij = e a i eb j an b, (6) where e b j = xa y i is the projection operator satisfying N a e a i = 0. On the screen, the change of energy has two sources. One is due to variation of the energy density σ, the other is due to energy flowing out the patch to other parts of the screen, Figure 1 An energy flux δe passing through an open patch on the holographic screen. given by the energy flux j. The energy variation on the patch is then given by δe = (u i τ ij ξ j )da t+dt t m i τ ij ξ j dydt = (M i )τ ij ξ j hdz 2 = D i (τ ij ξ j )dadt, (7) where the first term in the first equality is due to change of the density and the second term is the energy flow through the patch boundary parameterized by y (see Figure 2). These two terms can be naturally written in a uniform form, since the boundary of the patch consists of two space-like surfaces (d at t and t + dt), and a time-like boundary. h is the determinant of the reduced metric on, D i is the covariant derivative on. M i is a unit vector in and is normal to. Let us choose a suitable function f (y i ) = c on to denote the boundary, and then we have M i = γ ij D j f D j f D j f. (8) Note that when M i is along the direction of dy 0 (dt) itbecomes u i, and when along the direction of dy i (dy 1, dy 2 )it becomes m i. For a reason to be clear later, we focus on a quasi-static process in the following derivations. Recall that in eq. (2) we use f (x a ) = c to denote the holographic screen. In the quasi-static limit, f (x a ) is independent of time, so N a (0, 1 f, 2 f, 3 f ). Note that ξ a (1, 0, 0, 0) in the quasi-static limit. Thus we have N a ξ a 0. The Killing vector ξ i on can be induced from the Killing vector ξ a in the bulk M in the quasi-static limit: ξ i = ξ a e a i, D (iξ j) = a (ξ b N c ξ c N b )e a (i eb j) = K ijn a ξ a 0. (9) From eq. (5) and the Gauss-Codazzi equation R ab N a e b i = D j (K j i Kγ j i), one can rewrite eq. (7) as: δe = [nr ab ξ i e a i Nb ξ i D i (nk + q)]dadt = [nr ab ξ a N b ξ a a (nk + q)]dadt, (10) wherewehaveusedtheformulasξ a = ξ i e a i, ξi D i f = ξ a a f in the quasi-static limit. It should be stressed that the second term ξ a a (nk + q) in eq. (10) can not be written in the form ξ a N b B ab (B ab is independent of ξ a and N b ). For details, please refer to Appendix.

3 Miao R X, et al. Sci China-Phys Mech Astron March (2012) Vol. 55 No are satisfied and so the surface stress tensor τ ij is the Brown- York stress tensor eq. (13). Then, eq. (14) is expected to be Figure 2 Left panel: The first term (u i τ ij ξ j )da t+dt t in eq. (7) is due to change of the density. Right panel: The second term m i τ ij ξ j dydt is the energy flow (denoted by the arrows) through the patch boundary parameterized by y. Given that ξ a, N b are independent vectors and g ab ξ a N b = 0, equating eqs. (1) and (10) results in nr ab + fg ab = T ab, q = nk, (11) where f is an arbitrary function. We note that the second equation is a consequence of the fact that the second term on the R.H.S. of eq. (10) must vanish. This result tells us that the surface stress tensor is the same as the Brown-York surface stress tensor, but we have not made use of any action. Using the energy conservation equation a T ab = 0, we obtain f = n( R 1 2 +Λ). Defining the Newton s constant as G = 8πn, we then get the Einstein equations R ab R 2 g ab +Λg ab = 8πGT ab. (12) Substituting n = 1 8πG and q = nk into eq. (5), we obtain τ ij = 1 8πG (Kij Kγ ij ), (13) which is just the quasi-local stress tensor of Brown-York defined in ref. [6]. Now we have derived the Einstein equations and the Brown-York stress tensor from our holographic program. In the above derivations, for simplicity, we have imposed the quasi-static condition N a ξ a = 0. We now briefly discuss the relation between the quasi-static condition and the holographic principle. Let