JHEP04(2011)029 Springer April 7, 2011 : March 19, 2011 October 22, 2010 : : Published Accepted 10.1007/JHEP04(2011)029 Received doi: Published for SISSA by 1001.0785 Gauge- correspondence, Models of Quantum Gravity Starting from first principles and general assumptions we present a heuristic [email protected] Institute for , UniversityValckenierstraat 65, of 1018 Amsterdam, XE, Amsterdam, The E-mail: Open Access auggests that it is actually the law ofKeywords: inertia whose origin is entropic. ArXiv ePrint: argument that shows that Newton’s lawspace of emerges gravitation through naturally a arises holographic incaused scenario. a by theory changes Gravity in is in which identifiedrelativistic the with information generalization an associated of entropic with the force tions. the presented positions arguments When directly of space leads material is to bodies. emergent the even A Einstein Newton’s equa- law of inertia needs to be explained. The Abstract: Erik Verlinde On the origin of gravity and the laws of Newton JHEP04(2011)029 So far, 1 19 21 15 14 12 11 15 18 9 20 – 1 – 5 10 ]. 7 – 1 23 20 8 6 24 3 1 Gravity dominates at large distances, but is very weak at small scales. In fact, its basic An incomplete list of references includes [ 6.1 The end6.2 of gravity as Implications a for6.3 fundamental force theory and Black relation hole6.4 with horizons AdS/CFT revisited Final comments 5.1 The law5.2 of inertia and Towards a the5.3 derivation equivalence of principle the The Einstein force equations on a collection of particles at arbitrary locations 4.1 The Poisson4.2 equation for general The matter gravitational distributions force for arbitrary particle locations 3.1 Force and3.2 inertia Newton’s law3.3 of gravity Naturalness and3.4 robustness of the Inertia derivation and the Newton potential 1 laws have only beenconsiderably tested harder up to to combinequest distances with for of quantum unification the mechanics of ordermay than gravity therefore of all with not a the be these millimeter. the other other right forces. forces approach. Gravity of It is The is Nature, also known at to lead a to microscopic many level, problems, paradoxes enced by everything that carriesof an space-time. energy, and is The intimatelybasic universal connected equations nature with closely the of resemble structure the gravitythere laws is has of not also thermodynamics been and demonstrated a hydrodynamics. by clear the explanation fact for this that resemblance. its 1 Introduction Of all forces of Nature gravity is clearly the most universal. Gravity influences and is influ- 5 The equivalence principle and the Einstein equations 6 Conclusion and discussion 4 Emergent gravity for general matter distributions 3 of the laws of Newton Contents 1 Introduction 2 JHEP04(2011)029 ] and the AdS/CFT correspon- 3 , 1 ]. The strongest supporting evidence for the 9 , – 2 – 8 ]. These theoretical developments indicate that at least part of the microscopic 10 In this paper we present a holographic scenario for the emergence of space and address Usually holography is studied in relativistic contexts. However, the gravitational force The concept of holography appears to be much more general, however. For instance, The most important assumption will be that the information associated with a part The universality of gravity suggests that its emergence should be understood from Many physicists believe that gravity, and space-time geometry are emergent. Also explained again. the origins of gravity anding inertia, from which are first connected principles,temperature, by using the it only equivalence is principle. shown space that Start- independent Newton’s concepts laws appear like naturally energy, and entropy practically and unavoidably. is also present intherefore the also non-relativistic naturally world. explain whylaw The this of force origin gravitation. appears of the gravity,to In way whatever be it fact, it does, re-derived, when is, and because space should obeysforce standard is Newton are concepts emergent, far like also from position,alongside obvious. the velocity, acceleration, with Hence, other mass space in laws and itself. such of a Even Newton a setting have basic the concept laws like of inertia mechanics is have to not appear given, and needs to be space. In allredshifts, these and cases related thecorrect to emergence there of a should exist the coarse awith holographic graining general gravity. direction framework procedure. that is describes accompanied how If by space all emerges together these combined ideas are in the AdS/CFTholographic correspondence version of one the renormalization group. canideas Similarly, move in that black the the hole physics information boundary thereabout can exist be inwards accelerated stored by observers, on one exploiting stretched can horizons. a in Furthermore, principle by thinking locate holographic screens anywhere in of space obeys theholographic principle [ comes fromdence black [ hole physics [ degrees of freedom can be representedor holographically on either horizons. on the boundary of space-time its location, in whateverof form entropy. the microscopic Changes theoryOur in likes aim this to is entropy have to when it,of show matter gravity. measured that in is this terms displaced force, leads given certain to reasonable a assumptions, reaction takes force. the form gravity and those thatemerge don’t. from It a therefore microscopic description provides that evidence doesn’t for know the aboutgeneral fact principles its that that existence. gravity are independent can theory. of the specific In details this ofinformation. the paper underlying More we microscopic precisely, will it argue is that the the amount of central information notion associated needed with to matter derive and gravity is we still have to figure out what the stringstring theoretic theory solution and teaches its us. relatedParticularly developments important have clues given come several fromstring indications the correspondence. in AdS/CFT, This this or correspondence direction. more leads to generally, a the duality open/closed between theories that contain and puzzles. has to a certain extent solved some of these, but not all. And JHEP04(2011)029 3 -axis. x we extend 5 . For definiteness, 1 . In section 4 , as shown in figure F – 3 – with an exposition of the concept of entropic force. Section . 2 6 By using tweezers one can pull the endpoints of the polymer apart, and bring it out of Perhaps the best known example is the elasticity of a polymer. A single polymer We start in section The holographic principle has not been easy to extract from the laws of Newton and The presented ideas are consistent with our knowledge of string theory, but if correct A crucial ingredient is that only a finite number of degrees of freedom are associated statistical tendency to returnforce, in to this a case maximal the elastic entropy force. state translatesits into equilibrium a configuration by macroscopic anwe external keep force one end fixed, say at the origin, and move the other endpoint along the monomer can freely rotate arounddirection. the Each points of of attachment theseis and configurations direct immersed has itself the into in same a anysince energy. spatial heat these When bath, are the entropically it polymermolecule favored. likes molecule is to There short are put compared many itself to more into when such a it configurations randomly is when stretched coiled the into configuration an extended configuration. The force. Entropic forcesphysics. occur Big typically colloid infor molecules macroscopic instance, suspended experience systems entropic in such forcesphenomenon a as due driven to thermal in by excluded environment an colloid volume of entropic effects. or force. smaller Osmosis bio- is particles, another molecule can be modeled by joining together many monomers of fixed length, where each An entropic force isdegrees an of effective freedom macroscopic by forceis the that expressed statistical originates in tendency in terms to a ofscopic increase system entropy dynamics. differences, its with and entropy. many In is The particular, independent force of there equation the is details no of fundamental the field micro- associated with an entropic these results to theare relativistic presented case, in section and derive the Einstein equations. The2 conclusions Entropic force tion of gravity as beingemergent due scenario. to the exchange of closed strings hasillustrates to the be explained main from heuristicization an argument to arbitrary in matter a distributions simple is non explained relativistic in section setting. Its general- Einstein, and is deeply hiddenthat within these them. well known Conversely, laws starting comeof from out. gravity holography, to we By find holography, reversing one the logic obtains that a lead simpler people picture from ofthey the should what laws have gravity important is. implications for this theory as well. In particular, the descrip- with a given spatial