Is Gravity an Entropic Force?

Transcription

1 Is Gravity an Entropic Force? San Gao Unit for HPS & Centre for Time, SOPHI, University of Sydney, Sydney, NSW 006, Australia; Abstract: Te remarkable connections between gravity and termodynamics seem to imply tat gravity is not fundamental but emergent, and in particular, as Verlinde suggested, gravity is probably an entropic force. In tis paper, we will argue tat te idea of gravity as an entropic force is debatable. It is sown tat tere is no convincing analogy between gravity and entropic force in Verlinde s example. Neiter olograpic screen nor test particle satisfies all requirements for te existence of entropic force in a termodynamics system. Furtermore, we sow tat te entropy increase of te screen is not caused by its statistical tendency to increase entropy as required by te existence of entropic force, but in fact caused by gravity. Terefore, Verlinde s argument for te entropic origin of gravity is problematic. In addition, we argue tat te existence of a minimum size of spacetime, togeter wit te Heisenberg uncertainty principle in quantum teory, may imply te fundamental existence of gravity as a geometric property of spacetime. Tis may provide a furter support for te conclusion tat gravity is not an entropic force. Keywords: gravity; termodynamics; entropic force; discrete spacetime PACS Codes: Dy; 04.0.Cv 1. Introduction It is still a controversial issue weter gravity is fundamental or emergent. Te solution of tis problem may ave important implications for a complete teory of quantum gravity. A remarkable indication for te nature of gravity comes from te deep study of black ole termodynamics, wic suggests tat general connections between gravity and termodynamics may exist [1 4]. Inspired by tese teoretical developments, Jacobson argued tat te Einstein equation can be derived from te proportionality of entropy and orizon area, togeter wit te first law of termodynamics, and e concluded tat te Einstein equation is a termodynamics equation of state [5]. Padmanaban furter sowed tat te equations of motion describing gravity in any diffeomorpism invariant teory can be given a termodynamic re-interpretation, wic is closely linked to te structure of action functional [6,7]. Tese results suggest tat gravity may be explained as an emergent penomenon and as a termodynamics or entropic origin (see, e.g., [8] for a review). Recently Verlinde proposed a new argument for emergent gravity, mainly based on te olograpic principle [9]. He argued and explicitly claimed tat gravity is an entropic force, wic is caused by a cange in te amount of information associated wit te positions of bodies of matter. Tis idea is interesting and, if rigt, may ave important implications for te origin of gravity and its unification wit te quantum. In tis paper,

2 we will critically examine te idea of gravity as an entropic force, focusing more on te pysical explanation.. Verlinde s Argument Verlinde s argument can be basically formulated as follows. Consider a small piece of a olograpic screen. A particle of mass m approaces it from te side at wic space as already emerged. First, it is assumed tat before te particle merges wit te microscopic degrees of freedom on te screen, it already influences te amount of information tat is stored on te screen, and te corresponding cange of entropy on te screen is: mc ΔS = πkb Δx (1) were Δ x is te displacement of te particle near te screen and comparable wit te Compton wavelengt of te particle, k B is te Boltzmann constant, c is te speed of ligt, and is Planck s constant divided by π. Next, it is assumed tat te olograpic principle olds, and te number of te used bits on te screen is: 3 Ac N = () G were A is te area of te screen, and G is a constant tat will be identified wit Newton s constant later. Tirdly, it is assumed tat te screen as a total energy E, wic is divided evenly over te bits N, and te temperature of te screen T is determined by te equipartition rule: 1 E = NkBT (3) Lastly, it is assumed tat te mass M, wic would emerge in te part of space enclosed by te screen, satisfies te relativistic mass-energy relation: E = Mc (4) How does force arise ten? Here Verlinde used an analogy wit osmosis across a semi-permeable membrane. Wen a particle as an entropic reason to be on one side of te membrane and te membrane carries a temperature, it will experience an effective entropic force equal to: FΔ x = TΔS (5) Ten e claimed tat in te above interacting process between a olograpic screen and a particle, te particle will also experience an entropic force in a similar way. Moreover, e sowed tat tis Mm force satisfies Newton s law of gravity, namely F = G, based on Equations (1) (5) and te area R relation A = 4πR. Inspired by tese interesting results, Verlinde tus concluded tat gravity is an entropic force. 3. Understanding Entropy Force In order to see weter gravity is an entropic force or not, we need to first understand te concept of an entropic force. An entropic force can be defined as an effective macroscopic force tat originates in