us try to see what will happen when we abandon the quasi-static condition N a ξ a = 0. The energy flux δe passing through an open patch on the holographic screen d =dadt becomes δe = T ab ξ a N b dadt = [T ab e a i ξi N b + T ab N a N b (N c ξ c )]dadt, (14) anewtermt ab N a N b (N c ξ c ) which is proportional to the pressure along the direction of N a appears. Since now we aim to study the relationship between the quasi-static condition and the holographic principle, not to derive the Einstein equations, for simplicity, we assume that the Einstein equations δe 1 = 8πG = [ ( R ab e a i ξi N b + R ab R ) ] 2 g ab N a N b (N c ξ c ) [ ξ j D i τ ij πG ( (3) R + K 2 K ij K ij )(N c ξ c ) We have used the Gauss-Godazzi equations dadt ] dadt. (15) R ab N a e b i = D j (K j i Kδ j i), (16) (2R ab Rg ab )N a N b = (3) R + K 2 K ij K ij, (17) where (3) R is the Ricci scalar on. Note that ξ i = e a i ξ a is on longer a Killing vector on the screen, since D (i ξ j) = K ij N a ξ a 0. If we still assume eq. (4) with ξ i being the projection of ξ a on, the change of energy on screen eq. (7) becomes δe = (u i τ ij ξ j )da t+dt t m i τ ij ξ j dydt = (M i )τ ij ξ j hdz 2 = D i (τ ij ξ j )dadt = ( ξ j D i τ ij τ ij D i ξ j )dadt [ = ξ j D i τ ij + 1 ] 8πG (K2 K ij K ij )(N c ξ c ) dadt. (18) Since (3) R K 2 + K ij K ij 0, the second term of eq. (15) is not equal to the second term of eq. (18) except when N c ξ c = 0. Of course, one can add some terms to eq. (18) to equate eqs. (15) and (18), but it is very unnatural. So if we expect the holographic principle to require that the energy flow through the holographic screen (defined by eq. (15)) and the change of energy on the holographic screen (defined by eq. (18)) be equal, the quasi-static condition (N a ξ a 0) is necessary. Finally, even without using the Einstein equation, the fact that the second term in eq. (14) is proportional to the transverse component of T ab tells us that we need to introduce an transverse term in the surface stress tensor. This is not holographic at all. 2 f(r) gravity from the holographic principle In this section, we shall derive the equations of f (R) gravity in the same manner as in ref. [4] and the previous section. For the same reason as in the previous section we impose the quasi-static condition (N a ξ a 0). The energy flux δe passing through an open patch on the holographic screen d =dadt and the change of energy on the screen take the same form as in eqs. (1) and (7): δe = T ab ξ a N b dadt, (19)

4 378 Miao R X, et al. Sci China-Phys Mech Astron March (2012) Vol. 55 No. 3 δe = (u i τ ij ξ j )da t+dt t m i τ ij ξ j dydt = (M i )τ ij ξ j hdz 2 = D i (τ ij ξ j )dadt. (20) The only difference is the assumption of surface stress tensor τ ij. As argued in sect. 1, τ ij should depend on the the extrinsic geometry of the screen, since it is a natural bridge relating the bulk M and the holographic screen. Instead of the simplest assumption eq. (5), we now assume more generally τ ij = nf (R)K ij + qγ ij, (21) where f (R) is a general function of the Ricci scalar R,andq is to be determined. The case f (R) = 1 is special and is what underlying the Einstein equations. Substituting eq. (21) into eq. (20), we obtain δe = D i (τ ij ξ j )dadt = [nf R ab N a ξ b nξ a K ab b f ξ a a (q + nf K)]dAdt = [nf R ab N a ξ b + nξ a a N b b f ξ a a (q + nf K)]dAdt = [nn a ξ b (R ab f a b f ) + ξ a a (nn b b f nf K q)]dadt. (22) In the above calculation, we have used the Gauss-Codazzi equation R ab N a e b i = D j (K j i Kγ j i), ξ a a R = 0, ξ a K ab = ξ a a N b and the quasi-static condition (N a ξ a 0). As is the case in sect. 1 the second term of eq. (22) ξ a a (nn b b f nf K q) does not contain term ξ a N b B ab,whereb ab is independent of ξ a and N b. For details, please refer to Appendix (just replace (nk + q)by(nf K + q nn