Where is hbar Hiding in Entropic Gravity?
WWhere is (cid:126)
Hiding in Entropic Gravity
Pisin Chen , , , ∗ and Chiao-Hsuan Wang , †
1. Department of Physics, National Taiwan University, Taipei, Taiwan 106172. Graduate Institute of Astrophysics, National Taiwan University, Taipei, Taiwan 106173. Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei, Taiwan 106174. Kavli Institute for Particle Astrophysics and Cosmology,SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, U.S.A.
The entropic gravity scenario recently proposed by Erik Verlinde reproduced the Newton’s lawof purely classical gravity yet the key assumptions of this approach all have quantum mechanicalorigins. So one naturally wonders: where is (cid:126) hiding in entropic gravity? To address this question, wefirst reformulate the entropic derivation of Newton’s gravitation force law to address a self-consistentapproach to the problem. Next we argue that as the concept of minimal length has been invoked inthe Bekenstein entropic derivation, the generalized uncertainty principle (GUP), which is a directconsequence of the minimal length, should be taken into consideration in the entropic interpretationof gravity. Indeed based on GUP it has been demonstrated that the black hole Bekenstein entropyarea law must be modified not only in the strong but also in the weak gravity regime where in theweak gravity limit the GUP modified entropy exhibits a logarithmic correction. In the weak gravitylimit, such a GUP modified entropy exhibits a logarithmic correction term. When applying it to theentropic interpretation, we demonstrate that the resulting gravity force law does include sub-leadingorder correction terms that depend on (cid:126) . Such deviation from the classical Newton’s law may serveas 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
The issue of how gravity and thermodynamics are cor-related has been studied for decades, triggered by theseminal discovery by Bekenstein[1, 2] on the area-law ofblack hole (BH) entropy and temperature. After Hawk-ing’s discovery of the BH evaporation and the interpre-tation of its temperature as the thermal temperatureof blackbody radiation[3], considerable efforts have beenmade to find the statistical interpretation of the propor-tionality of black hole entropy and its horizon area. See[4] and [5], for example, for a review. By now a well-accepted view is that the black hole entropy is associatedwith the external thermal state perceived by an observeroutside the event horizon who has no access to the BHinterior. Namely, the correlation between the degreesof freedom on opposite sides of the horizon results in amixed state for observation from the outside, i.e., the‘entanglement entropy’[6, 7], which depends upon theboundary properties and will be discussed more in thelater sections of this paper.The inversion of the logic that describes gravity as anemergent phenomenon was first proposed by Sakharov[8], who suggested that gravity is induced by quantumfield fluctuations. Invoking the area scaling property ofentanglement entropy, Jacobson in 1995 [9] used basiclaws of thermodynamics to derive Einstein equations. In ∗ Electronic address: [email protected] † Electronic address: [email protected] his perspective, Einstein equations are now an equationof state rather than a fundamental theory. More ideas onemergent gravity have been recently proposed (See, forexample, [10–13]).Similar to Jacobson’s derivation of Einstein equa-tions through thermodynamic, Verlinde treated gravityas an entropic force analogous to the restoring force of astretched elastic polymer driven by the system’s tendencytowards the maximization of entropy [13], and interest-ingly the Newton’s law of gravitation was shown to arise.To arrive at the Newton’s force law of gravity throughthe first law of thermodynamic
F dx = T dS , Verlindefirst invoked the Compton wavelength of the test par-ticle to find the change of entropy with respect to itsdisplacement. He then invoked the holographic principle[14–16] and the equipartition theorem to define the tem-perature experienced by the test particle. One cannotbut notices that all these building blocks have quantummechanical origin, or more specifically the presence of (cid:126) . Yet all the (cid:126) ’s just get subtly cancelled and at theend a purely classical Newton’s law has emerged. So onenaturally wonders: where is (cid:126) hiding in entropic gravity?There have been previous works aiming at finding theentropic corrections to Newton’s law but however unsat-isfactory. We will discuss with them in later sections.We here argue that as there have already existed aminimal length scale in the entropic derivation, the gen-eralized uncertainty principle (GUP) should be taken intoconsideration under this minimal length. Indeed basedon GUP it has been demonstrated that the black holeBekenstein