3 3 a termodynamics system by its statistical tendency to increase entropy. As a typical example, let us see te elasticity of a polymer molecule, wic was also discussed by Verlinde [9]. A polymer molecule is a large molecule composed of repeating structural units, typically connected by covalent cemical bonds. Te simplest polymer arcitecture is a linear cain, and it can be modeled by joining togeter many monomers of fixed lengt, were eac monomer can freely rotate around te points of attacment and direct itself in any spatial direction. Eac of tese configurations as te same energy. Wen te polymer molecule is immersed into a eat bat, it tends to put itself into a randomly coiled configuration since tis configuration as iger entropy. Tere are many more suc configurations wen te molecule is sort compared to wen it is stretced into an extended configuration. Te statistical tendency to reac a maximal entropy state ten generates te elastic force of a polymer, wic is a typical entropic force. We can furter determine te entropic force by introducing an external force F to pull te polymer out of its equilibrium state and ten examining te balance of forces. For example, one can fix one endpoint of te polymer at te origin and pull te oter endpoint of te polymer apart along te x-axis. Te entropy of te system can be written as S( E, x) = kb log Ω( E, x), were Ω ( E, x) denotes te volume of te configuration space for te entire system as a function of te total energy E of te eat bat and te average stretc lengt of te polymer (in tis case te position x of te pulled endpoint). One can ten determine te entropic force by analyzing te micro-canonical ensemble given by Ω ( E + Fx, x), were te external force is introduced as an external variable dual to te lengt x of te polymer, and imposing te extremal condition for entropy. Tis gives F = T S / x, were te temperature T is defined by 1 / T = S / E. By te balance of forces, te entropic force, wic tries to restore te polymer to its equilibrium position, will be equal to te external force F. For te polymer te entropic force can be sown to obey Hooke s law, i.e., te entropic force is proportional to te stretced lengt. Te entropic force can also be understood in terms of te first law of termodynamics. Te principle is an expression of energy conservation, according to wic te increase in te internal energy of a system is equal to te amount of energy added by eating te system minus te amount lost as a result of te work done by te system on its surroundings. Wen te internal energy of a system does not cange during te studied process, te law can be simply written as FΔ x = TΔS. In te following we will analyze te entropic force of te polymer in terms of tis equation. We ave tree interacting systems in total, namely a eat bat, a polymer immersed in it, and an external system connected to te polymer. Tere is no entropic force wen te polymer is in its equilibrium state wit maximum entropy. Only wen te polymer leaves its maximum entropy state can an entropic force appear. We assume tat te internal energy of te polymer remains constant during te cange of its entropy. Wen te polymer is pulled out of equilibrium by an external force, te work done by te force will be equal to te energy increase of te eat bat according to te first law of termodynamics. In te l..s of te equation FΔ x = TΔS, F is te external force, Δ x is te displacement of te polymer, and FΔ x is te work done on te polymer by te external system. In te r..s of te equation, T is te temperature of te polymer, Δ S is te entropy decrease of te polymer, and TΔ S is te eat loss of te polymer. Te work is equal to te eat loss, and te internal energy of te polymer keeps constant. Moreover, te eat flows from te polymer to te eat bat, and te eat loss of te polymer is equal to te energy increase of te eat bat. As a result, te total energy of te tree systems is conserved [10]. In sort, energy flows from te external system to te polymer and

4 4 furter to te eat bat, and te corresponding causal cain is: External system Polymer Heat bat. During tis process, te entropy of te polymer decreases, and te entropy of te eat bat increases. Wen keeping te polymer at a fixed lengt te entropic force will be equal to te external force, wic is F = TΔS / Δx according to te above equation, and teir directions are opposite. Wen one lets te stretced polymer gradually return to its equilibrium position, wile allowing te force to perform work on te external system, te work done by te entropic force will be equal to te energy decrease of te eat bat according to te first law of termodynamics. In te l..s of te equation FΔ x = TΔS, F is te entropic force, Δ x is te displacement of te polymer, and FΔ x is te work done by te polymer on te external system. In te r..s of te equation, T is te temperature of te polymer, Δ S is te entropy increase of te polymer, and TΔ S is te eat of te polymer extracted from te eat bat. Te work is equal to te eat gain, and te total energy of te tree systems is also conserved. During tis process, te entropy of te polymer increases, and te entropy of te eat bat decreases. Moreover, energy flows from te eat bat to