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Home , Entropic force, Entropic gravity, Erik Verlinde

SLAC-PUB-15931 Where is ~ Hiding in Entropic ?

Pisin Chen1,2,3,4∗ and Chiao-Hsuan Wang1,3† 1. Department of , National Taiwan University, Taipei, Taiwan 10617 2. Graduate Institute of Astrophysics, National Taiwan University, Taipei, Taiwan 10617 3. Leung Center for and Particle Astrophysics, National Taiwan University, Taipei, Taiwan 10617 4. Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, U.S.A. (Dated: December 15, 2011) The scenario recently proposed by reproduced the Newton’s law of purely classical gravity yet the key assumptions of this approach all have quantum mechanical origins. This is atypical for emergent phenomena in physics, where the underlying, more fundamental physics often reveals itself as corrections to the leading classical behavior. So one naturally wonders: where is ~ hiding in entropic gravity? To address this question, we first revisit the idea of holographic screen as well as and its variation law in order to obtain a self-consistent approach to the problem. Next we argue that when dealing with issues the generalized uncertainty principle (GUP) should be the more appropriate foundation. Indeed based on GUP it has been demonstrated that the Bekenstein entropy area law must be modified not only in the strong but also in the weak gravity regime. In the weak gravity limit, such a GUP modified entropy exhibits a logarithmic correction term. When applying it to the entropic interpretation, we demonstrate that the resulting gravity law does include sub-leading order correction terms that depend on ~. Such deviation from the classical Newton’s law may serve as a probe to the validity of the entropic gravity postulate.

PACS numbers: 03.65.Ud, 04.50.Kd, 04.70.Dy, 89.70.Cf

I. INTRODUCTION his perspective Einstein equations are now an equation of state rather than a fundamental theory. More ideas on emergent gravity have been recently proposed (See, for The issue of how gravity and are cor- example, [10–13]). related has been studied for decades, triggered by the seminal discovery by Bekenstein[1, 2] on the area-law of Similar to Jacobson’s derivation of Einstein equa- black hole (BH) entropy and temperature. After Hawk- tions through thermodynamic, Verlinde treated gravity ing’s discovery of the BH evaporation and the interpre- as an analogous to the restoring force of a tation of the its temperature as the thermal temperature stretched elastic driven by the system’s tendency of blackbody radiation[3], considerable efforts have been towards the maximization of entropy [13], and interest- made to find the statistical interpretation of the propor- ingly the Newton’s law of gravitation was shown to arise. tionality of black hole entropy and its horizon area. See To arrive at the Newton’s force law of gravity through [4] and [5], for example, for a review. By now a well- the first law of thermodynamic F dx = T dS, Verlinde accepted view is that the black hole entropy is associated first invoked the Compton wavelength of the test par- with the external thermal state perceived by an observer ticle to find the change of entropy with respect to its outside the event horizon who has no access to the BH displacement. He then invoked the interior. Namely, the correlation between the degrees [14–16] and the to define the tem- of freedom on opposite sides of the horizon results in a perature experienced by the test particle. One cannot mixed state for observation from the outside, i.e., the but notices that all these building blocks have quantum ‘entanglement entropy’[6, 7], which depends upon the mechanical origin, or more specifically the presence of ~. boundary properties and will be discussed more in the Yet all the ~’s just get subtly cancelled and at the end a later sections of this paper. purely classical Newton’s law has emerged. This is rather atypical for emergent phenomena in physics, where the The inversion of the logic that describes gravity as an underlying, more fundamental physics often reveals itself emergent phenomenon was first proposed by Sakharov as corrections to the leading classical behavior. So one [8], who suggested that gravity is induced by quantum naturally wonders: where is hiding in entropic gravity? field fluctuations. Invoking the area scaling property of ~ entanglement entropy, Jacobson in 1995 [9] used basic There have been previous works aiming at finding the laws of thermodynamics to derive Einstein equations. In entropic corrections to Newton’s law but however unsat- isfactory: Santos et al. [17] and Ghosh [18] suggested the corrected to Newton’s grativy force law when the depen- dence on the uncertainty in position is included, which ∗Electronic address: [email protected] is bothersome; both Modesto [19] and Setare [20] sug- †Electronic address: [email protected] gested the sub-leading terms of force law rather than an

