Nonhomogeneous Cooling, Entropic Gravity and MOND Theory
aa r X i v : . [ h e p - t h ] J un Nonhomogeneous Cooling, Entropic Gravityand MOND Theory
Jorge Ananias Neto ∗ Departamento de F´ısica, ICE,Universidade Federal de Juiz de Fora, 36036-900,Juiz de Fora, MG, Brazil
Abstract
In this paper, using the holographic principle, a modified equipar-tition theorem where we assume that below a critical temperaturethe energy is not equally divided on all bits, and the Unruh temper-ature, we derive MOND theory and a modified Friedmann equationcompatible with MOND theory.
PACS number: 04.20.Cv, 04.50.KdKeywords: Entropic gravity, MOND theory ∗ e-mail: jorge@fisica.ufjf.br Introduction
The use of models belonging to the different areas of physics has been highlyefficient in order to produce important theories. Gravity is a good exam-ple where pioneering works of Bekenstein[1], Hawking[2] and Unruh[3] haveshown a deep connection between thermodynamics and relativity. Takingadvantage of this result, E. Verlinde[4] by using basically the holographicprinciple and the equipartition theorem has derived the usual gravitationalfield. Certainly, this is an important and intriguing result. Before the workof Verlinde, some authors have also investigated gravity from a thermody-namics point of view. Among them we can cite the works of Jacobson[5] andPadmanabhan[6]. Related ideas to the Padmanabhan’s works can also befound in Cantcheff[7] . It is opportune to notice that Banerjee and Majhi[8]have also given a statistical interpretation of gravity.Having in mind a possible connection between gravity theories and solidstate models, Gao[9], in the context of Verlinde’s formalism, has observedthat at low temperatures the equipartition theorem must be corrected ac-cording to the Debye solid model[10]. As an important consequence, thegravitational force is modified. More specifically, Li and Chang[11] haveobtained the modified Newtonian dynamics (MOND theory[12]) from theone dimensional Debye model. Here, we must mention that Modesto andRandono[13] have also pointed out the possibility of obtaining MOND the-ory from an area entropy relation in the Verlinde’s formalism. Kiselev andTimofeev[14] have also related MOND theory with the collective motions ofholographic screen bits. The MOND theory, in the context of cosmology, hasbeen successful in explaining most of the observed galaxies rotation curve.2s the equipartition law of energy plays an important role in the entropicgravity formalism, then our purpose in this paper is to modify the equiparti-tion theorem by adopting an alternative method to the Debye model. Duringthe cooling of the system we will consider that part of the bits acquires zeroenergy without the system to have still reached the zero temperature. Inprinciple, there is a critical temperature where this phenomenon starts tooccur. When we subtract the number of bits with zero energy from the totalnumber of bits in the equipartition formula, the MOND theory is obtained.There are several possibilities of introducing inhomogeneity in the Verlinde’sformalism and our motivation in choosing a particular form, Eq.(8), was toderive MOND theory. In order to clarify the exposition of the subject, thispaper is organized as follows: in Section 2 we give a short review of Verlinde’sformalism where the usual Newton’s law of gravity is derived. In Section 3we derive the MOND theory from the nonhomogeneous cooling of bits. InSection 4 we derive the MOND theory equivalent of the Friedmann equa-tions also from the nonhomogeneous cooling of bits. In Section 5 we makeour concluding remarks.