b b f )). Since ξ a, N b are independent vectors and g ab ξ a N b = 0, equating eqs. (19) and (22) results in q = nn b b f nf K, f R ab a b f + Fg ab = 1 n T ab, (23) where F is an arbitrary function. Using a T ab = 0, we get ( b F = b c c f f ). (24) 2 Thus, F = f f 2 +Λ. Define the Newton s constant as G = 1 8πn, and we then obtain equations of motion of the f (R) gravity f R ab f 2 g ab a b f + g ab f +Λg ab = 8πGT ab. (25) Substituting q = 1 8πG (Nb b f f K) into eq. (21), we get the surface stress tensor of f (R) gravity τ ij = 1 8πG [ f (R)(K ij Kγ ij ) + N c c f γ ij ]. (26) Now, we have obtained the f (R) equations and surface stress tensor from our holographic program. Let us continue to understand the physical meaning of the surface stress tensor eq. (26). It is well known that the action of f (R) gravity S = 1 dx 4 gf(r) + S M (g ab,ψ) (27) 2κ is equivalent to the Einstein gravity with a scalar field S = dx [ R 4 g 2κ 1 ] 2 a φ a φ V( φ) ( + S M e 2κ/3 g ab,ψ ), (28) if we perform conformal transformation g ab = f g ab and 3 φ = 2κ ln f, V( φ) = Rf f. The surface stress tensor for 2κ( f ) 2 metric g ab is τ ij = 1 κ ( K ij K γ ij ). (29) Note that Ñ a = ( f ) 1/2 N a, Ñ a = ( f ) 1/2 N a, γ ab = f γ ab and Γ d cb =Γd cb + Cd cb,wherecd cb = 1 2 f (δ d c b f + δ d b c f g cb g de e f ). After some calculation, we find τ ij = ( f ) 1/2 (K ij Kγ ij ) + ( f ) 1/2 γ ij N d d f κ κ = ( f ) 1/2 τ ij. (30) It is interesting that the surface stress tensors τ ij and τ ij of eq. (26) are related exactly by an conformal factor ( f ) 1/2. Let us move on to understand this conformal factor ( f ) 1/2. Firstly, let us derive some useful formulas. T ab M = 2 1 δs M g δ g = 1 ab f T ab M, T φ ab = a φ b φ 1 2 g ab g cd c φ d φ g ab V( φ), Ñ a = ( f ) 1/2 N a, γ ab = g ab Ñ a Ñ b = f γ ab, (31) M i = ( f ) 1/2 M i, h ij = γ ij M i M j = f h ij, ξ a = ξ a. We shall prove the last equation ξ a = ξ a below. Assume ξ a being a Killing vector of the metric g ab,wehave then L ξ g ab = ξ c c g ab + ( a ξ c )g cb + ( b ξ c )g ca = 0, (32) L ξ g ab = ξ c c ( f g ab ) + f ( a ξ c )g cb + f ( b ξ c )g ca = g ab ξ c c f + f (ξ c c g ab + ( a ξ c )g cb + ( b ξ c )g ca ) = 0, (33) where we have used the identity ξ a a R = 0. Now, it is clear that ξ a is also a Killing vector of g ab. The energy flux δe passing through the holographic screen is δẽ = ( T M ab + T φ ab )ξa Ñ b γdx 2 dt

5 Miao R X, et al. Sci China-Phys Mech Astron March (2012) Vol. 55 No = T ab M ξa Ñ b γdx 2 dt = T ab ξ a N b γdx 2 dt = δe. (34) Above we have used again the identity ξ a a R = 0andthe quasi-static limit (N a ξ a 0), so that T φ ab ξa Ñ b = 0. The change of energy on the screen is δẽ = M hdz i τ ij ξ j 2 = M i ( f ) 1/2 τ ij ξ j hdz 2. (35) Equating eqs. (34) and (35), we find δe = M i ( f ) 1/2 τ ij ξ j hdz 2 = M i τ ij ξ j hdz 2. (36) It is clear that ( f ) 1/2 τ ij plays the rule of τ ij. The above discussion is a check that from the holographic principle we have derived the correct surface stress tensor (26) of f (R) gravity, and helps us to gain some insight into the physical meaning of f (R) surface stress tensor (26): It is related with its conformal counterpart by an appropriate conformal factor τ ij = ( f ) 1/2 τ ij. 3 Maxwell equations from the holographic principle In this section, we shall derive the Maxwell equations by a similar holographic approach as in the above two sections. However, there are two main differences. First, instead of using conservation of energy, we use conservation of charge to derive the Maxwell equations. We calculate the charge passing through the holographic screen in the bulk M and the charge change on the screen, respectively, and equate them as dictated by the holographic principle. With an appropriate assumption of the current on the screen, we can obtain the Maxwell