volume,equivalent to as the dictated matter, by is theto distributed holographic a evenly principle. temperature. over the The Thedisplacement degrees energy, product of of that of freedom, matter is and the is thusIn temperature shown leads this to and way be the we change equal find in to that entropy the Newton’s due law work to of done the gravity by emerges the in gravitational a force. surprisingly simple fashion. JHEP04(2011)029 (2.2) (2.3) 2 and pulled out T required to keep F . . T x B /k ) . can be deduced from the saddle i . F x of the heat bath and the position x ) (2.1) E + h ∂x ∂S E E T ( E, x − B = e k ) · T F should be equal to the entropic force, that log Ω( E, x that depends on of the polymer. The force F B ) denotes the volume of the configuration space const k Ω( is introduced in the partition function E – 4 – x . The entropic force points the other way. , F F E, x ∼ − ) = ∂S ∂E dEdx Z E, x = dependence is entirely a configurational effect: there is ( S 1 T polymer x for a given temperature ) = F x T,F ( Z is Boltzman’s constant and Ω( . A free jointed polymer is immersed in a heat bath with temperature B k It is interesting to study the energy and entropy balance when one gradually lets the In the canonical ensemble the force We like to thank B. Nienhuis and M. Shigemori for enlightening discussions on the following part. 2 of the second endpoint. The external system. By energybeen conservation, extracted this from work the mustFor be heat an equal bath. infinite to heat The the baththe entropy energy this polymer. of that would the Hence, has be in heat by this bath the situation will same the hence amount total be as entropy reduced. the will remain increase constant. in entropy of This example makes cleartive, that at at least a when macroscopicmicroscopic the level meaning, temperature an however, is and entropic kept is force constant. emergent. can The be corresponding conserva- potentialpolymer has return no to its equilibrium position, while allowing the force to perform work on an tries to restore theby polymer the facts to that itsproportional it equilibrium to points the position. in temperature. thelaw An For direction the entropic of polymer force the increasing force is entropy, can and, recognized be secondly, shown that to it obey is Hooke’s the polymer at a fixedpoint length equations By the balance of forces, the external force as an external variable dual to the length where for the entire system asx a function of theno total microscopic energy contribution to the energy Figure 1 of its equilibrium state by an external force The entropy equals JHEP04(2011)029 ), and imposes that the F x, x ). However, it illustrates that + 2.3 E ) = 0 (2.4) F x, x – 5 – + and examines the balance of forces. Specifically, E ] ( F S 16 d dx can be viewed as the energy that was put into the system F x In the following our discussion of the polymer is used as a metaphor to Another example of an emergent adiabatic force is the reaction force of a fast system This can be studied in more detail in the micro-canonical ensemble, because it takes the impossible to localize a particlefact, with points infinite and precision continuous atOne coordinates a could assume point should that of eventually information a arise ismodel). continuum stored only space. in But as points if In of derived all aobtain concepts. the discretized associated space a (like information holographic in would description, a be lattice and without in duplication, fact, one one would would not not recover gravity. 3 Emergence of the lawsSpace of is Newton in theof first particles. place Space amatter. device is Given introduced therefore that to literally the describe a maximal allowed the storage information positions space is and for finite movements information for each associated part with of space, it is energies below the given stateas remains constant. the This entropic leadsconcept force to exactly with in a that a similar expression ofbetween thermodynamic adiabatic an situation. reaction entropic forces force.be with In explained entropic this A in forces, a more paper and future detailed the we publication relevance explanation [ will for of merge gravity, the will this connection information. Specifically, one mayof think the about phase this space amount occupied of by information the as microscopic the states. when volume it is influenced bythis a slowly force varying system. can In be the Born-Oppenheimer obtained approximation by requiring that the phase space volume of all states with force. Our aim is tosee argue that that gravity the is also same anparticular, kind entropic the force of notions in of reasonings this entropy sense. apply andas Indeed, to temperature we a used gravity, will in with way this some paperdegrees of slight should of modifications. be characterizing freedom interpreted the and In the amount energy of costs that information is associated associated with with changes the in this microscopic amount of the total energy isbut reduced that when the entropy the stays polymer the slowly same. returns InNote to this added. sense its the equilibrium forceillustrate position, acts that adiabatically. changes in the amount of information, measured by entropy, can lead to a One easily verifies that thismicro-canonically leads the to the temperature sameenergy is equations dependent. in ( The term generalby position pulling the dependent, polymer out and of the its equilibrium force position. also This equation tells us therefore that total energy into account includingone that again of introduces the heat an bath.one external force considers To determine the the micro-canonical entropicentropy ensemble force, is given extremal. by Ω( This gives JHEP04(2011)029 (3.1) . ~ mc = x when ∆ – 6 – . B 2 πk = 2 S ∆ that approaches it from the side at which space time has already emerged. attached to a fictitious “string” that is lowered towards a . Just be- m ]from which he obtained his famous entropy formula. He considered a particle with 1 m Motivated by Bekenstein’s argument, let us postulate that the change of entropy as- We want to mimic this reasoning not near a black hole horizon, but in flat non- Let us also assume that (like in AdS/CFT) there is one special direction corresponding Thus we are going to assume that information is stored on surfaces, or screens. Screens The situation is depicted in figure sociated with the information on the boundary equals such as theformulated positions in of term particles. ofthat microscopic we degrees The can of imagine physics freedom. thatseparates of the its The two associated the region. holographic information other principle Eventuallyfreedom, can the but tells side be particle before us mapped it merges of does with on so the these to it microscopic already the screen degrees influences screen the is of that amount of still its associated information. led him to his area law for the blackrelativistic hole space. entropy. Soof we mass consider aConcretely small this piece means that of on a one holographic side screen, the and physics is a described particle in macroscopic variables fore the horizon theof particle the is black dropped hole in.gas can of Due be particles, to made this the factBekenstein arbitrarily infinite would solved small, redshift lead this the classically. to by mass problems Ifthe arguing increase with horizon, one that the it would when second is take a lawand considered a of particle horizon to thermodynamics. thermal is area be by one part a Compton of small wavelength the amount, from black which he hole. identified Therefore, with it one increases bit of the information. mass This 3.1 Force and inertia Our starting assumptionment is [ directly motivatedmass by Bekenstein’s original thought experi- from which space is derived.notion We will of assume time, that the andenergy, microscopic its and theory by dynamics has employing a is techniques wellbasic time of defined ingredients statistical translation together physics, invariant. with temperature. This the These entropy allows will associated one be with to the the define amount of information. or anything familiar. The microscopic details are irrelevant forto us. scale or a coarse graining variablespace of is the microscopic emergent. theory. ThisOn So is the one the direction side in screens which we that imagethe store that other we the side are information of using are the the like variables screen associated stretched we with horizons. describe space everything and still time. in On terms of the microscopic data separate points, and in thisthat way move are from the one naturallocation side place particles to to store is the information stored other.is about in particles given Thus discrete we by bits imagineinformation some on that that the unknown is this screens. stored rules, information on about The it. which the dynamics Hence, can it on be does each not thought screen have to of be as given by a a local way field of theory, processing the JHEP04(2011)029 (3.5) (3.3) (3.4) (3.2) ). was chosen to be of the 3.3 S , let us imagine splitting the m ) and ( ) for ∆ 3.2 3.2 x. , ∆ a S. c ~ ~ ∆ mc π 1 T B 2 . Since entropy and mass are both additive, ma. x = πk = = – 7 – x , will become apparent soon. Let us rewrite this T F π ∆ = 2 B . F k x ) together with ( S . It is picked precisely in such a way that one recovers π ∆ 3.4 . A particle with mass approaches a part of the holographic screen. The screen bounds denotes the acceleration. Let us take this as the temperature associated with the a bits on the screen.given Now form, it including is the factor clearthe of why 2 second the law equation of ( Newton as is easily verified by combining ( related. Namely, astemperature Unruh showed, an observer in an accelerated framewhere experiences a This is the entropicnon force. vanishing temperature. Thus, Fromacceleration. Newton’s in law order we Of to know course, that have it a a force is non leads well zero to known a force, that non we zero acceleration need and to temperature have are a closely associated change in entropyit after a is shift natural ∆ todoes expect force that arise? the The entropy basicmembrane. change idea When is should a to be particle use proportional has thethe an analogy to membrane with entropic carries the reason osmosis a mass. across to temperature, a be How semi-permeable on it one will side experience of an the effective membrane force and equal to screen is linear in the displacement ∆ To understand why itparticle is into also two proportional or to more the lighter mass sub-particles. Each sub-particle then carries its own the emerged part ofspace space, that which has contains not yet the emerged, particle, as and well stores as data some part thatThe of reason describe the for the emerged putting part space. informula of in the the factor slightly of more 2 general form by assuming that the change in entropy near the Figure 2 JHEP04(2011)029 . G (3.7) (3.8) (3.6) ) for the 3.2 that is required T . The temperature is N ), since we don’t need it. It only 3.4 T . . and inserts the expression for the number 2 3 2 B . It is natural to assume that this number ~ E N G . Next one uses the postulate ( Ac πR Nk Mc A 1 2 = – 8 – = = 4 = N E present in the system. Let us now just make the A and . Eventually this constant is going to be identified E ). The key statement is simply that we need to have G E . And not as usual, as the temperature caused by M a 3.2 . In fact, in a theory of emergent space this how area may be A in terms of T ) should be read as a formula for the temperature 3.4 . Even though the mass is not directly visible in the emerged space, its presence 3 represents the mass that would emerge in the part of space enclosed by the screen, M The rest is straightforward: one eliminates Suppose there is a total energy One can think about the boundary as a storage device for information.Let Assuming us denote the number of used bits by Equation ( is noticed though the energy that is distributed ofof the bits the to screen. determine change of entropy to determine the force. Finally one inserts Here see figure then determined by the equipartition rule as the average energy per bit. After this we need only one more equation. It is: existence a gravitational force, oneSo, can the simply only regard assumption thisNothing made equation more. here as is the that definition the of number of bits issimple proportional assumption to that the the area. energy is divided evenly over the bits will be proportional to the area. So we write where we introduced a newwith Newton’s constant constant, of course. But since we have not assumed anything yet about the that the holographic principleis holds, proportional the to maximal the storage area defined: space, each or fundamental total bit number occupies of by definition bits, one unit cell. specifically, let us assumethe it following is it is a best sphereserved to as with forget a already about further emerged the motivationa for Unruh space temperature ( law on in ( the order towe outside. have need a to For force. know where Since the we want temperature to comes understand from. the origin of gravity, to cause anan acceleration acceleration. equal to 3.2 Newton’s law ofNow gravity suppose our boundary is not infinitely extended, but forms a closed surface. More JHEP04(2011)029 still (3.9) S that would emerge in the part M . This fact could therefore be true even . 2 R E/A Mm G – 9 – to get the right identification with Newton’s constant. = 2 3 F − − d d 1 2 near a spherical holographic screen. The energy is evenly m with the same conclusion. But how robust and natural are these 3 ) includes a factor 3.6 . A particle with mass dimensions ( d Perhaps the least obvious assumption is equipartition, which in general holds only for These equations do not just come out by accident. It had to work, partly for dimen- In 3 of freedom according torandomly chosen some among non all trivialto bits, be distribution one proportional function. expects to the thewhen When energy equipartition energy per the is change unit not lost associated area strictly with bits obeyed. ∆ are resulting force takes theto form arbitrary of dimensions Newton’s law.heuristic In arguments? fact, this reasoning can be generalized free systems. But how essential is it? Energy usually spreads over the microscopic degrees 3.3 Naturalness and robustnessOur of starting the point was derivation thatingredients space were has that one emergent (i)number holographic of there direction. degrees is of The freedom additional a areis proportional change evenly to of distributed the area entropy over of in these the screen, degrees the and of emergent (iii) freedom. the direction energy (ii) After that the it is unavoidable that the that lead to black holereversed thermodynamics these and arguments. the holographicorigin But principle. of the gravity: In a logic it sense is isnot we an been clearly have entropic made force! different, before. That and If is sheds the true, new main this statement, light should which have on is profound the new consequences. and has We have recovered Newton’s law of gravitation, practically from firstsional principles! reasons, and also because the laws of Newton have been ingredients in the steps Figure 3 distributed over the occupied bits,of and space is surrounded equivalent by to the the screen. mass and one obtains the familiar law: JHEP04(2011)029 , ~ is 2 ) ~ /R (3.11) (3.10) 0 R while the screen R when it approaches the m . That is all there is to it. ), we can express the entropy change . x x . In a certain way it seems more natural T. 2 ), is equal to the physical acceleration 3.4 follows from , while in our identifications the entropy c ∆ B m 2 a n a 3.4 n k B 1 2 ) is exactly valid. The main content of this k = – 10 – = in this non relativistic context? It was necessary 3.1 2 c S ), and use ( n mc ∆ 3.2 ) is the basic assumption from which everything else follows, and the displacement ∆ eventually drops out of the most important formulas. So, in , the number of bits variable that is needed for dimensional reasons. It can therefore into an energy, which provides the heat bath required for the ~ m T 3.1 B M k 2 1 . The number of bits on the screen is multiplied by a factor ( is related to the scalar quantity auxiliary < R is related to the vector quantity x 0 ∆ T R S/ So let reconsider what happens to the particle with mass If we would move further away from the screen, the change in entropy will in general Since the postulate ( Why do we need the speed of light . In fact, so far our discussion was quasi static, so we have not determined yet how to , that was basically introduced by hand in ( ¨ When we insert this intoin equation terms ( of the acceleration as to have it the other way around. screen. Here it shouldhence merge it with will the bebit microscopic made carries up degrees an out of energy of freedom the on same the bits screen, as and those that live on the screen. Since each a x connect space at different times.temperature In fact, it may appear somewhatgradient counter-intuitive ∆ that the small area, but spreads overhow the it screen. is The distributed, next even section for contains general a matter3.4 proposal configurations. for precisely Inertia andTo the complete Newton the potential derivation of the laws of Newton we have to understand why the symbol no longer be given by theis same moved rule. to Suppose thewhile particle stays the at temperature radius is dividedmultiplied by by the that same factor, factor. andthis since Effectively, situation this it the means drops that information out, only the associated resulting with force the will particle stay the is same. no In longer concentrated in a just serves as an be chosen at will,equation is and therefore defined simply soproportional that to that there the ( is mass an entropy change perpendicular to the screen let us discuss its meaningwhen one in shifts more by detail. oneintroduce Why Compton Planck’s does wave constant length? the in entropy the Inlaws precisely first fact, place, of change one since like may Newton. the this wonder only whyprinciple Indeed, aim we one was needed could to to multiply derive it the with classical any constant and still obtain the same result. Hence, to translate the mass entropic force. In the non-relativistichas setting to take this into heat account bathits that is the location infinite, heat under but bath the in loses influenceas principle or of we one gains an will energy external when see. force. the particle This changes will lead to relativistic redshifts, JHEP04(2011)029 (3.12) as the entropic ma = F . 