entropy area law must be modified not onlyin the strong but also in the weak gravity regime [17]. Inthe weak gravity limit, such a GUP modified Bekenstein a r X i v : . [ g r- q c ] N ov entropy exhibits a logarithmic correction. Such a log-correction is consistent with similar conclusions drawnfrom string theory, AdS/CFT correspondence, and loopquantum gravity considerations [18–21]. When applyingit to the entropic derivations, we demonstrate that theresulting entropic gravity does include sub-leading ordercorrection terms that depend on (cid:126) .The organization of this paper is as follows. To addressthe question we posted, we first set up the key ingredi-ents of entropic gravity framework toward the derivationof weak-field-limit gravity force law in section II, wherethe concept of entanglement entropy is introduced andthe entropic derivation of Newton’s gravitation force lawreformulated in a self-consistent approach. Also in thissection the entropy content in the system is clarified, theentropy variation law re-derived, and a new reasoning forBH temperature based not on the equipartition theorembut on the blackbody radiation is introduced. In sec-tion III we invoke the generalized uncertainty principle,which leads to a corrected form of black hole tempera-ture and entropy. The modification of the informationcontent so provided by GUP will result in the revision ofthe force law. In section IV we briefly review and com-ment on some previous efforts aiming at finding the quan-tum corrections– that is, the missing (cid:126) ’s– in the entropicgravity scenario. We then repeat the steps of Verlinde’s,but with the entropy variation law and the temperatureredefined by the GUP corrected entropy. We arrive atan exact force law of gravity at the end, and this exactforce law recovers not only the classical Newton’s law butalso the sub-leading order quantum correction terms inthe weak-field limit. In section V, conclusions and com-ments are made about the implications of our findings.We suggest that the resulting deviation from the classicalNewton’s law may serve as a probe to the validity of theentropic gravity postulate. II. ENTROPIC DERIVATION OF NEWTON’SGRAVITY
In this section we will reformulate the derivation ofclassical Newton’s law of gravity through thermodynam-ics and the leading behavior of holographic entanglemententropy. Some formulas will resemble that invoked byVerlinde in section 3.2 of Ref.[13], but the rationale be-hind his and ours are different. Our most crucial depar-ture from Verlinde’s previous work is that in our per-spective, this entropic gravity approach is not an emer-gent description of gravity. This point will be elaboratedmore in the last section.
A. Holographic entanglement entropy
In the derivation of the entropic gravity force law, webelieve that one should invoke the concept of entangle-ment entropy [22–26] to find the information content of the system. The entanglement entropy is a quantummechanical quantity that measures the correlation be-tween a subsystem A and its complementary subsystemB. When the world is divided into two subsystems, thetotal Hilbert space can be written as H tot = H A ⊗ H B .If an observer can access the entire system, then the to-tal entropy of the system is the quantum version of theclassical Gibb’s entropy, S = − k B (cid:80) i P i ln ( P i ), here P i the probability for a given state i , i.e., the von Newmannentropy for a statistical state in H tot with density ma-trix ρ tot : S ( ρ tot ) = − k B Tr( ρ tot ln ρ tot ) [22]. For an ob-server who can only access the information of subsystemA, she will feel as if the state is described by a reduceddensity matrix ρ A = Tr B ρ tot , where the trace is a par-tial trace over all eigenstates in H B for the total densitymatrix. The entanglement entropy is thus defined as thevon Neumann entropy for the reduced density matrix ρ A : S A = − k B Tr A ( ρ A ln ρ A ). If the total state is entangled,that is, if it is not factorizable as | Ψ tot (cid:105) = | Ψ A (cid:105) ⊗ | Ψ B (cid:105) ,then the entanglement entropy is non-vanishing even ifthe total state is a pure state with zero entropy [23].It can be shown by straight-forward calculations thatthe entanglement entropy of subsystem A is equal to thatof subsystem B if the total state is pure [23]. Srednicki[4] pointed out that with the property S A = S B , theentanglement entropy for a pure state, which we oftenreferred to as the unique ground state of the total sys-tem, should only depend on the properties shared by thetwo regions. Therefore, it is expected that the leadingbehavior for pure ground state of a quantum field systemscales as the boundary area rather than the volume of thesubsystems. This area-scaling leading behavior of the en-tanglement entropy has been revealed in various physicalsystems such as the quantum critical phenomena [28], ex-plicit calculations of quantum field systems [6], and theAdS/CFT correspondence in string theory [29, 30]. Thisarea-scaling property of entanglement entropy is referredto as the holographic entanglement entropy: for a quan-tum field theory in a space that is divided by a surface Σinto two regions, the entanglement entropy for the groundstate of the field is S E = Area(Σ) c k B (cid:126) G + subleading terms . (1)Here Area(Σ) is the area of the surface, and G is Newton’sconstant (see also [27] for a review).The condition of the holographic entanglement entropyhas been demonstrated for minimal surfaces with vanish-ing extrinsic curvature [6, 28–30]. Some special casesof nonvanishing extrinsic curvature such as 2-sphere and2-d cylinder also possess this property [31] The entan-glement entropy can be renormalized by fixing the cutofflength of the theory at Planck Length L p . Because thisentropy of entanglement is associated with the quantumground state, some refer to it as the entropy of the fun-damental degrees of freedom for the underlying quantumfield theory across the boundary, others may call it theentanglement entropy on the boundary surface.The original motivation for the entanglement entropywas to give a statistical explanation for Bekenstein en-tropy in black hole thermodynamics. The entanglemententropy in quantum gravity has been known as the quan-tum corrections to black hole entropy from matter fields[6, 7, 15, 27]. Some further pointed out that the blackhole entropy is a pure entanglement entropy if the entiregravitational action is ‘induced’ by the quantum fluctu-ations inside and outside the event horizon [7, 15, 27].Thus the black hole entropy is provided by this correla-tion between the degrees of freedoms on opposite sides ofthe horizon. An observer outside the event horizon with-out the access to what happens inside will experience athermal state associated to this entanglement entropy.We should note that the entanglement entropy is notexactly proportional to the area; only the leading orderterm follows the Bekenstein’s law: S = A (cid:14) L p . The cor-rection terms for the entanglement will be discuss later insection III, and the fact that simple entropy-area relationis only valid in the leading order will be emphasized toretrieve the missing (cid:126) factor in weak field entropic gravityhypothesis. In this section we will only treat the entropyfollowing Bekebstein’s law without any extra terms, as isthe case in Verlinde’s scenario. B. Entropic gravity scenario with Bekenstein formof entropy
1. Entropic gravity system
In the entropic interpretation of gravity, a sphericalscreen is invoked with radius R that centers at a massivesource M and separates the universe into two regions, oneinside the sphere and the other outside. A test particlewith mass m is placed just outside the spherical screen,see FIG. 1. The spirit of this entropic gravity systemis that for the test particle outside the sphere, it willinteract thermodynamically with the screen on which theinformation of the massive source is registered. If thevariation in the entropy occurs as the test particle moves,the test particle will then confronted a restoring forceaccording to the first law of thermodynamics: F dx = T dS . To find the form of this restoring force causedby the system’s tendency toward the maximization ofentropy, one first has to know how the entropy varies inresponse to the displacement of the test particle. If thetemperature can also be determined, then putting thesetogether one can arrive at the entropic force law. We willshow that if the entanglement entropy on the sphere isnormalized by Planck Length and we consider only itsleading order behavior following the Bekenstein law, therestoring force will have the same form of Newton’s lawof gravity in the end.We note here that Verlinde called this spherical screena ‘holographic screen’, on which the information contentobeys the holographic principle so that the informationinside the screen is registered by the number of bits that
FIG. 1: Verlinde’s system: a massive source M is encoded bya spherical screen with radius R , and test particle m is placedjust outside the screen.FIG. 2: Our system: a massive source M located at the originis encoded by a spherical screen with radius R . A test particle m is placed outside the screen at a distance r from the origin. is proportional to its surface area. The use of the holo-graphic principle here, however, is ambiguous or evenmisleading because the holographic principle only sug-gests an inequality in the information content [14–16].According to this principle only the black hole horizonwould saturate the upper-bound of the inequality and re-covers Bekenstein’s area law. Under this light, it wouldbe more appropriate to refer to this assignment as ‘theholographic formula for entanglement entropy’.