equations. Second, we do not need the quasi-static condition (N a ξ a 0), in fact we make no use of a Killing vector in our derivations of the Maxwell equations, since a Killing vector is related to energy instead of charge. Consider the electric charge δq passing through an open patch on the holographic screen d =dadt, δq = J a N a dadt, (37) where J a is the electric current of matter in the bulk M. Assume the surface electric current on the screen be j i,andthen the charge change on the screen is δq = u i j i t+dt t m i j i hdydt = M i j i hdy 2 = D i j i dadt, (38) where the first term in the first equality is due to change of charge density and the second term is the electric flow through the patch boundary parameterized by y. Again these two terms can be naturally written in a uniform form Mi j i hdy 2. Applying Stokes s Theorem, we get the last equality. According to the holographic principle, we should equate eq. (37), the charge flow through the patch of the holographic screen, and eq. (38), the charge change on this patch, yielding D i j i = N a J a. (39) So far, we have not made any assumption about the form of j i. Now, let us gain some insight into the form of j i from eq. (39). According to the holographic principle, we expect to derive the bulk equations of motion from eq. (39), which implies that j i = e a i j a should linearly depend on N a. Thus, the general form for j a is j a = A ab N b + A abc b N c +... (40) And j a must satisfy the following conditions N a j a = 0, D i j i = γ ab a j b = N a J a, (41) where γ ac is the projection operator defined as γ ac = g ac N a N c. For simplicity, we consider the simplest assumption j a = A ab N b. The conditions eq. (41) become A ab N a N b = 0, (γ ac a A cb )N b + γ c aa cb a N b = N b J b. (42) From the last equation of eq. (42), we have (γ ac a A cb + J b )N b = 0, γ c a A cb a N b = 0. (43) Since γa c a N b = K cb is symmetrical in c and b, wederive γcγ a b da ad = γb aγd c A ad from γaa c cb a N b = 0. As we can change the direction of N a arbitrarily with changing the screen, taking γc aγd b A ad = γ a b γd c A ad and A ab N a N b = 0into account, we find that A ab is antisymmetric. Thus, the first equation of eq. (43) becomes (γ ac a A cb + J b )N b = ( a A a b + J b)n b (N a a A cb )N c N b So for arbitrary N b,weget = ( a A a b + J b)n b = 0. (44) a A ab = J b, (45) with A ab an antisymmetric tensor and the surface electric current j a = A ab N b on the screen. Note that conservation of charge a J a = 0 is satisfied automatically for antisymmetric A ab, a J a = a b A ab = 1 2 ( a b b a )A ab = R ab A ab = 0. (46) The above approach can be directly extended to the case of magnetic charge. Assume the magnetic current in the bulk

6 380 Miao R X, et al. Sci China-Phys Mech Astron March (2012) Vol. 55 No. 3 and on the screen to be Jm a = 0and j a m = B ab N b, respectively. Withthesameprocedurewearriveat B ab = B ba, a B ab = Jm b = 0. (47) Since the magnetic current Jm b = 0 in the bulk, it is expected that j a m = Bab N b is not an independent physical quantity. It should either vanish or be related to the surface electric current j a on the screen. In view of the electromagnetical duality, it is natural to assume that the surface electric current and magnetic current on the screen are related with each other by the Hodge duality B ab = 1 2 ɛabcd A cd. (48) Then, eq. (47) becomes ɛ abcd b A cd = 0, (49) which is just the Bianchi identity. The general solution to the above equation is A cd = d A c c A d with A c an arbitrary vector field. Renaming A cd by F dc, we summarize our results. The surface electric current and magnetic current on the screen are j a = F ba N b, j a m = 1 2 ɛbacd F cd N b. (50) The equations of motion in the bulk M are a F ab = J b, [a F bc] = 0, (51) where F ab = a A b b A a and [ ] denotes complete antisymmetrization. The above equations are just the Maxwell equations. Applying the formulas F ab F bc = 1 4 F Fδ c a, F ab F bc F ab F bc = 1 2 F2 δ c a, (52) we obtain the following interesting identities jj M = 1 4 F F, j 2 j 2 M = 1 2 F2, (53) where F ab is the Hodge duality of F cd, F ab = 1 2 ɛabcd F cd. Note that the L.H.S. of eq. (53) are physical quantities on the screen while the R.H.S. of eq. (53) contains only physical quantities in the bulk which are independent of the direction of the screen N a,so j a and j a M contain all the information of the bulk (F 2 and FF). That is a representation of the holographic principle. Now, we have derived the Maxwell equations from the holographic principle and an appropriate assumption for the relationship between the surface electric current and magnetic current on the holographic screen. 4 Conclusions We have derived f (R) gravity and the Maxwell equations from the holographic program we proposed in ref. [4]. We find the surface stress tensor and surface electric current, surface magnetic current for f (R) gravity and Maxwell s theory, respectively. It is interesting to extend our holographic approach to more general higher derivative gravity, and investigate the corresponding thermodynamics on a time-like holographic screen. It should be mentioned that in sect. 3 we only find the simplest solution to eqs. (40) and (41), but whether there are other solutions and corresponding holographic electromagnetic theories is a topic of interest. We hope we will gain more insight into these problems in the future. Appendix In this appendix, we shall prove that the second term ξ a a (nk + q) in eq. (10) does not contain the term ξ a N b B ab, where B ab is independent of ξ a and N b.ifξ a a (nk + g) does include such terms, (nk + g) must depend on N b linearly, nk + g = A b N b + A bc b N c +... (a1) Thus, ξ a a (nk + g) = ξ a N b a A b + ξ a A b a N b + (ξ d d A ab ) a N b... = ξ a N b a A b + ξ d a N b ( d A ab + g da A b ) +... (a2) Taking into consideration the fact that eq. (1) does not contain the term ξ d a N b and that R ab is symmetrical, equating eqs. (1) and (10) yields a A b = b A a, (a3) D i A jk + γ ij A k = e d i ea j eb k ( da ab + g da A b ) = 0. (a4) From the above equations, we derive D i A + A i = 0, D i A i k + 4A k = 0, A = A jk γ jk, (a5) D i A jk D j A ik = 0, D i A D j A j i = D i A + 4A i = 0. (a6) From the first equation in eq. (a5) and the last equation in eq. (a6), we get A i = 0, so A a = e a i Ai + N a (N b A b ) = N a (N b A b ). Now, we can prove that the first term of eq. (a2) vanishes: ξ a N b a A b = ξ a N b b A a = N b b (ξ a A a ) N b A a b ξ a = N b b (ξ i A i ) (N c A c )N b N a b ξ a = 0. (a7) Thus, the second term ξ a a (nk + g) in eq. (10) does not contain the term ξ a N b B ab. We must require (nk + g) = 0in order to equate eqs. (1) and (10). This research was supported by the National Natural Science Foundation of China (Grant Nos , A050207, and ) and the National Basic Research Program of China (Grant No. 2007CB815401). 1 Verlinde E P. On the origin of gravity and the laws of Newton. J High Energy Phys, 2011, 1104: Padmanabhan T. Thermodynamical aspects of gravity: New insights. Rep Prog Phys, 2010, 73: ; Padmanabhan T. Equipartition of energy in the horizon degrees of freedom and the emergence of gravity. Mod Phys Lett A, 2010, 25: Jacobson T. Thermodynamics of space-time: The Einstein equation of state. Phys Rev Lett, 1995, 75: Gu W, Li M, Miao R X. A new entropic force scenario and holographic thermodynamics. Sci China-Phys Mech Astron, 2011, 54: Li M, Pang Y. A no-go theorem prohibiting inflation in the entropic force scenario. Phys Rev D, 2010, 82: Brown J D, York J W. Quasilocal energy and conserved charges derived from the gravitational action. Phys Rev D, 1993, 47: