# Partition Function of the Schwarzschild Black Hole

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Stretched Horizon

#### 2.2. Energy

#### 2.3. Microscopic Properties

## 3. The Partition Function

#### 3.1. Counting of States

#### 3.2. The Partition Function

## 4. Energy vs. Temperature

## 5. Phase Transition and the Hawking Effect

**Figure 1.**The average energy $\overline{E}$ $(=EK(T\left)\right)$ of the Schwarzschild black hole per a constituent of its stretched horizon as a function of the absolute temperature T, when the number of the constituents of the stretched horizon is $N=100$. The absolute temperature T has been expressed in the units of ${T}_{C}$, and the average energy $\overline{E}$ in the units of ${k}_{B}{T}_{C}$. If $T<{T}_{C}$, $\overline{E}$ is effectively zero, which means that the constituents of the stretched horizon (except one) are in vacuum. When $T={T}_{C}$, the curve $\overline{E}=\overline{E}\left(T\right)$ is practically vertical, which indicates a phase transition at the temperature $T={T}_{C}$. During this phase transition the constituents of the stretched horizon are excited from the vacuum to the second excited states. The latent heat per a constituent corresponding to this phase transition is $\overline{L}=2(ln2){k}_{B}{T}_{C}\approx 1.4{k}_{B}{T}_{C}$. When $T>{T}_{C}$, the curve $\overline{E}=\overline{E}\left(T\right)$ is approximately linear.