Work supported by US Department of under contract DE-AC02-76SF00515 and HEP. 2 exact form while the former considered only the correc- II. ENTROPIC GRAVITY tions to entropic variation without noting that the in- formation content and therefore the temperature is also A. Holographic principle and holographic affected, and the latter failed to introduce the right form entanglement entropy of GUP corrected entropy into consideration. We argue that when dealing with quantum gravity issues the gen- 1. Holographic principle eralized uncertainty principle (GUP) should be the more appropriate foundation. Indeed based on GUP it has Holographic principle [14, 15], which is developed from been demonstrated that the black hole Bekenstein en- black hole thermodynamics, plays a key role in Verlinde’s tropy area law must be modified not only in the strong entropic interpretation of gravity. It dictates that the de- but also in the weak gravity regime [21]. In the weak grees of freedom in a d+2 dimensional quantum gravity gravity limit, such a GUP modified Bekenstein entropy system can be seen as encoded on its d+1dimensional exhibits a logarithmic correction. Such a log-correction boundary like a holographic image. This principle orig- is consistent with similar conclusions drawn from inates from the Bekenstein-Hawking formula where the theory, AdS/CFT correspondence, and loop quantum BH entropy is proportional to the surface area of its event gravity considerations [22–25]. When applying it to the horizon: entropic derivations, we demonstrate that the resulting 2 2 3 entropic gravity does include sub-leading order correction 4πkBGM M kBc terms that depend on . SB = = 4π 2 = A. (1) ~ ~c Mp 4~G Here A is the surface area of black hole event horizon and p Mp = ~c/G is the Planck mass. For the second law of thermodynamics to hold, that is, the entropy in the universe to be non-decreasing, the holographic principle suggests that the information con- tent S of an enclosed region should be no larger than the Bekenstein-Hawking entropy defined by its sur- face area[2, 16]:

k k c3 The organization of this paper is as follows. To address B B S ≤ 2 A = A = SB. (2) the question we posted, we first revisit the ideas of holo- 4Lp 4~G graphic screen as well as entropy and its variation law in p 3 order to obtain a self-consistent approach to the problem Here Lp = G~/c is the . This implies in section II. We set up the key ingredients of entropic that the number of fundamental degrees of freedom is gravity framework toward the derivation of weak-field- related to the surface area in spacetime. Note that the limit gravity force law. Holographic principle, which we black hole information paradox, that is, the information deem misapplied by Verlinde, is illustrated and the con- seems to be lost as all physical states evolve into the same cept of entanglement entropy is introduced to support final state of black hole, is also resolved by this principle. the validity of the derivation. We then revisit Verlinde’s An extensive illustration of the holographic principle is previous and make some elaboration for a more con- given by Bousso [16]. crete foundation of this entropic gravity, clarification of the idea of holographic screen and re-derivation of en- tropy variation law are made especially. In section III 2. Holographic entanglement entropy we bring in the generalized uncertainty principle, which leads to a corrected form of black hole temperature and In the derivation of the entropic gravity force law, entropy. The modification of the information content so the maximum value of the holographic entropy is used provided by GUP will result in the revision of the force rather than the inequality, which is unjustified. To treat law. In section IV we repeat the steps of Verlinde’s, but the problem more properly, one should instead invoke with the entropy variation law and the temperature form the concept of entanglement entropy [26–30]. The en- redefined by the GUP corrected entropy. We arrive at tanglement entropy is a quantum mechanical quantity an exact force law of gravity at the end, and this exact that measures the correlation between a subsystem A force law recovers not only the classical Newton’s law but and its complementary subsystem B. When the world also the sub-leading order quantum correction terms in is divided into two subsystems, the total Hilbert space the weak-field limit. In section V, conclusions and com- can be written as Htot = HA ⊗ HB. If an observer can ments are made about the implications of our findings. access the entire system, then the total entropy of the We suggest that the resulting deviation from the classical system is the quantum version of the classical Shannon P Newton’s law may serve as a probe to the validity of the entropy, H = −kB i Pi ln (Pi), here Pi the probabil- entropic gravity postulate. ity for a given state i, i.e., the von Newmann entropy 3 for a statistical state in Htot with density matrix ρtot: thermal state associated to this entanglement entropy. S(ρtot) = −kBTr(ρtot ln ρtot) [26]. For an observer who We should note that the entanglement entropy