# Effects of Quantum Gravity on Thermodynamic Quantities of Gases around a Novel Neutral Four-Dimensional Gauss–Bonnet Black Hole

## Abstract

**:**

## 1. Introduction

## 2. Four-Dimensional Gauss–Bonnet Gravity Space-Time and Field Equations

## 3. Thermodynamic Quantities

## 4. Discussion and Conclusions

_{2}→ 0 and η(r, p) → p − s. Then, the contribution of the spin only comes from the spin state, p = −s.

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Li, Z.H. State equations for massless spin fields near the event horizon in Schwarzschild spacetime. Class. Quantum Grav.
**2004**, 21, 1181. [Google Scholar] - Li, Z.H. Effect of spin on thermodynamical quantities around Reissner–Nordstroem black holes. Chin. Phys. Lett.
**2005**, 22, 1321. [Google Scholar] - Boulware, D.G. Quantum field theory in Schwarzschild and Rindler spaces. Phys. Rev. D
**1975**, 11, 1404. [Google Scholar] [CrossRef] - Mi, L.Q.; Li, Z.H. Thermodynamical quantities around a RNAdS black hole. Chin. Phys.
**2006**, 15, 1184. [Google Scholar] - Li, G.Q. Thermodynamic quantities for gases in static spherically symmetric backgrounds possessing a horizon. JETP Lett.
**2007**, 86, 153. [Google Scholar] [CrossRef] - Li, G.Q. Effect of spin on the thermodynamical quantities around dilatonic black hole. Europhys. Lett.
**2007**, 77, 10001. [Google Scholar] [CrossRef] - Li, G.Q. Spin-Dependence of Thermodynamic Quantities Around A Horowitz–Strominger Black Hole. Mod. Phys. Lett. A
**2008**, 23, 437. [Google Scholar] [CrossRef] - Li, G.Q. Subleading terms of thermodynamic quantities around static spherical black holes. Chin. Phys. B
**2009**, 18, 66. [Google Scholar] - Li, G.Q.; Xiao, S.F. State equations for massless spin fields in static spherical spacetime filled with quintessence. Gen. Relativ. Gravit.
**2010**, 42, 1719. [Google Scholar] [CrossRef] - Mead, C.A. Possible connection between gravitation and fundamental length. Phys. Rev. D
**1964**, 135, 849. [Google Scholar] [CrossRef] - Kempf, A.; Mangano, G.; Mann, R.B. Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D
**1995**, 52, 1108. [Google Scholar] [CrossRef] [PubMed] - Chang, L.N.; Minic, D.M.; Okamura, N.; Takeuchi, T. Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem. Phys. Rev. D
**2002**, 65, 125028. [Google Scholar] [CrossRef] - Gross, D.J. High-energy symmetries of string theory. Phys. Rev. Lett.
**1988**, 60, 1229. [Google Scholar] [CrossRef] [PubMed] - Maggiore, M. The algebraic structure of the generalized uncertainty principle. Phys. Lett. B
**1993**, 319, 83–86. [Google Scholar] [CrossRef] - Padmanabhan, T. Limitations on the operational definition of spacetime events and quantum gravity. Class. Quant. Grav.
**1987**, 4, L107. [Google Scholar] [CrossRef] - Amati, D.; Ciafaloni, M.; Veneziano, G. Can spacetime be probed below the string size? Phys. Lett. B
**1989**, 216, 41–47. [Google Scholar] [CrossRef] - Maggiore, M. Quantum groups, gravity, and the generalized uncertainty principle. Phys. Rev. D
**1994**, 49, 5182. [Google Scholar] [CrossRef] - Scardigli, F. Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment. Phys. Lett. B
**1999**, 452, 39. [Google Scholar] [CrossRef] - Kempf, A.; Mangano, G. Minimal length uncertainty relation and ultraviolet regularization. Phys. Rev. D
**1997**, 55, 7909. [Google Scholar] [CrossRef] - Gross, D.J.; Mende, P.F. String theory beyond the Planck scale. Nucl. Phys. B
**1988**, 303, 407. [Google Scholar] [CrossRef] - Garay, L.J. Quantum gravity and minimum length. Int. J. Mod. Phys. A
**1995**, 10, 145. [Google Scholar] [CrossRef] - Hossenfelder, S.; Bleicher, M.; Hofmann, S.; Ruppert, J.; Scherer, S.; Stöcker, H. Signatures in the Planck regime. Phys. Lett. B
**2003**, 575, 85. [Google Scholar] [CrossRef] - Bambi, C.; Urban, F.R. Natural extension of the generalized uncertainty principle. Class. Quant. Grav.
**2008**, 25, 095006. [Google Scholar] [CrossRef] - Brau, F.J. Minimal length uncertainty relation and the hydrogen atom. Phys. A
**1999**, 32, 7691. [Google Scholar] [CrossRef] - Magueijo, J.; Smolin, L. Lorentz invariance with an invariant energy scale. Phys. Rev. Lett.
