# Is Gravity Entropic Force?

## Abstract

**:**

## 1. Introduction

## 2. Relations between Gravity and Thermodynamics

^{μ}is the unit normal vector field to the 3 dimension hypersurfaces ∑ and $\mathcal{E}$ = Γ

^{2}(ρ+p) − p with Γ the Lorentz factor. According to the scalar Gauss relation, one can get

_{ij}the extrinsic curvature tensor of ∑, and K the trace of the K

_{ij}. Combining Equations (8) and (10), we obtain the expression of ρ + p in general relativity [26]

_{m}is the Lie derivative along

**m**of any vector tangent to ∑ and D

_{i}is the Levi–Civita connection associated with the metric of the 3 dimension hypersurfaces ∑. Then Equation (11) can be expressed with three dimension spacial geometrical quantities as [26]

_{μν}= η

_{μν}+h

_{μν}, with h

_{μν}≪ 1. For matter with p ≃ 0 (the pressure of a body becomes important when its constituent particles are traveling at speeds close to that of light, which we can exclude from the Newtonian limit by hypothesis), such as dust or dark matter, we have the Poisson equation ∇

^{2}ϕ = 4πρ with ϕ = −h

_{00}/2. Taking into account Gibbs–Duhem relation (5) and the hypothesis (14), we obtain

## 3. Conclusions

_{gravitational source}= (ρ + p)

_{thermal source}, thermal quantities, such as entropy, temperature, and chemical potential, can induce gravitational effects, or gravity can induce thermal effects. For Newtonian approximation, the gravitational potential is related to the temperature, entropy, chemical potential, and particle number, which implies that gravity is entropic force only for systems with constant temperature and zero chemical potential. For general case, gravity is not an entropic force. Whether the results obtained here can be generalized to the case of modified gravity, such as F(R) gravity [10] and F( ) gravity [18], is worthy of investigation. All the analyses have been carried out without assuming a specific expression of temperature or horizon. For a static system at thermal equilibrium in general relativity, the temperature of the perfect fluid may take the form, $T\sqrt{-{g}_{00}}=const.$, which is called the Tolman temperature [34–36]. Whether the temperature in Equation (1) can be taken as Tolman temperature is also worthy of further investigation. The results we obtained confirm that there is a profound connection between gravity and thermodynamics.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Yang, R.
Is Gravity Entropic Force? *Entropy* **2014**, *16*, 4483-4488.
https://doi.org/10.3390/e16084483

**AMA Style**

Yang R.
Is Gravity Entropic Force? *Entropy*. 2014; 16(8):4483-4488.
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**Chicago/Turabian Style**

Yang, Rongjia.
2014. "Is Gravity Entropic Force?" *Entropy* 16, no. 8: 4483-4488.
https://doi.org/10.3390/e16084483