is not can only access the information of subsystem A, she will exactly proportional to the area; only the leading order 2 feel as if the state is described by a reduced density ma- term follows the Bekenstein’s law: S = A /4 Lp. The cor- trix ρA = TrBρtot, where the trace is over all eigenstates rection terms for the entanglement will be discuss later in in HB for the total density matrix. The entanglement section III, and the fact that simple entropy-area relation entropy is thus defined as the von Neumann entropy for is only valid in the leading order will be emphasized to the reduced density matrix ρA: SA = −kBTrA (ρAlnρA). retrieve the missing ~ factor in weak field entropic gravity If the total state is entangled, that is, if it is not factor- hypothesis. In this section we will only treat the entropy izable as |Ψtoti = |ΨA i ⊗ |ΨBi, then the entanglement following Bekebstein’s law without any extra terms, as is entropy is non-vanishing even if the total state is a pure the case in Verlinde’s scenario. state with zero entropy [27]. It can be shown by straight-forward calculations that the entanglement entropy of subsystem A is equal to that B. Verlinde’s entropic gravity scenario of subsystem B if the total state is pure [27]. Srednicki [4] pointed out that with the property SA = SB, the In this subsection we briefly review how Verlinde ar- entanglement entropy for a pure state, which we often rives at classical Newton’s law of gravity through ther- referred to as the unique ground state of the total sys- modynamics and the holographic principle, and we will tem, should only depend on the properties shared by the clarify some points in his approach. two regions. Therefore, it is expected that the leading behavior for pure ground state of a quantum field sys- tem scales as the boundary area rather than the volume 1. Meaning of holographic screen of subsystems. This area-scaling leading behavior of the entanglement entropy has been revealed in various physi- In Verlinde’s picture, there is a spherical screen with cal systems such as the quantum critical phenomena [31], radius R which centers at the massive source M and sep- explicit calculations of quantum field systems [6], and arates the universe into two components, one inside the the AdS/CFT correspondence in [32, 33]. sphere and the other stays outside. A particle of mass This property of entanglement entropy is referred to as m is placed just outside the spherical screen, see FIG. 1. the holographic entanglement entropy: for a gravity field The spirit of this entropic gravity system is that for the theory, a given boundary ∂B that divides the field theory test particle outside the sphere, it will interact thermo- into B and B’ components, the entanglement entropy for dynamically with the screen on which the information of the ground state of the field is the massive source is registered. If the variation in the 3 entropy occurrs as the test particle moves, the test par- Area(∂B)c kB SE = + subleading terms. (3) ticle will then confronted a restoring force according to 4 G ~ the first law of thermodynamics: F dx = T dS. To find The entanglement entropy can be renormalized by fix- the form of this restoring force caused by the system’s ing the cutoff length of the theory at Planck Length Lp. tendency toward the maximization of entropy, one first Here Area(∂B) is the area of the minimal surface on the has to know how the entropy varies in response to the boundary ∂B, and G is the Newton’s constant (see also displacement of the test particle. If the temperature can [34] for a review). Because this entropy of entanglement also be determined, then putting these together one can is associated with the quantum ground state, some refer arrive at the entropic force law. to it as the entropy of the fundamental degrees of free- Verlinde called this spherical screen a ‘holographic dom for the underlying quantum field theory across the screen’, on which the information content obeys holo- boundary, others may call it the entanglement entropy graphic principle. He uses this holographic principle ar- on the boundary surface. gument to make the suggestion that the information in- The original motivation for the entanglement entropy side the screen is distributed over the number of bits was to give a statistical explanation for Bekenstein en- proportional to its surface area. The use of holographic tropy in black hole thermodynamics. The entanglement principle here, however, is unclear and misleading. As entropy in quantum gravity has been known as the quan- discussed earlier, the holographic principle only suggests tum corrections to black hole entropy from matter fields an inequality in the information content. According to [6, 7, 15, 34]. Some further pointed out that the black this principle only the black hole horizon would saturate hole entropy is a pure entanglement entropy if the entire the upper-bound and recovers Bekenstein’s area law. We gravitational action is ‘induced’ by the quantum fluctu- believe what Verlinde really intended to convey is that ations inside and outside the event horizon [7, 15, 34]. the microscopic degrees of freedom can be represented Thus the black hole entropy is provided by this correla- holographically on the boundary. Thus the appropri- tion between the degrees of freedoms on opposite sides of ate terminology should be ‘the holographic formula for the horizon. An observer outside the event horizon with- entanglement entropy’ rather than the holographic prin- out the access to what happens inside will experience a ciple itself. We therefore clarify the meaning of ‘holo- 4

