# Entropic Forces and Newton’s Gravitation

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## Abstract

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## 1. Introductory Materials

#### 1.1. A First Quantization Procedure

#### 1.2. Gravitation as an Entropic Force

#### 1.3. Main Effects of a First Quantization of Entropic Gravity (EG)

#### 1.4. Present Goals

#### 1.5. Structure of This Manuscript

## 2. Quantifying a Boson-Boson Interaction

#### 2.1. The Ensuing Potential Function

#### 2.2. A Taylor Approximation for $V\left(r\right)$

## 3. Solving the Schrödinger Equation

#### 3.1. ${V}_{1}$ Discussion

#### 3.2. ${V}_{2}$ Discussion

- ${R}_{l2}\left({r}_{0}\right)=0$,
- ${R}_{l2}^{{}^{\prime}}\left({r}_{0}\right)=0$,
- ${R}_{l2}\left({r}_{1}\right)=0$, and
- ${R}_{l2}^{{}^{\prime}}\left({r}_{1}\right)=0$.

#### 3.3. ${V}_{3}$ Discussion

#### 3.3.1. Analysis of the Case $E<0$

#### 3.3.2. Analysis of the Case $E>0$

## 4. A Dark Matter (DM) Model

#### 4.1. DM and Entropic Gravity

#### 4.2. Our DM-Model: Rough Numerical Estimates

- (2) We saw above that the energy equivalent of the total dark mass in the observable Universe is $K=2.96\times {10}^{84}$ eV [27].
- (3) We verify now that, indeed, $|{V}_{0}|$ is very small, as had been anticipated in Sect. 3 above. Its magnitude is $\frac{8.7}{{N}^{\frac{5}{6}}}\times {10}^{-23}$ eV Thus (setting $N=1$), we have $|{V}_{0}|<9.8\times {10}^{-22}$ eV, which is negligible compared to the first sum that appears in (17), where we have for the ground state ${E}_{0,0}$ a value ${\chi}_{0,0}=\pi $ and ${r}_{0}={10}^{-10}$. Thus, we immediately get ${E}_{0,0}=9.75\times {10}^{4}$ eV.
- (4) Therefore the number N of axions in the observable universe becomes approximately $2K/{E}_{0,0}\sim 6\times {10}^{79}$, if, as assumed here, the energy ${E}_{0,0}$ would be the sole origin of dark matter.

## 5. Summary

- Our path began with the Gupta-Feynman suggestion of looking for quantum states of gravity.
- This search was implemented by using Verlinde’s idea of gravity as an emergent statistical force, which lead then to
- a gravitation interaction functional form that differs from Newton’s for distances smaller than 25 microns.
- This Verlinde functional form was introduced as the potential term of a Schrödinger equation.
- the equation was solved, so that its eigenstates became determined,
- thus realizing Gupta-Feynman’s aspirations.
- We analyzed the pertinent eigenvalues and on such a basis made conjectures regarding dark matter.

