# Black Hole Horizons and Thermodynamics: A Quantum Approach

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## Abstract

**:**

## 1. Introduction and Summary

## 2. Quantum Structures on the Horizon of a Black Holes

#### 2.1. Geometrical Background and its Quantum Interpretation

**(C1)**it has the general structure (1) and in particular it enjoys ${\mathbb{S}}^{2}$-spherical symmetry, Σ being tangent to the associated Killing fields,

**(C2)**it solves equations (7) with $C>0$ (that is the mass M) fixed a priori in some way depending, at quantum level, on a quantum reference state as we shall discuss shortly, and

**(C3)**$\mathbb{F}\cup \mathbb{P}$ is a bifurcate Killing horizon with bifurcation surface Σ.

**admissible null global frames**those coordinate frames. It is simply proved that the following is the most general transformation between pairs of admissible null global frames provided η transforms as a scalar field (as we assume henceforth):

#### 2.2. The Field ρ and the Interplay with ϕ on Killing Horizons

**Remark**. The reader should pay attention to the used notation. ρ should be viewed as a function of both the points of $\mathbb{F}\times \mathbb{P}$ and the used chart. If the chart $\mathcal{C}$ is associated with the coordinate frame ${x}^{+},{x}^{-}$, an appropriate notation to indicate the function representing ρ in $\mathcal{C}$ could be $\rho \left(\mathcal{C}\right|{x}^{+},{x}^{-})$ or ${\rho}_{\mathcal{C}}({x}^{+},{x}^{-})$. However we shall use the simpler, but a bit miss-understandable, notation $\rho ({x}^{+},{x}^{-})$. As a consequence, the reader should bear in his/her mind that, in general

**Theorem 1**. Working in a fixed admissible null global frame ${x}^{\pm}$ on $\mathbb{M}$, if ${\rho}_{+}={\rho}_{+}\left({x}^{+}\right)$, ${\rho}_{-}={\rho}_{-}\left({x}^{-}\right)$ are smooth bounded-below functions, there is a unique metric which satisfies (C1), (C2), (C3) (with assigned mass $M>0$) and such that ${\rho |}_{\mathbb{F}}={\rho}_{+}$ and ${\rho |}_{\mathbb{P}}={\rho}_{-}$.

**Proof**. First of all we prove that if $d{s}^{2}$, $\tilde{d{s}^{2}}$ are solutions of Einstein equation on $\mathbb{M}$ satisfying (C1), (C2), (C3) (with a fixed value of the mass), they coincide if the restrictions of the respectively associated functions ρ, $\tilde{\rho}$ to $\mathbb{F}\cup \mathbb{P}$ coincide working in some admissible null global frame ${x}^{\pm}$.

#### 2.3. Quantization

**Remark**. Generally speaking, in quantizing gravity one has to discuss how the covariance under diffeomorphisms is promoted at the quantum level. In our picture, the relevant class of diffeomerphisms is restricted to the maps (17). Their action at the quantum level is implemented through (21) and (22) in terms of a class of automorphisms of the algebra generated by the quantum fields $\widehat{\varphi}$ and $\widehat{\rho}$. However, introducing quantum states, the picture becomes more subtle. We shall shortly see that such a symmetry will be broken, the remaining one being described by $PSL(2,\mathbb{R})$ or a subgroup.

#### 2.4. Properties of ${\Psi}_{\zeta}$ and ${\widehat{\varphi}}_{\zeta}$: Spontaneous Breaking of Conformal Symmetry, Hawking Temperature, Bose-Einstein Condensate

**Theorem 2**. If $\zeta \ne 0$, there is no unitary representation of the whole group $PSL(2,\mathbb{R})$ which unitarily implements the action of $PSL(2,\mathbb{R})$ on the field ${\widehat{\varphi}}_{\zeta}$ (33) referred to ${\Psi}_{\zeta}$. Only the subgroup associated with $\mathcal{D}$ admits unitary implementation

**Sketch of Proof**. If $\zeta \in \mathbb{R}$ is fixed arbitrarily, and ω varies in the class of the admissible real forms used to smear the field operator, the class of all of unitary operators in the Fock space based on ${\Psi}_{\zeta}$