FIG. 1: Verlinde’s system: a massive source M is encoded by a spherical screen with radius R, and test particle m is placed FIG. 2: Fursaev’s system: two infinite surface B and B , just outside the screen. 1 2 with their z coordinates fixed, are placed around a massive source M. Outside the sphere is a test particle m whose displacement will affect the area of the surfaces. graphic screen’ as the ‘minimal surface in the quantum gravity spacetime’ [35], on which the holographic prop- erty of entanglement entropy holds as in Eq.(3), and the length away. How, therefore, would the horizon react to degrees of freedom is now proportional to the area of the infinitesimal displacement within this uncertainty? this minimal surface. The entropy on the screen is the A more crucial issue of this argument toward the en- entanglement entropy associated with the separation of tropy variation law has to do with the possible inconsis- the spacetime regions defined by this screen. tency in Verlinde’s approach. There are two equations corresponding to the nature of entropy. One is the en- tropy variation law in Eq.(4) and the other is the Beken- 2 2. Entropy variation law stein law S = A /4 Lp, which was implicitly used through the holographic formula of entanglement. A priori, these With the meaning of the holographic screen and the in- two formulas may or may not be compatible. This ten- formation content clarified we now proceed to see how the sion was also been pointed out by others [19, 20]. While entanglement entropy changes as the test particle moves. Verlinde conjectured the entropy variation law indepen- We first review Verlinde’s conjecture of the entropy vari- dently of the definition of entropy itself, we should point ation law, in which we find some inconsistencies. Then out that these two equations must be compatible since we review the consistency check made by Fursaev [36] both of them are tightly related to the underlying form through the metric calculation where two infinite surfaces of entropy. In other words, as a whole serve as the holographic screen. We then make 3 our calculation for spherical holographic screens follow- kBc mc ∆S = ∆A = −2πkB ∆x (5) ing the same spirit as Fursaev. We will demonstrate these 4~G ~ three approaches to entropy variation law by Verlinde, must hold under the entropic gravity framework. There- Fursaev and us, respectively. fore, the change in surface area should be described as a. Verlinde’s approach Motivated by Bekenstein’s argument “When a particle is one Compton wavelength mG ∆A = −8π ∆x (6) from the horizon, it is considered to be part of the black c2 hole,” Verlinde proposed that the entropy on the screen decreases by 2πkB when the test particle m moves one when the test particle m moves a distance ∆x away from Compton wavelength away from the screen. Further as- the normal to the screen, if the entropy variation is cor- suming that the entropy varies linearly within small dis- rect. tance, Verlinde assumes that the variation of entropy as- Whereas Verlinde gave the argument associated with sociated with a small displacement ∆x of the test particle entropy without guaranteeing the compatibility, we will m away from the screen is show that Eq.(6) can be calculated straight-forwardly in the weak gravity limit based on the knowledge of the ∆x mc spacetime, without the ∆S = −2πkB = −2πkB ∆x. (4) b. Fursaev’s approach Fursaev studied the dynam- λm ~ ics of the holographic minimal surface by the displace- One difficulty with this argument is that Bekenstein’s ment of test particle under the gravity in the weak field idea involves the nature of quantum uncertainty in dis- limit [35]. In his approach two infinite surfaces B1 and tance. That is, when the particle is located within one B2 are located at z = z1 and z = z2, respectively. These Compton wavelength from the horizon, quantum me- two surfaces separate the universe into two regions: a chanics dictates that it is indistinguishable whether the subspace between the two surfaces and its complement particle is right on the horizon or one Compton wave- on the outside. The massive source M is placed between 5 the spheres while a test particle is on the outside near one of the surfaces, see FIG. 2. The spacetime metric with a massive source placed at the origin, in the weak gravity limit, is