## 6. Entropy vs. Horizon Area

## 7. Discussion

## References and Notes

- Hawking, S.W. Particle creation by black holes. Commun. Math. Phys.
**1975**, 43, 199. [Google Scholar] [CrossRef] - Brown, J.D.; York, J.W. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev.
**1993**, D47, 1407. [Google Scholar] [CrossRef] - A somewhat related idea has been considered by J. D. Bekenstein and G. Gour [31].
- ’t Hooft, G. Dimensional reduction in quantum gravity. arXiv, 1993; arXiv:gr-qc/9310026v2. [Google Scholar]
- Susskind, L.; Thorlacius, L.; Uglum, J. The stretched horizon and black hole complementarity. Phys. Rev.
**1993**, D48, 3743. [Google Scholar] [CrossRef] - Susskind, L. The world as a hologram. J. Math. Phys.
**1995**, 36, 6377. [Google Scholar] [CrossRef] - Birrell, N.D.; Davies, P.C. Quantum Fields in Curved Space; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar]
- Bekenstein, J.D. Black holes and entropy. Phys. Rev.
**1973**, D7, 2333. [Google Scholar] [CrossRef] - Wald, R.M. General Relativity; The University of Chicago Press: Chicago, IL, USA, 1984. [Google Scholar]
- Bekenstein, J.D. The quantum mass spectrum of the kerr black hole. Lett. Nuovo Cim.
**1974**, 11, 467. [Google Scholar] [CrossRef] - Bekenstein, J.D.; Mukhanov, V.F. Spectroscopy of the quantum black hole. Phys. Lett.
**1995**, B360, 7. [Google Scholar] [CrossRef] - Peleg, Y. The spectrum of quantum dust black holes. Phys. Lett.
**1995**, B356, 462. [Google Scholar] [CrossRef] - Barvinsky, A.; Kunstatter, G. Exact physical black hole states in Generic 2-D dilaton gravity. Phys. Lett.
**1996**, B389, 231. [Google Scholar] [CrossRef] - Hod, S. Gravitation, the quantum and Bohr’s correspondence principle. Gen. Rel. Grav.
**1999**, 31, 1639. [Google Scholar] [CrossRef] - Bekenstein, J.D. The case for discrete energy levels of a black hole. Int. J. Math. Phys.
**2002**, A17S1, 21. [Google Scholar] - Louko, J.; Mäkelä, J. Area spectrum of the Schwarzschild black hole. Phys. Rev.
**1996**, D54, 4982. [Google Scholar] [CrossRef] - Mäkelä, J.; Repo, P. Quantum-mechanical model of the Reissner-Nordström black hole. Phys. Rev.
**1998**, D57, 4899. [Google Scholar] [CrossRef] - Mäkelä, J.; Repo, P.; Luomajoki, M.; Piilonen, J. Quantum-mechanical model of the Kerr-Newman black hole. Phys. Rev.
**2001**, D64, 415. [Google Scholar] [CrossRef] - Mäkelä, J. Black holes as atoms. Found. Phys.
**2002**, 32, 1809. [Google Scholar] [CrossRef] - Mäkelä, J. Black hole spectrum: Continuous or discrete? Phys. Lett.
**1997**, B390, 125. [Google Scholar] [CrossRef] - Hod, S. A note on the quantization of a multi-horizon black hole. Class. Quant. Grav.
**2007**, 24, 4871. [Google Scholar] [CrossRef] - Zhang, B.; Cai, Q.-y.; You, L.; Zhan, M.S. Hidden messenger revealed in Hawking radiation: A resolution to the paradox of black hole information loss. Phys. Lett.
**2009**, B675, 98. [Google Scholar] [CrossRef] - Singleton, D.; Vagenas, E.C.; Zhan, T.; Ren, J. Insights and possible resolution to the information loss paradox via the tunneling picture. J. High Energy Phys.
**2010**, 1008, 089. [Google Scholar] [CrossRef] - Mandl, F. Statistical Physics; John Wiley and Sons Ltd.: Bristol, UK, 1983. [Google Scholar]
- It is interesting that, up to an unimportant numerical factor 2 ln2, this expression for energy is the same as the one used as a starting point in a sketch for an entropic theory of gravity by Verlinde in [32].
- Wheeler, J.A. Information, physics, quantum: The search for links. In Complexity, Entropy and the Physics of Information; Zurek, W., Ed.; Addison-Wesley: Redwood City, CA, USA, 1990. [Google Scholar]
- See, for example, [33]. For the counting of states of a black hole in string theory see, for example, [34].
- Hawking, S.W. The path integral approach to quantum gravity. In General Relativity: An Einstein Centenary Survey; Hawking, S.W., Israel, W., Eds.; Cambridge University Press: New York, NY, USA, 1979. [Google Scholar]
- Unruh, W.G. Notes on black hole evaporation. Phys. Rev.
**1976**, D14, 5670. [Google Scholar] [CrossRef] - For the similarities between the Hawking and the Unruh effects see, for example, [35].
- Bekenstein, J.D.; Gour, G. Building blocks of a black hole. Phys. Rev.
**2002**, D66, 024005. [Google Scholar] [CrossRef] - Verlinde, E. On the origin of gravity and the laws of Newton. J. High Energy Phys.
**2011**, 1104, 29. [Google Scholar] [CrossRef] - Ashtekar, A.; Baez, J.; Corichi, A.; Krasnov, K. Quantum geometry and black hole entropy. Phys. Rev. Lett.
**1998**, 80, 904. [Google Scholar] [CrossRef] - Strominger, A.; Vafa, C. Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett.
**1996**, B379, 99. [Google Scholar] [CrossRef] - Peltola, A. Local approach to Hawking radiation. Class. Quant. Grav.
**2009**, 26, 035014. [Google Scholar] [CrossRef] - Spiegel, M.R. Schaum’s Outline Series: Mathematical Handbook of Formulas and Tables; McGraw-Hill: New York, NY, USA, 1968. [Google Scholar]

## Appendix

## A. Calculation of the Partition Function

## B. Properties of the Partition Function Near the Characteristic Temperature

© 2011 by the authos; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Mäkelä, J.
Partition Function of the Schwarzschild Black Hole. *Entropy* **2011**, *13*, 1324-1354.
https://doi.org/10.3390/e13071324

**AMA Style**

Mäkelä J.
Partition Function of the Schwarzschild Black Hole. *Entropy*. 2011; 13(7):1324-1354.
https://doi.org/10.3390/e13071324

**Chicago/Turabian Style**

Mäkelä, Jarmo.
2011. "Partition Function of the Schwarzschild Black Hole" *Entropy* 13, no. 7: 1324-1354.
https://doi.org/10.3390/e13071324