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This content was downloaded from IP address 131.169.4.70 on 28/11/2017 at 11:06 JCAP10(2012)012 hysics P le ed strongly on the ic ity perspective. Thus t educe to their respective ar gh, we find that the origin RIAAM) Maragha, y of statistical mechanics we 10.1088/1475-7516/2012/10/012 tanding the theoretical origin he study for Poisson equation limit of very low temperature. nces, Shiraz University, holographic screen induced by gy in statistical thermodynamics e regarded as a thermodynamical doi: strop c A [email protected] , osmology and and osmology C and K. Rezazadeh Sarab a,b modified gravity, gravity

Verlinde’s proposal on the entropic origin of gravity is bas [email protected] rnal of rnal

ou An IOP and SISSA journal An IOP and 2012 IOP Publishing Ltd and Sissa Medialab srl Center for Excellence in Astronomy and Astrophysics (CEAA- P.O. Box 55134-441, Iran Physics Department and BiruniShiraz Observatory, College 71454, of Iran Scie Department of Physics, ShahidP.O. Bahonar Box University, 76175, Kerman, Iran E-mail: b c a c Received June 4, 2012 Accepted September 13, 2012 Published October 8, 2012 Abstract. assumption that the equipartitionthe law mass of energy distribution holds ofknow on that the the the system. equipartition lawInspired However, of by from energy the Debye does the model not for theor and hold the adopting in equipartition the the law viewpoint of that ener system, gravitational systems we can modify b Einsteinand field modified equations. Newtonian We also dynamicsof perform (MOND). the t Interestingly MOND enou our theory study can may be fillof in understood MOND the from theory. gapstandard Debye existing In gravitational entropic in equations. the the grav limit literature of unders high temperatureKeywords: our results r A. Sheykhi Einstein equations and MONDfrom theory Debye entropic gravity J JCAP10(2012)012 ] 1 2 5 7 8 2 (1.2) (1.1) n arise ]. In 1995 Jacobson [ 1 [ tropy and the temperature force caused by the changes equations should encode in- T dS cond order partial differential ed strongly on the assumption ween horizon thermodynamics c screen induced by the mass the entropic force has raised a mpt people to take a statistical ntities such as horizon area and = ntropy and the horizon area to- ection between gravitational dy- l bodies. According to Verlinde ion of state for the and tion to thermodynamic behavior. very least offers a strong analogy o the temperature, his derivation opens a new window s such as and temperature. dE x, ] who claimed that the laws of gravity are . 3 △ x S ~ m . Following Jacobson, however, several recent △ △ π T approaches a holographic screen from a distance – 1 – T dS = = 2 m = F S △ δQ ] and references therein). 10 – 4 = 1 for simplicity, through this paper. The entropic force ca c = B k Verlinde’s proposal on the entropic origin of gravity is bas A next great step put forwarded by Verlinde [ , the change of entropy on the holographic screen is x surface gravity are related to theThe thermodynamic first quantitie law oftogether with black the hole energy thermodynamics (mass) of implies the that black the hole satisfy en 1 Introduction Thermodynamics of black holes reveals that geometrical qua 3 MOND theory from Debye4 entropic gravity Poisson equation from Debye5 entropic force Conclusions Contents 1 Introduction 2 Einstein equations from Debye entropic gravity Verlinde’s derivation of Newton’s lawwith of a well gravitation understood at statistical the to mechanism. understand Therefore, gravity t fromlot the of first attention principles. recently The (see study [ on where we have set that the equipartition law of energy holds on the holographi not fundamental and in particular they emerge as an entropic in the information associatedproposal with when a the test positions△ particle of with materia mass put forwarded a new stepEinstein and equation suggested for that the theHe spacetime disclosed hyperbolic metric that has se the a Einsteinin predisposi field particular equation is it just can angether be equat with the derived fundamental from relation the proportionality of e investigations have shown that therenamics is and indeed horizon a thermodynamics. deeper Theand conn deep gravitational connection bet dynamics, helpformation