 2GM   2GM  ds2 = − 1 − c2dt2 + 1 + dr2 + r2dΩ2 rc2 rc2  2GM   2GM  = − 1 − c2dt2 + 1 + dx2 + dy2 + dz2 , ρc2 ρc2 (7) where ρ = r 1 − GM/rc2 = px2 + y2 + z2. Now in Fursaev’s system there are a massive source M at (x, y, z) = (0, 0, 0) and a test particle m at (0, 0, r0). FIG. 3: The presence of the test particle should affect the above metric, which can be viewed as a back-reaction. The resultant metric becomes c. Our approach Although Fursaev has successfully   2 2GM 2Gm 2 2 reproduced the entropy variation law that is consistent ds = − 1 − 2 − 2 c dt c ρ c ρ0 with the Bekenstein law of entropy, his derivation is only  2GM 2Gm valid for the special case of infinite surfaces. The more + 1 + + dx2 + dy2 + dz2 . (8) physically relevant geometry should be a sphere. We c2ρ c2ρ 0 therefore follow Fursaev’s appraoch but apply it to the 2 p 2 2 2 variation of the surface area of a sphere, i.e., a spherical Here ρ = r 1 − GM/rc = x + y + z and ρ0 = p 2 2 2 holographic screen, to recheck Verlinde’s set up of the (x − x0) + (y − x0) + (z − z0) . This new metric entropic gravity. makes intuitive sense based on the consideration. Consider a gravitational source of mass M located at r = 0, the spacetime metric in the weak field approxima- A small segment of area on one infinite surface Bk, 2 2 2 tion is with k = 1, 2, is described as da = gxxdx gyydy . The total area of surface Bk is therefore  2GM   2GM  ZZ   ds2 = − 1 − c2dt2 + 1 + dr2 + r2dΩ2 2GM 2Gm rc2 rc2 Ak = dxdy 1 + 2 + 2 . (9) c ρk c ρk,0  2GM   2GM  = − 1 − c2dt2 + 1 + dρ2 + ρ2dΩ2 , p 2 2 2 ρc2 ρc2 Here ρk,0 = (x − x0) + (y − x0) + (zk − z0) ,ρk = p 2 2 2 (12) x + y + zk . As the distance between the test particle and the sur- 2 2 2 2 2 face changes by an amount ∆r ≈ ∆z0, the surface area where ρ = r 1 − GM/rc and dΩ = dθ + sin θdφ . will vary by an amount, to the leading order, In the system of interest, we have a massive source M ZZ at (x, y, z) = (0, 0, 0) and a test particle m at (0, 0, r0). 2Gm∆z0 ∂ (1 /ρk,0 ) ∆Ak = 2 dxdy A sphere of radius r = R is a minimal surface that cuts c ∂z0 ZZ the universe into two complementary regions and also 2Gm∆z0 (z0 − zk) separates M and m into these two regions of the universe, = − 2 dxdy 3 c ρk,0 see FIG.3. Similar to Fursaev’s case, the area of the 2 2Gm∆z Z ∞ Z 2π (z − z ) surface no longer equals to 4πR because of the slight = − 0 ududψ 0 k c2 (u2 + (z − z ) 2) 3/2 warpage of the metric induced by the presence of the u=0 ψ k 0 test particle. The metric in this system becomes 4πGm∆z 4πGm∆r = − 0 = − , (10) c2 c2 ! 2GM 2Gm where the change of variables with x − x = ucosψ and ds2 = − 1 − − c2dt2 0 2 2p 2 2 ρc c ρ0 + ρ − 2ρ0ρcosθ y − y0 = usinψ have been made. The total variation of ! the area is then equal to 2GM 2Gm + 1 + + dρ2 + ρ2dΩ2 , 8πGm∆r ρc2 c2pρ 2 + ρ2 − 2ρ ρcosθ ∆A = ∆A + ∆A = − . (11) 0 0 1 2 c2 (13) With this, the entropy variation law Eq.(4) is successfully 2 2 reproduced. where ρ = R 1 − GM/Rc and ρ0 = r0 1 − GM/r0c . 6

The surface area of the sphere is therefore 3. Temperature ! Z 2GM 2Gm A = ρ2sinθdθdφ 1 − − Once the entropy variation with the displacement is de- 2 2p 2 2 c ρ c ρ0 + ρ − 2ρ0ρcosθ termined, we only need to define the temperature as the final step towards the entropic gravity force law. Verlinde 2 8πGMρ =4πρ − 2 + Am . (14) utilized the idea of hologram by assuming that the total c number of bits of information is proportional to the area Am is the surface area deviation due to the metric cor- of the holographic screen A, which is a consequence of rection induced by the test particle m: the entanglement entropy obeying Bekenstein law. That is, the degrees of freedom on the screen is proportional Z 2π Z 1 2Gmρ2 to its surface area Am = − dφ dcosθ 2p 2 2 0 −1 c ρ0 + ρ − 2ρ0ρcosθ Ac3 1 N = . (20) −4πGmρ2  −1 p  G = ρ 2 + ρ2 − 2ρ ρcosθ ~ c2 ρ ρ 0 0 0 −1 We comment again that this equation is satisfied because ( 8πGmρ2 of the holographic formula for entanglement entropy, not 2 , ρ0 > ρ , = c ρ0 (15) by taking the upper limit of the holographic bound. 8πGmρ , ρ < ρ . c2 0 The temperature on the surface is determined by the equipartition rule where the rest energy of mass M is Keeping the leading order in GmR/c2 and GMR/c2, we distributed evenly over the occupied bits, find

2 2 1 2 8πGmR E = Mc = NkBT. (21) r0 > R : A = 4πR + 2 , 2 c r0 8πGmR The underlying physics for this equipartition-determined r < R : A = 4πR2 − . (16) 0 c2 temperature derivation is that the test particle m out- side the screen interacts with the massive source M in- While to the leading order the surface area A is equal directly through thermodynamic effects near the screen, 2 to 4πR , the leading behavior of the surface area varia- with the information of the mass M being registered on tion induced by an infinitesimal displacement of the test this boundary surface [35]. We now arrive at the expres- 2 2 particle at r0 is instead contributed by the 8πGmR /c r0 sion of the temperature term. (Here we assume that the test particle is outside the sphere.) Now we would like to see how this infinitesi- 2Mc2 2G~ M mal displacement of the test particle m affects the surface T = = . (22) NkB ckB A area of the sphere due to the spacetime warpage. When the test particle makes a small displacement ∆r0 away When the test particle makes a small displacement ∆x from the sphere, the area will change by an amount from the screen, the entropy on the screen changes by an amount ∆S and therefore the test particle will face an ∂A 8πGmr2 effective macroscopic restoring force originated from the ∆r0 = − 2 2 ∆ r0 . (17) ∂r0 c r0 system’s tendency to increase its entropy. This “entropic force law” should thus follow the first law of thermody- Therefore if the entropy on the surface follows the namics: Bekenstein’s law, Eq.(1), then the entropy variation given by the displacement of the test particle should be F ∆x = T ∆S . (23)