2 c . 2 ∆Φ associated with the particle, we succeeded Φ B n k −∇ − – 11 – = = a S n ∆ from the equations, which in view of our earlier comment is ~ We start from microscopic information. It is assumed to be stored on holographic Therefore, space can not just emerge by itself. It has to be endowed by a book keeping Thus we conclude that acceleration is related to an entropy gradient. This will be one Φ. And the resulting entropic force is calledscreens. gravity. Note that information hasby a coarse natural graining, inclusion property: oneachieved by reduces through forgetting the certain averaging, bits, amount a of block information. spin transformation, This integrating coarse out, graining can or be some other details of the information,by nor the the exact entropy, dynamics, andfunction only the of the the energy amount location that of of information is the given matter associated distribution, with itdevice it. will that lead keeps to If track anturns the of entropic out, force. the entropy that in amount changes a of as non information a relativistic for situation this a device given is energy provided by distribution. Newton’s potential It phase space is forgotten.configuration, Hence, which there measures will the amount be of ato microscopic the finite information macroscopic that entropy is observer. associated made with In invisible matter. each general, matter this The amount microscopic will dependfrom dynamics on a by the macroscopic which distribution of point this the of information view. is processed Fortunately, to looks determine random the force we don’t need the 4 Emergent gravity forSpace general emerges matter at distributions a macroscopicout level a only large after number coarse of graining. microscopic One degrees forgets of or freedom. integrates A certain part of the microscopic variable, like the (renormalizationwe group) propose scale a in AdS/CFT. holographictential Indeed, precisely scenario in plays for that the the role.distributions next emergence and section This arbitrary of allows positions space us inpresented to in arguments. a generalize which natural our way, the discussion and Newton to give additional other po- support mass for the This allows us to express the change in entropy in the concise way We thus reach thedepletion important of conclusion the that entropy per the bit. Newton It potential is Φ therefore natural keeps to track identify of it the with a coarse graining of our main principles:in inertia rest is because a consequence therethe of are Newton the no potential fact entropy Φ that gradients. and a write particle Given the in this acceleration rest fact as will it a stay gradient is natural to introduce force. But, by introducing thein number making of the bits identifications moreIn natural fact, in we terms have of eliminated a their good scalar thing. versus vector character. By combining the above equations one of course again recovers JHEP04(2011)029 ). 3.12 1. − = 2 , as can be seen from ( 2 /c /c – 12 – . 4 is a dimensionless number that is always between zero and one, and Φ of the Newton potential. In other words, the holographic screens 2 ∇ /c 2Φ − . The holographic screens are located at equipotential surfaces. The information on the We want to determine the gravitational force by using virtual displacements, and The amount of coarse graining is measured by the ratio Φ A priori there is no preferred holographic direction in flat space. However, this is where The coarse grained data live on smaller screens obtained by moving the first screen distribution in space. All microscopicto states the that same lead macroscopic to state. theof same The microscopic mass entropy for states distribution each belong that of flow these to state is the defined same as macroscopiccalculating the state. number the associated change in energy. So, let us freeze time and keep all the matter where all bitsat have black been hole maximally horizons. coarse grained.4.1 Thus the The foliation Poisson naturally equationConsider stops for a general microscopic matter state, distributions which after coarse graining corresponds to a given mass correspond to equipotential surfaces.that This leads screens to may a break wellspace. defined up This foliation into of is disconnected space, depicted except parts in figure that each encloseNote different that regions of is only equal to one on the horizon of a black hole. We interpret this as the point a “coarse graining” variable, something like the cut-off scalewe of use our the observation system about on thethe Newton the amount potential. screens. of It coarse is graining theby on natural variable the the that screens. measures gradient Therefore, the holographic direction is given further into the interior ofis the replaced space. by the Thenested emerged information or part that foliated description of is of space removed spacewords, by by between having just coarse the surfaces like contained graining two within in screens. surfaces. AdS/CFT, In there In other is this one way emerging one direction gets in a space that corresponds to screens is coarsemaximum grained coarse in graining the happens at direction black of hole horizons, decreasing when values 2Φ ofrenormalization the group procedure. Newton At potential eachof step Φ. the one obtains original The a microscopic further data. coarse grained version Figure 4 JHEP04(2011)029 (4.2) (4.3) (4.4) (4.5) (4.1) ) is con- x ( ρ ). Our aim is to ~r ( ρ . After inserting our M ). It is takes the form of 3.7 . ) . dA. ~r ( · ), namely by Φ Φ kc ∇ T dN. dA. 3.4 should change accordingly. This condition ~ ∇ S πG ρ ~ , for example, by first putting it close to the 3 S π Z c 1 S Z G 2 M B k ) is generalized to = – 13 – = ) = 4 1 1 2 πG ~r 4 T 3.6 = corresponding to an equipotential surface with fixed dN B Φ( = 2 k S E ∇ M . i ~r . We assume that the entire mass distribution given by 0 and positions is again expressed in terms of the total enclosed mass i m E The next ingredient is the density of bits on the screen. We again assume that these First let us identify the temperature. We do this by taking a test particle and moving We choose a holographic screen can only hold in general if the potential Φ satisfies the Poisson equation We conclude that by making natural identificationsdensity for on the temperature the and holographic the information screens, that the laws of gravity come out in a straightforward This should hold for arbitraryis screens added given to by the equipotential region surfaces.screen enclosed and by When the then a screen pushing bit of it mass across, the mass It is an amusingenergy exercise to workidentifications for out the the left consequence hand side of one this obtains a relation. familiar relation: Of Gauss’s course, law! the Now let us imposean the integral analogue expression of for the the equipartition energy relation ( Here the derivative is takenNote in at the this direction point ofwe the Φ don’t outward is know pointing just yet normal introduced whether to it as the satisfies a screen. an device equation tobits that describe are relates the uniformly it distributed, to local and the acceleration, mass so but distribution. ( it close to the screen,discussion and we measuring define the temperature local analogous acceleration. to Thus, ( motivated by our earlier volume. To explain theis force performed on by the the particles,multiplied force we by and the again temperature. show need thatthe to The it temperature determine difference is on with the the the naturally workeral spherically screen that written not symmetric is as case in not the is equilibrium. necessarily change that unit constant. Nevertheless, in area. Indeed, one entropy the can situation locally is define in temperature gen- and entropy per obtain the force thatmasses the matter distribution exerts on a collectionNewton of potential test Φ particles with tained inside the volume enclosed by the screen, and all test particles are outside this at fixed locations. Hence, it is described by a static matter density JHEP04(2011)029 i ) in δ~r ) to 3.2 (4.6) δs 3.2 in the already i ~r of the positions i δ~r 3

m 1 ! r r 1 δ

) (4.7) m

3 i ) that the Newton potential ! ~r F

1 − ! F 3.12

~r ( δ i ·∇ i dN δ~r 2 i are located at arbitrary points Φ c δ i 2 m due to gravity are determined by the virtual work m B i i k . We noted in ( X ~ – 14 – F S = 2 πG m of the particles. δs i