te polymer and furter to te external system, and te corresponding causal cain is Heat bat Polymer External system. By te equation FΔ x = TΔS we can also find te entropy force is F = TΔS / Δx. Before ending tis section we will stress several important features of entropic force, wic are relevant to te analysis of Verlinde s argument tat follows. First, entropic force results entirely from te statistical tendency of a termodynamics system to increase its entropy, not from any energy effect. Te energy of te system is conserved wen te entropic force is in action. In tis sense, entropic force as a purely entropic origin. But ow can a force be generated witout any energy effect? Tis leads us to te second important feature of entropic force, namely tat te existence of a eat bat is indispensable for entropic force. Altoug an entropic force is independent of te details of te microscopic dynamics, its existence depends on te existence of interaction between te microscopic components of te studied system and environment or eat bat. For example, tere is no entropic force for an isolated polymer in a vacuum. Due to te interaction, te system is constantly subject to random collisions from microscopic components of te eat bat, and eac of tese collisions sends te system from its current microscopic state to anoter. Tis will lead te system to its maximum entropy state, te state wit te maximum number of microscopic states, and te statistical tendency to reac a maximum entropy state ten generates an entropic force. In most familiar situations, te interaction as an eletromagnetic origin. In tis sense, it may be not wolly rigt to say tat entropic force as a purely entropic origin. Tirdly, te magnitude of an entropic force is only related to te properties of te system and te surrounding eat bat, e.g., te size of te system and te temperature of te eat bat. For example, te entropic force of a polymer is proportional to its stretced lengt and te temperature of te eat bat. In particular, te magnitude of an entropic force is not related to te properties of any external system. For instance, te entropic force of a polymer is irrelevant to te mass of te external system connected to te polymer. Lastly, an entropic force always points in te direction of increasing entropy. An external force exerted on a system can point in te direction of decreasing its entropy, wile te entropic force generated by te system must point in te direction of increasing its entropy. Wen te system reaces its maximum entropy, te entropic force becomes zero. 4. Wy Gravity Is Not An Entropy Force

5 5 After we ave understood wat an entropic force is, we can next examine Verlinde s argument for te entropic origin of gravity. In is example of gravity, tere are two systems, namely a olograpic screen and a test particle. Verlinde argued tat te gravitational interaction between te olograpic screen and te particle is an entropic force. In order to see weter an entropic force exists in te example and weter gravity is an entropic force, we need to, parallel to te polymer example, answer te following two questions concerning Verlinde s example: Wic is te eat bat? And wic is te polymer? Option 1: Verlinde s answer. According to Verlinde, te olograpic screen serves as te eat bat, and te particle can be regarded as te end point of te polymer tat is gradually allowed to go back to its equilibrium position ([9], p. 5). He furter suggested tat te particle can be tougt of as being immersed in te eat bat representing te screen in te olograpic description, and by te time tat te particle reaces te screen it will become part of te termal state, just like te polymer. As already admitted by Verlinde, owever, it is not appropriate to view te screen as a eat bat. Te reason is tat te screen is not exactly at termal equilibrium ([9], p. 6). If assuming te screen at an equipotential surface is in equilibrium, ten te entropy needed to get te Unru temperature will appear to be very ig and violate te Bekenstein bound tat states tat a system contained in region wit radius R and total energy E cannot ave entropy larger tan ER. Verlinde tried to solve tis problem by rescaling te value of Planck s constant. Tis rescaling would affect te values of te entropy and te temperature in opposite directions: T will get multiplied by a factor, and S will be divided by te same factor. He also briefly proposed anoter possible solution based on a description tat uses weigted average over many screens wit different temperatures. Altoug Verlinde admitted tat tere is someting to be understood, e still concluded tat gravity is an entropic force. In te following, we will point out some oter problems of Verlinde s analogy between gravity and entropic force. First, altoug te particle can be regarded as a polymer in some sense, it seemingly as no well-defined temperature and entropy. As a result, it seems meaningless to talk about te entropy increase of te polymer-like particle or its statistical tendency to increase entropy, but te latter is required by te existence of an entropic force as we ave seen in te polymer example. Furtermore, even