# Entropy and Gravitation—From Black Hole Computers to Dark Energy and Dark Matter

## Abstract

**:**

## 1. Introduction

## 2. Quantum Fluctuations of Spacetime

#### 2.1. Gedanken Experiment

#### 2.2. Mapping the Geometry of Spacetime

## 3. Clocks, Computers and Black Holes

## 4. Dark Energy

#### 4.1. Spacetime Foam and Dark Energy

#### 4.2. Dark Energy as Quanta of Infinite Statistics

## 5. From Causal-Set Theory and Unimodular Gravity to Space-Time Foam

## 6. Dark Matter

#### 6.1. From Gravitational Thermodynamics /Entropic Gravity to MDM

- (I)
- Newton’s 2nd law $\overrightarrow{F}=m\overrightarrow{a}$:
- (a)
- Verlinde uses the first law of thermodynamics to propose the concept of entropic force ${F}_{entropic}=T\frac{\Delta S}{\Delta x}.$
- (b)
- Then he invokes Bekenstein’s original arguments concerning the entropy S of black holes: $\Delta S=2\pi {k}_{B}\frac{mc}{\hslash}\Delta x$.
- (c)
- Finally he applies the formula for the Unruh temperature, ${k}_{B}T=\frac{\hslash a}{2\pi c},$ associated with a uniformly accelerating (Rindler) observer.

- (II)
- Newton’s law of gravity $a=GM/{r}^{2}$:
- (a)
- Verlinde considers an imaginary quasi-local (spherical) holographic screen of area $A=4\pi {r}^{2}$ with temperature T.
- (b)
- Then he uses equipartition of energy $E=\frac{1}{2}N{k}_{B}T$ with $N=A{c}^{3}/(G\hslash )$ being the total number of degrees of freedom (bits) on the screen.
- (c)
- Finally he applies the Unruh temperature formula and $E=M{c}^{2}$.

#### 6.2. Quanta of MDM Obey Infinite Statistics

#### 6.3. Observational Tests of MDM

## 7. Turbulence and Spacetime Foam

## 8. Summary and Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Energy-Momentum Fluctuations and Possible Tests of Spacetime Foam