2 ! F δ~r

Φ is determined by solving the variation of the Poisson ) = 4 δ ~r 0 Φ( Φ δ 2 = ∇ Φ . A general mass distribution inside the not yet emerged part of space enclosed by the Φ is the response of the Newton potential due to the shifts δ Let us now determine the entropic forces on the particles. The combined work done One can verify thatwhen with one this of identification the one particles indeed approaches reproduces the the screen. entropyby shift all ( of the forcesHowever, on we the need test particles to is express determined by in the terms first law of of the thermodynamics. local temperature and entropy variation. where of the particles. Toequation be specific, the entropy density locally onΦ the keeps screen track of thethe changes of change information of per entropy unit density bit. is Hence, the right identification for 4.2 The gravitational forceThe for next issue arbitrary is particle to locations obtainpoints the outside force the acting on screen. matterthis For particles situation. this that What are we located is need at theof a arbitrary entropy the generalization change particles? due of to There the arbitrary is first infinitesimal only postulate displacements one ( natural choice here. We want to find the change fashion. Note that tothe obtain screen Newtonian is gravity one uniform, has andof to that these assume assumption the that leads energy the to follows bit from another density an form on of equipartition. gravity. Changing either Figure 5 screen. A collection of testemerged particles space with outside masses the screen.done after The infinitesimal forces displacement JHEP04(2011)029 , 0 i as ~ F (4.8) (4.9) M . To see the emergence a ξ dA  Φ δ ∇ Φ − T δs Φ S ∇ Z Φ δ = i in our arguments, it is a logical step to try S without changing the forces on the particles. Z c δ~r – 15 – · S i 1 πG ~ F 4 Φ is sourced by only the particles outside the screen, δ i = X i δ~r · i ~ F i X The forces we obtained are independent of the choice of the location of the screen. We 5.1 The law ofConsider a inertia static and background with the aof global equivalence time inertia principle like and Killing vector thefield equivalence with principle, the one temperature has andusual to the geodesic relate entropy motion gradients. the of particles choice In can of particular, be understood we this like as Killing being to vector the see result that of the an entropic force. geometries, since a prioricurvatures there would are not no be preferred present.relativity choices emerges Specifically, of from we similar coordinates, would reasoningsthat like nor as this to a in is see reason the possible. how why previous Einstein’s But section. general first We we will study indeed the show origin of inertia and the equivalence principle. Since we made useand generalize of our the discussion speedscopic to theory of a knows relativistic light about situation.symmetry. Lorentz symmetry, So This or means let even we us has havewith assume the to emergent that space-time combine Poincar´egroup as quite the time naturally a micro- and leads global space to general into coordinate one invariance geometry. and curved A scenario consistent with idea thatscale a that particular controls the locationin coarse corresponds particular graining to the of an forces, the should arbitrary microscopic be choice data. independent of The of the that macroscopic choice. physics, 5 The equivalence principle and the Einstein equations the remaining integral just gives zero. could have chosen any equipotentialthe surface, ones and described we by woulddimensional the obtain analysis. laws the The of same invariance under values Newton. the for choice That of equipotential all surface of is this very much works is not just a matter of which holds for anyverified location by of using the Stokeswhen screen the theorem outside screen and is the theand chosen mass pull Laplace at distribution. it a equation. out equipotential of This surface. the The is integral. To second easily Since see term this, simply vanishes replace Φ by Φ The variation of theLaplacian. Newton The potential rest can of be theapplied proof obtained to is gravity. from a The the straightforward basic application Greens identity of function one electrostatics, for needs but to the then prove is To see that thishas indeed to gives the use gravitational thea force electrostatic in mass analogy. the surface Namely, most density one general over can case, the redistribute one the screen simply entire mass Hence, JHEP04(2011)029 (5.2) (5.3) (5.1) ], 11 in mind, since that can be expressed explicit. 2 b ~ u a ∇ and a is measured with respect u G . T ≡ b b ξ To make this explicit, let us a a to define a foliation of space, 5 ∇ φ a ξ φ 2 . − . ) e . In a general relativistic setting force φ. b a b ξ ξ m = 0 = a ∇ φ. b ξ a b b ξ − b N φ ∇ −∇ e and to log( π + – 16 – = ~ 2 S 1 2 b b ξ a a = = ∇ , a , because the temperature T φ b φ ξ e φ as − are put equal to one, but we will keep b e on the screen is now in analogy with the non relativistic B ξ k = T = 0, which we will take to be at infinity. b the natural generalization of Newton’s potential is [ u φ of the particle and its acceleration and 4 c a ]. . One finds that the acceleration can again be simply expressed the u 11 φ normal to the screen b represents the redshift factor that relates the local time coordinate to that N φ e . So we can turn it into a scalar quantity by contracting it with a unit outward S To find the force on a particle that is located very close to the screen, we first use The local temperature The four velocity Just like in the non relativistic case, we like to use We want to show that the redshift perpendicular to the screen can be understood In In this subsection and the next we essentiallyIn this follow entire Wald’s section book it on will general be relativity very useful (pg. to 288–290). keep the polymer example of section 4 5 again the same postulate as in section two. Namely, we assumeWe use that a the notation change in of which entropy will make the logic of our reasoning very clear. situation defined by Here we inserted a redshiftto factor the reference point at infinity. Note that just likescreen in the nonpointing vector relativistic situation the acceleration is perpendicular to and the definition of gradient in terms of the Killing vector We can further rewrite the last equation by making use of the Killing equation consider the force thatis acts on less a clearly particle defined,mation. of since mass it But can by be usingthe transformed the concept away of time-like by force Killing a [ vector general one coordinate can transfor- given an invariant meaning to and put our holographicsince screens in at this surfaces case of themicroscopic constant entire data redshift. screen on uses This the the is screen same a can time natural be coordinate. choice, done So usingmicroscopically the signals processing as that of originating travel the from without time the delay. entropy gradients. Its exponent at a reference point with JHEP04(2011)029 3.3 (5.6) (5.4) (5.5) , should be ) is entirely force acting is due to the x 2 ( φ that contains a e φ force statistically E microscopic entropic ). This fixes the equilibrium ) 5.5 . 5.3 φ ) in a microcanonical form. Let a = 0 , ) and the 5.5 ∇  a φ x a . The additional factor ( N φ me ~ m ma − m, x π ) = also includes the energy of the particle. The 2 x = ( − φ F E S e – 17 – a = + ∇ S E ) for the polymer, discussed in section T a ) by making use of the identifications of the tempera- ∇ S . Here = , an entropic force can be determined micro-canonically 2.4 a a 5.6 a 2 x d F dx force, parametrized by at position has again dropped out. ~ for a displacement by one Compton wavelength normal to the screen. m ) tells us that the entropy remains constant if we move the particle and π external 5.6 ) be the total entropy associated with a system with total energy a We have arrived at equation ( But it is important to realize that the redshift must be seen as a consequence of the Equation ( As we explained in the section It is instructive to rewrite the force equation ( E, x ( entropy gradient and notredshifts the can other be way interpretedfield around. in or due The the to equivalence emergent thein principle space fact the tells that time relativistic one us as considers setting, that both either an but emergent accelerated due phenomena. neither frame. to view Both a views is gravitational are microscopic. equivalent Acceleration and gravity are ture and entropy variations in spaceWe time. should But have actually started we fromin should terms have the gone of microscopics the them. other and Wesomewhat way. defined chose contrived. not the to space follow dependent such a concepts presentation, since it might have appeared simultaneously reduce its energy byis the redshift very factor. light, This andprobe is does true of only not the when disturb the emergentfixed particle the geometry. by other the This other energy also matter