if te particle as appropriate temperature and entropy, tey will be very different from tose of te screen, and tus te resulting entropic force for te polymer-like particle will be different from Newton s gravity. In fact, te temperature and entropy of te particle does not appear in Verlinde s derivation, were tere are only te temperature and entropy of te screen. Secondly, during te process of te polymer-like particle going back to its equilibrium position ([9], p. 5), energy flows into te eat-bat-like screen. Tis is also inconsistent wit te energy-flow feature of entropic force; energy sould flow out of te eat bat for suc a process. Tis leads us to te tird problem. As we ave known from te polymer example, during te process of a polymer returning to its equilibrium position, it is te entropy of te polymer tat increases, wile te entropy of te eat bat decreases. However, te entropy of te screen as a eat bat increases wen te particle as a polymer returns to its equilibrium position. Terefore, tere is also an obvious inconsistency in te analogy. To sum up, Verlinde s answer is problematic. It is improper to view te olograpic screen as a eat bat and te particle a polymer. Moreover, no entropic force exists wen assuming tis view. Option : Te screen is taken as a eat bat, and te particle is taken as an external system.

6 6 Can an entropic force exist for tis option? Te answer is negative too. As we ave seen from te analysis of Option 1, tere is no convincing analogy between te screen and a eat bat. Tus, an environment suitable for te existence of entropic force is not available eiter for tis option. Besides, te gravitational interaction is related to te mass of te external particle, wile entropic force sould only depend on te properties of te system (in tis case te eat bat) and be irrelevant to te mass of any external particle. As a result, gravity cannot be an entropic force no matter weter entropic force exists for tis option. Option 3: Te screen is taken as a polymer, and te particle is taken as an external system. Tis option is suggested by te observation tat te system wose entropy increases during te interacting process is te screen, not te particle, and it is te polymer, not te eat bat, tat increases its entropy during similar process in te polymer example. Tus te screen is more like a polymer tan like a eat bat. Besides, tis option is also supported by te above observation tat te gravitational interaction undergone by te particle points in direction of te entropy increase of te screen, wic is consistent wit te direction feature of entropic force. However, gravity cannot be an entropic force for tis option eiter wen considering te magnitude feature of entropic force. Te gravitational interaction between te screen and te particle is related to te masses of bot systems, wile te entropic force originating from te polymer-like screen is irrelevant to te properties of te particle taken as an external system. In fact, tere is no entropic force eiter for tis option, as tere is no eat bat ere. As we ave pointed out in te last section, a eat bat is indispensable for te existence of an entropic force. Option 4: Te screen and te particle are taken as te two ends of a polymer. Tis option is consistent wit te magnitude feature of an entropic force; te gravitational interaction between te screen and te particle is related to te properties of bot systems, wile te entropic force of a polymer also depends on te properties of te wole polymer, in tis case including bot te screen and te particle. However, tere is an obvious objection to tis option, namely tat tere is no termal equilibrium between te screen and te particle. Moreover, as we ave argued before, it is also problematic to view te particle as a polymer. Besides, since tere is no eat bat, tere is no entropic force eiter for tis option. Note tat altoug constant quantum vacuum fluctuations exist in te emergent space between te screen and te particle, te temperature of tis Minkowski vacuum is precisely zero, and te in-between space is not a eat bat. Even if we assume te existence of an in-between gravitational field beforeand, and furter regard it as an approximate eat bat, an environment suitable for te existence of entropic force is still unavailable. Te reason is tat te gravitational field is not in equilibrium wit te screen and particle. Moreover, since te average temperature of te field is in general muc lower tan tat of te screen or orizon, energy cannot spontaneously flow from te field to te screen due to a pure entropic reason. After examining all tese possible options, we find tat tere is no convincing analogy between gravity and entropic force in Verlinde s example. Neiter te olograpic screen nor te particle satisfies all te requirements for te existence of entropic force in a termodynamics system. Lastly, we will present a general objection to viewing gravity as an entropic force in Verlinde s example. Te objection concerns te energy increase of te screen. During te interaction between te screen and te particle, te energy of te screen also increases along wit its entropy increase [11].

7 7 Moreover, te amount of its entropy increase just corresponds to te amount of its energy increase. Tis already indicates tat gravity cannot be a pure entropic effect, because, unlike te polymer example, energy is obviously input to te screen during te process, especially in te same amount as required by te increase of entropy. We can strengten tis conclusion by furter analyzing te causal relationsip between energy increase and entropy increase. It is well known tat energy increase can cause entropy increase, but te (spontaneous) entropy increase cannot cause energy increase (on te contrary, energy will decrease if te entropy-increasing system does work. Wen no work is done by te entropy-increasing system, its energy is conserved in case of no input energy). Terefore, te entropy increase of te screen is in fact caused by its energy increase, not by a statistical tendency to increase entropy, wic is essentially required for te existence of an entropic force. As a result, te interaction between te screen and te particle is not an entropic effect, and gravity is not an entropic force eiter [1]. Te entropy increase of te screen results from its energy increase. Were does te increased energy come from ten? Wen assuming a gravitational field exists between te screen and te particle as usual, it is easy to answer tis question. It is te gravitational field tat provides te increased (matter) energy for te screen [13]. In oter words, te energy flow originates from te work done by te gravitational field troug te force, gravity [14]. Tis is essentially different from te process of energy gain of te polymer. In te polymer example, te energy gain of te polymer comes from te eat bat troug a pure termodynamics process. Can a similar termodynamics process explain te energy increase of te screen? Te answer is negative. As we ave argued in te analysis of Option 4, tere is no well-defined eat bat ere. Even if te gravitational field can be taken as an approximate eat bat, its average temperature will be in general muc lower tan tat of te screen or orizon, and tus energy cannot spontaneously flow from te field to te screen due to a pure entropic reason. In a word, te increased energy of te screen can only come from te work done by te gravitational field. Altoug te above argument seems reasonable, one question still needs to be answered before we can reac a definite conclusion, namely wy te force F, derived from te formula FΔ x = TΔS, is just gravity for te interacting process between a olograpic screen and a particle wen assuming te mc formula ΔS = πk B Δx as Verlinde did ([9], p. 7). In fact, tis result can be readily understood after we ave known tat te entropy increase of te screen results from its energy increase and te energy Mm comes from te work done by te gravitational field. Since F = G (Newton s law of gravity) and R 1 a 1 GM mc T = = (Unru s formula), tere must exist te relation ΔS = πk B Δx in πk B c πk B cr accordance wit te formula FΔ x = TΔS. Ten it is not surprising tat wen assuming mc ΔS = πkb Δx (and Unru s formula or olograpic principle), Newton s law of gravity naturally follows [15]. It is wort noting tat tis argument does not depend on te distance between te screen and te mcr particle, and te particle needs not to be near te screen. Te general formula is ΔS = π kb Δx, R were R is te radius of te sperical screen, and R is te distance between te particle and te

8 8 center of te screen [16]. Terefore, wy Verlinde s entropic force is gravity is because it is just te work done by gravity tat results in te increase in entropy [17]. To sum up, altoug Verlinde s derivation is rigt, it does not prove in pysics tat gravity is an entropic force [18]. Moreover, a detailed analysis sows tat gravity is not an entropic force in te gravitational system e considered. In particular, te gravitational interaction between a olograpic screen and a test particle is caused neiter by te entropy increase of te screen nor by its statistical tendency to increase entropy; rater, te entropy increase of te screen is caused by gravity [19]. 