The holographic screen at low temperatures
aa r X i v : . [ h e p - t h ] J a n The holographic screen at low temperatures
V.V.Kiselev, S.A.Timofeev
Russian State Research Center “Institute for High Energy Physics”,Pobeda 1, Protvino, Moscow Region, 142281, RussiaMoscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141701, Russia
A permissible spectrum of transverse vibrations for the holographic screen modifies both a dis-tribution of thermal energy over bits at low temperatures and the law of gravitation at small accel-erations of free fall in agreement with observations of flat rotation curves in spiral galaxies. Thismodification relates holographic screen parameters in de Sitter space-time with the Milgrom accel-eration in MOND.
PACS numbers: 04.20.Cv, 04.50.Kd, 95.35.+d
I. INTRODUCTION
The holographic approach [1–3] within the thermodynamical formulation of gravity [4–14] became logically accom-plished in papers by E.Verlinde [15] and T.Padmanabhan [16, 17] as well as in further developments, modificationsand applications of their ideas in various aspects [18–55].In [56] we have connected the entropic force acting on a probe particle near a holographic screen with its surfacetension p A . Therefore, we have calculated the surface density of holographic entropy s A = d S/ d A as follows from theequipartition rule, so that s A = 14 G , (1)whereas the variation of screen energy E is given by the following relation:d E = T d S − p A d A, (2)with A denoting the screen area, while the surface tension is equal to p A = − T s A . (3)However, according to the Nernst heat theorem one should expect that the surface entropy density tends to zeroat zero temperature: s A → T →
0, because dynamical degrees of freedom are frozen out at absolute zero.Furthermore, according to [57] the derivative of pressure with respect to the temperature at a constant volume shouldtend to zero, so that in the case under consideration one gets (cid:18) ∂p A ∂T (cid:19) A → , at T → . (4)From (1) and (3) it follows that (cid:18) ∂p A ∂T (cid:19) A = − G = const. (5)That is in conflict with the Nernst heat theorem, and the same concerns for expression (1) determining the entropydensity, which is independent of the temperature.Obviously, one could satisfy the Nernst heat theorem if the degrees of freedom on the holographic screen are frozenout when approaching the absolute zero. Therefore, the equipartition distribution of energy is modified as E = 12 T N E = 12 T N B ( T ) , (6)where E = M is the gravitating mass inside the screen, and the number of “bits” N is equal to the number of Planckcells on the screen area N = AG , (7) The Newton gravitation constant G = ℓ , where ℓ Pl = 1 /m Pl is the Planck length inverse to the Planck mass. Here we put c = 1, ~ = 1 and k B = 1. while B ( T ) → , at T → . In [58] the following expression has been heuristically suggested B ( T ) = D ( x ) , (8)where D ( x ) is the one-dimensional Debye-function D ( x ) = 1 x x Z z d z e z − , (9)at x = T D /T . The new empirical parameter T D occurs to be comparable with the current value of Hubble constant H . At high temperatures T ≫ T D the equipartition rule is valid since D (0) = 1, but at low temperatures T ≪ T D the Debye function gets the linear behavior, namely D ( x ) x →∞ ≈ TT , T = 6 π T D . (10)It gives the modification of Newton gravitation law for a body of mass M , so that in the framework of holographicformulation the acceleration of free fall a = 2 πT becomes equal to a = GM a r , (11)where a = 2 πT is the Milgrom acceleration phenomenologically introduced in the MOND paradigm (modifiedNewtonian dynamics) [59] to explain the flat rotation curves of spiral galaxies, when the velocity of peripheral starsis asymptotically expressed in terms of the mass of visible matter M in the galaxy according to the formula v = GM a , (12)empirically known as the Tully-Fisher law.Note that the authors of [58] have not physically validated both the Debye factor and the connection between theDebye temperature and Hubble constant, though they have noted that the notion of Debye correction is meaningfulfor solid states, but not for holographic screens, since the factor describes the wide interval of temperatures fromsound vibrations to