In this section we will briefly show, in a quantitative manner, the steps usedby Verlinde in order to derive the gravitational acceleration. We will beginby considering a spherical surface as the holographic screen with a particleof mass M positioned in its center. A holographic screen can be thought asa storage device for information. The number of bits is then assumed to be3roportional to the area A of the holographic screen N = Ac G ¯ h , (1)where A = 4 πr . The energy of the particle is E = M c . (2)An important assumption in the Verlinde’s formalism is that the total energyof the bits on the screen is given by the equipartition law of energy E = 12 N k B T. (3)Assuming that the energy of the particle inside the holographic screen isequal to the equipartition law of energy M c = 12 N k B T, (4)and using Eq.(1) together with the Unruh temperature formula k B T = 12 π ¯ hac , (5)we are then able to derive the well known (absolute) gravitational accelerationformula a = GMr . (6)4 The MOND Theory and Modified EntropicForce
The success of MOND theory is due to its ability, in principle as a phe-nomenological model, to explain the most of the rotation curve of galaxies.MOND theory reproduces the well known Tully-Fisher relation[15] and canalso be an alternative to the dark matter model. Basically this theory is amodification of Newton’s second law in which can be described as F = m µ (cid:18) aa (cid:19) a, (7)where µ ( x ) is a function with the following properties: µ ( x ) ≈ x >> µ ( x ) ≈ x for x << a is a constant. There are different interpolationfunctions for µ ( x )[16]. However, it is believed that the main implicationscaused by the MOND theory do not depend on the specific form of thesefunctions. Therefore, for simplicity, it is usual to assume that the variationof µ ( x ) between the asymptotic limits occurs abruptly at x = 1 or a = a .We will begin our proposal by considering that, below a critical temper-ature, the cooling of the holographic screen is not homogeneous. We choosethat the fraction of bits with zero energy is given by the formula N N = 1 − TT c . (8)For T ≥ T c we have N = 0. Thus, T c is a critical temperature where, belowthis, the zero energy phenomenon for some bits starts to occur. Equation(8) is a usual relation of the critical phenomena and second order phase5ransitions theory. The number of bits with energy different of zero for agiven temperature T < T c is N − N = N TT c . (9)Substituting Eq.(9) in the equipartition law of energy, we get E = 12 N TT c k B T. (10)Then, following the Verlinde’s formalism where we have used Eqs.(1), (2),(5) and (10), we derive, for T < T c , the MOND theory for Newton’s law ofgravitation a (cid:18) aa (cid:19) = G Mr , (11)where a = 2 πc k B T c ¯ h . (12)Using a ≈ − m s − we get T c ≈ − K , an extremely low temperaturewhich is far from the usual temperatures observed in our real world. There-fore, below the critical temperature, T c , we have obtained as a consequenceof MOND theory that the usual Newton’s second law is no longer valid.At this point we would like to stress that the hypothesis of nonhomo-geneous cooling of bits can be justified by making use of thermostatistic6rguments. Below the critical (low) temperature, the thermal bath does notensure an exact application of the equipartition theorem for all bits, i.e., thethermalization of the system does not guarantee that all bits have the sameenergy. This result frequently occurs in many different physical systems.As an example, we can mention the well known coupled harmonic oscillatorsmodel[17] where the energy is not be shared between the independent normalmodes of this system.The concept of nonhomogeneous cooling of bits allows us to imagine acurious hypothetical case that is when all bits acquire zero energy at thesame time for a particular acceleration or specific critical temperature dif-ferent of zero. In the language of critical phenomena, we can say that thiscase is similar to the first order phase transition. Then, using the abruptenergy change of the bits and Eqs.(1), (2), (4) and (5), we obtain that thegravitational acceleration has an interesting behavior that is not approach-ing to zero for very large distances but tends to a constant value. If thisphenomenon occurs, we expect that the asymptotic value of the accelerationmust be very small when compared with values of our everyday world andcertainly it is very difficult of being measured by the current experimentaltechniques. In this section we will derive a modified Friedmann equation compatible withMOND theory. We use the same steps of Cai, Cao and Ohta[18] who have7btained the Friedmann equations from entropic force (similar results can befound in references [19], [20] and [21]). The only difference is that we willintroduce the concept of nonhomogeneous cooling of bits in the equipartitionlaw of energy as we did in the previous section. We will begin by consideringa flat Friedmann-Robertson-Walker (FWR) universe whose metric is ds = − dt + a ( t ) (cid:16) dr + r d Ω (cid:17) . (13)We suppose that in the FWR universe the matter source, for simplicity, is aperfect fluid with the stress-energy tensor given by T µν = ( ρ + pc ) u µ u ν + pc g µν , (14)where ρ is the mass density, p is the pressure and u µ = (1 , , ,
