# Entropic gravity

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Verlinde's statistical description of gravity as an entropic force leads to the correct inverse square distance law of attraction between classical bodies.

Entropic gravity is a theory in modern physics that describes gravity as an entropic force—not a fundamental interaction mediated by a quantum field theory and a gauge particle (like photons for the electromagnetic force, and gluons for the strong nuclear force), but a consequence of physical systems' tendency to increase their entropy. The proposal has been intensely contested in the physics community but it has also sparked a new line of research into thermodynamic properties of gravity.

## Origin

The thermodynamic description of gravity has a history that goes back at least to research on black hole thermodynamics by Bekenstein and Hawking in the mid-1970s. These studies suggest a deep connection between gravity and thermodynamics, which describes the behavior of heat. In 1995 Jacobson demonstrated that the Einstein field equations describing relativistic gravitation can be derived by combining general thermodynamic considerations with the equivalence principle.[1] Subsequently, other physicists, most notably Thanu Padmanabhan, began to explore links between gravity and entropy.[2][3]

## Erik Verlinde's theory

In 2009, Erik Verlinde disclosed a conceptual model that describes gravity as an entropic force.[4] On 6 January 2010 he published a preprint of a 29-page paper titled On the Origin of Gravity and the Laws of Newton.[5] The paper was published in the Journal of High Energy Physics in April 2011.[6] Reversing the logic of over 300 years, it argued (similar to Jacobson's result) that gravity is a consequence of the "information associated with the positions of material bodies". This model combines the thermodynamic approach to gravity with Gerard 't Hooft's holographic principle. It implies that gravity is not a fundamental interaction, but an emergent phenomenon which arises from the statistical behavior of microscopic degrees of freedom encoded on a holographic screen. The paper drew a variety of responses from the scientific community. Andrew Strominger, a string theorist at Harvard said “Some people have said it can’t be right, others that it’s right and we already knew it — that it’s right and profound, right and trivial."[7]

In July 2011 Verlinde presented the further development of his ideas in a contribution to the Strings 2011 conference, including an explanation for the origin of dark matter.[8]

Verlinde's article also attracted a large amount of media exposure,[9][10] and led to immediate follow-up work in cosmology,[11][12] the dark energy hypothesis,[13] cosmological acceleration,[14][15] cosmological inflation,[16] and loop quantum gravity.[17] Also, a specific microscopic model has been proposed that indeed leads to entropic gravity emerging at large scales.[18]

## Derivation of the law of gravitation

The law of gravitation is derived from classical statistical mechanics applied to the holographic principle, that states that the description of a volume of space can be thought of as ${\displaystyle N}$ bits of binary information, encoded on a boundary to the region, a surface of area ${\displaystyle A}$. The information is evenly distributed on the surface and each bit is stored on an elementary surface of area

${\displaystyle N=A/\ell _{\mathrm {P} }^{2}}$

where ${\displaystyle \ell _{\mathrm {P} }}$ is the Planck length. The statistical equipartition theorem relates the temperature ${\displaystyle T}$ of a system with its average energy

${\displaystyle E={\frac {1}{2}}Nk_{\text{B}}T}$

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant. This energy is identified with a mass ${\displaystyle M}$ by the mass–energy equivalence relation

${\displaystyle E=Mc^{2}}$.

The effective temperature experienced by a uniformly accelerating detector in a vacuum field is given by the Unruh effect. This temperature is

${\displaystyle T={\frac {\hbar a}{2\pi ck_{\text{B}}}},}$

where ${\displaystyle \hbar }$ is the reduced Planck constant, and ${\displaystyle a}$ is the local acceleration, which is related to a force ${\displaystyle F}$ by Newton's second law of motion

${\displaystyle F=ma}$.

By assuming that the holographic screen is a sphere of radius ${\displaystyle r}$, its surface is given by

${\displaystyle A=4\pi r^{2}}$,

and one derives from these principles Newton's law of universal gravitation

${\displaystyle F=G{\frac {mM}{r^{2}}}}$.

## Criticism and experimental tests

Entropic gravity, as proposed by Verlinde in his original article, reproduces Einstein field equations and, in a Newtonian approximation, a 1/r potential for gravitational forces. Since it does not make new physical predictions, it can not be falsified with existing experimental methods, at this time, any more than Newtonian gravity and general relativity.

Even so, entropic gravity in its current form has been severely challenged on formal grounds. Matt Visser, professor of mathematics at Victoria University of Wellington, NZ in "Conservative Entropic Forces" [19] has shown that the attempt to model conservative forces in the general Newtonian case (i.e. for arbitrary potentials and an unlimited number of discrete masses) leads to unphysical requirements for the required entropy and involves an unnatural number of temperature baths of differing temperatures. Visser concludes:

There is no reasonable doubt concerning the physical reality of entropic forces, and no reasonable doubt that classical (and semi-classical) general relativity is closely related to thermodynamics [52–55]. Based on the work of Jacobson [1–6], Thanu Padmanabhan [7– 12], and others, there are also good reasons to suspect a thermodynamic interpretation of the fully relativistic Einstein equations might be possible. Whether the specific proposals of Verlinde [26] are anywhere near as fundamental is yet to be seen — the rather baroque construction needed to accurately reproduce n-body Newtonian gravity in a Verlinde-like setting certainly gives one pause.

