# Quantum Capacity and Vacuum Compressibility of Spacetime: Thermal Fields

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## Abstract

**:**

Contents | ||

1. | Introduction | 2 |

1.1. Energy and Pressure Densities of Quantum Fields and Fluctuations........................................................................................................ | 2 | |

1.2. Physical Contexts of Quantum Capacitance and Negative Compressibility............................................................................................. | 3 | |

1.3. Methods, Findings and Organization............................................................................................................................................................ | 4 | |

2. | 2d: Thermal Casimir and Einstein Cylinder | 5 |

2.1. Low Temperature Expansion: Approaching R^{1} × S^{1}.................................................................................................................................... | 6 | |

2.2. High Temperature Expansion: Approaching S^{1} × R^{1}................................................................................................................................... | 10 | |

3. | 4d: Einstein UniverseS^{1} × S^{3} | 13 |

3.1. Low Temperature Expansion....................................................................................................................................................................... | 14 | |

3.2. High Temperature Expansion...................................................................................................................................................................... | 16 | |

4. | Even Spatial Dimensions:S^{1} × S^{2} and S^{1} × S^{4} | 19 |

4.1. Low Temperature Expansion....................................................................................................................................................................... | 19 | |

4.2. High Temperature Expansion...................................................................................................................................................................... | 21 | |

5. | Conclusions and Discussions.................................................................................................................................................................................. | 24 |

References............................................................................................................................................................................................................................. | 26 |

## 1. Introduction

#### 1.1. Energy and Pressure Densities of Quantum Fields and Fluctuations

**Importance of Quantum Field Fluctuations Phenomena**

**Thermodynamics of quantum fields reflecting the properties of spacetime**

#### 1.2. Physical Contexts of Quantum Capacitance and Negative Compressibility

**Nuclear compressibility in quantum hadrodynamics**

**What does the negative compressibility of a 2-dim electron gas reveal?**

#### 1.3. Methods, Findings and Organization

## 2. 2d: Thermal Casimir and Einstein Cylinder

#### 2.1. Low Temperature Expansion: Approaching ${R}^{1}\times {S}^{1}$

#### 2.2. High Temperature Expansion: Approaching ${S}^{1}\times {R}^{1}$

## 3. 4d: Einstein Universe ${S}^{1}\times {S}^{3}$

#### 3.1. Low Temperature Expansion

#### 3.2. High Temperature Expansion

## 4. Even Spatial Dimensions: ${S}^{1}\times {S}^{2}$ and ${S}^{1}\times {S}^{4}$

#### 4.1. Low Temperature Expansion

#### 4.2. High Temperature Expansion

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Note

1 | Relevant to the underlying theme of our inquiry is how geometry and topology play a role in determining the thermodynamic properties of spacetimes, and to what extent we can derive these properties from the thermodynamic properties of quantum fields. A beautiful example of the former approach is Wald’s proof of the Bekenstein-Hawking (B-H) entropy as the Noether charge of diffeomorphism symmetry [11]. An example of the latter approach (taken by many particle/field theorists) is Frolov and Fursaev’s [12] derivation of the B-H entropy the spacetime by working with the thermal field outside of the black hole horizon. Here, we are not saying anything, or implying to say that our work can reveal anything, about the microscopic structrues of spacetime. That belongs to the realm of quantum gravity in its multifarious forms, formulations, approaches and emphasis. To name just two familiar historical proposals. In string theory Strominger and Vafa [13] proposed the B-H entropy can be derived by counting the degeneracy of BPS solition bound states. In loop quantum gravity, Ashtekar et al. [14] showed that the constant of proportionality in the B-H entropy formula is determined by the Immirzi parameter, which fixes the spectrum of the area operator. |

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$\mathit{d}=2$ | 3 | 4 | 5 | |
---|---|---|---|---|

$\rho $ | $-\pi /6L$ | $-0.0105478/{a}^{3}$ | Equation (73) | $-0.00820215/{a}^{5}$ |

S | $1-\frac{1}{2}\mathrm{ln}\left(\mu {\beta}^{2}\right)$ | $1-\frac{1}{2}\mathrm{ln}\left(\mu {\beta}^{2}\right)$ | $1-\frac{1}{2}\mathrm{ln}\left(\mu {\beta}^{2}\right)$ | $1-\frac{1}{2}\mathrm{ln}\left(\mu {\beta}^{2}\right)$ |