about to horizon understand thermodynamics. whyphysics These point the results of field pro view on gravity. in the direction of increasing entropy and is proportional t JCAP10(2012)012 (2.2) (2.3) (2.1) ravity. ravitation, Einstein ] studied three dimen- ded and forms a closed 11 theory is discussed in the rmation, i.e. a holographic rive Einstein field equations e dimensional equipartition ature of the system is much Friedmann equations. Such ot hold in the limit of very avitations and modify them Gao [ n holographic screen interact h bit on holographic screen is in all range of temperatures is o obtain the modified Einstein laws of gravity accordingly. In ld). on’s law of gravitation, Einstein maximal storage space, or total finish our paper with conclusions ach bit on the holographic screen IV is devoted to the derivation of ystems should be modified in the emperature. Hence, it is expected niverse without invoking any kind cal thermodynamics origin. Thus, dy, 1 , ) ) is the one dimensional Debye function x − y ( x . y ( . However, from the theory of statistical ~ e , A D x G A 0 . It was demonstrated that the Debye model is NT NTD Z = 2 1 D – 2 – 1 2 1 x T N = = ≡ ≪ ) present in the system. Let us now just make the simple E E T x ( E D ]. In this paper we use the Debye model to modify the entropic g , is proportional to the area 11 N This paper is structured as follows. In the next section we de We consider a system that its boundary is not infinitely exten Suppose there is a total energy It is important to note that Verlinde got the Newton’s law of g We find that this modifiedaccordingly. entropic force affects on the law of gr defined as from Debye entropicframework gravity. of Debye The entropic gravity theoreticalthe in Poisson origin section equation from III. of Debye Section entropic MOND which force scenario. appear in We section V. 2 Einstein equations fromFollowing Debye Verlinde’s entropic scenario, gravity gravityany may modification have a of statisti statisticalthis mechanics section should we use modifyequations. the the modified equipartition law of energy t surface. We canscreen. take Assuming the that boundarynumber the as of holographic a bits principle storage holds, device the for info where T is the temperature of the screen and assumption that the energy is divided evenly over the bits. E has one dimensional degreelaw of of freedom, energy. hence The we equipartition can law use of the energy on which is valid distribution of the system, namely, sional Debye model andmodification modified can the interpret entropic the current forceof acceleration and dark of henece energy the u [ that the equipartition lawlimit of of energy very for low the temperature gravitational (or s very weak gravitational fie very successful in interpreting the physics at the very low t equations and Poisson equationfree with of the interaction. assumptioneach that It others. eac should In such be case,equation more and one Poisson could general equation anticipate must that that be the the modified. Newt bits For example, o low temperature. By lowsmaller temperature, than we Debye mean temperature, that i.e. the temper mechanics we know that the equipartition law of energy does n JCAP10(2012)012 , is M a (2.5) (2.8) (2.9) (2.6) (2.4) (2.7) = N (2.10) (2.13) (2.11) (2.12) E ecause , because the φ e ) may be written = 0, which we will 2.6 φ potential holographic s and energy, th the Newton’s potential ral relativity and for it we represents the redshift factor mperature, we use the Unruh φ , point with e ve as a closed equipotential surface ab dA. ) . S 2 x dS . Therefore eq. ( . / ( ) 1 dA, φ 2 , x ) ) / ) ( φ, 1 φ x a φD a ) ( a D ξ a b φ ∇ a φ, , φ , b ∇ ξ ∇ a b ξ a a a a φ π D TD ∇ − T N a ~ 2 ∇ T ( ∇ φ S N φ −∇ b ∇ φ e I S = ( ≡ e – 3 – Ln I ∇ π φ = ~ ) in a more general form, ~ ( S 1 2 . We inserted a redshift factor 2 e T x 1 b b G I π a ξ 1 2 2.2 ~ = πG = = 2 8 a 1 φ = T πG = − N 4 M T = ) as = ), we get ], we can obtain M 2.9 M 2.5 12 ) in eq. ( ), we can rewrite eq. ( 2.9 is measured with respect to the reference point at infinity. B 2.1 T is the unit outward pointing vector that is normal to the equi is the Debye temperature. Using the equivalence between mas and time like Killing vector is a global time like Killing vector. The exponent a is the