For the derivation of Einstein's equations from an entropic gravity perspective, Tower Wang shows in [20] that the inclusion of energy-momentum conservation and cosmological homogeneity and isotropy requirements severely restrict a wide class of potential modifications of entropic gravity, some of which have been used to generalize entropic gravity beyond the singular case of an entropic model of Einstein's equations. Wang asserts that

As indicated by our results, the modified entropic gravity models of form (2), if not killed, should live in a very narrow room to assure the energy-momentum conservation and to accommodate a homogeneous isotropic universe.

### Entropic gravity and quantum coherence

Another criticism of entropic gravity is that entropic processes should, as critics argue, break quantum coherence. Experiments with ultra-cold neutrons in the gravitational field of Earth are claimed to show that neutrons lie on discrete levels exactly as predicted by Schrödinger equation considering the gravitation to be a conservative potential field without any decoherent factors. Archil Kobakhidze argues that this result disproves entropic gravity.[21][22] Luboš Motl gives popular explanations of this position in his blog.[23][24]

It was argued in favor of entropic gravity that such inconsistencies may be removed.[25]

## References

1. ^ Jacobson, Theodore (4 April 1995). "Thermodynamics of Spacetime: The Einstein Equation of State". Phys. Rev. Lett. 75 (7): 1260–1263. arXiv:gr-qc/9504004. Bibcode:1995PhRvL..75.1260J. doi:10.1103/PhysRevLett.75.1260.
2. ^ Padmanabhan, Thanu (26 November 2009). "Thermodynamical Aspects of Gravity: New insights". Rep. Prog. Phys. 73 (4): 6901. arXiv:0911.5004. Bibcode:2010RPPh...73d6901P. doi:10.1088/0034-4885/73/4/046901.
3. ^ Mok, H.M. (13 August 2004). "Further Explanation to the Cosmological Constant Problem by Discrete Space-time Through Modified Holographic Principle". arXiv:physics/0408060 [physics.gen-ph].
4. ^ van Calmthout, Martijn (12 December 2009). "Is Einstein een beetje achterhaald?". de Volkskrant (in Dutch). Retrieved 6 September 2010.
5. ^ Verlinde, Eric (6 January 2010). "On the Origin of Gravity and the Laws of Newton". arXiv:1001.0785 [hep-th].
6. ^ E.P. Verlinde. "On the Origin of Gravity and the Laws of Newton". JHEP. arXiv:1001.0785. Bibcode:2011JHEP...04..029V. doi:10.1007/JHEP04(2011)029.
7. ^ Overbye, Dennis (12 July 2010). "A Scientist Takes On Gravity". The New York Times. Retrieved 6 September 2010.
8. ^ E. Verlinde, The Hidden Phase Space of our Universe, Strings 2011, Uppsala, 1 July 2011.
9. ^ The entropy force: a new direction for gravity, New Scientist, 20 January 2010, issue 2744
10. ^ Gravity is an entropic form of holographic information, Wired Magazine, 20 January 2010
11. ^ Fu-Wen Shu; Yungui Gong (2010). "Equipartition of energy and the first law of thermodynamics at the apparent horizon". arXiv:1001.3237 [gr-qc].
12. ^ Rong-Gen Cai; Li-Ming Cao; Nobuyoshi Ohta (2010). "Friedmann Equations from Entropic Force". Phys. Rev. D 81 (6). arXiv:1001.3470. Bibcode:2010PhRvD..81f1501C. doi:10.1103/PhysRevD.81.061501.
13. ^ It from Bit: How to get rid of dark energy, Johannes Koelman, 2010
14. ^ Easson; Frampton; Smoot (2010). "Entropic Accelerating Universe". Phys. Lett. B 696 (3): 273–277. arXiv:1002.4278. Bibcode:2011PhLB..696..273E. doi:10.1016/j.physletb.2010.12.025.
15. ^ Yi-Fu Cai; Jie Liu; Hong Li (2010). "Entropic cosmology: a unified model of inflation and late-time acceleration". Phys. Lett. B 690 (3): 213–219. arXiv:1003.4526. Bibcode:2010PhLB..690..213C. doi:10.1016/j.physletb.2010.05.033.
16. ^ Yi Wang (2010). "Towards a Holographic Description of Inflation and Generation of Fluctuations from Thermodynamics". arXiv:1001.4786 [hep-th].
17. ^ Lee Smolin (2010). "Newtonian gravity in loop quantum gravity". arXiv:1001.3668 [gr-qc].
18. ^ Jarmo Mäkelä (2010). "Notes Concerning "On the Origin of Gravity and the Laws of Newton" by E. Verlinde". arXiv:1001.3808 [gr-qc].
19. ^ Visser, Matt. "Conservative entropic forces". arXiv:1108.5240.
20. ^ Wang, Tower. "Modified entropic gravity revisited". arXiv:1211.5722.
21. ^ Kobakhidze, Archil. "Gravity is not an entropic force". arXiv:1009.5414.
22. ^ Kobakhidze, Archil. "Once more: gravity is not an entropic force". arXiv:1108.4161.
23. ^ Motl, Luboš. "Why gravity can't be entropic". The Reference Frame. Retrieved 10 March 2015.
24. ^ Motl, Luboš. "Once more: gravity is not an entropic force". The Reference Frame. Retrieved 29 April 2015.
25. ^ Chaichian, Masud; Oksanen, Markku; Tureanu, Anca (2012). "On entropic gravity: the entropy postulate, entropy content of screens and relation to quantum mechanics". Physics Letters B 712 (3). arXiv:1109.2794. Bibcode:2012PhLB..712..272C. doi:10.1016/j.physletb.2012.04.065.