independent oscillations of lattice notes in the solid state bodies, when the spectrum of vibrationsis well approximated by the Debye formula. In addition, the introduction of one-dimensional Debye function in [58]is the guess beyond any reasonable speculations, and it raises a question on the dimension of holographic screen.Here we consider collective vibrations of holographic screen on the basis of its surface tension. We find the im-possibility of propagation of 2-dimensional sound waves with both polarization and wave vector lying in the planeof holographic screen. So, the only permissible collective motion of bits on the holographic screen is given by theone-dimensional scaling of a holographic screen as whole in the transverse direction to the screen plane . We analyzethe frequency spectrum of such motion and validate an appropriate thermal correction to the energy distributionlaw using the Debye method, i.e. a cutoff in the frequency spectrum. We investigate a physical interpretation for anumber of vibrational modes of the holographic screen and its connection to the scale hierarchy: the Planck scale,the Hubble rate and a sub-Planckian scale which is of the order of the grand unification scale in the particle physics. II. COLLECTIVE MOTIONS OF HOLOGRAPHIC SCREEN “BITS”
The sound speed squared c s in the medium is determined by the adiabatic derivative of pressure with respect tothe energy density. In the case of holographic screen we get c s = (cid:18) ∂p A ∂ρ A (cid:19) S , (13) In [60] Y.Bekenstein has constructed a relativistic Lagrangian formalism consistent with MOND. Surface oscillations of the holographic screen without gradients of density, pressure or temperature are possible with wave vectorstangent to the screen plane (see Appendix A). But because of holographic principle, these oscillations correspond to specific vibrationsof gravitating matter inside the screen, that breaks its initial spatial symmetry, for example, the spherical symmetry of field for a massivepoint-like source. where the surface energy density is denoted by ρ A = d E/ d A at high temperatures, when the equipartition rule isvalid, and it takes the form ρ A = 12 G T = 2
T s A . (14)Taking into account the expression for the surface tension in (3) we find that the speed squared for sound on theholographic screen is negative c s = − , (15)whereas this expression is valid at low temperatures too because thermal correction does not change the relationbetween the energy density and surface tension p A = − ρ A .Thus, the sound wave propagation along the screen with the polarization in the screen plane is absolutely impossible.This fact can be predicted from the definition of the holographic screen as the surface with a constant accelerationof free fall and, hence, with a constant temperature. Therefore, a temperature gradient is forbidden, hence, gradientsof the energy density and pressure do not propagate. The relation between the surface tension and surface energydensity confirms the validity of such the consideration in respect to the sound waves.The only permissible collective motion is the uniform extension or contraction of holographic screen as a whole inthe transverse direction to the screen, i.e. the scaling transformation depending on the time: r a ( t ) r , where thescale factor a ( t ) completely determines the dynamics of such motion with single degree of freedom. In Appendix A weconsider the transverse oscillations of the screen propagating along the screen. However, such oscillation with a wavevector lying in the screen plane are unavoidably accompanied by a motion of matter inside the holographic surface.In the case of spherically symmetric matter the vibrational wave vector should be directed orthogonally to the screen,only.It is easy to calculate the spectrum of such transverse motion because the scale factor is determined by the Universeevolution. Since we are interested of the present evolution, we can use the parametrization for the Hubble constant H in the form of H ( t ) = H (cid:18) Ω M a ( t ) + Ω Λ (cid:19) , a (0) = 1 , (16)where the energy balance in the flat Universe for fractions of the matter Ω M and the cosmological constant Ω Λ givesΩ M + Ω Λ = 1, and numerically Ω Λ ≈ .