natural generalization of Newton’s potential in gene denotes the acceleration. The acceleration has relation wi ], a D S ξ a φ is related to the temperature N T 12 ] x 3 normal to the equipotential holographic screen, for it we ha temperature where the integrationtemperature is formula over on the the holographic holographic screen. screen, For te where screen where and as well as eq. ( Therefore we can rewrite eq. ( where as [ where and in it may be written as have [ that relates the local time coordinate to that at a reference Substituting eq. ( or in other words, a closed surface of constant redshit Following the same logic of [ where take to be at infinity. We choose the holographic screen JCAP10(2012)012 ) Σ, d a 2.13 (2.15) (2.21) (2.20) (2.16) (2.19) (2.18) (2.14) (2.17) n − = a Σ d ) for eq. ( a 2.14 . Σ d ,  a )] , ab x Σ a g Σ ( d Σ d T D b )] d c 1 2 ξ x )] a ( ∇ x a . − n c ( D rk of entropic gravity scenario. ξ ing from considering the Debye Σ Σ )] b , a ab kes theorem ( D d d a x b ∇ T b ( ∇ )] b Σ ξ )  ∇ ξ x d D a b b is equal to -1 or 1 if the hypersurface ( a c ξ n ξ ab , b . ε a D ∇ ∇ a πG ξ a  c B c ξ , b ∇ is the two dimensional boundary of the ξ − + b a Σ ab ∇ Σ = 0 a ( ], c d ∇ g a R + d φ S ξ a ξ ∇ )] 2 a Σ ) = 8 b T a b 12 − ξ b x [ ξ − Z b x ξ 1 2 , and get ( a ∇ e εn ( ∇ ). In the last line we have used a φ = b ∇ 2 ab ξ D − ∇ D b = – 4 – b T − b ]. On the other hand, according to the Stokes c ∇ = 2 2.8 ξ + ξ )+ )+ e ) a ab a a ∇ b ∇ x x x 13 T ab ) c Σ ξ ( ( ( − ∇ ∇ ξ a x d  [ a ) ( a D D D dS b b b ∇ x ∇ ( ∇ D − Z ab ξ ξ ∇ [ [ b ξ D ab ab B Σ Σ Σ φ Z Z Z S R R ab 2 = 2 [ [ I − R Σ Σ ), we find [ e can be expressed as an integral over the enclosed volume of 1 1 1 Z Z M πG πG πG Σ 4 4 4 ] − Z 2.20 M − − − 1 1 12 πG πG ab 4 4 1 πG R = = = ) 4 − − x is a directed surface element on Σ and for it we have ] ( M = = = a ) and ( 13 D Σ M d 2.19 is the two-surface element [ is an antisymmetric tensor field and is the unit normal of the hypersurface Σ and ab ab a dS B n On the other hand, where is spacelike or timelike, respectively. Now we apply the Sto where hypersurface Σ. where The above equation iscorrection the modified to Einstein the equations equipartition result law of energy in the framewo Equating eqs. ( because the hypersurface Σ is spacelike. where in the last step we have used the Killing equation, and get theorem, we have [ where in the second line we have used eq. ( certain components of stress energy tensor which is implied by the Killing equation for Now we use the relation [ JCAP10(2012)012 x 1. g > ≪ ] and (2.22) (2.25) (2.23) (2.24) , then y 16 D , thus for D T T >T ), leads to ) and as a result, ≫ 2.3 2.21 T he flat rotational curves he strength of the gravi- in the high temperatures he Debye temperature, is 1 and consequently rature, i.e. an be detectable practically tends to zero and it becomes ≪ bye temperature (very strong rst specify the Debye function e Debye acceleration, i.e. e we have assumed the general y predicts that objects that are x se we have an define the Debye acceleration . he correction term only plays role ld equations as expected.  ify the gravitational field equations. nde’s approach, any modification of e that the rotational velocity curves . ab g ). The resulting solutions should be T = 1 . ]. Among them are the Oort discrepancy 2 1 in the integral of eq. ( 2.21 D ), we have dy 14 y T − T, x π 0 π 2.3 ~ ~ ab 2 2 Z 1+ T ]. These observations are in contradiction with 1 x – 5 –  ≈ = = 17 y g ≈ D e g πG ) x ( = 8 D ab R ) = 1) in the modified Einstein equations ( ], the velocity dispersions of dwarf Spheroidal galaxies [ x ( 15 D is the norm of the gravitational acceleration. Therefore, t g in the definition of the Debye function ( It is instructive to examine the modified Einstein equations y ) and then try to solve the field equations ( x . In other words, the limit of high temperatures compared to t ( D and g corresponding to the strong gravitational fields. In this ca Substituting this result ( tational field is proportionalrelating to to the the temperature. Debye Also, temperature we as c where This equation is nowequipartition valid law of for energy. all Therefore, rangefirst we principles see of such that as temperature, in equipartition Verli law sinc The of energy question will mod whether theor