5. Furter Discussions Te connections between gravity and termodynamics seem so remarkable tat one cannot elp conjecturing tat gravity as a termodynamics or entropic origin. From a general point of view, owever, tis opinion is at least debatable. To begin wit, te temperature and entropy of various orizons are all derived from te vacuum fluctuations of quantum fields in curved spacetime. Ten ow can one re-ascribe tese emergent properties to te termodynamics of spacetime itself? It seems tat tere is a uge gap between tem in pysics. Next, altoug te existing arguments based on termodynamical analysis can derive te Einstein equation [0], tey do it only wit te elp of te principle of equivalence togeter wit some oter assumptions. In oter words, tey can answer ow matter curves spacetime only after assuming matter indeed curves spacetime. Tey do not explain wy matter curves spacetime or wy gravity is a curved spacetime penomenon, wic is te very nature of gravity according to general relativity. Only after one explains tis particular nature, can one understand te origin of gravity and answer weter gravity is emergent or not. In te following, we will present a tentative answer. It is argued tat te existence of a minimum size of spacetime, togeter wit te Heisenberg uncertainty principle in quantum teory, migt elp to explain wy matter curves spacetime. According to te Heisenberg uncertainty principle in quantum mecanics we ave: Δx Δp Te momentum uncertainty of a particle, Δ p, will result in te uncertainty of its position, Δ x. Tis poses a limitation on te localization of a particle in te nonrelativistic domain. Tere is a more strict limitation on Δ x in te relativistic domain. A particle at rest can only be localized witin a distance of te order of its reduced Compton wavelengt, namely: (6) Δ x m c (7) were m 0 is te rest mass of te particle. Te reason is tat wen te momentum uncertainty Δ p is greater tan m 0 c te energy uncertainty Δ E will exceed m 0c, but tis will create a particle anti-particle pair from te vacuum and make te position of te original particle invalid. It ten follows tat te minimum localization lengt of a particle at rest can only be te order of its reduced Compton wavelengt. Using te Lorentz transformation, te minimum localization lengt of a particle moving wit (average) velocity v is: 0

9 Δ x or mc 9 c Δ x (8) E were m = m0 / 1 v / c is te relativistic mass of te particle, and E = mc is te total energy of te particle. Tis means tat wen te energy uncertainty of a particle is of te order of its (average) energy, it as te minimum localization lengt. Note tat Equation (8) also olds true for particles wit zero rest mass suc as potons. Te above limitation is valid in continuous spacetime; wen te energy and energy uncertainty of a particle bot become arbitrarily large, its localization lengt Δ x can still be arbitrarily small. However, te existence of a minimum size of spacetime will demand tat te localization of any particle sould ave a minimum value L U, namely Δ x sould satisfy te limiting relation: Δ x L U (9) In order to satisfy tis relation, te r..s of Equation (8) sould at least contain anoter term proportional to te (average) energy of te particle, namely in te first order of E it sould be: c LU E Δ x + (10) E c Tis new inequality, wic can be regarded as one form of te generalized uncertainty principle [1], can satisfy te limitation relation imposed by te discreteness of spacetime. It means tat te localization lengt of a pointlike particle as a minimum value L U. How to understand te new term demanded by te discreteness of spacetime ten? Obviously it indicates tat te (average) energy of a particle increases te size of its localized state, and te increase is proportional to te energy. Since tere is only one particle ere, te increase of its localization lengt cannot result from any interaction between it and oter particles suc as electromagnetic interaction. Besides, since te increased part, wic is proportional to te energy, is very distinct from te original quantum part, wic is inverse proportional to te energy, it is a reasonable assumption tat te increased localization lengt does not come from te quantum motion of te particle eiter. As a result, it seems tat tere is only one possibility left, namely tat te (average) energy of te particle influences te geometry of its background spacetime and furter results in te increase of its