${C}_{V}$ | 1 | 1 | 1 | 1 |

${C}_{P}$ | 1 | 1 | 1 | 1 |

P | $-\pi /6{L}^{2}$ | $-0.0053/{a}^{3}$ | Equation (74) | $-0.0021/{a}^{5}$ |

${\kappa}_{T}$ | $-3{L}^{2}/\pi $ | $-126{a}^{3}$ | Equation (76) | $-390{a}^{5}$ |

${\kappa}_{S}$ | $-3{L}^{2}/\pi $ | $-126{a}^{3}$ | Equation (76) | $-390{a}^{5}$ |

$\alpha $ | $\sim {e}^{-2\pi \beta /L}$ | $\sim {e}^{-\sqrt{2}\beta /a}$ | $\sim {e}^{-\sqrt{3}\beta /a}$ | $\sim {e}^{-2\beta /a}$ |

$\mathit{d}=2$ | 3 | 4 | 5 | |
---|---|---|---|---|

$\rho $ | $\pi /6{\beta}^{2}$ | $\zeta \left(3\right)/\pi {\beta}^{3}$ | ${\pi}^{2}/30{\beta}^{4}$ | $3\zeta \left(5\right)/{\pi}^{2}{\beta}^{5}$ |

S | $\pi L/3\beta $ | $6\zeta \left(3\right){a}^{2}/{\beta}^{2}$ | $4{\pi}^{4}{a}^{3}/45{\beta}^{3}$ | $10\zeta \left(5\right){a}^{4}/{\beta}^{4}$ |

${C}_{V}$ | $\pi L/3\beta $ | $12\zeta \left(3\right){a}^{2}/{\beta}^{2}$ | $4{\pi}^{4}{a}^{3}/15{\beta}^{3}$ | $40\zeta \left(5\right){a}^{4}/{\beta}^{4}$ |

${C}_{P}$ | $-{\pi}^{2}{L}^{2}/9{\beta}^{2}$ | $-108{\left(\zeta \left(3\right)\right)}^{2}{a}^{4}/{\beta}^{3}$ | $32{\pi}^{6}{a}^{5}/75{\beta}^{5}$ | $600{\left(\zeta \left(5\right)\right)}^{2}{a}^{6}/\zeta \left(3\right){\beta}^{5}$ |

P | $\pi /6{\beta}^{2}$ | $\zeta \left(3\right)/2\pi {\beta}^{3}$ | ${\pi}^{2}/90{\beta}^{4}$ | $3\zeta \left(5\right)/4{\pi}^{2}{\beta}^{5}$ |

${\kappa}_{T}$ | $-\beta L$ | $-12\pi \beta {a}^{2}$ | $108{\beta}^{2}{a}^{2}$ | $16{\pi}^{2}{\beta}^{3}{a}^{2}/\zeta \left(3\right)$ |

${\kappa}_{S}$ | $3{\beta}^{2}/\pi $ | $4\pi {\beta}^{3}/3\zeta \left(3\right)$ | $135{\beta}^{4}/2{\pi}^{2}$ | $16{\pi}^{2}{\beta}^{5}/15\zeta \left(5\right)$ |

$\alpha $ | $-\pi L/3$ | $-18\zeta \left(3\right){a}^{2}/\beta $ | $24{\pi}^{2}{a}^{2}/5\beta $ | $60\zeta \left(5\right){a}^{2}/\zeta \left(3\right)\beta $ |

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Cho, H.-T.; Hsiang, J.-T.; Hu, B.-L.
Quantum Capacity and Vacuum Compressibility of Spacetime: Thermal Fields. *Universe* **2022**, *8*, 291.
https://doi.org/10.3390/universe8050291

**AMA Style**

Cho H-T, Hsiang J-T, Hu B-L.
Quantum Capacity and Vacuum Compressibility of Spacetime: Thermal Fields. *Universe*. 2022; 8(5):291.
https://doi.org/10.3390/universe8050291

**Chicago/Turabian Style**

Cho, Hing-Tong, Jen-Tsung Hsiang, and Bei-Lok Hu.
2022. "Quantum Capacity and Vacuum Compressibility of Spacetime: Thermal Fields" *Universe* 8, no. 5: 291.
https://doi.org/10.3390/universe8050291