#### Appendix A.1. Energy-Momentum Fluctuations

#### Appendix A.2. Possible Ways to Test Spacetime Foam

## Appendix B. Infinite Statistics

## References

- Lloyd, S.; Ng, Y.J. Black Hole Computers. Sci. Am.
**2004**, 291, 52–61. [Google Scholar] [CrossRef] [PubMed] - Wheeler, J.A. Relativity, Groups and Topology; DeWitt, B.S., DeWitt, C.M., Eds.; Gordon & Breach: New York, NY, USA, 1963; p. 315. [Google Scholar]
- Hawking, S.W.; Page, D.N.; Pope, C.N. Quantum Gravitational Bubbles. Nucl. Phys.
**1980**, 170, 283–306. [Google Scholar] [CrossRef] - Ashtekar, A.; Rovelli, C.; Smolin, L. Weaving a Classical Geometry with Quantum Threads. Phys. Rev. Lett.
**1992**, 69, 237–240. [Google Scholar] [CrossRef] [PubMed] - Ng, Y.J.; van Dam, H. Limit to Spacetime Measurement. Mod. Phys. Lett. A
**1994**, 9, 335–340. [Google Scholar] - Ng, Y.J.; van Dam, H. Remarks on Gravitational Sources. Mod. Phys. Lett. A
**1995**, 10, 2801–2808. [Google Scholar] [CrossRef] - Salecker, H.; Wigner, E.P. Quantum Limitations of the Measurement of Space-Time Distances. Phys. Rev.
**1958**, 109, 571–577. [Google Scholar] [CrossRef] - Karolyhazy, F. Gravitation and Quantum Mechanics of Macroscopic Objects. Il Nuovo Cimento
**1966**, A 42, 390–402. [Google Scholar] [CrossRef] - Sasakura, N. An Uncertainty Relation of Space-Time. Prog. Theor. Phys.
**1999**, 102, 169–179. [Google Scholar] [CrossRef] [Green Version] - Margolus, N.; Levitin, L.B. The Maximum Speed of Dynamical Evolution. Physica D
**1998**, 120, 188–195. [Google Scholar] [CrossRef] - ’t Hooft, G. Salamfestschrift; Ali, A., Ellis, J., Randjbar-Daemi, S., Eds.; World Scientific: Singapore, 1993; p. 284. [Google Scholar]
- Susskind, L. The World as a Hologram. J. Math. Phys. (N. Y.)
**1995**, 36, 6377–6396. [Google Scholar] [CrossRef] - Maldacena, J. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys.
**1998**, 2, 231–252. [Google Scholar] [CrossRef] - Gambini, R.; Pullin, J. Holography in Spherically Symmetric Loop Quantum Gravity. Int. J. Mod. Phys. D
**2008**, 17, 545–549. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black Holes and Entropy. Phys. Rev.
**1973**, D 7, 2333–2346. [Google Scholar] [CrossRef] - Hawking, S. Particle Creation by Black Holes. Comm. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] [CrossRef] - Ng, Y.J. From Computation to Black Holes and Space-time Foam. Phys. Rev. Lett.
**2001**, 86, 2946–2949, Erratum in**2002**, 88, 139902. [Google Scholar] [CrossRef] [PubMed] - Barrow, J.D. Wigner Inequalities for Black Holes. Phys. Rev. D
**1996**, 54, 6563–6564. [Google Scholar] [CrossRef] [PubMed] - Arzano, M.; Kephart, T.W.; Ng, Y.J. From Spacetime Foam to Holographic Foam Cosmology. Phys. Lett.
**2007**, B 649, 243–246. [Google Scholar] [CrossRef] - Maziashvili, M. Space-Time in Light of Karolyhazy Uncertainty Relation. Int. J. Mod. Phys. D
**2007**, 16, 1531–1539. [Google Scholar] [CrossRef] - Ng, Y.J. Holographic Foam, Dark Energy and Infinite Statistics. Phys. Lett. B
**2007**, 657, 10–14. [Google Scholar] [CrossRef] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiattia, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, R.P.; et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron. J.
**1998**, 116, 1009–1038. [Google Scholar] [CrossRef] [Green Version] - Fischler, W.; Susskind, L. Holography and Cosmology. arXiv
**1998**, arXiv:hep-th/9806039. [Google Scholar] - Easther, R.; Lowe, D. Holography, Cosmology, and the Second Law of Thermodynamics. Phys. Rev. Lett.
**1999**, 82, 4967–4970. [Google Scholar] [CrossRef] [Green Version] - Ng, Y.J. Proceedings of the Fortieth Karpacz Winter School on Theoretical Physics; Kowalski-Glikman, J., Amelino-Camelia, G., Eds.; Springer: Berlin, Germany, 2005; p. 321. [Google Scholar]
- Doplicher, S.; Haag, R.; Roberts, J. Local Observables and Particle Statistics I. Commun. Math. Phys.