2 ∆A 2πkBr mc With Eqs.(4)–(23) and that the area of the spherical ∆S = kB 2 = − 2 ∆r0 . (18) 2 4lp r0 ~ screen equals to 4πR , we finally obtain the entropic force law which is identical to Newton’s force law of gravity, When the center of the test particle is just outside the sphere, that is, R ≈ r , with R − r  Gm/c2 to satisfy GMm 0 0 F = − . (24) the weak field condition, the entropy variation on the R2 sphere becomes The minus sign in this force law shows that the entropic mc ∆r0 force is oriented opposite to the direction of distance’s ∆S = −2πkB ∆r0 = −2πkB , (19) increase, just as in Newton’s view of the gravitational ~ λm force that is attractive between two massive sources. with λm = ~/mc the Compton wavelength of the par- While Newton’s force law of gravitation seems to ticle. We have thus obtained the entropy variation law appear elegantly through this entropic derivation, we suggested by Verlinde explicitly and consistently without should emphasize again that both the entropy varia- invoking the ambiguous Compton wavelength argument. tion formula and the temperature formula involve an ~, 7 which indicates their quantum origin. Both these two ~’s The integration constant of the integral is fixed by setting are originated from the information content of the holo- the entropy to zero at the final remnant state. Thus in graphic screen, where one comes out of the direct calcula- the large mass limit we have tion of the entropy formula while the other emerges from 2  2  the distribution of the number of bits on the surface. The M M SGUP =4πkB 2 − πkBLog 2 + const. + ..., complete cancellation between these two ~’s was due to Mp Mp the coincidence that both the number of bits, N, and the A  A  Bekenstein law are straight-forwardly proportional to the =kB − πkBLog + const. + ... , (28) 4L 2 L 2 surface area of the holographic screen, which was fortu- p p itous. We will argue in the next section that the entropy which recovers Bekenstein entropy as MP /M goes to of entanglement is not exactly proportional to the area. zero. As demonstrated in Ref.[21], the generalized uncertainty The general correction terms to the semiclassical area principle (GUP) implies a corrected formula for entan- law of black hole entropy has been extensively studied. A glement entropy not only in the strong gravity but also logarithmic term as the leading correction for black hole in the weak gravity regime. entropy has been deemed universal by considerations in string theory and approaches, see for example [22–25]. We now treat GUP as a fundamen- III. GENERALIZED UNCERTAINTY PRINCIPLE tal assumption to provide the correct form of entropy, which is in agreement with the well-supported logarith- mic subleading behavior. That the standard Heisenberg uncertainty principle, As BH entropy is just the entanglement entropy on which is deduced under the flat, Minkowski spacetime, the BH horizon, we assert that under GUP the area- must be modified, or generalized, when the effect of grav- dependence of entanglement entropy is now expressed in ity becomes strong, or the spacetime becomes warped, the correct form as has long been suggested since 1950s. Since 1980s, GUP acquires additional theoretical support from the string " 2 r 2 # AkB 16πLp 16πLp theory’s perspective [37–41]. One important implication SGUP = 1 − + 1 − 8L 2 A A of GUP is that the standard forms of Bekenstein entropy p " r !# and Hawking temperature no longer hold as the size of A 16πL 2 −2πk Log √ 1 + 1 − p , black hole approaches the Planck length [21]. A direct B 4 πL A consequence of this GUP modified BH entropy is that p the BH evaporation process will come to a stop when its (29) Schwarzschild radius approaches the Planck length. As and in the asymptotic limit of large black hole, its per- a result there should be a BH remnant at Planck mass turbative expansion behaves as Eq.