0) is thefour velocity of the fluid measured by a comoving observer. We choose asthe holographic screen a spherical surface with the radius ˜ r = a ( t ) r . Con-sequently, the acceleration in which will be used in the Unruh temperatureformula is a r = ¨ ar . The mass inside the holographic screen is given by theTolman-Komar mass relation M = 2 Z V dV (cid:18) T µν − T g µν (cid:19) u µ u ν = 4 π a r ( ρ + 3 pc ) . (15)At this point we are ready to apply the Verlinde’s formalism with the fractionbits number with zero energy, Eq.(8), being used in the equipartition law ofenergy. Combining Eqs.(1), (2), (5), (10) and (15), we get8 a = − (cid:20) πGa ρ + 3 pc ) ar (cid:21) , (16)which is the acceleration equation for the dynamical evolution of the FWRuniverse compatible with MOND theory. The parameter a is the same asdefined in Eq.(12). The minus sign in Eq.(16) is due to our considerationthat the acceleration is caused by the matter inside the holographic screen.Due to the algebraic form of Eq.(16), at first, we can not integrate it andconsequently we can not write an expression for ˙ a , as usually is done for theFriedmann equations. What it is amazing in the entropic formalism is that weare deriving a modified Friedmann equation compatible with MOND theoryby a thermostatistic approach and not by a modified relativistic theory[22]which would, in principle, necessary to derive an exact result.Rewriting Eq.(16) in terms of ˜ r = a ( t ) r , we get¨˜ r = − (cid:20) πGa ρ + 3 pc )˜ r (cid:21) , (17)which is the same expression obtained by Sanders[23] in another context. In this work, we use the holographic principle and a modified equipartitionlaw of energy, where we have introduced the notion of nonhomogeneous cool-ing of bits, in order to derive MOND theory and a modified Friedmann equa-tion compatible with MOND theory. The key point of our work is that, belowa critical temperature, the energy of the particle inside the holographic screen9s not distributed equally on all bits. This hypothesis leads to a change inthe equipartition energy formula where now it is proportional to the squareof temperature. Using the Unruh temperature we are then able to deriveMOND theory. To finish, we would like to point out that an important re-sult of our work is the reinterpretation of MOND’s parameter a , Eq.(12), interms of the Planck constant, and a critical temperature where below this,the nonhomogeneous bits distribution of energy begins to occur. The author thanks M. Botta Cantcheff for sending reference[7], B. R. Majhifor sending reference [8] and V. V. Kiselev for sending reference[14].
References [1] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).[2] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).[3] W. G. Unruh, Phys. Rev. D 14, 870 (1976).[4] E. Verlinde, arXiv: hep-th/1001.0785.[5] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).[6] T. Padmanabhan, Class. Quantum Grav. 21, 4485 (2004); Mod. Phys.Lett. A 25, 1129 (2010).[7] M. Botta Cantcheff, arXiv: hep-th/0408151.108] R. Banerjee and B. R. Majhi, Phys. Rev. D 81, 124006 (2010).[9] C. Gao, Phys. Rev.D 81, 087306 (2010).[10] See for example C. Kittel, Introduction to Solid State Physics, eighthedition (Wiley, New York, 2004)[11] X. Li and Z. Chang, arXiv: hep-th/1005.1169.[12] M. Milgrom, ApJ 270, 365 (1983); ApJ 270, 371 (1983); ApJ 270, 384(1983).[13] L. Modesto and A. Randono, arXiv: hep-th/1003.1998.[14] V. V. Kiselev and S. A. Timofeev, arXiv: hep-th/1009.1301.[15] R. B. Tully and J. R. Fisher, Astr. Ap. 54, 661 (1977).[16] B. Famaey, G. Gentile and J. P. Bruneton. Phys. Rev. D 75, 063002(2007). H. S. Zhao and B. Famaey, ApJ 638: L9 (2006).[17] L. E. Reichl, A Modern Course in Statistical Physics, second edition (Wiley, New York, 1998)[18] R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81, 061501(R) (2010).[19] T. Padmanabhan, arXiv: gr-qc/1001.3380.[20] F. W. Shu and Y. Gong, arXiv: gr-qc/1001.3237.[21] A. Sheykhi, Phys. Rev. D 81, 104011 (2010).1122] J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004); Erratum-ibid. D 71,069901 (2005).[23] R. H. Sanders, Mon. Not. Roy. Astron. Soc. 296, 1009 (1998).12 r X i v : . [ h e p - t h ] J un Nonhomogeneous Cooling, Entropic Gravityand MOND Theory
Jorge Ananias Neto ∗ Departamento de F´ısica, ICE,Universidade Federal de Juiz de Fora, 36036-900,Juiz de Fora, MG, Brazil
Abstract
In this paper, by using the holographic principle, a modifiedequipartition theorem where we assume that below a critical tempera-ture the energy is not equally divided on all bits, and the Unruh tem-perature, we derive MOND theory and a modified Friedmann equationcompatible with MOND theory. Furthermore, we rederive a modifiedNewton’s law of gravitation by employing an adequate redefinition ofthe numbers of bits.