localization lengt. We can also give an estimate of te strengt of tis influence in terms of te new L term U LU TU E. Tis term sows tat te energy E will lead to an lengt increase Δ L. Tis E c furter implies tat te energy E contained in a region wit size L will cange te proper size of te region to: LU TU E L L + (11) Tis means tat a flat spacetime will be curved by te energy contained in it. Wen te energy is equal to zero or tere are no particles, te background spacetime will not be canged. Since wat canges spacetime ere is te average energy, tis relation between energy and proper size increase cange is irrelevant to te quantum fluctuations. Te above analysis based on te quantum uncertainty principle and te discreteness of spacetime migt provide a possible basis for te Einstein equivalence principle. It implies tat gravity is

10 10 essentially a geometric property of spacetime, wic is determined by te energy density contained in tat spacetime, not only for macroscopic objects but also for microscopic particles. Moreover, te Einstein gravitational constant can also be determined in terms of te minimum size of discrete spacetime []. Te result is: L U T κ = π U (1) Note tat tis formula itself seems to also suggest tat gravity originates from te discreteness of spacetime (togeter wit te quantum principle tat requires 0 ). In continuous spacetime were T U = 0 and L U = 0, we ave κ = 0, and tus gravity does not exist. It sould be stressed tat te existence of a minimum size of spacetime as been widely argued and acknowledged as a model-independent result of te proper combination of quantum mecanics and general relativity (see, e.g., [3] for a review). Te model-independence of te argument for te discreteness of spacetime strongly suggests tat discreteness is a more fundamental feature of spacetime. Terefore, it seems appropriate to analyze te implication of spacetime discreteness for te origin of gravity as above. Certainly, if spacetime itself is emergent, ten gravity must be also emergent as it is essentially a curved spacetime penomenon. But even so, tey sould ave corresponding microscopic elements in te pre-spacetime teory. On te oter and, as we ave argued above, gravity is probably fundamental in te emergent spacetime. Te argument not only olds true for microscopic particles, but also may apply to te bits living on a olograpic screen as well. Tis may provide a furter support for te conclusion tat gravity is not an entropic force. Acknowledgments I am very grateful to Sabine Hossenfelder, Cristian Wütric, Dean Rickles and Huw Price for elpful discussions and suggestions. I am also grateful to te Special Issue Guest Editor Jacob D. Bekenstein and two anonymous reviewers for teir constructive comments. Tis work was supported by te Postgraduate Scolarsip in Quantum Foundations provided by te Unit for History and Pilosopy of Science and Centre for Time (SOPHI) of te University of Sydney. References and Notes 1. Bardeen, J.M.; Carter, B; Hawking, S.W. Te four laws of black ole mecanics. Commun. Mat. Pys. 1973, 31, Bekenstein, J.D. Extraction of energy and carge from a black ole. Pys. Rev. D 1973, 7, Bekenstein, J.D. Black oles and entropy. Pys. Rev. D 1973, 7, Hawking, S.W. Particle creation by black oles. Commun. Mat. Pys. 1975, 43, Jacobson, T. Termodynamics of spacetime: Te Einstein equation of state. Pys. Rev. Lett. 1995, 75, Padmanaban, T. Entropy density of spacetime and termodynamic interpretation of field equations of gravity in any diffeomorpism invariant teory. arxiv 009, arxiv: [gr-qc]. 7. Padmanaban, T. Entropy density of spacetime and gravity: A conceptual syntesis. Int. Jour. Mod.Pys. D 009, 18,