**1971**, 23, 199–230. [Google Scholar] [CrossRef] - Doplicher, S.; Haag, R.; Roberts, J. Local Observables and Particle Statistics II. Commun. Math. Phys.
**1974**, 35, 49–85. [Google Scholar] [CrossRef] - Greenberg, O.W. Example of Infinite Statistics. Phys. Rev. Lett.
**1990**, 64, 705–708. [Google Scholar] [CrossRef] - Jejjala, V.; Kavic, M.; Minic, D. Fine Structure of Dark Energy and New Physics. Adv. High Energy Phys.
**2007**, 21586, 2007. [Google Scholar] [CrossRef] - D’Olivo, J.C.; Nahmad-Achar, E.; Rosenbaum, M.; Ryan, M.P.; Urrutiaet, L.F., Jr. Relativity and Gravitation: Classical and Quantum. In Proceedings of the 7th Latin American Symposium on Relativity and Gravitation (SILARG VII), Cocoyoc, Mexico, 2–8 December 1990. [Google Scholar]
- Van der Bij, J.J.; van Dam, H.; Ng, Y.J. The Exchange of Massless Spin-Two Particles. Physica
**1982**, A116, 307–320. [Google Scholar] [CrossRef] - Anderson, J.L.; Finkelstein, D. Cosmological Constant and Fundamental Length. Am. J. Phys.
**1971**, 39, 901. [Google Scholar] [CrossRef] - Ng, Y.J.; van Dam, H. Possible solution to the cosmological-constant problem. Phys. Rev. Lett.
**1990**, 65, 1972–1974. [Google Scholar] [CrossRef] - Ng, Y.J.; van Dam, H. A small but nonzero cosmological constant. Int. J. Mod. Phys. D
**2001**, 10, 49–55. [Google Scholar] [CrossRef] - Ng, Y.J. Selected topics in Planck-scale physics. Mod. Phys. Lett. A
**2003**, 18, 1073–1098. [Google Scholar] [CrossRef] - Barrow, J.D.; Shaw, D. New Solution of the Cosmological Constant Problems. Phys. Rev. Lett.
**2011**, 106, 101302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Henneaux, M.; Teitelboim, C. The cosmological constant and general covariance. Phys. Lett. B
**1989**, 222, 195. [Google Scholar] [CrossRef] - Baum, E. Zero cosmological constant frm minimum action. Phys. Lett. B
**1983**, 133, 185–186. [Google Scholar] [CrossRef] - Hawking, S.W. The cosmological constant is probably zero. Phys. Lett. B
**1984**, 134, 403–404. [Google Scholar] [CrossRef] - Tulin, S.; Yu, H.B. Dark Matter Self-interactions and Small Scale Structure. Phys. Rept.
**2018**, 730, 1–57. [Google Scholar] [CrossRef] - Milgrom, M. A Modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J.
**1983**, 70, 36365–36370. [Google Scholar] [CrossRef] - Tully, R.B.; Fisher, J.R. A New method of determining distances to galaxies. Astron. Astrophys.
**1977**, 54, 661–673. [Google Scholar] - Milgrom, M.; Sanders, R.H. Rings and shells of dark matter as MOND artifacts. Astrophys. J.
**2008**, 678, 131–143. [Google Scholar] [CrossRef] - McGaugh, S.; Lelli, F.; Schombert, J. Radial Acceleration Relation in Rotationally Supported Galaxies. Phys. Rev. Lett.
**2016**, 117, 201101. [Google Scholar] [CrossRef] [PubMed] - Ho, C.M.; Minic, D.; Ng, Y.J. Cold Dark Matter with MOND Scaling. Phys. Lett. B
**2010**, 693, 567–570. [Google Scholar] [CrossRef] - Ho, C.M.; Minic, D.; Ng, Y.J. Quantum Gravity and Dark Matter. Int. J. Mod. Phys. D
**2011**, 20, 2887–2893. [Google Scholar] [CrossRef] - Cadoni, M.; Casadio, R.; Giusti, A.; Mück, W.; Tuveri, M. Effective Fluid Description of the Dark Universe. Phys. Lett. B
**2018**, 776, 242. [Google Scholar] [CrossRef] - Jacobson, T. Thermodynamics of space-time: The Einstein equation of state. Phys. Rev. Lett.