76. In the epoch of cosmological constant the Hubble constant is equal to H Λ = H p Ω Λ , (17)and the scale factor behaves like a ( t ) = e H Λ t , t ∈ ( −∞ , . (18)The spectrum density ν ( ω ) = Z −∞ d t e − i ωt a ( t ) = 1 H Λ − i ω (19)gives the number of the scale-vibrational modes for the holographic screen N G ( ω ) in the frequency range (0 , ω ) N G ( ω ) = ω Z − ω d ω π | ν ( ω ) | = ω Z d ωπ p H + ω . (20)Notice that at small frequencies ω ≪ H Λ the frequency density becomes constant, and the number of modes isequal to N G ( ω ) ≈ ωπH Λ ≪ . (21)The contribution of such vibrations to the free energy F can be written as F = − T X bits , ω ln Z ( ω ) , (22)where the sum is taken over both “bits” on the screen and low frequency modes of oscillations, if N G ≪
1, while Z ( ω ) = (cid:16) − e − ω/T (cid:17) − . (23)At high temperatures ω ≪ T the equipartition rule is reproduced becauseln Z ≈ ln T − ln ω, and X bits , ω ln Z ( ω ) ≈ −
12 ˜ N bits N G ( h ln ω i − ln T )where the average value is defined as X ω ln ω = N G h ln ω i , so that the energy E = F − ∂ F /∂ ln T is equal to E = 12 T N, (24)where N = ˜ N bits N G is the total number of “bits” equal to the number of Planckian cells of the screen area, and ˜ N bits is the number of “bits” per the scale-vibrational mode of the screen.At low temperatures T ≪ ω by introducing the frequency cutoff ω < T D , formula (22) gives F = 12 T ˜ N bits T D Z d ωπH Λ ln (cid:16) − e − ω/T (cid:17) , (25)hence, the energy equals E = 12 T N D (cid:18) T D T (cid:19) , (26)where again N = ˜ N bits N G and N G = N G ( T D ), so that E ≈ N T T , T ≪ T . Obviously, the formulas (25) and (26) can be applicable not only when T ≪ T D , but in the full temperature range,if N G ≪ | ν ( ω ) | depends on the frequency).We find the relation T = 6 π H Λ N G = 6 π p Ω Λ H N G , (27)wherefrom it follows that T ≪ H in consistency with the empirical data setting a ≈ H / π , so that N G ≈ π √ Ω Λ ∼ − . (28)One can see that the quantity N G is the new fundamental parameter in the framework of thermodynamical andholographical gravity. III. DISCUSSION AND CONCLUSION
In our approach the thermodynamical “bit” on the holographic screen occupies the Planckian area A Pl = ℓ Pl × ℓ Pl ,defining the quantum of area. Note that a binding energy of the screen “bits” is perhaps of the order of Planck scaleitself. To speculate on the notion of surface, one should require that energy fluctuations in the transverse directionto the screen should be much less than the Planck energy.On the other hand, the “bits” on the screen experience the collective transverse motion as a whole. Then, thequantity defined by Λ G = N G ℓ Pl , represents the density of vibrational modes per the Planckian length. By dimension, it is the energy of transversemotion for the area quantum or “bit”. Therefore, the surface structure is rigid or stable, if Λ G ≪ m Pl .Thus, we draw the conclusion that the stability of holographic screen as the surface demands N G ≪
1, while bythe order of magnitude “the transverse energy of the Planckian quantum area” is Λ G ≈ N G m Pl ∼ − GeV, i.e.it is close to the scale of the grand unification theory (GUT).For the Milgrom acceleration we obtain a ≈ p Ω Λ H Λ G m Pl . (29)Thus, the basic parameter of the MOND involves the scale of vacuum energy (the cosmological constant), Planckmass and the scale closed to the GUT energy.In this paper we have given the justification for the modification of the equipartition rule for the “bits” on theholographic screen at low temperatures due to the proper description for the low-frequency collective transverseoscillations of screen at the presence of cosmological constant. The requirement on the stability of holographic screenwith respect to such oscillations is reduced to the introduction of energy scale which should be substantially less thanthe Planck energy. Then, the corrected low-temperature distribution of thermal energy on the holographic screenis reduced to the modified gravitational law like MOND at accelerations of free fall less than the critical Milgromacceleration. The Milgrom acceleration has been empirically introduced in the MOND paradigm in order to describethe flat rotation curves in spiral galaxies and to explain the Tully-Fisher law connecting the visible mass of galaxy tothe star velocity in the region dominating by the dark matter halo. Phenomenologically, observational data give thescale of transverse oscillating energy for the screen close to the scale of grand unification theory.Another treatment of Milgrom acceleration in terms of entropic force has been recently presented in [61]. Acknowledgments
This work was partially supported by grants of Russian Foundations for Basic Research 09-01-12123 and 10-02-00061, Special Federal Program “Scientific and academics personnel” grant for the Scientific and Educational Center2009-1.1-125-055-008, ant the work of T.S.A. was supported by the Russian President grant MK-406.2010.2.