modified not needs term more inD investigations Einstein in equation the c future. One needs to fi Therefore, in thegravitational temperatures fields), extremely one larger obtains than the standard the Einstein De fie 3 MOND theory fromModified Debye Newtonian entropic dynamics gravity (MOND)of spiral was galaxies. proposed A toof great explain all variety spiral t of galaxies observations tend indicat to some constant value [ the prediction of Newtonian theory because Newtonian theor the one dimensional Debye function reduces to in the disk ofthe Milky flat Way [ rotation curves of spiral galaxiesfar [ from the galaxy center have lower velocities. checked with experiments or observations.in It very is low clear temperature, thatflat. in t which the curvature of spacetime the norm of the gravitational acceleration is larger than th Therefore we can use the approximation limit. According to the Unruh temperature formula we have Therefore, if the temperature is larger than the Debye tempe JCAP10(2012)012 ’ is a (3.7) (3.2) (3.3) (3.6) (3.1) (3.5) (3.4) , i.e., 2 ]. This 18 m/s ), and define 10 − 2.4 in 1983 [ owever its theoretical as the boundary of the , i.e. eq. ( R e dark matter hypothesis. x 1) if the acceleration ‘ on of Newtonian dynamics le to show that the MOND us distance; the rotational-curve served anomalous rotational- y. If we compare this equation ’ (which is normally taken as e between mass and energy as ravity. This derivation further n of ≪ a ( lish a gravitational theory which e nonluminous matters. Another 0 . At large distance, at the galaxy a a ), 2 0 . a a . function as = (  . , 2 m/s 0 . 2 a a 2 a R 2 , GM 6 10 2 R . R GM π = GM − D 0 R = ~ GM T a  a eff π 2 = 6  a )= 12 π D 0 = 10  x )= a ( – 6 – a 0 0 ≡ ≡ = ), the above equation may be written as x a 2 6 a a ( 0 π x  ’ is extremely small, smaller than 10  a TD 2.6 0 a  . Consequently, the velocity of star on circular orbit a aD a π 0 ~ a a 2 a  aD ) gives ) = 0 a a 3.3 ( , we obtain 2 ), we see that we can define πR 3.1 = 4 A ) as effective kinematic acceleration = 1 for usual-values of accelerations and , hence the function /r 0 2 The most widely adopted way to resolve these difficulties is th Although MOND theory can explain the flat rotational curve, h Again, we consider a spherical holographic screen with radi a v as ≪ = 0 a out skirt, the kinematical acceleration ‘ extremely low, lower than a critical value Using the Unruh temperature formula ( Also, if we use the Unruh temperature formula in the definitio This is the MOND theorywith resulting well-known from eq. Debye entropic ( gravit a where Using the above result in eq. ( from the galaxy-center is constantis and flat, does as not it depend observed. on the a It is assumed that allapproach visible is stars are the surroundedtheory by MOND massiv appears theory to whichvelocity. be was highly In suggested successfulthrough fact, by for modification M. the explaining in Milgrom MOND the the theory kinematical ob is acceleration term (empirical) ‘ modificati origin remains un-known. Thus, itcan is results well MOND motivated to theorytheory estab can naturally. be In extractedsupport this completely from the section, the viability we Debye of are entropic Debye g ab entropic gravity formalism. then we obtain system. Combining eqs. (3)well and as (4), relation and using the equivalenc JCAP10(2012)012 (4.1) (3.9) (3.8) (4.3) (4.2) (3.10) ere are , we have corresponding to an S smaller than the Debye ure its local acceleration. Debye entropic gravity. In rt. Thus the origin of the tions similar to those of tational fields, e.g. at large creen nding the theoretical origin of ) reduces to . to account the Debye correction 2 x , we get 3.6 perature of the holographic screen, relative to the Debye temperature. π 6 . We assume that the entire mass 0 ) = 1. We conclude that in the limit ≈ φ . 0 a . | a D dy ( φ 2 . 