**Theorem 3**. Every state ${\Psi}_{\zeta}$ (including $\zeta =0$), restricted to the algebra of observables localized at ${\mathbb{F}}_{>}$, is invariant under the transformations generated by $\mathcal{D}={\partial}_{v}$ and it is furthermore thermal with respect to the time v with inverse temperature $\beta =2\pi $. As a consequence, adopting the physical “time coordinate” $\zeta v$ which accounts for the actual size of the Black hole (enclosed in the parameter ζ), the inverse temperature β turns out to be just Hawking’s value ${\beta}_{H}=8\pi M$.

#### 2.5. Properties of Υ and $\widehat{\rho}$: Feigin-Fuchs Stress Tensor

**Theorem 4**. The state Υ restricted to the algebra of observables localized at ${\mathbb{F}}_{>}$ turns out to be a thermal (KMS) state with respect to ${\partial}_{\zeta v}$ at Hawking temperature.

**Theorem 5**. Working in coordinates $V,s$ and referring to the representation of $\widehat{\rho}$ based on Υ:

## 3. Thermodynamical Quantities: Free Energy and Entropy

## 4. Final Comments and Open Issues

## Acknowledgements

## References

- Bekenstein, J.D. Black holes and the second law. Lett. Nuovo Cim.
**1972**, 4, 737–740. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and entropy. Phys. Rev.
**1973**, D7, 2333–2346. [Google Scholar] [CrossRef] - Hawking, S.W. Particle creation by black holes. Comm. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] [CrossRef] - Padmanabhan, T. Gravity and the Thermodynamics of Horizons. Phys. Rep.
**2005**, 406, 49–125, arXiv:gr-qc/0311036v2. [Google Scholar] [CrossRef] - ’t Hooft, G. Dimensional Reduction in Quantum Gravity. arXiv:gr-qc/9310026v2.
- ’t Hooft, G. The scattering matrix approach for the quantum black hole, an overview. Int. J. Mod. Phys.
**1996**, A11, 4623–4688. [Google Scholar] [CrossRef] - Susskind, L. The World as a Hologram. J. Math. Phys.
**1995**, 36, 6377–6396. [Google Scholar] [CrossRef] - Padmanabhan, T. Holographic Gravity and the Surface term in the Einstein-Hilbert Action. Braz. J. Phys.
**2005**, 35, 362–372, arXiv:gr-qc/0412068v3. [Google Scholar] [CrossRef] - Wald, R.M. The Thermodynamics of Black Holes. Living Rev. Relativ.
**2001**, 4, 6. [Google Scholar] [CrossRef] - Fursaev, D.V. Can one understand black hole entropy without knowing much about quantum gravity? Phys. Part. Nucl.
**2005**, 36, 81–99. [Google Scholar] - Guido, D.; Longo, R.; Roberts, J.E.; Verch, R. Charged sectors, spin and statistics in quantum field theory on curved spacetimes. Rev. Math. Phys.
**2001**, 13, 125–198. [Google Scholar] - Moretti, V.; Pinamonti, N. Bose-Einstein condensate and Spontaneous Breaking of Conformal Symmetry on Killing Horizons. J. Math. Phys.
**2005**, 46, 062303. [Google Scholar] [CrossRef] - Moretti, V. Bose-Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons II. J. Math. Phys.
**2006**, 47, 032302. [Google Scholar] [CrossRef] - Dappiaggi, C.; Moretti, V.; Pinamonti, N. Rigorous steps towards holography in asymptotically flat spacetimes. Rev. Math. Phys.
**2006**, 18, 349–416. [Google Scholar] [CrossRef] - Moretti, V. Uniqueness theorems for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary. observable algebra correspondence. Commun. Math. Phys.
**2006**, 268, 727–756, arXiv:gr-qc/0512049v2. [Google Scholar] [CrossRef] - Moretti, V. Quantum out-states states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property. Commun. Math. Phys.