**1995**, 75, 1260–1263. [Google Scholar] [CrossRef] [PubMed] - Verlinde, E.P. On the Origin of Gravity and the Laws of Newton. JHEP
**2011**, 1104, 029. [Google Scholar] [CrossRef] - Padmanabhan, T. Emergent Gravity Paradigm: Recent Progress. Mod. Phys. Lett. A
**2015**, 30, 1540007. [Google Scholar] [CrossRef] - Chakraborty, S.; Padmanabhan, T. Thermodynamical interpretation of the geometrical variables associated with null surfaces. Phys. Rev. D
**2015**, 92, 104011. [Google Scholar] [CrossRef] - Davies, P.C.W. Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A
**1975**, 8, 609–616. [Google Scholar] [CrossRef] - Unruh, W.G. Notes on black hole evaporation. Phys. Rev. D
**1976**, 14, 870. [Google Scholar] [CrossRef] - Deser, S.; Levin, O. Accelerated detectors and temperature in (anti)-deSitter spaces. Class. Quant. Grav.
**1997**, 14, L163–L168. [Google Scholar] [CrossRef] - Jacobson, T. Comment on ‘Accelerated detectors and temperature in anti-de Sitter spaces’. Class. Quant. Grav.
**1998**, 15, 251–253. [Google Scholar] [CrossRef] - Ho, C.M.; Minic, D.; Ng, Y.J. Dark Matter, Infinite Statistics and Quantum Gravity. Phys. Rev. D
**2012**, 85, 104033. [Google Scholar] [CrossRef] - Blanchet, L. Gravitational polarization and the phenomenology of MOND. Class. Quant. Grav.
**2007**, 24, 35293540. [Google Scholar] [CrossRef] - Gibbons, G.W. Aspects of Born-Infeld theory and string/M theory. Rev. Mex. Fis.
**2003**, 49S1, 19–29. [Google Scholar] - Edmonds, D.; Farrah, D.; Ho, C.M.; Minic, D.; Ng, Y.J.; Takeuchi, T. Testing MONDian Dark Matter with Galactic Rotation Curves. ApJ
**2014**, 793, 41. [Google Scholar] [CrossRef] - Edmonds, D.; Farrah, D.; Ho, C.M.; Minic, D.; Ng, Y.J.; Takeuchi, T. Testing modified dark matter with galaxy clusters: Does dark matter know about the cosmological constant? Int. J. Mod. Phys. A
**2017**, 32, 1750108. [Google Scholar] [CrossRef] - Cadoni, M.; Casadio, R.; Giusti, A.; Tuveri, M. Emergence of a Dark Force in Corpuscular Gravity. Phys. Rev. D
**2018**, 97, 044047. [Google Scholar] [CrossRef] - Edmonds, D.; Farrah, D.; Minic, D.; Ng, Y.J.; Takeuchi, T. Modified Dark Matter: Relating Dark Energy, Dark Matter and Baryonic Matter. Int. J. Mod. Phys. D
**2018**, 27, 1830001. [Google Scholar] [CrossRef] - Tolman, R.C. On the Weight of Heat and Thermal Equilibrium in General Relativity. Phys. Rev.
**1930**, 35, 904–924. [Google Scholar] [CrossRef] [Green Version] - Tolman, R.C.; Paul Ehrenfest, P. Temperature Equilibrium in a Static Gravitational Field. Phys. Rev.
**1930**, 36, 1791–1798. [Google Scholar] [CrossRef] [Green Version] - Sanders, R.H. Resolving the virial discrepancy in clusters of galaxies with modified newtonian dynamics. Astrophys. J.
**1999**, 512, L23. [Google Scholar] [CrossRef] - Ng, Y.J.; Edmonds, D.; Farrah, D.; Minic, D.; Takeuchi, T.; Ho, C.M. Modified Dark Matter. In Proceedings of the 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG14) (In 4 Volumes), Rome, Italy, 12–18 July 2015; Bianchi, M., Jantzen, R.T., Ruffini, R., La Sapienza, U.R., Eds.; World Scientific: Singapore, 2017; p. 3942. [Google Scholar]
- Chu, X.; Garcia-Cely, C.; Murayama, H. Puffy Dark Matter. arXiv