distributions. means in the that It system. redshift simply function serves as a One easily verifies thatpoint this where leads the to thebalance same each equation other. ( here! Again we The stress analogy with theobvious now. equation point ( that there is no by adding by handthis an situation external this force condition term, looks and like impose that the entropy is extremal. For S particle with mass entropy will in generalin also this dependent discussion. on many other parameters, but we suppress these This is indeed theposition correct near gravitational the force screen,tivistic that as analogue is of measured Newton’s required from law toredshift. the of keep inertia reference Note a that point particle at at infinity. fixed It is the rela- Hence, where the minus sign comes fromoutside the to fact that the the inside. entropy increasesapply The when here we comments cross as made from well. the on The the entropic validity force of now this follows postulate from in ( section at the screen is 2 JHEP04(2011)029 ) 5.9 (5.7) (5.8) (5.9) (5.10) (5.11) which is a ξ a b R − = b ξ a ∇ a ) can alternatively be re- ∇ ) is indeed known to be the 5.9 d 5.9 ξ c ∇ dV dA b · abcd ξ  a φ b n ∇ we obtain ~ ab dx φ T dN e dA G R ∧ S a S Z Σ dN = Z Z as 1 2 dx – 18 – a S 1 = ξ dN πG and 1 Z πG . The particular combination of the stress energy 4 4 . This leads to an expression for the mass in terms ab T M a T 1 = ξ πG = 8 M M = M . Hence T 1 2 drops out, as expected. The equation ( is distributed over all the bits. Again by equipartition each bit carries a ~ M . We assume that it is enclosing a certain static mass configuration with total ). Following the same logic as before, let us assume that the energy associated φ . The bit density on the screen is again given by 4.2 M First we make use of the fact that the Komar mass ( An analogous question was addressed by Jacobson for the case of null screens. By of the Ricci tensor. From this identity one canthat get the to left an hand side integratedcomponents can form of be of stress expressed the energy as Einstein an tensor equations, integral by over noting the enclosed volume of certain Next one uses Stokesimplied theorem by and the subsequently Killing the equation relation for adapting his reasoning toKomar this mass, situation, it and isEinstein combining straightforward equations. it to We with will construct Wald’s present an exposition a argument of sketch of that the this. naturallyexpressed in leads terms to of the the Killing vector reasoning, however, we aretemperature coming and from the the number othersatisfies any side. of field bits We equations. on madeis The sufficient the identifications key to for screen. question derive the at the But this full point Einstein we equations. is don’t whether know the yet equation whether ( it Note that again natural generalization of Gauss’s lawprecisely to Komar’s General definition Relativity. Namely, of thestatic the right curved mass hand space side contained is time. inside an It arbitrary can volume be inside derived any by assuming the Einstein equations. In our mass unit equal to After inserting the identifications for as in ( with the mass We would like to extendobtain our the derivation Einstein of equations. theanalogous We fashion. laws will of present Let gravity us a to again sketchredshift consider of the a how relativistic holographic this case, screen can and mass on be closed surface done of in constant a very 5.2 Towards a derivation of the Einstein equations JHEP04(2011)029 ], is 7 a n (5.12) and S , this will effect i δx . But that still leaves a dV n b . By following the steps ξ a m n ab R Σ Z ) by performing independent steps . In particular also in its norm, the a 5.9 ξ 1 πG 4 dependence of the Komar mass that the i = dependences, simply due to changes in the x i x dV b ξ – 19 – a n . a  ξ ) directly from ( ab ) for all these Killing vectors, and for arbitrary screens. away from the screen and the mass distribution. The T g i of the particles. 5.12 2 1 x i 5.12 x − . By requiring that it holds for arbitrary screens implies that ab a T ξ  . This means that we can vary the normal Σ S Z 2 . If one moves the particles by a virtual displacement φ e Next let us assume that in addition to this The virtual displacements can be performed quasi-statically, which means that the To try and get to the Einstein equations we now use a similar reasoning as Jacobson [ We arrived at the equation ( entropy of the entire systemamount has of also information. explicit These arewish to de determine. difference that We will willis now lead give based to a on natural the a prescription entropic maximal for force entropy the that principle entropy we dependence and that indeed gives the right forces. Namely, assume Killing vector itself is stillIn present. fact, also Its the norm, spatialtry or metric to the may solve redshift be these factor, dependences, affectedthe may by since Einstein change, this that however. equations. displacement. would But bebe We are we impossible a not can due function going simply to of to the the use positions non the linearity fact of that the Komar mass is going to Komar definition of theKomar mass mass will depends again on be theredshift choice useful factor of for Killing this vector the purpose. definition of The the definition Komartemperature of mass on of the the the matter screen configuration will inside be the directly screen. affected In by fact, a the change in the redshift factor. making this sketchy derivation into a complete proof5.3 of the Einstein equations. The forceWe on close a this collection of sectionparticles particles with at at explaining arbitrary arbitrary the locations locations way the entropic force acts on a collection of like Minkowski space, we cana choose small approximate local time part likeKomar of Killing integral the vectors. will screen, jump Now anddescribed consider precisely impose above by that then the when leads corresponding to matterThis mass ( crosses kind of it, reasoning the should value be of sufficient the to obtain the full Einstein equations. It is worth boundary is given by a contraction with the Killing vector except now applied to time-likeand screens. look also Let at us consider very short a time very small scales. region Since of locally space every time geometry looks approximately on both the left anda the time right like hand Killing side.also vector It the is derived integrands for onthe a both general Einstein static equation. sides background are with In equal. fact, we This can gives choose us the surface only Σ a in certain many component ways, of as long as its servation laws of the tensorsrelation that occur in the integrals. This leads finally to thewhere integral Σ is theits three normal. dimensional volume bounded by the holographic screen tensor can be fixed by comparing properties on both sides, such as for instance the con- JHEP04(2011)029 i δx (5.13) ], which 12 ) of the positions i -th particle due to and in addition on x i ( M M  i , x ) i x ( M S = 0 (5.14) = S dependence in the entropy. The maximal  i i i ∇ x δx T + i – 20 – + , x M ) i i ∇ δx + i x ( M S . But since the Komar mass should be regarded as a function i Of course, Einstein’s geometric description of gravity is beautiful, and in a certain way Since the Komar integral is defined on the holographic screen, it is clear that like in the x , there will be an explicit and an implicit i in summarizing many aspects ofa a community, physical have problem. been Presumablybeing so this fundamental. reluctant explains But why to we, it givegeometry. as is up Einstein inevitable the tied we geometric these dowant formulation two to so. concepts of understand If together, one gravity gravity or and as is the both emergent, other have so at to is a be space more given time fundamental up level. if we still seen as a fundamentalsuch force. as thermodynamics The and similarities hydrodynamics, withanalogies. have other been known mostly It emergent regarded phenomena, is asfinally just time suggestive do we away with not gravity only as notice a fundamental the force. analogy,compelling. and Geometry talk appeals about to the the visual similarity, part but of our minds, and is amazingly powerful The ideas and resultsdiscuss presented and in attempt this to paper answer lead some of to these. many6.1 questions. First we In present The this our end main sectionGravity of conclusion. we has gravity given as many a hints fundamental of force being an emergent phenomenon, yet up to this day it is derivative gravity theories by makingis use perfectly of suitable Wald’s for Noether this charge purpose. formalism [ 6 Conclusion and discussion non relativistic case the forcenot is tried expressed to as make anwas this integral very representation over more general, this concrete. screen andEinstein as Finally, did equations. well. we not note We So really have that it use this should argument the be straightforward precise to form generalize of this the reasoning Komar