Appendix A: Transverse oscillations running along the screen
Let us derive the speed of propagation for the transverse oscillations on the holographic screen with wave vectorin the screen plane. The screen can be considered as a membrane with the constant surface tension σ = − p A = T s A and the surface density of energy ρ A = E/A = 2
T s A in the approximation of high temperatures. Let us regard thesinusoidal wave running in the direction of axis X (Fig.1).At the initial time the wave equation reads off y ( x ) = A sin (cid:18) πxλ (cid:19) , where A is a small wave amplitude, λ is a wave length. Let us select the element of the wave between x and x ′ coordinates. Forces F and F ′ are equal to each other by magnitude σL , where L is a wave width in the directionperpendicular to the wave vector. But the directions of those forces are tangent to the screen and, hence, differentbecause the points of action slightly differ. Denote the angles between axes X and forces F, F ′ by α and α ′ , respectively, FIG. 1: Lateral oscillations of the screen. and put ∆ x = x ′ − x . Then the mass of selected element equals m = ρ A ∆ xL , where we ignore the slope of the waveto axis X , because the wave amplitude is small. Similarly, we neglect the net force along axis X . The net force alongaxis Y equals σL (sin α − sin α ′ ) = σL d y ( x )d x ∆ x = − σL ∆ x (cid:0) πλ (cid:1) y . Immediately, we obtain the equation of oscillations ρ A ∆ xL ¨ y = − σL ∆ x (cid:18) πλ (cid:19) y ⇒ ¨ y + c T y = 0 . So, we straightforwardly get the square of the wave speed c T = σρ A = 12 . Notice, that considered oscillations occur at zero gradients of the energy density and pressure, i.e. it is not a sound.Moreover, since the holographic screen is the surface with a constant acceleration of free fall, these oscillations takeplace in connection with proper oscillations of the matter inside the screen. [1] G. ’t Hooft, “Dimensional reduction in quantum gravity,” arXiv:gr-qc/9310026;[2] L. Susskind, “The World as a hologram,”
J. Math. Phys. , 6377 (1995) arXiv:hep-th/9409089.[3] R. Bousso, “The holographic principle,” Rev. Mod. Phys. , 825 (2002) arXiv:hep-th/0203101.[4] J. D. Bekenstein, “Black Holes And The Second Law,” Lett. Nuovo Cim. , 737 (1972);[5] J. D. Bekenstein, “Black Holes And Entropy,” Phys. Rev. D , 2333 (1973);[6] J. D. Bekenstein, “Generalized Second Law Of Thermodynamics In Black Hole Physics,” Phys. Rev. D , 3292 (1974).[7] S. W. Hawking, “Particle Creation By Black Holes,” Commun. Math. Phys. , 199 (1975);[8] J. B. Hartle and S. W. Hawking, “Path Integral Derivation Of Black Hole Radiance,” Phys. Rev. D , 2188 (1976).[9] S. W. Hawking, “Black Holes And Thermodynamics,” Phys. Rev. D , 191 (1976);[10] G. W. Gibbons and S. W. Hawking, “Cosmological Event Horizons, Thermodynamics, And Particle Creation,” Phys. Rev.D , 2738 (1977);[11] S. W. Hawking and G. T. Horowitz, “The Gravitational Hamiltonian, action, entropy and surface terms,” Class. Quant.Grav. , 1487 (1996) arXiv:gr-qc/9501014.