**2019**, arXiv:1901.00075. [Google Scholar] - Kolmogorov, A.N. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR
**1941**, 30, 299–303. [Google Scholar] [CrossRef] - Kolmogorov, A.N. Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR
**1941**, 32, 16–18. [Google Scholar] [CrossRef] - Jejjala, V.; Minic, D.; Ng, Y.J.; Tze, C.H. Turbulence and holography. Class. Quant. Grav.
**2008**, 25, 225012. [Google Scholar] [CrossRef] - Unruh, W. Dumb holes and the effects of high frequencies on black hole evaporation. Phys. Rev. D
**1995**, 51, 2827–2838. [Google Scholar] [CrossRef] - Wheeler, J.A. Geons. Phys. Rev.
**1955**, 97, 511–536. [Google Scholar] [CrossRef] - Strominger, A. Black Hole Statistics. Phys. Rev. Lett.
**1993**, 71, 3397–3400. [Google Scholar] [CrossRef] - Volovich, I.V. D-branes, Black Holes and SU(∞) Gauge Theory. arXiv
**1996**, arXiv:hep-th/9608137. [Google Scholar] - Maldacena, J.; Susskind, L. Cool horizons for entangled black holes. Fortsch. Phys.
**2013**, 61, 781. [Google Scholar] [CrossRef] - Adler, S.L. Quantum Theory as an Emergent Phenomenon; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Singh, T.P. Proposal for a new quantum theory of gravity. arXiv
**2019**, arXiv:1910.06350. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Ellis, J.; Mavromatos, N.E.; Nanopoulos, D.V.; Sarkar, S. Tests of Quantum Gravity from Observations of γ-Ray Bursts. Nature
**1998**, 393, 763–765. [Google Scholar] [CrossRef] - Ng, Y.J.; Christiansen, W.A.; van Dam, H. Probing Planck-scale Physics with Extragalacic Sources? Astrophys. J.
**2003**, 591, L87–L90. [Google Scholar] [CrossRef] - Lieu, R.; Hillman, L.W. The Phase Coherenece of Light from Extragalactic Sources—Direct Evidence Against First Order Planck Scale Fluctuations in Time and Space. Astrophys. J.
**2003**, 585, L77–L80. [Google Scholar] [CrossRef] - Christiansen, W.A.; Ng, Y.J.; van Dam, H. Probing Spacetime Foam with Extragalactic Sources. Phys. Rev. Lett.
**2006**, 96, 051301. [Google Scholar] [CrossRef] [Green Version] - Perlman, E.S.; Rappaport, S.A.; Christiansen, W.A.; Ng, Y.J.; DeVore, J.; Pooley, D. New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations. Astrophys. J.
**2015**, 805, 10. [Google Scholar] [CrossRef] - Perlman, E.S.; Rappaport, S.A.; Ng, Y.J.; Christiansen, W.A.; DeVore, J.; Pooley, D. New constraints on quantum foam models from X-ray and gamma-ray observations of distant quasars. In Proceedings of the 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG14) (In 4 Volumes), Rome, Italy, 12–18 July 2015; Bianchi, M., Jantzen, R.T., Ruffini, R., La Sapienza, U.R., Eds.; World Scientific: Singapore, 2017; p. 3935. [Google Scholar]
- Amelino-Camelia, G. Limits on the Measurability of Space-Time Distances in the Semi-Classical Approximation of Quantum Gravity. Mod. Phys. Lett. A
**1994**, 9, 3415–3422. [Google Scholar] [CrossRef] - Amelino-Camlia, G. An Interferometric Gravitational Wave Detector as a Quantum-Gravity Apparatus. Nature
**1999**, 398, 216–218. [Google Scholar] [CrossRef] - Ng, Y.J.; van Dam, H. Measuring the Foaminess of Spacetime with Gravity-Wave Interferometers. Found. Phys.
**2000**, 30, 795–805. [Google Scholar] [CrossRef] - Fredenhagen, K. On the Existence of Antiparticles. Commun. Math. Phys.
**1981**, 79, 141–151. [Google Scholar] [CrossRef]

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Ng, Y.J.
Entropy and Gravitation—From Black Hole Computers to Dark Energy and Dark Matter. *Entropy* **2019**, *21*, 1035.
https://doi.org/10.3390/e21111035

**AMA Style**

Ng YJ.
Entropy and Gravitation—From Black Hole Computers to Dark Energy and Dark Matter. *Entropy*. 2019; 21(11):1035.
https://doi.org/10.3390/e21111035

**Chicago/Turabian Style**

Ng, Y. Jack.
2019. "Entropy and Gravitation—From Black Hole Computers to Dark Energy and Dark Matter" *Entropy* 21, no. 11: 1035.
https://doi.org/10.3390/e21111035