to higher integral, or the of the Komar mass,microscopic reason simply for because the force.the of energy In energy of analogy the conservation. with heat theby bath. polymer, But redshifts the Its whose again, Komar dependence microscopic mass on thisdue represents origin the is to position lies entropic of not in effects. the the the other depletion particles is of caused the energy in the heat bath By working out this conditionone and finds singling out the equations implied by each variation The point is thatthe the mass first distribution term inside represents the the screen. force This that force acts is on indeed equal the to minus the derivative the x entropy principle implies that these two dependences should cancel. So we impose that the entropy may be written as a function of the Komar mass JHEP04(2011)029 ]. But we have added an important element 14 – 21 – ]. Also other authors have proposed that gravity has an entropic 7 The open/closed string and AdS/CFT correspondences are manifestations of the The presented arguments have admittedly been rather heuristic. One can not expect The derivation of the Einstein equations presented in this paper is analogous to previ- The results of this paper suggest gravity arises as an entropic force, once space and Our description was clearly motivated by the AdS/CFT correspondence. The gravi- UV/IR connection that isthat deeply short engrained and within long string distancedistance physics theory. can phenomenon This not that be connection seen clearly implies that as knows totally Newton’s about decoupled. constant short Gravity is is distanceString a physics, a long theory since measure invalidates it for the is the evident ”general number wisdom” of underlying microscopic the degrees Wilsonian of freedom. effective field If gravity is justis an seen entropic force, as then anwe what integral do does part know of this about string say dualitiespled between about theory, open closed that string string string can theory? theories theories notcorrespondence. that Gravity that be contain don’t. gravity taken and out A decou- just particularly like important that. example But is the AdS/CFT existing ideas and aremore supported by supporting several evidence pieces fromhole of physics. evidence. string In theory, the the following AdS/CFT we gather correspondence,6.2 and black Implications for string theory and relation with AdS/CFT entropy differences is now clear. It has to, becauseotherwise, this given is the what fact causesnot that motion! exist we to are begin enteringheuristic with. an level unknown The of territory profound reasoning. in nature which The of space assumptions these does we questions made in our have been view justifies natural: the they fit with that is new. Insteadwe of uncovered only what focussing is the on origin theWe of identified equations force that a and govern inertia cause, the inwhatever a a gravitational way mechanism, context field, defined, in for which and gravity. space athe is It emerging. consequence microscopic is of level. driven the by statistical differences The averaged in random reason entropy, dynamics why in at gravity has to keep track of energies as well as quences for cosmology. For instance,lead to the many way new redshifts insights. arise from entropy gradients could ous works, in particular [ or thermodynamic origin, see for instance [ atoms. We probably wouldand have closed been strings equally are resistant not in much accepting different, the but obvious. wetime just Gravity themselves have have not emerged. yetemergent got If used phenomena, the to this gravity the and shouldgravity idea. space have plays time important a can implications central indeed role. for be many explained It areas as would in be which especially interesting to investigate the conse- tational side of this dualitya is macroscopic usually emergent seen description, aswe which independently by understood defined. chance it we But happenedresist in as making our to being the view know the analogy it about with is elasticity dual before a using of stress situation tensors in a in which microscopic a we continuous theory would medium have without half developed a a gravity. century theory before for We knowing can’t about JHEP04(2011)029 – 22 – . The open and closed string cut offs are related by the 6 . The microscopic theory in a) is effectively described by a string theory consisting of The holographic scenario discussed in this paper has certainly be inspired by the way The AdS/CFT correspondence has an increasing number of applications to areas of The reason why the Wilsonian argument fails, is that it makes a too conservative freedom on these screensThe represented emerged as part open ofUV strings space cut defined off, is as with occupiedUV/IR shown a correspondence: by in cut closed pushing figure the off strings,cut open in which string off the are cut towards off UV. also the to defined IR, the with UV and forces a vice the closed versa. string The value of the cut offs is determined by the extra dimension. No clearThe explanation exists entropic of mechanism where described thissystems, gravitational and in force explain this comes the from. paper emergence of should gravity. be applicableholography works to in AdS/CFT these and open physical the closed string holographic correspondences. screens In can string language, be identified with D-, and the microscopic degrees of physics in which gravity is notused present as at tools a to fundamental level. calculatefails. physical Gravitational quantities techniques The in are latest a of regimeone these where doubts the that developments in microscopic is these description the situationsnot gravity application emerges in to only the as condensed an same matter effective theory. description. space No It as arises the microscopic theory, but in a holographic scenario with one tree level contributions due toresponsible the for exchange gravity. of Thisover close interaction all string is, states, quantum however, which contributions equivalentlygravity among represented of is other by the also are the supported open sum by string. string theory. In this sense the emergent nature of completely true, the macroscopic physics would be insensitive to the shortassumption distance physics. about thestring asymptotic theory growth the of numberout the of indeed high number energy leads of open to states string long at states range high is effects. such energies. that Their integrating In one them loop amplitudes are equivalent to the Figure 6 open and closed strings as shown in b). Both typestheory, of namely strings that are cut integrating off outterms in in short the the distance UV. effective degrees action, of most freedom of only which become generate irrelevant local at low energies. If that were JHEP04(2011)029 are 3 – 23 – that led Bekenstein to his entropy law is surprisingly 3 increases by this same additional amount. Consistency of the M By pulling out the particle a bit further, one changes its energy by a small amount Upon closer inspection Bekenstein’s reasoning can be used to show that gravity be- Of course, there is no polymer in the gravity system, and there appears to be no direct In this way, the open/closed string correspondence supports the interpretation of grav- be important, however, since thereThe is equations not should a therefore natural preferred not distance depend that on one small can variationsequal choose. of to this the distance. workblack done by hole, the the gravitational mass force. If one then drops the particle into the comes an entropic force nearexactly the valid. horizon, He and argued that that thehorizon one equations has at presented to in a choose section a distancethat location of slightly the away about from particle the the black andcorresponds order hole the to of choosing black the a hole Compton holographic have wave screen. length, become The where one precise we system. location declare of Let this us screen can say not this location dual to a thermal stateoperator on that the is boundary, while graduallyhorizon the being it particle has thermalized. is become representedis part By as clearly of a the entropic delocalized the time in thermal nature, thatsystem state, and to the just is its particle like the state the consequence reaches of polymer. of the maximal a entropy. This statistical phenomenon process that drives the contact between the particleone and of the the black dimensions hole.the is particle But can emergent. here behole. we In thought are of This the fact as ignoring holographic is being the description particularly immersed fact obvious of in in that this the the same context heat of bath process, AdS/CFT, representing in the which black a black hole is Hawking and others discovered the lawsexperiment of black mentioned hole in thermodynamics. section Insimilar fact, to the thought the polymer problem.can be The black thought hole of serves asits as the equilibrium the end situation. heat point bath, of while the the