Entropic Forces and Newton’s Gravitation
Abstract
: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
3. Solving the Schrödinger Equation
3.1. Discussion
3.2. Discussion
- ,
- ,
- , and
- .
3.3. Discussion
3.3.1. Analysis of the Case
3.3.2. Analysis of the Case
4. A Dark Matter (DM) Model
4.1. DM and Entropic Gravity
4.2. Our DM-Model: Rough Numerical Estimates
- (1) Calling to the axion mass, we have milli electron volt [25].
- (2) We saw above that the energy equivalent of the total dark mass in the observable Universe is eV [27].
- (3) We verify now that, indeed, is very small, as had been anticipated in Sect. 3 above. Its magnitude is eV Thus (setting ), we have eV, which is negligible compared to the first sum that appears in (17), where we have for the ground state a value and . Thus, we immediately get eV.
- (4) Therefore the number N of axions in the observable universe becomes approximately , if, as assumed here, the energy 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 in a suitable fashion so as to obtain analytical solutions for the Schrödinger equation of potential . It is of the essence to realize that 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 component of ). 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 , 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
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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
Plastino A, Rocca MC. Entropic Forces and Newton’s Gravitation. Entropy. 2020; 22(3):273. https://doi.org/10.3390/e22030273
Chicago/Turabian StylePlastino, Angelo, and Mario Carlos Rocca. 2020. "Entropic Forces and Newton’s Gravitation" Entropy 22, no. 3: 273. https://doi.org/10.3390/e22030273
APA StylePlastino, A., & Rocca, M. C. (2020). Entropic Forces and Newton’s Gravitation. Entropy, 22(3), 273. https://doi.org/10.3390/e22030273