11 11 8. Padmanaban, T. Termodynamical aspects of gravity: New insigts. arxiv 009, arxiv: [gr-qc]. 9. Verlinde, E.P. On te origin of gravity and te laws of Newton. arxiv 010, arxiv: [ep-t]. 10. For an infinite eat bat, te corresponding entropy increase of te eat bat will be equal to te entropy decrease of te polymer, and te total entropy will be also conserved. Tus te entropic force is conservative [9]. 11. In te polymer example, te energy of te polymer keeps constant wen te entropic force does work (te energy for te work comes from te surrounding eat bat). Tis dissimilarity reconfirms te conclusion tat te screen is not like a polymer. 1. Interestingly, [5] sowed tat if gravity is an entropic force as Verlinde argued, ten te Coulomb force sould be also an entropic force. But it is well accepted tat te Coulomb force is a fundamental interaction transferred by virtual potons. Besides, te directions of te entropic force and te Coulomb force are opposite for two carges wit different signatures. Tis as been identified as te problem of negative electromagnetic temperature [5,6]. As we tink, tese apparent contradictions also suggest tat te idea of gravity as an entropic force is probably wrong. In addition, tis result can also be taken as a support for our conclusion tat it is gravity (and te Coulomb force) tat result in te entropy increase of te screen, not te contrary. 13. Note tat te energy of te screen only depends on te mass inside it and is irrelevant to te mass of te external particle. Moreover, te entropy of te screen does not include te entropy of gravitational field, and it is only te entropy of matter. 14. As a result, te energy and entropy of te gravitational field correspondingly decrease during te process. 15. It as been sown tat tis consistency also as a matematical origin, and it is a consequence of te specific properties of solutions to te Poisson equation [6]. 16. Verlinde also discussed tis large-distance situation and implicitly presented te rigt formula ([9], p.10). 17. Verlinde also admitted tat wy is equations come out is because te laws of Newton ave been ingredients in te steps tat lead to black ole termodynamics and te olograpic principle (see [9], p.9). However, as we ave argued, is attempt to reverse tis argument was not successful. 18. It as been recently claimed tat Verlinde s idea is supported by a matematical argument based on a discrete group teory [7]. As we tink, altoug te teory migt provide a possible matematical formulation of te olograpical principle, it does not necessarily entail tat gravity is an entropic force in pysics. Besides, it is far from clear weter tis formulation can naturally lead to general relativity and quantum field teory as two proper approximations. 19. As we tink, our pysical analysis also applies to te similar arguments proposed by oter autors (e.g., [5 8]). Like Verlinde, Jacobson did not explicitly state te causal relationsip between energy flux and entropy cange eiter, toug is analysis was more rigorous tan Verlinde s [5]. It seems tat Jacobson assumed te rigt causal cain, i.e., energy flux entropy cange, as e said te entropy is proportional to te orizon area and te area increase of a portion of te orizon will be proportional to te energy flux across it. However, e also reaced a similar conclusion tat te Einstein equation is a termodynamics equation of state [5].

12 1 0. Tis fact sould not surprise us very muc, as te termodynamics of gravitational systems suc as a black ole are just derived in terms of general relativity and quantum field teory. 1. Te argument ere migt be regarded as a reverse application of te generalized uncertainty principle (see, e.g., [3,4]). But it sould be stressed tat te existing arguments for te principle are based on te analysis of measurement process, and teir conclusion is tat it is impossible to measure positions to better precision tan a fundamental limit. On te oter and, in te above argument, te position uncertainty or localization lengt of a particle is objective and real, and te discreteness of spacetime requires tat te objective lengt as a minimum value, wic is independent of measurement.. Gao, S. Wy gravity is fundamental. arxiv 010, arxiv: [pysics.gen-p]. 3. Garay, L.J. Quantum gravity and minimum lengt. Int. J. Mod. Pys. A 1995, 10, Adler, R.J.; Santiago, D.I. On gravity and te uncertainty principle. Mod. Pys. Lett. A 1999, 14, Wang, T. Te Coulomb Force as an Entropic Force. arxiv 010, arxiv: [ep-t]. 6. Hossenfelder, S. Comments on and Comments on Comments on Verlinde s paper On te Origin of Gravity and te Laws of Newton. arxiv 010, arxiv: [gr-qc]. 7. Winkelnkemper, H.E. AP Teory V: Termodynamics in Topological Disguise, Gravity from Holograpy and Entropic Force as Dynamic Dark Energy. Available online: ttp:// (accessed on 7 April 011); Preprint, February 011.