1 | ) turns into the standard Newtonian dy- . ∇ R πT | φ. φ GM − 2 y ~ 2 ) in two limits of temperatures. First, we π 3.6 ∇ ∇ y , we have 2 R | = e GM x = − ~ 3.6 ), we obtain  – 7 – ∞ D = = 0 = T 0 a T 3.6 a )=1. Thus eq. ( a Z a T x  1 x ( ≡ a we have also D x 0 )= a x ( ) in eq. ( ≫ D 3.9 ) have the same behavior and become equal to 1. a x ( ) we have 0 a ) is contained inside the volume enclosed by the screen and th x ( , that is to say in the weak gravitational fields. In this limit ≫ ρ D ). Let us examine eq. ( ) and a T x ( 1 ( 3.1 ≪ D ), and the Debye function can be expanded as ≪ T 0 a x both ≪ 0 a a The second limit corresponds to the temperatures extremely 1 ( ≫ a ≫ Substituting this relation into Unruh temperature formula temperature, x some test particles outside this volume.we To take identify the a tem testThe local particle acceleration and is move related it to close the to Newton potential the as screen and meas distribution given by equipotential surface with fixed Newtonian potential If we use the approximation ( In what followsfunction we in show eq. that ( this function satisfies the condi Therefore, the Newtoniandistance dynamics from is the modifiedMOND galaxy for theory weak center, can be gravi namely understoodthis at completely way in we the the fill galaxyMOND framework in of theory. out the ski gap existing in the literature understa 4 Poisson equation fromFinally, Debye we obtain entropic the force modifiedto Poisson equation the by equipartition taking in law of energy. We choose a holographic s Therefore, for strong gravitational fields, eq. ( of consider the limit corresponding toIn the this temperatures large case namics. As we discussed, for Using the above equation in the definition of JCAP10(2012)012 (4.6) (4.8) (4.4) (4.5) (4.7) , we have the S on the holographic ) = 1. In this case x ( NT 1 2 D 2782-77. ic gravity scenario. Since = ion is modified accordingly. mperatures extremely larger energy, we see that not only e galaxy flat rotation curves, y enough, we found that the E ge of temperatures. For high ng the modified gravitational rm of the Einstein equations, know from statistical mechan- tained the Einstein equations, mical system and taking into ence in Astronomy and Astro- ion to the equipartition law of d Newton’s law of gravitation. low temperatures and it should ance. This result is impressive s), the obtained modified equa- ions. The results obtained here y as . dV. ] A. ) φ x ( . φ.d ∇ ) as 1) and hence ) ) ∇ x x dV. inside the closed surface ) ( ( 4.4 πGρ ) x ≪ x ( D [ ( ) for the number of bits on the holographic M . x D ρ πGρ ] = 4 ∇ S V 2.1 φ I – 8 – Z V = 4 Z ∇ ) = φ 1 πG x 2 ( 1 4 πG M ) is the Debye function. Following Verlinde’s strategy ∇ 4 D x [ = . ( = ∇ D M M ), we get ), after using eq. ( 4.6 ), where x 2.5 ( ) and ( NTD 2 1 4.5 ) in eq. ( = 4.2 E ) reduces to the standard Poisson equation, 4.7 Equating eqs. ( relation eq. ( On the other hand, for the mass distribution This is the modifiedtemperatures. Poisson equation i.e. which is strong valid gravitational in field all ( ran Inserting ( screen induced by theNewton’s mass law distribution of of gravitation the andics system, the that Poisson and the equation. ob equipartition But lawbe we of corrected. energy does In notenergy this hold as paper, at very we considered the Debye correct screen, we obtain Thus, considering theaccount gravitational the system Debye as modelEinstein for a equation the thermodyna and modified MOND equipartitionClearly theory law the but of modification also of the Poisson Poisson equation equat leads to5 modifie Conclusions In his work, Verlinde applied the equipartition law of energ Using the divergence theorem we can rewrite eq. ( MOND theory and theorigin modified of Poisson MOND equation. theorythe can Interestingl MOND be theory understood isthus an from acceptable the the theory Debye studies for entrop and on explanation show of its that th theoretical thefield origin approach equations here is from is Debye of model. powerfulthan great enough We the import also for Debye showed derivi that temperaturetions in (very the turn strong te into gravitational theirfurther field support respective the well-known viability standard of equat Verlinde’s formalism. Acknowledgments This work has beenphysics supported of financially IRAN by Center (CEAAI- for RIAAM) Excell under research project No. 1/ on the entropic origin of gravity, we obtained the modified fo JCAP10(2012)012 , nn ] e Gen. , tropic ] pace-times ]; (2011) 455 (2010) 017 ]; , 56 08 (2010) 104011 (2010) 124006 , (2011) 029 , SPIRE ]. te IN [ SPIRE arXiv:1002.1062 ]; 04 [ ]; ]; , IN [ D 81 D 81 el JCAP , , arXiv:1003.1998 SPIRE , , ]. 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