PACS number: 04.20.Cv, 04.50.KdKeywords: Entropic gravity, MOND theory ∗ e-mail: jorge@fisica.ufjf.br Introduction
The use of models belonging to the different areas of physics has been highlyefficient in order to produce important theories. Gravity is a good exam-ple where pioneering works of Bekenstein[1], Hawking[2] and Unruh[3] haveshown a deep connection between thermodynamics and relativity. Takingadvantage of this result, E. Verlinde[4] by using basically the holographicprinciple and the equipartition theorem has derived the usual gravitationalfield. Certainly, this is an important and intriguing result. Before the workof Verlinde, some authors have also investigated gravity from a thermody-namics point of view. Among them we can cite the works of Jacobson[5] andPadmanabhan[6]. We should also mention that related ideas to the Padman-abhan’s works can also be found in Cantcheff[7]. In addition, Banerjee andMajhi[8] have also given a statistical interpretation of gravity.Having in mind a possible connection between gravity theories and solidstate models, Gao[9], in the context of the Verlinde’s formalism, has observedthat at low temperatures the equipartition theorem must be corrected ac-cording to the Debye’s solid model[10]. As an important consequence, thegravitational force is modified. It is opportune to mention here that the en-ergy of a system in the Debye’s model goes to zero when the temperaturereaches zero. However, in the Gao’s approach, the energy of the gravitationalsystem remains constant. This behavior indicates that there is a possible dif-ference between the Gao’s procedure and the usual Debye’s model. Li andChang[11], in an interesting work, have obtained the modified Newtonian dy-namics (abbreviated by MOND and originally proposed by Milgrom[12]) fromthe one dimensional Debye model. Also, in the Verlinde’s picture, Modesto2nd Randono[13] have pointed out the possibility of obtaining MOND the-ory from an area entropy relation and Kiselev and Timofeev[14] have relatedMOND theory with the collective motions of the holographic screen bits.MOND theory, in the context of cosmology, has been successful in explain-ing most of the observed galaxies rotation curve. As the equipartition lawof energy plays an important role in the entropic gravity formalism then ourpurpose in this paper is to modify the equipartition theorem by adopting analternative method to the Debye model. In the Debye’s solid model the atomsare arranged in a crystal lattice structure where each can be considered as aharmonic oscillator. In our scheme, based on models of critical phenomena,during the cooling of the system, we assume that, at first, the bits are notclustered in a lattice structure. We will consider that part of the bits acquireszero energy without the system to have still reached the zero temperature.In principle, there is a critical temperature where this phenomenon starts tooccur. When we subtract the number of bits with zero energy from the totalnumber of bits in the equipartition formula, the MOND theory is obtained.There are several possibilities of introducing inhomogeneity in the Verlinde’sformalism and our motivation in choosing a particular form, Eq.(8), was toderive MOND theory. In the last part of this work, we rederive a modifiedform of the gravitational force, previously obtained by Sheykhi[15], by usingthe equipartition theorem with an appropriate redefinition of the number ofbits. 3
Entropic Gravitational Acceleration
In this section we will briefly show, in a quantitative manner, the steps usedby Verlinde[4] in order to derive the gravitational acceleration. We will beginby considering a spherical surface as the holographic screen with a particleof mass M positioned in its center. A holographic screen can be thought asa storage device for information. The number of bits is then assumed to beproportional to the area A of the holographic screen N = Al p , (1)where A = 4 πr and l p = G ¯ hc . The term bit signifies the smallest unit ofinformation in the holographic screen. The energy of the particle is E = M c . (2)An important assumption in the Verlinde’s formalism is that the total energyof the bits on the screen is given by the equipartition law of energy E = 12 N k B T. (3)It is important to mention that for black hole space-time Eq.(3) can beproved without assuming a priori the validity of this equation[8]. Assumingthat the energy of the particle inside the holographic screen is equal to theequipartition law of energy 4 c = 12 N k B T, (4)and using Eq.