(28). and size left behind. Based on GUP, it was found that the modified BH temperature is of the form [21] IV. QUANTUM EFFECTS IN ENTROPIC " r # GRAVITY Mc2 M 2 T = 1 − 1 − p (25) GUP 4πk M 2 B In Verinde’s entropic gravity scenario, the purely clas- for a Schwazschild black hole of mass M. In the large sical Newton’s force law of gravitation is derived based on mass limit, i.e., MP /M  1, the BH temperature is a quantum-mechanical and thermodynamical setup. By keeping track of the underlying quantum dynamics, we 2 2  2 4  c MP MP MP now invoke generalized uncertainty principle to uncover TGUP = 1 + 2 + 4 + ... , (26) 8πkBM 4M 8M the missing quantum contribution in entropic gravity. Again we consider a spherical holographic screen, which agrees with the standard Hawking temperature in whose information content is defined by the GUP cor- leading order. rected entanglement entropy, encoding a massive source As the black hole temperature has been modified, the M at the center and a test particle m placed just outside entropy obtained by S = R c2T dM is also corrected to this spherical surface of radius R. The restoring force act- the form different from the simple Bekenstein entropy ing on the test particle m as it moves a little bit through expression: the first law of thermodynamics will be found. ( 2 2 r 2 ! First of all, the entropy variation law is directly af- M Mp Mp SGUP = 2πkB 1 − + 1 − fected by the GUP corrected form. Under GUP, the en- M 2 M 2 M 2 p tropy varies with the surface area as " r 2 !# ) M Mp −Log 1 + 1 − . (27) ∂SGUP M M 2 ∆S = ∆A , (30) p ∂A 8 while the definition of ∆A remains the same as in Eq.(6). In our attempt of seeking the missing ~’s, we reinvesti- Next we determine the temperature on the screen. We gated all the fundamental assumptions in the existing see that Eq.(20) is proposed on the prerequisite that the derivations of entropic gravity. Within Verlinde’s en- information of the space is proportional to the surface tropic gravity derivation, two ingredients involving en- area of the screen. Under the GUP framework the en- tropy formula have been invoked without the guaran- tropy is no longer proportional to the area, so we compare tee of their mutual compatibility. We applied Fursaev’s Eq.(20) and Bekenstein entropy to arrive at the form for consistency-check procedure to reproduce the leading or- the number of bits on the screen as der entropy variation in Verlinde’s setup of spherical holographic screen. While our approach manages to S N = GUP . (31) avoid the compatibility issue, there is a price to pay. In 4kB our alternative approach we have introduced the concept of spacetime metric and its deformation due to the pres- We again apply the equipartition formula Eq.(21) to ence of a massive object, which implicitly assumed the determine the temperature on the screen and find knowledge of , the standard theory of 2Mc2 Mc2 classical gravity. Yet the very attempt of entropic gravity T = = . (32) Nk 2S is to deduce it from and statistical B GUP physics alone without any prior knowledge of gravity. We Finally, using the first law of thermodynamics we arrive are therefore at risk of a circular logic in our approach if at the modified gravity force law: gravity is to be interpreted as an emergent phenomenon. In this regard a more cogent and consistent argument 2[α(1 + η) − 2(2 + η)] without involving any gravity-related concept is needed F = F . GUP N   1  towards an alternative entropy variation law, in order η(1 + η) −4 + α(1 + η) − 4Log 2 α(1 + η) (33) to assert the validity of the entropic framework of grav- ity as an emergent phenomena. By the similar token,  2 Here FN = GmM R is Newton’s gravitational the existing derivations of entropic gravity also faces the force law, and we have introduced symbols η = similar issue since Newton’s constant has been invoked p  3 2 1 − 4G~ /c3 R2 and α = G~ c R to simplify the ex- as a fundamental constant from the outset instead of be- pression. ing a secondary, derived parameter from the theory as it p 3 In the large distance limit where R  Lp = G~/c should if gravity is to be an emergent phenomenon.  3 2 and therefore α = G~ c R  1, we can expand the Under this light one can instead view our derivation of force to the third order of α as the entropic gravity not as an emergent phenomenon but as a means to deduce the ‘quantum gravity force law’ via  2  2 FGUP = FN 1 + α [2 − Logα] + α 4 − 5Logα + (Logα) the quantization of the information content on the mini- +α3 7 − 18Logα + 8(Logα)2 − (Logα)3 + ... . mal surfaces in units of Planck area provided by GUP as (34) well as the spacetime warpage effect in the presence of a massive particle provided by general relativity. It is clear that this GUP-based force law recovers the Although there are still rooms to improve in this line classical Newton’s gravitational force law in the infinite of approach to gravity, we have provided an exact form distance limit, while some subleading quantum correc- of quantum corrected entropic gravity force law based on tions is present as long as α is finite. On the other hand the assumption of GUP as a fundamental input. Such these correction terms go to zero in the classical limit as quantum corrections, though minute, may serve as a ~ vanishes. These α-dependent terms, we conclude, are probe to examine the concreteness of the entropic grav- where ~ is hiding in entropic gravity. ity interpretation in the the experimentally measurable scale of large distance and weak gravity limit.