2. Entropy variation law
With the system set up, we now proceed to see how theentanglement entropy changes as the test particle moves.Consider a gravitational source of mass M located at r = 0. The spacetime metric in the weak field approxi-mation is ds = − (cid:18) − GMrc (cid:19) c dt + (cid:18) GMrc (cid:19) dr + r d Ω = − (cid:18) − GMρc (cid:19) c dt + (cid:18) GMρc (cid:19) (cid:0) dρ + ρ d Ω (cid:1) , (2)where ρ = r (cid:0) − GM/rc (cid:1) and d Ω = dθ + sin θdφ .In the system of interest, there is a massive source M located at ( x, y, z ) = (0 , ,
0) and a test particle m located at (0 , , r ). A 2-sphere with radius r = R sur-rounding M is a surface that possesses the holographicproperty of entanglement entropy, which partitions theuniverse into two complementary regions to which M and m separately belong (see FIG.3). The area of the surfaceno longer equals to 4 πR because of the slight warpageof the metric induced by the presence of the test particle.The metric in this system becomes ds = − (cid:32) − GMρc − Gm/c (cid:112) ρ + ρ − ρ ρ cos θ (cid:33) c dt + (cid:32) GMρc + 2 Gm/c (cid:112) ρ + ρ − ρ ρ cos θ (cid:33) (cid:0) dρ + ρ d Ω (cid:1) , (3)where ρ = R (cid:0) − GM/Rc (cid:1) and ρ = r (cid:0) − GM/r c (cid:1) .The surface area of the sphere is therefore A = (cid:90) ρ sin θdθdφ (cid:32) − GMc ρ − Gm/c (cid:112) ρ + ρ − ρ ρ cos θ (cid:33) . (4)Keeping the leading order in GmR/c and GM R/c ,we find r > R : A = 4 πR − πGM Rc + 8 πGmR c r ,r < R : A = 4 πR − πGM Rc − πGmRc . (5)We see that while to the leading order the surfacearea A is equal to 4 πR , its correction induced by thepresence of the test particle at r is contributed by the8 πGmR /c r term. (Here we assume that the test par-ticle is outside the sphere.) Now we like to see how aninfinitesimal displacement of the test particle m wouldfurther affect the surface area of the sphere. When thetest particle makes a small displacement ∆ r away fromthe sphere, the area will change by an amount ∂A∂r ∆ r = − πGmR c r ∆ r . (6)Therefore if the entropy on the holographic screen followsthe Bekenstein’s law, then the entropy variation inducedby the displacement of the test particle should be∆ S = k B ∆ A l p = − πk B R r mc (cid:126) ∆ r . (7)When the test particle is just outside the sphere, thatis, R ≈ r , but with R − r (cid:29) Gm/c to satisfy theweak field condition, the entropy variation on the spherebecomes ∆ S = − πk B mc (cid:126) ∆ r . (8) This entropy variation law coincides with that con-jectured by Verlinde. Verlinde’s argument relies on thequantum uncertainty of a particle’s position, that is,the position of a particle is indistinguishable within oneCompton wavelength from the horizon. Since the preciselocation of the particle is unresolved within one Comptonwavelength, how, therefore, would the horizon be able toreact to the infinitesimal displacement within this uncer-tainty?A more critical issue of Verlinde’s argument towardthe entropy variation law has to do with the possible in-consistency in his approach. There are two equationscorresponding to the nature of entropy. One is the en-tropy variation law and the other is the Bekenstein law: S = A /4 L p , which was implicitly used through the holo-graphic formulation of entanglement.Prior to Verlinde’s conjecture, there existed a met-ric calculation of entropy variation law given by Fursaev[32, 33]. Fursaev used two infinite surfaces as the screensto divide the spacetime. However his derivation is onlyvalid for the special case of infinite surfaces. We believethat a sphere is a physically more suitable geometry, sinceone can introduce a uniform temperature more naturallyon a sphere than a infinite surface. Our work thereforefollows Fursaev’s approach but apply it to the variationof the surface area of a sphere. Through that we manageto reproduce the entropy variation law suggested by Ver-linde without the need to invoke his ambiguous Comptonwavelength argument.