[12] T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. , 1260 (1995)arXiv:gr-qc/9504004.[13] T. Padmanabhan, “Thermodynamical Aspects of Gravity: New insights,” Rept. Prog. Phys. , 046901 (2010)arXiv:0911.5004 [gr-qc];[14] T. Padmanabhan, “Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity,”arXiv:1003.5665[gr-qc].[15] E. P. Verlinde, “On the Origin of Gravity and the Laws of Newton,” arXiv:1001.0785 [hep-th].[16] T. Padmanabhan, “Equipartition of energy in the horizon degrees of freedom and the emergence of gravity,”arXiv:0912.3165 [gr-qc].[17] T. Padmanabhan, “Gravitational entropy of static spacetimes and microscopic density of states,” Class. Quant. Grav. ,4485 (2004) arXiv:gr-qc/0308070.[18] R. Banerjee and B. R. Majhi, “Statistical Origin of Gravity,” arXiv:1003.2312 [gr-qc];[19] R. Banerjee, B. R. Majhi, S. K. Modak, S. Samanta, “Smarr Formula and Killing Symmetries for Black Holes in ArbitraryDimensions,” arXiv:1007.5204 [gr-qc]; [20] Y.-X. Liu, Y.-Q. Wang and S.-W. Wei, “Temperature and Energy of 4-dimensional Black Holes from Entropic Force”,arXiv:1002.1062[hep-th].[21] F. Piazza, “Gauss-Codazzi thermodynamics on the timelike screen,” arXiv:1005.5151 [gr-qc][22] R. G. Cai, L. M. Cao and N. Ohta, “Friedmann Equations from Entropic Force,” Phys. Rev. D , 061501 (2010)arXiv:1001.3470 [hep-th];[23] F. W. Shu and Y. Gong, “Equipartition of energy and the first law of thermodynamics at the apparent horizon,”arXiv:1001.3237 [gr-qc];[24] M. Li and Y. Wang, “Quantum UV/IR Relations and Holographic Dark Energy from Entropic Force,” Phys. Lett. B ,243 (2010) arXiv:1001.4466 [hep-th];[25] Y. Wang, “Towards a Holographic Description of Inflation and Generation of Fluctuations from Thermodynamics,”arXiv:1001.4786 [hep-th];[26] D. A. Easson, P. H. Frampton and G. F. Smoot, “Entropic Accelerating Universe,” arXiv:1002.4278 [hep-th];[27] U. H. Danielsson, “Entropic dark energy and sourced Friedmann equations,” arXiv:1003.0668 [hep-th];[28] J. W. Lee, “Zero Cosmological Constant and Nonzero Dark Energy from Holographic Principle,” arXiv:1003.1878 [hep-th];[29] Y. F. Cai, J. Liu and H. Li, “Entropic cosmology: a unified model of inflation and late-time acceleration,” arXiv:1003.4526[astro-ph.CO];[30] A. Sheykhi, “Entropic Corrections to Friedmann Equations,” arXiv:1004.0627 [gr-qc];[31] R. G. Cai and S. P. Kim, “First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker uni-verse,”
JHEP , 050 (2005) arXiv:hep-th/0501055;[32] S. W. Wei, Y. X. Liu and Y. Q. Wang, “Friedmann equation of FRW universe in deformed Horava-Lifshitz gravity fromentropic force,” arXiv:1001.5238 [hep-th];[33] M. Li and Y. Pang, “A No-go Theorem Prohibiting Inflation in the Entropic Force Scenario,” arXiv:1004.0877 [hep-th];[34] D. A. Easson, P. H. Frampton, G. F. Smoot, “Entropic Inflation,” arXiv:1003.1528 [hep-th][35] S. W. Wei, Y. X. Liu and Y. Q. Wang, “Friedmann equation of FRW universe in deformed Horava-Lifshitz gravity fromentropic force,” arXiv:1001.5238 [hep-th].