- We started by accepting Verlinde’s suggestion that gravitation emerges from an entropic information measure S.
- We have approximated above $V\left(r\right)$ in a suitable fashion so as to obtain analytical solutions for the Schrödinger equation of potential $V\left(r\right)={V}_{1}+{V}_{2}+{V}_{3}$. It is of the essence to realize that ${V}_{1}\left(r\right)$ in Equation (5) supports bound states of the Schrödinger equation. Their associated self-energies provides then an as-yet unaccounted-for energy source.
- To repeat, the novelty of our treatment emerges at very short distances (the ${V}_{1}$ component of $V\left(r\right)$). The low-lying Schrödinger quantum states provide a novel energy-source, not accounted for previously. The pertinent energy eigenvalues yield, via Einstein’s relation energy $=m{c}^{2}$, a significant quantity of matter, that we might identify as dark one, of the order of five times the extant quantity of ordinary matter. As a matter of fact, one can limit oneself to the energy of the ground state of our Schrödinger equation to account for the extant amount of dark matter in the observable Universe.
- As just an illustration of the above line of reasoning, we considered a hypothetical dark matter model based on three hypotheses. The model involves a conjectural dark matter generating mechanism, working through (mutually) gravitationally interacting axions, in which the entropic (2-body) gravity potential according to Verlinde, emerges from a gas of axions.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gupta, S.N. Quantization of Einstein’s Gravitational Field: Linear Approximation. Proc. Pys. Soc. A
**1956**, 65, 161. [Google Scholar] [CrossRef] - Gupta, S.N. Quantization of Einstein’s Gravitational Field: General Treatment. Proc. Pys. Soc. A
**1952**, 65, 608. [Google Scholar] [CrossRef] - Gupta, S.N. Supplementary Conditions in the Quantized Gravitational Theory. Phys. Rev.
**1968**, 72, 1303. [Google Scholar] [CrossRef] - Feynman, R.P. Quantum Theory of Gravitation. Acta Phys. Pol.
**1963**, 24, 697. [Google Scholar] - Verlinde, E. On the Origin of Gravity and the Laws of Newton. arXiv
**2011**, arXiv:1001.0785. [Google Scholar] [CrossRef] [Green Version] - Plastino, A.; Rocca, M.C.; Ferri, G.L. Quantum treatment of Verlinde’s entropic force conjecture. Physica A
**2018**, 511, 139. [Google Scholar] [CrossRef] [Green Version] - Overbye, D. A Scientist Takes On Gravity. The New York Times. 12 July 2010. Available online: http://www.physicsland.com/Physics10_files/gravity.pdf (accessed on 12 July 2010).
- Makela, J. Notes Concerning “On the Origin of Gravity and the Laws of Newton” by E. Verlinde. arXiv
**2010**, arXiv:1001.3808v3. [Google Scholar] - Lee, J. Comments on Verlinde’s entropic gravity. arXiv
**2010**, arXiv:1005.1347. [Google Scholar] - Kiselev, V.V.; Timofeev, S.A. The surface density of holographic entropy. Mod. Phys. Lett. A
**2010**, 25, 2223. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Statistical mechanics of gravitating systems: An Overview. arXiv
**2008**, arXiv:0812.2610v2. [Google Scholar] - Guseo, R. Diffusion of innovations dynamics, biological growth and catenary function. Physica A
**2016**, 464, 1. [Google Scholar] [CrossRef] [Green Version] - Verlinde, E. Emergent Gravity and the Dark Universe. arXiv
**2017**, arXiv:1611.02269. [Google Scholar] [CrossRef] - Verlinde, E. The Hidden Phase Space of our Univers. Available online: http://www2.physics.uu.se/external/strings2011/presentations/5%20Friday/1220_Verlinde.pdf (accessed on 1 July 2011).
- Plastino, A.; Rocca, M.C. Statistical Mechanics-Based Schrodinger Treatment of Gravity. Entropy
**2019**, 21, 682. [Google Scholar] [CrossRef] [Green Version] - Lemons, D.S. A Student’s Guide to Entropy; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Smullin, S.J.; Geraci, A.A.; Weld, D.M.; Kapitulnik, A. Testing Gravity at Short Distances. In Proceedings of the SLAC Summer Institute on Particle Physics (SSI04), Menlo Park, CL, USA, 2–13 August 2004. [Google Scholar]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series and Products; Academic Press: New York, NY, USA, 1980. [Google Scholar]
- Majumdar, D. Dark Matter: An Introduction; CRC Press: New York, NY, USA, 2015. [Google Scholar]
- Ceyhan, F.A. Dark Matter as an Emergent Phenomenon of Entanglement. Available online: http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_Spring2018/Files/ceyhan.pdf (accessed on 13 May 2018).
- Peccei, R.D.; Quinn, H.R. CP Conservation in the Presence of Pseudoparticles. Phys. Rev. Lett.
**1977**, 38, 1440. [Google Scholar] [CrossRef] [Green Version] - Peccei, R.D.; Quinn, H.R. Constraints imposed by CP conservation in the presence of pseudoparticles. Phys. Rev. D
**1977**, 16, 1791. [Google Scholar] [CrossRef] - Peccei, R.D. The Strong CP Problem and Axions. In Axions: Lecture Notes in Physics; Springer: Berlin, Germany, 2008. [Google Scholar]
- Wilczek, F. Time’s (Almost) Reversible Arrow. Available online: https://www.ias.edu/news/wilczek-quanta-time (accessed on 7 January 2016).
- Borsanyi, S.; Fodor, Z.; Guenther, J.; Kampert, K.-H.; Katz, S.D.; Kawanai, T.; Kovacs, T.G.; Mages, S.W.; Pasztor, A.; Pittler, F.; et al. Calculation of the axion mass based on high-temperature lattice quantum chromodynamics. Nature
**2016**, 539, 69. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bergstrom, L. Dark matter candidates. New J. Phys.
**2009**, 11, 105006. [Google Scholar] [CrossRef] - Brooks, J. Galaxies and Cosmology; Archived 14 July 2014; Wayback Machine: San Francisco, CA, USA, 2014. [Google Scholar]
- Van Raamsdonk, M. Building up spacetime with quantum entanglement. Gen. Rel. Gravit.
**2010**, 42, 2323. [Google Scholar] [CrossRef] - Lashkari, N.; McDermott, M.B.; Van Raamsdonk, M. Gravitational dynamics from entanglement thermodynamics. J. High Energy Phys.
**2014**, 4, 195. [Google Scholar] [CrossRef] [Green Version] - Resconi, G.; Licata, I.; Fiscaletti, D. Unification of quantum and gravity by nonclassical information entropy space. Entropy
**2013**, 15, 3602. [Google Scholar] [CrossRef] [Green Version] - Ver Steeg, G.; Menicucci, N.C. Entangling power of an expanding universe. Phys. Rev. D
**2009**, 79, 044027. [Google Scholar] [CrossRef] [Green Version] - Zizzi, P.A. Holography, quantum geometry, and quantum information theory. Entropy
**2000**, 2, 39. [Google Scholar] [CrossRef] [Green Version] - Zizzi, P. Entangled spacetime. Mod. Phys. Letts. A
**2018**, 33. [Google Scholar] [CrossRef]

**Figure 1.**The orange curve (L) represents the exact potential given by (3) divided by $A=G{m}^{2}/{r}_{2}$. It is finite at the origin! (no UV troubles at all then). The violet curve (H) represents the long-range (Newtonian) approximate potential ${V}_{3}/A$ given by (6), where the horizontal coordinate $x=r/{r}_{2}$ and ${r}_{2}={(a/b)}^{1/3}$.

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Plastino, A.; Rocca, M.C.
Entropic Forces and Newton’s Gravitation. *Entropy* **2020**, *22*, 273.
https://doi.org/10.3390/e22030273

**AMA Style**

Plastino A, Rocca MC.
Entropic Forces and Newton’s Gravitation. *Entropy*. 2020; 22(3):273.
https://doi.org/10.3390/e22030273

**Chicago/Turabian Style**

Plastino, Angelo, and Mario Carlos Rocca.
2020. "Entropic Forces and Newton’s Gravitation" *Entropy* 22, no. 3: 273.
https://doi.org/10.3390/e22030273