3. Temperature
Once the entropy variation associated with the dis-placement of the test particle is established, the onlyremaining task is to define the temperature as the fi-nal step towards the entropic gravity force law. Herewe suggest a heuristic derivation of Hawking tempera-ture for Schwarzschild black hole in terms of its massand entanglement entropy. In terms of black hole ther-modynamics, the Hawking temperature can be viewedas the blackbody radiation temperature associated withits evaporation. In this regard, the averaged energy is2 . k B T per photon based on statistical mechanics. Thedegrees of freedom for a black hole is N = S B /k B (see, forexample, Ref.[16], such argument based on holographicprinciple). We suppose that these degrees of freedom, N,are associated with the number of the blackbody photonsand that the total energy of the blackbody radiation inturn takes up the entire rest mass energy M c of the BH.Thus the temperature can be written as T = M c aS B , (9)where a is a constant of order one. To reproduce thecorrect form of Hawking temperature, we fix the coeffi-cient and arrive at a form for the temperature followingBekenstein’s law of BH entropy: T = M c S B = 2 G (cid:126) ck B MA . (10)We now arrive at the same form of temperature as pro-posed by Verlinde. However, his derivation invokes theequipartition theorem of energy, which is a classical con-cept and is therefore unjustified in the entropic gravityframework, where the entropy has already involved (cid:126) , anindication of the quantum nature of the formulation.With T and the relation between ∆ S and ∆ x fixed,we are now ready to the form of entropy. When thetest particle makes a small displacement ∆ x relative tothe screen, the entropy on the screen will change by anamount ∆ S according to Eq.(8). The test particle willtherefore experience a restoring force originated from thesystem’s tendency to increase its entropy. Unlike therestoring force of a stretched polymer which has two pos-sible directions due to a finite nonvanishing equilibriumposition, the entropic gravity force has only one direc-tion, which corresponds to bringing two massive objectscloser to each other. This “entropic force law” shouldthus follow the first law of thermodynamics: F ∆ x = T ∆ S . (11)Following Eqs.(8)–(11) and equating the area of thespherical screen to 4 πR in the leading order approxi-mation, we finally obtain the entropic force law that isidentical to Newton’s force law of gravity, F = − GM mR . (12)The minus sign in this force law indicates that the en-tropic force is oriented opposite to the direction of the dis-placement, just as in Newton’s view of the gravitationalforce that is attractive between two massive sources.While Newton’s force law of gravitation seems toemerge elegantly through this entropic reasoning, weshould emphasize again that both the entropy varia-tion formula and the temperature formula involve an (cid:126) ,which manifests their quantum origin. Both these two (cid:126) ’s are originated from the information content of theholographic screen, where one comes out of the directcalculation of the entropy formula and the other emergesfrom the distribution of the degrees of freedom on thesurface. The complete cancellation between these two (cid:126) ’swas due to the coincidence that both the degrees of free-dom, N , and the Bekenstein law are straight-forwardlyproportional to the surface area of the holographic screen,which was fortuitous. We will argue in the next sectionthat the entropy of entanglement is not exactly propor-tional to the area. As demonstrated in Ref.[17], the gen-eralized uncertainty principle (GUP) implies a correctedformula for entanglement entropy not only in the stronggravity but also in the weak gravity regime.In the derivation of the entropic gravity, the actualform of the entropy is a key ingredient. Extra care must therefore be taken in the determination of the BH en-tropy. With this in mind we emphasize that the holo-graphic formulation of entanglement entropy is based ona cutoff length of the same order of the Planck length.This introduction of the cutoff length implies the exis-tence of a minimal length scale that is essential in theentropic interpretation of gravitational force. The stan-dard Heisenberg uncertainty principle, which is deducedunder the Minkowski spacetime, must be modified, orgeneralized, when the spacetime cannot be reduced in-definitely but is subject to some minimal length scale[34]. Originally suggested in 1960s [35] based purely onthe considerations of GR, GUP acquires additional the-oretical support from string theory’s perspective [36–40]since 1980s. III. GENERALIZED UNCERTAINTYPRINCIPLE
One important implication of GUP is that the standardforms of Bekenstein entropy and Hawking temperatureno longer hold as the size of a black hole approaches thePlanck length [17]. A direct consequence of this GUPmodified BH entropy is that the BH evaporation pro-cess will come to a stop when its Schwarzschild radiusapproaches the Planck length. As a result the Hawkingevaporation should leave behind a BH remnant at Planckmass and size.Based on GUP, it was found that the modified BHtemperature is of the form [17] T GUP = M c πk B (cid:34) − (cid:114) − M p M (cid:35) (13)for a Schwarzschild black hole of mass M . In the largemass limit, i.e., M P /M (cid:28)
