SLAC-PUB-15931 Where is ~ Hiding in Entropic Gravity? Pisin Chen1;2;3;4∗ and Chiao-Hsuan Wang1;3y 1. Department of Physics, National Taiwan University, Taipei, Taiwan 10617 2. Graduate Institute of Astrophysics, National Taiwan University, Taipei, Taiwan 10617 3. Leung Center for Cosmology 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 entropic gravity scenario recently proposed by Erik Verlinde 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 entropy and its variation law in order to obtain a self-consistent approach to the problem. Next we argue that when dealing with quantum gravity issues the generalized uncertainty principle (GUP) should be the more appropriate foundation. Indeed based on GUP it has been demonstrated that the black hole 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 force 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 thermodynamics 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 entropic force analogous to the restoring force of a tation of the its temperature as the thermal temperature stretched elastic polymer 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 holographic principle interior. Namely, the correlation between the degrees [14{16] and the equipartition theorem 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- yElectronic address: [email protected] gested the sub-leading terms of force law rather than an Work supported by US Department of Energy 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 string 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 spacetime 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 Planck length. 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 work 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.