**2002**, 88, 190403. [Google Scholar] [CrossRef] [PubMed] - Maggiore, M. A generalized uncertainty principle in quantum gravity. Phys. Lett. B
**1993**, 304, 65. [Google Scholar] [CrossRef] - Cortes, J.L.; Gamboa, J. Quantum uncertainty in doubly special relativity. Phys. Rev. D
**2005**, 71, 065015. [Google Scholar] [CrossRef] - Glavan, D.; Lin, C. Einstein-Gauss-Bonnet gravity in four-dimensional spacetime. Phys. Rev. Lett.
**2020**, 124, 081301. [Google Scholar] [CrossRef] - Aoki, K.; Gorji, M.A.; Mizuno, S.; Mukohyama, S. A consistent theory of D → 4 Einstein-Gauss-Bonnet gravity. Phys. Lett. B
**2020**, 810, 135843. [Google Scholar] [CrossRef] - Aoki, K.; Gorji, M.A.; Mukohyama, S. Inflationary gravitational waves in consistent D → 4 Einstein-Gauss-Bonnet gravity. JCAP
**2021**, 2021, 54. [Google Scholar] [CrossRef] - Lu, H.; Pang, Y. Horndeski gravity as D → 4 limit of Gauss-Bonnet. Phys. Lett. B
**2020**, 809, 135717. [Google Scholar] [CrossRef] - Hennigar, R.A.; Kubizňák, D.; Mann, R.B.; Pollack, C.J. On taking the D → 4 limit of Gauss-Bonnet gravity: Theory and solutions. High Energy Phys.
**2020**, 7, 27. [Google Scholar] [CrossRef] - Qiao, X.Y.; Liang, O.Y.; Wang, D.; Pan, Q.Y.; Jing, J.L. Holographic superconductors in 4D Einstein-Gauss-Bonnet gravity. JHEP
**2020**, 192. [Google Scholar] [CrossRef] - Guo, M.Y.; Li, P.C. Innermost stable circular orbit and shadow of the 4 D Einstein–Gauss–Bonnet black hole. Eur. Phys. J. C
**2020**, 80, 588. [Google Scholar] [CrossRef] - Wei, S.W.; Liu, Y.X. Extended thermodynamics and microstructures of four-dimensional charged Gauss-Bonnet black hole in AdS space. Phys. Rev. D
**2020**, 101, 104018. [Google Scholar] [CrossRef] - Fernandes, P.G.S. Charged black holes in AdS spaces in 4D Einstein Gauss-Bonnet gravity. Phys. Lett. B
**2020**, 805, 135468. [Google Scholar] [CrossRef] - Hegde, K.; Kumara, A.N.; Rizwan, C.L.A.; Ajith, K.M.; Ali, M.S. Thermodynamics, Phase Transition and Joule Thomson Expansion of novel 4-D Gauss Bonnet AdS Black Hole. arXiv
**2020**, arXiv:2003.08778. [Google Scholar] - Konoplya, R.A.; Zinhailo, A.F. Quasinormal modes, stability and shadows of a black hole in the 4D Einstein–Gauss–Bonnet gravity. Eur. Phys. J. C
**2020**, 80, 1049. [Google Scholar] [CrossRef] - Newman, E.; Penrose, R. An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys.
**1962**, 3, 566. [Google Scholar] [CrossRef] - García, A.; Macías, A. Black Holes as Exact Solutions of the Einstein-Maxwell Equations of Petrov Type D; Springer Press: Berlin/Heidelberg, Germany, 1998. [Google Scholar]
- Teukolsky, S.A.; Press, W.H. Perturbations of a rotating black hole. III-Interaction of the hole with gravitational and electromagnetic radiation. Astrophys. J.
**1974**, 193, 443–461. [Google Scholar] [CrossRef] - Hooft, G.T. On the quantum structure of a black hole. Nucl. Phys. B
**1985**, 256, 727. [Google Scholar] [CrossRef] - Misner, C.W.; Thorne, K.S.; Wheeler, J.A. Gravitation; Freeman: San Francisco, CA, USA, 1973. [Google Scholar]
- Tolman, R.C. Relativity, Thermodynamics and Cosmology; Oxford University Press: Oxford, UK, 1934. [Google Scholar]
- Unruh, W.G.; Wald, R.M. Acceleration radiation and the generalized second law of thermodynamics. Phys. Rev. D
**1982**, 25, 942. [Google Scholar] [CrossRef] - Li, L.X.; Liu, L. Properties of radiation near the black-hole horizon and the second law of thermodynamics. Phys. Rev. D
**1992**, 46, 3296. [Google Scholar] [CrossRef] [PubMed]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, G.
Effects of Quantum Gravity on Thermodynamic Quantities of Gases around a Novel Neutral Four-Dimensional Gauss–Bonnet Black Hole. *Universe* **2023**, *9*, 253.
https://doi.org/10.3390/universe9060253

**AMA Style**

Li G.
Effects of Quantum Gravity on Thermodynamic Quantities of Gases around a Novel Neutral Four-Dimensional Gauss–Bonnet Black Hole. *Universe*. 2023; 9(6):253.
https://doi.org/10.3390/universe9060253

**Chicago/Turabian Style**

Li, Guqiang.
2023. "Effects of Quantum Gravity on Thermodynamic Quantities of Gases around a Novel Neutral Four-Dimensional Gauss–Bonnet Black Hole" *Universe* 9, no. 6: 253.
https://doi.org/10.3390/universe9060253