1, the BH temperature is T GUP = c M P πk B M (cid:20) M P M + M P M + ... (cid:21) , (14)which agrees with the standard Hawking temperature atthe leading order.Since the black hole temperature has been modified,the entropy obtained by S = (cid:82) dM c /T must also becorrespondingly modified to a form different from thesimple Bekenstein entropy expression: S GUP = 2 πk B (cid:40) M M p (cid:32) − M p M + (cid:114) − M p M (cid:33) − log (cid:34) MM p (cid:32) (cid:114) − M p M (cid:33)(cid:35) (cid:41) . (15)The integration constant of the integral is fixed by settingthe entropy to zero at the final remnant state. Thus inthe large mass limit we have S GUP =4 πk B M M p − πk B log (cid:18) M M p (cid:19) + const. + ... , = k B A L p − πk B log (cid:18) AL p (cid:19) + const . + ... , (16)which recovers Bekenstein entropy as M P /M goes tozero.The correction to the semiclassical area law of blackhole entropy has been extensively studied. For examplea generic logarithmic term as the leading correction toblack hole entropy has been found universal up to a co-efficient of order unity based on string theory and loopquantum gravity considerations, see for example [18–21].Such logarithmic correction also appears in the entangle-ment entropy on minimal surfaces and some special non-minimal cases such as the 2-sphere in flat space [31]. Herewe treat GUP as a basic assumption to provide the cor-rect form of holographic entanglement entropy becauseof its fundamentalness when dealing with issues relatedto quantum gravity. It is interesting to note that theGUP-based correction to the entanglement entropy hasits IR limit that is in agreement with the well-supportedlogarithmic sub-leading corrections deduced from stringtheory or loop quantum gravity. However a fundamen-tal difference between the GUP and other approaches isthat the GUP correction to the entanglement entropy asshown in Eq.(15) is an exact form, valid for both the UVand the IR limits. Therefore this GUP corrected form ofentropy is also valid in the UV limit, which will be use-ful in our future work to extend our result to the stronggravity regime.As the BH entropy is precisely the entanglement en-tropy on the BH horizon, we assert that under GUPthe area-dependence of entanglement entropy is now ex-pressed in the correct form as S GUP = Ak B L p (cid:34) − πL p A + (cid:114) − πL p A (cid:35) − πk B log (cid:34) A √ πL p (cid:32) (cid:114) − πL p A (cid:33)(cid:35) , (17)which reduced to Eq.(16) in the large BH mass limit. IV. QUANTUM EFFECTS IN ENTROPICGRAVITYA. Quantum corrections to entropic gravity: otherapproaches
There have been previous efforts aiming at finding theentropic corrections to Newton’s law but however unsat-isfactory. Santos et al. [41] used the uncertainty princi-ple to postulate a corrected entropy variation law and ob-tained the uncertainty in Newton’s law of gravity. Ghosh [42] further extended the previous work by using the ideaof GUP. However, their approaches led to a force law thatdepends on the uncertainty in position, which is bother-some.Modesto et al. [43] introduced a log correction anda volume-scale correction term to the area-entropy rela-tion, and arrived at a modified gravitation force law. Wehere point out that when the entropy-area law is changed,the number of bits is no longer simply inverse propor-tional to the Planck area. As a consequence, the temper-ature defined by equipartition rule will also be modified.Modesto et al. considered only the corrections to en-tropic variation law without noticing that the form oftemperature should also be corrected. Setare et al. [44]revisited Modesto’s idea and modified Newton’s gravita-tional force law via GUP and self-gravitational correc-tions. Setare et al. modified both entropy and tempera-ture but failed to introduce, in our opinion, the right formof GUP-modified entropy, which should follow the well-acknowledged logarithmic form of correction. Anotherthing we should note here is that both Modesto et al.and Setare et al. suggested only the sub-leading correc-tions of the force law rather than an exact form. Nicolini[45] obtained an exact form of the corrected entropic forcelaw via noncommutative gravity and ungraviton correc-tions, but he also did not take into consideration theeffect of modified information content on the definitionof the temperature.Here we argue that one can trace the quantum ef-fects in the entropic gravity scenario by invoking an ex-act form of GUP corrected entropy formula. This GUPcorrected entropy formula has a universally-accepted log-arithmic leading-order correction, and its deviation fromthe Bekenstein law will affect the entropic variation equa-tion as well as the temperature and therefore will lead usto a quantum corrected gravitational force law.