polymer particle that is gradually allowed to go back to and closed strings are not much different, but6.3 we just have Black to hole getWe horizons used saved to perhaps revisited the the clearest idea. argumentvery for last. the The fact first that cracks gravity in is the an fundamental entropic nature force of to gravity the appeared when Bekenstein, defined fundamental theory. Butonly in present our as view macroscopic concept. gravitybefore and It we closed just understood strings happened it that are couldresist we emergent be already making and knew obtained the about from analogy gravity a withelasticity microscopic a using theory stress situation without tensors in it. in whichatoms. a We we can’t continuous would We medium probably have half developed would a a have century theory been before for equally knowing about resistant in accepting the obvious. Gravity leads to the emergencethe of existence space of gravity and or gravity. closedas strings Note, an is effective however, not description. that assumed microscopically: from our they are point emergent ofity view as an entropic force. Yet, many still see the closed string side of these dualities is a well location of the screen. Integrating out the open strings produces the closed strings, and JHEP04(2011)029 m (6.1) the mass x , must be precisely H T due to a change in the . Again the formulas work BH S ma = . F . 3 BH ∂x ∂S H translates into the gravitational force. It is T S = – 24 – ∇ T = gravity F F translates into the inertial force S ∇ . T 3 = F of the particle to the horizon. This fact is surely known, and probably just x The horizon is only a special location for observers that stay outside the black hole. The mass is defined in terms the energy associated with the particle’s holographic Suppose we take the screen even further away from the horizon. The same argument . The previous arguments make clear that near the horizon the equations are valid with ~ perpendicular to the screen is just some scale variable associated with the holographic identified with the actual Planck constant. However, we have no direct confirmation or 6.4 Final comments This brings us to aof somewhat subtle and not yet~ fully understood aspect. Namely, the role weak. Therefore, thisteaches thought us experiment about is the naturetells not of us just space about teaching time, the us andhorizon, cause and the about of reach origin inertia. black exactly of the We holes. gravity.be same can Or conclusion. such do It that more In the precisely, this same it caseout thought the as experiment correspondence in for rules section must a Rindler on the other, muststraightforward be to see such that that and this the indeed entropy works change and are that exactly the as equations given for inThe the section black temperature hole can be arbitrarily large and the gravitational force at its horizon arbitrarily gradually lower the particle. towards Of the course, locationthe this of thermalization can the process. be oldthat nothing We screen there else must is behind then conclude an the an emergent thatthe new entropic entropic scale the force one variable effect, only acting. and the microscopic energy In force on explanation fact, is the the is one simply correspondence side, rules due and between to the emergent coordinate image, which presumablybecause is it a is nearequilibrium, still thermal just slightly state. like away the fromin It polymer. the energy is that black One not is hole may exactly holographically horizon. then thermal, dual ask: to however, We what the have is work pulled the done it cause when out of in of the the change emergent space we should have a holographicthe representation perpendicular on direction the has nox screen. microscopic meaning, In nor this acceleration.image system of The there the coordinate particle. is Itsit force interpretation means in as have an an coordinate emergent is space. a derived concept: this is what regarded as a consistency check. But let us takeapplies, it but one we or can twothe steps also particle further. choose to to theprocess ignore location would the happen fact of in that the the the previous part screen screen, of has the space moved, one behind and closer the lower new to screen the location, horizon. and hence This it Hawking entropy, when multiplied withequal to the the Hawking work temperature done by gravity. Hence, The derivative of thedistance entropy is defined as the response of laws of black hole thermodynamics implies that the additional change in the Bekenstein JHEP04(2011)029 ] S 15 (6.2) can not E by a function ~ and total energy R gets multiplied by a factor, while T is in equilibrium, the entropy needed 0 dA. S Z ~ 3 . This would not effect the final outcome for G c ~ 4 – 25 – = S ) is given by the Bekenstein-Hawking formula, including . The reason for this discrepancy may be that Bekenstein’s 3.4 ER . An interesting thought is that fluctuations may turn out to be more ~ 4, / ) of the Newton potential on the screen. This rescaling would affect the values of the One of the main points of this paper is that the holographic hypothesis provides a It is well known that Newton was criticized by his contemporaries, especially by Hooke, Does this view of gravity lead to predictions? The statistical average should give the But there may also be other ways to reconcile these statements, for example by making 0 (Φ natural mechanism for gravity todegrees of emerge. freedom associated It with allowsvolume one direct can material be “contact” body mapped interactions and on between another,for the since Newton’s same gravity all and holographic bodies Hooke’s screen. inside elasticity are Onceof a surprisingly this these similar. is rivals We done, would suspect the that have mechanisms neither been happy with this conclusion. force. Ironically, this is preciselyas the fundamental, reason why while Hooke’s Newton’s elasticthan force gravitational three is centuries. force nowadays not What has seen universe Newton maintained is did that holographic. not status know, Holography and foras is certainly absurd also more Hooke as an didn’t, an hypothesis, is action of that at course, the a and distance. may appear just effective value of pronounced for weak gravitationalneed fields a between better small understanding bodies of of the matter. theory to But turnthat clearly, his this we law into of a gravity prediction. acts at a distance and has no direct mechanical cause like the elastic Even then the essence ofand our inertia conclusions are would entropic not forces. change, which is theusual fact laws, that hence gravity onedepends has on to study the the effective fluctuations temperature, in which the may gravitational not force. be Their universal size and depends on the f entropy and the temperature inwill opposite be directions: divided by theis same something factor. to Since be aa priori understood. description we can that In not uses fact, exclude weighted this there average possibility, are there over even many other screens modifications with possible, different like temperatures. argument does not holdof for the the fact holographic that degrees we of are freedom far on from equilibrium. theuse screen, of or the because freedomthe to force, rescale nor the the value fact of that it is entropic. In fact, one can even multiply the factor 1 This value for this entropy appearsthat to states be that very high, ahave and system an violates entropy contained the larger Bekenstein in than bound region [ with radius is no horizon presentslightly at different all. there. The In first fact,have is well that there defined one are temperatures, is and not reasons clearly exactlythe to are at screen believe in thermal thermal equilibrium. at that equilibrium. an Horizons theto If equations equipotential get one work surface the assumes that Unruh with temperature Φ ( = Φ proof that the same is true when we go away from the horizon, or eespecially when there JHEP04(2011)029 A 73 ] ] Int. J. ]. , , ]. J. Phys. ]. ]. , Phys. Rev. (1975) 199 SPIRES , [ SPIRES 43 [ SPIRES [ (1993) 3427 hep-th/9409089 SPIRES Rept. Prog. Phys. [ [ hep-th/9711200 , D 48 (1976) 870 (1995) 6377 (1973) 2333 D 14 (1998) 231] [ gr-qc/9310026 2 , 36 hep-th/0302219 Phys. Rev. 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Maldacena, R.M. Wald, R.M. Wald, T. Jacobson, G. ’t Hooft, L. Susskind, P.C.W. Davies, W.G. Unruh, T. Damour, J.M. Bardeen, B. Carter and S.W. Hawking, S.W. Hawking, J.D. Bekenstein, [7] [8] [9] [4] [5] [6] [2] [3] [1] [16] [13] [14] [15] [10] [11] [12] Attribution Noncommercial License whichand permits reproduction any in any noncommercial medium, use, provided distribution, the original author(s)References and source are credited. This work is partly supportedR. by Stichting Dijkgraaf, FOM. P. I McFadden, like G. toM. ’t thank Shigemori, J. Hooft, K. de B. Papadodimas, Boer, Nienhuis, and B. J.-P. H. Chowdhuri, van Verlinde derOpen for Schaar, discussions Access. and and especially comments. Acknowledgments