(1) together with the Unruh temperature formula k B T = 12 π ¯ hac , (5)we are then able to derive the well known (absolute) gravitational accelerationformula a = GMr . (6) The success of MOND theory is due to its ability, in principle as a phe-nomenological model, to explain the most of the rotation curve of galaxies.MOND theory reproduces the well known Tully-Fisher relation[16] and canalso be an alternative to the dark matter model. However, it is opportuneto mention that MOND theory can not explain the temperature profile ofgalaxy clusters and presents some trouble when confronting with cosmology.For details, see, for example, references [17] and [18]. Basically this theoryis a modification of Newton’s second law in which the force can be describedas 5 = m µ (cid:18) aa (cid:19) a, (7)where µ ( x ) is a function with the following properties: µ ( x ) ≈ x >> µ ( x ) ≈ x for x << a is a constant. There are different interpolationfunctions for µ ( x )[19]. However, it is believed that the main implicationscaused by the MOND theory do not depend on the specific form of thesefunctions. Therefore, for simplicity, it is usual to assume that the variationof µ ( x ) between the asymptotic limits occurs abruptly at x = 1 or a = a .We will begin our proposal by considering that, below a critical temper-ature, the cooling of the holographic screen is not homogeneous. We choosethat the fraction of bits with zero energy is given by the formula N N = 1 − TT c , (8)where N is the total number of bits given by the formula (1), N is the numberof bits with zero energy and T c is the critical temperature. For T ≥ T c wehave N = 0 and for T < T c the zero energy phenomenon for some bits startsto occur. Equation (8) is an usual relation of critical phenomena and secondorder phase transitions theory. The number of bits with energy different ofzero for a given temperature T < T c is N − N = N TT c . (9)Here, we are assuming that Eq.(1) is still valid below T c . Whereas now theenergy of the particle inside the holographic screen is equally distributed on6ll bits with nonzero energy and using relation (9) in the equipartition lawof energy, we get M c = 12 ( N − N ) k B T = 12 N TT c k B T. (10)Then, combining Eqs.(1), (5) and (10), we are capable of deriving, for T < T c ,the MOND theory for Newton’s law of gravitation a (cid:18) aa (cid:19) = G Mr , (11)where a = 2 πc k B T c ¯ h . (12)Using a ≈ − m s − we get T c ≈ − K , an extremely low temperaturewhich is far from the usual temperatures observed in our real world. Here, abrief comment on this extremely low temperature must be made. RewritingEq.(5) in the form T ≈ × − Km/s a, (13)then, we can observe that, at first, any model that combines the Unruh tem-perature with MOND theory ( a ≈ − m s − ) leads to the value of the7ritical temperature, T c , mentioned above. According to the Unruh effect,where an accelerating observer will observe a black-body radiation, our crit-ical temperature should be observed in an accelerated reference frame thatcan be the galaxies. Therefore, below the critical temperature, T c , we haveobtained as a consequence of MOND theory that the usual Newton’s secondlaw is no longer valid.We also want to make a comment on the possibility to occur phase tran-sitions in our model. The energy of the system, Eq.(10), is a constant value( E = M c ) as well as it is constant in the Debye’s model as mentioned in theintroduction. Then, the derivative of this expression with respect to tempera-ture, which is the specific heat, is zero. However, so that the phase transitiontakes place, it is necessary that the specific heat presents singularity at thecritical temperature. Therefore, if we use the behavior of the specific heatas the indicator of phase transition, we can conclude that, strictly speaking,there is no phase transition in our particular model[20].The hypothesis of nonhomogeneous cooling of bits can be justified bymaking use of thermostatistic arguments. Below the critical (low) tempera-ture, the thermal bath does not ensure an exact application of the equipar-tition theorem for all bits, i.e., the thermalization of the system does notguarantee that all bits have the same energy. This result frequently occursin many different physical systems. As an example, we can mention the wellknown coupled harmonic oscillators model[21] where the energy is not beshared between the independent normal modes of this system.The concept of nonhomogeneous cooling of bits allows us to imagine acurious hypothetical case that is when all bits acquire zero energy at the8ame time for a particular acceleration or specific critical temperature dif-ferent of zero. In the language of critical phenomena, we can say that thiscase is similar to the first order phase transition. Then, using the abruptenergy change of the bits and Eqs.(1), (2), (4) and (5), we obtain that thegravitational acceleration has an interesting behavior that is not approach-ing to zero for very large distances but tends to a constant value. If thisphenomenon occurs, we expect that the asymptotic value of the accelerationmust be very small when compared with values of our everyday world andcertainly it is very difficult of being measured by the current experimentaltechniques. In this section we will derive a modified Friedmann equation compatible withMOND theory. We use the same steps of Cai, Cao and Ohta[22] who haveobtained the Friedmann equations from entropic force (similar results canbe found in references [23] and [24]). The only difference is that we willintroduce the concept of nonhomogeneous cooling of bits in the equipartitionlaw of energy as we did in the previous section. We will begin by consideringa flat Friedmann-Robertson-Walker (FRW) universe whose metric is ds = − dt + a ( t ) (cid:16) dr + r d Ω (cid:17) . (14)9e suppose that in the FRW universe the matter source, for simplicity, is aperfect fluid with the stress-energy tensor given by T µν = ( ρ + pc ) u µ u ν + pc g µν , (15)where ρ is the mass density, p is the pressure and u µ = (1 , , ,