B. Entropic gravity under GUP
In Verinde’s entropic gravity scenario, the purely clas-sical Newton’s force law of gravitation is derived basedon a quantum-mechanical and thermodynamical setup.To keep track of the underlying quantum dynamics, wenow invoke generalized uncertainty principle to uncoverthe missing quantum contribution in entropic gravity.Again we consider a spherical holographic screen,whose information content is defined by the GUP cor-rected entanglement entropy, encoding a massive source M at the center and a test particle m placed just outsidethis spherical surface of radius R . The restoring force act-ing on the test particle m induced by the displacementfrom its (equilibrium) location will be derived based onthe first law of thermodynamics.First of all, the entropy variation law is directly af-fected by the GUP corrected form. Under GUP, the en-tropy varies with the surface area as∆ S = ∂S GUP ∂A ∆ A , (18)with ∆ A = − πGm/c as calculated before.Next we determine the temperature on the screen. Un-der the GUP framework, the number of bits on the screenwill become N = S GUP k B . (19)We again apply the same argument based on the meanenergy of blackbody photons and connect it with thecounting of degrees of freedom on the screen to arriveat the following form of temperature: T = 2 M c N k B = M c S GUP . (20)Finally, using the first law of thermodynamics we arriveat the modified gravity force law: F GUP = F N η ) − α (2 + η )) η (1 + η ) (cid:16) − α + (1 + η ) + 4 α log (cid:104) √ α η (cid:105)(cid:17) . (21)Here F N = GmM/R is Newton’s gravitationalforce law, and we have introduced symbols η = (cid:112) − G (cid:126) /c R and α = G (cid:126) /c R to simplify the ex-pression.In the large distance limit where R (cid:29) L p = (cid:112) G (cid:126) /c and therefore α = G (cid:126) /c R (cid:28)
1, we can expand theforce to the third order of α as F GUP = F N { α [2 − log α ] + α [4 − α + (log α ) ]+ α [7 −
18 log α + 8(log α ) − (log α ) ] + ... } . (22)It is clear that this GUP-based force law recovers theclassical Newton’s gravitational force law in the infinitedistance limit, while some subleading quantum correc-tions is present as long as α is finite. On the other handthese correction terms go to zero in the classical limit as (cid:126) vanishes. These α -dependent terms, we conclude, arewhere (cid:126) is hiding in entropic gravity. V. CONCLUSION AND DISCUSSIONS
In this paper we raised the question about where (cid:126) ishiding in entropic gravity. Through the reanalysis of thefundamental building blocks of entropic gravity, in par-ticular the holographic formulation of the entanglemententropy, we argued that the perfect cancellation of (cid:126) ’samong all the quantum mechanically motivated inputs isbroken if the more exact form of the BH entanglemententropy based on GUP is to replace the Bekenstein area law. Based on this we found, in the weak gravity limit,the hided (cid:126) ’s in the form of logarithmic corrections tothe classical Newton’s law, in Eq.(22).In our attempt of seeking the missing (cid:126) ’s, we refor-mulated the existing derivations of entropic gravity. InVerlinde’s entropic gravity derivation, two ingredientsinvolving entropy formula have been invoked withoutthe guarantee of their mutual compatibility. We ap-plied Fursaev’s procedure to reproduce the leading orderentropy variation in Verlinde’s setup of spherical holo-graphic screen.While our approach manages to avoid the compati-bility issue, there is a price to pay. In our alternativeapproach we have introduced the concept of spacetimemetric and its deformation due to the presence of a mas-sive object, which implicitly assumed the knowledge ofgeneral relativity, the standard theory of classical grav-ity. Yet the very attempt of entropic gravity is to deduceit from quantum mechanics and statistical physics alonewithout any prior knowledge of gravity. We are thereforeat risk of a circular logic in our approach if gravity isto be interpreted as an emergent phenomenon. In thisregard a more cogent and consistent argument withoutinvolving any gravity-related concept is needed towardsan alternative entropy variation law, in order to assertthe validity of the entropic framework of gravity as anemergent phenomena. By the similar token, the existingderivations of entropic gravity also faces the similar issuesince Newton’s constant has been invoked as a fundamen-tal constant from the outset instead of being a secondary,derived parameter of the theory as it should if gravity isto be an emergent phenomenon.Under this light one can instead view our derivationof the entropic gravity not as an emergent phenomenonbut as a means to deduce the ‘quantum gravity forcelaw’ via the quantization of the information content onthe surfaces in units of Planck area provided by GUP aswell as the spacetime warpage effect in the presence of amassive particle provided by general relativity.Although there are still rooms to improve in this lineof approach to gravity, we have provided an exact formof quantum corrected entropic gravity force law based onthe assumption of GUP as a fundamental input. Suchquantum corrections, though minute, may serve as aprobe to examine the concreteness of the entropic grav-ity interpretation in the the experimentally measurablescale of large distance and weak gravity limit.