V. CONCLUSION AND DISCUSSIONS

Acknowledgement In this paper we raised the question about where ~ is hiding in entropic gravity. Through the reanalysis of the fundamental building blocks of entropic gravity, in par- We thank Debaprasad Maity, Taotao Qiu, Yen-Chin ticular the meaning of holographic screen and its asso- Ong , Chien-I Chiang, Nian-An Tung and Jo-Yu Kao for ciated entanglement entropy, we argued that the perfect helpful and inspiring discussions. This research is sup- cancellation of ~ among all the quantum mechanically ported by the Taiwan National Science Council (NSC) originated inputs is broken if the more exact form of the under Project No. NSC98-2811-M-002-501, No. NSC98- BH entanglement entropy based on GUP is to replace 2119-M-002-001, and the US Department of Energy un- the Bekenstein area law. Based on this we found, in the der Contract No. DE-AC03-76SF00515. We would also weak gravity limit, the hided ~’s in the form of logarith- like to thank the NTU Leung Center for Cosmology and mic corrections to the classical Newton’s law, in Eq.(34). Particle Astrophysics for its support. 9

[1] J. D. Bekenstein, Phys. Rev. D.7, 2333 (1973). [26] M. A. Nielsen, and I. L. Chuang, Quantum Computation [2] J. D. Bekenstein, Phys. Rev. D.23, 287 (1981). and (Cambridge University Press, [3] S. W. Hawking, Comm. Math. Phys. 43, 199 (1975). 2000). [4] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993). [27] D. Janzing, Compendium of Quantum Physics, edited by [5] J. Eisert, M. Cramer, and M.B. Plenio, Rev. Mod. Phys. D. Greenberger, K. Hentschel,and F. Weinert (Springer 82, 277 (2010). Berlin Heidelberg, 2009) p.205-209. [6] D. Kabat, Nucl. Phys. B453, 281 (1995). [28] R. Horodecki, P. Horodecki, M. Horodecki, and K. [7] T. Jacobson, arXiv:gr-qc/9404039 (1994). Horodecki, Rev. Mod. Phys. 81, 865 (2009). [8] A.D. Sakharo, Sov. Phys. Dokl. 12, 1040 (1968). [29] P. Calabrese, J. Cardy, Int. J. Quant .Inf. 4:429,2006, [9] T. Jacobson, Phys.Rev.Lett.75, 1260 (1995). arXiv:quant-ph/0505193v1(2006). [10] G. E. Volovik, Proc. Sci., QG-Ph 043(2007). [30] L. Susskind, and J. Uglum, Phys. Rev. D 50, 2700 (1994). [11] L. Sindoni, S. Liberati and F. Girelli, AIP Conf. Proc. [31] D. V. Fursaev, Phys.Rev.D73:124025,2006, arXiv:hep- 1196, 258 (2009). th/0602134v2(2006). [12] Rept. Prog. Phys. 73, 046901 (2010). [32] S. Ryu, and T. Takayanagi, JHEP 0608, 045 (2006). [13] E. P. Verlinde, arXiv:1001.0785(2010). [33] H. Casini, M. Huerta, and R. C. Myers, JHEP 1105, 036 [14] G. ’t Hooft, arXiv:gr-qc/9310026v2 (1993). (2011). [15] L. Susskind, J. Math. Phys. 36, 6377 (1995). [34] T. Nishioka, S. Ryu, and T. Takayanagi, J. Phys. A: [16] R. Bousso, Rev. Mod. Phys. 74, 825 (2002). Math. Theor. 42, 504008 (2009). [17] M. A. Santos and I. V. Vancea, arXiv:1002.2454v3 [35] D. V. Fursaev, arXiv:1006.2623(2010). (2010). [36] D.V. Fursaev, JHEP 0609, 018 (2006). [18] S. Ghos, arXiv:1003.0285v3 (2010). [37] G. Veneziano, Europhys. Lett. 2, 199 (1986). [19] L. Modesto and A. Randono, arXiv:1003.1998v1 (2010). [38] D. J. Gross and P. F. Mende, Nucl. Phys. B303, 407 [20] M. R. Setare and D. Momeni, arXiv:1004.2794v1 (2010). (1988). [21] R. J. Adler, P. Chen, and D. I. Santiago, Gen. Relativ. [39] D. Amati, M. Ciafolini and G. Veneziano. Phys. Lett. Gravit. 33, 2101 (2001). B216, 41 (1989). [22] S. N. Solodukhin, Phys. Rev. D 57, 2410 (1998). [40] K. Konishi, G. Paffuti and P. Provero, Phys. Lett. B234, [23] M, Cadoni, and M. Melis, Entropy 2010, 12, 2244 (2010). 276 (1990). [24] R. K. Kaul, and P. Majumda, Phys. Rev. Lett. 84, 5255 [41] E. Witten, Phys. Today, Apr. 24 (1996). (2000). [25] A, Ghosh, and P. Mitra, Phys. Rev. D 71, 027502 (2005).