**2008**, 279, 31–75, arXiv:gr-qc/0610143v2. [Google Scholar] [CrossRef] - Dappiaggi, C.; Moretti, V.; Pinamonti, N. Cosmological horizons and reconstruction of quantum field theories. Commun. Math. Phys.
**2009**, 285, 1129–1163, arXiv:0712.1770v3 [gr-qc]. [Google Scholar] [CrossRef] - Dappiaggi, C.; Pinamonti, N.; Porrmann, M. Local causal structures, Hadamard states and the principle of local covariance in quantum field theory. Commun. Math. Phys.
**2010**, in press; arXiv:1001.0858v1 [hep-th]. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Anomaly induced effective actions in even dimensions and reliability of s-wave approximation. Phys. Lett. B
**1999**, 463, 57–62, arXiv:hep-th/9904146v2. [Google Scholar] [CrossRef] - Cavaglia, M.; Ungarelli, C. Quantum gravity corrections to the Schwarzschild mass. Phys. Rev. D
**2000**, 61, 064019, arXiv:hep-th/9912024v1. [Google Scholar] [CrossRef] - Grumiller, D.; Kummer, W.; Vassilevich, D.V. Dilaton Gravity in Two Dimensions. Phys. Rep.
**2002**, 369, 327–430. [Google Scholar] [CrossRef] - Giacomini, A. Description of Black Hole Microstates by Means of a Free Affine-Scalar Field. arXiv:hep-th/0405120v1.
- Schmidt, H.-J. A new proof of Birkhoff’s theorem. Gravity Cosmol.
**1997**, 3, 185–190. [Google Scholar] - Moretti, V.; Pinamonti, N. Holography and SL(2, ℝ) symmetry in 2D Rindler spacetime. J. Math. Phys.
**2004**, 45, 230–256. [Google Scholar] [CrossRef] - Moretti, V.; Pinamonti, N. Quantum Virasoro algebra with central charge c=1 on the horizon of a 2D-Rindler spacetime. J. Math. Phys.
**2004**, 45, 257–284. [Google Scholar] [CrossRef] - Moretti, V.; Pinamonti, N. Aspects of Hidden and manifest SL(2,R) symmetry in 2D near horizon black hole background. Nucl. Phys. B
**2002**, 647, 131–152. [Google Scholar] [CrossRef] - Haag, R. Local Quantum Physics: Fields, Particles, Algebras, 2nd ed.; Springer: Berlin, Germany, 1992. [Google Scholar]
- Bratteli, O.; Robinson, D.W. C* and W* Algebras, Symmetry Groups, Decomposition Of States. In Operator Algebras and Quantum Statistical Mechanics; Springer-verl: New York, NY, USA, 1979; Volume 1. [Google Scholar]
- Bratteli, O.; Robinson, D.W. Equilibrium states. Models in quantum statistical mechanics. In Operator Algebras and Quantum Statistical Mechanics; Springer: Berlin, Germany, 1996; Volume 2. [Google Scholar]
- Popov, V.N. Functional Integrals in Quantum Field Theory and Statistical Physics; D. Reidel Publishing Company: Boston, MA, USA, 1983. [Google Scholar]
- Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys.
**1999**, 71, 463–512. [Google Scholar] [CrossRef] - Di Francesco, P.; Mathieu, P.; Sénéchal, D. Conformal Field Theory; Springer: New York, NY, USA, 1997. [Google Scholar]
- Kac, V.G.; Raina, A.K. Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras; World Scientific: Singapore, 1987. [Google Scholar]

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Moretti, V.; Pinamonti, N.
Black Hole Horizons and Thermodynamics: A Quantum Approach. *Entropy* **2010**, *12*, 1833-1854.
https://doi.org/10.3390/e12071833

**AMA Style**

Moretti V, Pinamonti N.
Black Hole Horizons and Thermodynamics: A Quantum Approach. *Entropy*. 2010; 12(7):1833-1854.
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**Chicago/Turabian Style**

Moretti, Valter, and Nicola Pinamonti.
2010. "Black Hole Horizons and Thermodynamics: A Quantum Approach" *Entropy* 12, no. 7: 1833-1854.
https://doi.org/10.3390/e12071833