[36] C. Gao, “Modified Entropic Force,” arXiv:1001.4585 [hep-th];[37] S. Ghosh, “Planck Scale Effect in the Entropic Force Law,” arXiv:1003.0285 [hep-th];[38] I. V. Vancea and M. A. Santos, “Entropic Force Law, Emergent Gravity and the Uncertainty Principle,” arXiv:1002.2454[hep-th];[39] L. Modesto and A. Randono, “Entropic corrections to Newton’s law,” arXiv:1003.1998 [hep-th];[40] M. R. Setare and D. Momeni, “Revisiting the Entropic corrections to Newton’s law,” arXiv:1004.2794 [physics.gen-ph];[41] Y. Zhao, “Entropic force and its fluctuation in Euclidian quantum gravity,” arXiv:1002.4039 [hep-th].[42] H. Culetu, “Boundary stress tensors for spherically symmetric conformal Rindler observers,” arXiv:1001.4740 [hep-th];[43] H. Culetu, “Comments on ’On the Origin of Gravity and the Laws of Newton’, by Erik Verlinde,” arXiv:1002.3876 [hep-th];[44] J. W. Lee, “On the Origin of Entropic Gravity and Inertia,” arXiv:1003.4464 [hep-th];[45] I. V. Vancea and M. A. Santos, “Entropic Force Law, Emergent Gravity and the Uncertainty Principle,” arXiv:1002.2454[hep-th].[46] R. G. Cai, L. M. Cao and N. Ohta, “Notes on Entropy Force in General Spherically Symmetric Spacetimes,”
Phys. Rev.D , 084012 (2010) arXiv:1002.1136 [hep-th];[47] R. A. Konoplya, “Entropic force, holography and thermodynamics for static space-times,” arXiv:1002.2818 [hep-th];[48] P. Mahato, “Axial Current, Killing Vector and Newtonian Gravity,” arXiv:1004.1818 [gr-qc].[49] L. Smolin, “Newtonian gravity in loop quantum gravity,” arXiv:1001.3668 [gr-qc];[50] F. Caravelli and L. Modesto, “Holographic actions from black hole entropy,” arXiv:1001.4364 [gr-qc];[51] Y.-X. Liu, Y.-Q. Wang and S.-W. Wei, “Temperature and Energy of 4-dimensional Black Holes from Entropic Force”,arXiv:1002.1062[hep-th].[52] J. Makela, “Notes Concerning ’On the Origin of Gravity and the Laws of Newton’ by E. Verlinde (arXiv:1001.0785),”arXiv:1001.3808 [gr-qc].[53] A. Morozov, “Black Hole Motion in Entropic Reformulation of General Relativity,” arXiv:1003.4276 [hep-th].[54] Y. Tian and X. Wu, “Thermodynamics of Black Holes from Equipartition of Energy and Holography,” arXiv:1002.1275[hep-th].[55] Y. Tian and X. Wu, “Dynamics of Gravity as Thermodynamics on the Spherical Holographic Screen,” arXiv:1007.4331[hep-th][56] V. V. Kiselev and S. A. Timofeev, “The surface density of holographic entropy,” Mod. Phys. Lett. A , 2223 (2010)arXiv:1004.3418 [hep-th].[57] L. D. Landau, E. M. Lifshitz “Statistical Physics, Part 1”. Vol. 5 (3rd ed.). Butterworth-Heinemann (1980).[58] X. Li and Z. Chang, “Debye entropic force and modified Newtonian dynamics,” arXiv:1005.1169 [hep-th].[59] M. Milgrom, “A Modification Of The Newtonian Dynamics As A Possible Alternative To The Hidden Mass Hypothesis,”
Astrophys. J. , 365 (1983).[60] J. D. Bekenstein, “Relativistic gravitation theory for the MOND paradigm,”
Phys. Rev. D , 083509 (2004) [Erratum-ibid.D71