Acknowledgement
We thank Debaprasad Maity, Taotao Qiu, KeisukeIzumi, Yen-Chin Ong , Chien-I Chiang, Nian-An Tungand Jo-Yu Kao for helpful and inspiring discussions.This research is supported by the Taiwan National Sci-ence Council (NSC) under Project No. NSC98-2811-M-002-501, No. NSC98-2119-M-002-001, and the USDepartment of Energy under Contract No. DE-AC03-76SF00515. We would also like to thank the NTU LeungCenter for Cosmology and Particle Astrophysics for its support. [1] J. D. Bekenstein, Phys. Rev. D. , 2333 (1973).[2] J. D. Bekenstein, Phys. Rev. D. , 287 (1981).[3] S. W. Hawking, Comm. Math. Phys. , 199 (1975).[4] M. Srednicki, Phys. Rev. Lett. , 666 (1993).[5] J. Eisert, M. Cramer, and M.B. Plenio, Rev. Mod. Phys. , 277 (2010).[6] D. Kabat, Nucl. Phys. B453 , 281 (1995).[7] T. Jacobson, arXiv:gr-qc/9404039 (1994).[8] A.D. Sakharo, Sov. Phys. Dokl. , 1040 (1968).[9] T. Jacobson, Phys.Rev.Lett. , 1260 (1995).[10] G. E. Volovik, Proc. Sci., QG-Ph 043(2007).[11] L. Sindoni, S. Liberati and F. Girelli, AIP Conf. Proc. , 258 (2009).[12] Rept. Prog. Phys. , 046901 (2010).[13] E. P. Verlinde, JHEP , 029 (2011).[14] G. ’t Hooft, arXiv:gr-qc/9310026v2 (1993).[15] L. Susskind, J. Math. Phys. , 6377 (1995).[16] R. Bousso, Rev. Mod. Phys. , 825 (2002).[17] R. J. Adler, P. Chen, and D. I. Santiago, Gen. Relativ.Gravit. , 2101 (2001).[18] S. N. Solodukhin, Phys. Rev. D , 2410 (1998).[19] M, Cadoni, and M. Melis, Entropy , 12, 2244 (2010).[20] R. K. Kaul, and P. Majumda, Phys. Rev. Lett. , 5255(2000).[21] A, Ghosh, and P. Mitra, Phys. Rev. D , 027502 (2005).[22] M. A. Nielsen, and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,2000).[23] D. Janzing,
Compendium of Quantum Physics , edited byD. Greenberger, K. Hentschel,and F. Weinert (SpringerBerlin Heidelberg, 2009) p.205-209.[24] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, Rev. Mod. Phys. , 865 (2009). [25] P. Calabrese, J. Cardy, Int. J. Quant .Inf. 4:429,2006,arXiv:quant-ph/0505193v1(2006).[26] L. Susskind, and J. Uglum, Phys. Rev. D , 2700 (1994).[27] T. Nishioka, S. Ryu, and T. Takayanagi, J. Phys. A:Math. Theor. , 504008 (2009).[28] D. V. Fursaev, Phys. Rev. D , 12402 (2006).[29] S. Ryu, and T. Takayanagi, JHEP , 045 (2006).[30] H. Casini, M. Huerta, and R. C. Myers, JHEP , 036(2011).[31] S. N. Solodukhin, Phys. Lett. B , 305-309 (2008).[32] D. V. Fursaev, arXiv:1006.2623 (2010).[33] D. V. Fursaev, Phys. Rev. D , 124002 (2008).[34] S. Hossenfelder, Phys. Rev. D , 105013(2006); S.Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S.Scherer, and H. Stcker, Phys. Lett. B , 85-99 (2003).[35] C. A. Mead, Phys. Rev. , B849 (1964); C. A. Mead , 990 (1966).[36] G. Veneziano, Europhys. Lett. , 199 (1986).[37] D. J. Gross and P. F. Mende, Nucl. Phys. B303 , 407(1988).[38] D. Amati, M. Ciafolini and G. Veneziano. Phys. Lett.
B216 , 41 (1989).[39] K. Konishi, G. Paffuti and P. Provero, Phys. Lett.
B234 ,276 (1990).[40] E. Witten, Phys. Today, Apr. 24 (1996).[41] M. A. Santos and I. V. Vancea, arXiv:1002.2454v3(2010).[42] S. Ghos, arXiv:1003.0285v3 (2010).[43] L. Modesto and A. Randono, arXiv:1003.1998v1 (2010).[44] M. R. Setare and D. Momeni, arXiv:1004.2794v1 (2010).[45] P. Nicolini, Phys. Rev. D82