# On the Thermodynamic Origin of Gravitational Force by Applying Spacetime Entanglement Entropy and the Unruh Effect

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## Abstract

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## 1. Introduction

## 2. Vacuum Quantum Fluctuations and the Unruh Effect for Minkowski Spacetime and Curved Spacetime

#### 2.1. The Unruh Effect for an Accelerating Particle in Minkowski Spacetime

#### 2.2. The Unruh Effect for Curved Spacetime

#### 2.3. The Unruh Effect for Gravitational Field ${h}_{\mu \nu}$

- ${T}_{U}$ should be regarded as a peak value of a local temperature distribution in an inertial frame of reference.
- Besides the case of an electrically-charged particle usually considered for the Unruh effect, the particle may have other types of charges. Hence, ${T}_{U}$ may also mean the temperature for other gauge fields, such as the gravitational field. Because the gravitational field is universal for any particle, Equation (7) can be applied to the gravitational field. In this paper, the Unruh temperature is considered mainly for the gravitational field.

## 3. Finite Spacetime Temperature Distribution Due to Matter

#### 3.1. The Spacetime Quantum Fluctuations

#### 3.2. Spacetime Entanglement Entropy and Spacetime Temperature

## 4. Newtonian Gravitational Force Derived by the Consideration of Local Spacetime Thermal Equilibrium

#### 4.1. Spacetime Thermal Equilibrium

#### 4.2. Quasi-Static Process to Determine the Direction of Gravitational Force

#### 4.3. Free-Fall Motion

## 5. Relativistic Formula of the Spacetime Temperature ${T}_{M}$ of a Classical Particle

## 6. Potential Application to Modified Gravity

^{2}[44], which means that the Compton wavelength of the graviton would be at least $1.6\times {10}^{16}$ m. This suggests that our theory will not give a significant modification to massive gravity for the long-wave mode below $1.6\times {10}^{16}$ m. However, when the cosmology evolution is addressed, we cannot exclude the possibility of significant modification due to massive gravity. Another possibility is the modification of the massive gravity to black holes [45,46,47,48,49,50], which has seen intensive studies in the last few years. Near the horizon of a black hole, the Unruh effect has close connection with Hawking radiation, and it would be interesting to consider the gravitational radiation in the Unruh effect and the relevant spacetime temperature in this work.

## 7. Summary and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Liu, S.; Xiong, H. On the Thermodynamic Origin of Gravitational Force by Applying Spacetime Entanglement Entropy and the Unruh Effect. *Entropy* **2019**, *21*, 296.
https://doi.org/10.3390/e21030296

**AMA Style**

Liu S, Xiong H. On the Thermodynamic Origin of Gravitational Force by Applying Spacetime Entanglement Entropy and the Unruh Effect. *Entropy*. 2019; 21(3):296.
https://doi.org/10.3390/e21030296

**Chicago/Turabian Style**

Liu, Shujuan, and Hongwei Xiong. 2019. "On the Thermodynamic Origin of Gravitational Force by Applying Spacetime Entanglement Entropy and the Unruh Effect" *Entropy* 21, no. 3: 296.
https://doi.org/10.3390/e21030296