# Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 3.**Metric function $g\left(r\right)$ for a regular cosmological black hole with the de Sitter center.

**Figure 4.**Metric function $g\left(r\right)$ for space-time with the de Sitter center asymptotically de Sitter for $r\to \infty $.

## 2. Basic Features of Space-Time

## 3. Thermodynamics of Horizons

#### 3.1. Thermodynamics of the Black Hole Horizon

#### 3.2. The Case of Global Temperature for an Observer between ${r}_{b}$ and ${r}_{c}$

## 4. Evolution of a Black Hole During Evaporation

## 5. Summary and Discussion

## Acknowledgment

## References and Notes

- Bardeen, J.M.; Carter, B.; Hawking, S.W. The four laws of black hole mechanics. Commun. Math. Phys.
**1973**, 31, 161–170. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and entropy. Phys. Rev.
**1973**, 7, 2333–2346. [Google Scholar] [CrossRef] - Hawking, S.W. Black-hole evaporation. Nature
**1974**, 248, 30–31. [Google Scholar] [CrossRef] - Hawking, S.W. Particle creation by black holes. Commun. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] [CrossRef] - Bekenstein, J.D. Generalized second law of thermodynamics in black hole physics. Phys. Rev.
**1974**, 9, 3292–3300. [Google Scholar] [CrossRef] - Bekenstein, J.D. Statistical black-hole thermodynamics. Phys. Rev.
**1975**, 12, 3077–3085. [Google Scholar] [CrossRef] - Hawking, S.W. Black holes and thermodynamics. Phys. Rev.
**1976**, 13, 191–197. [Google Scholar] [CrossRef] - Wald, R.M. Quantum Field Theory in Curved Space and Black Hole Thermodynamics; University of Chicago Press: Chicago, IL, USA, 1994. [Google Scholar]
- Gibbons, G.W.; Hawking, S.W. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D
**1977**, 15, 2738–2751. [Google Scholar] [CrossRef] - Bousso, R. Positive vacuum energy and the N-bound. JHEP
**2000**, 0108, 038:1–038:23. [Google Scholar] [CrossRef] - Bousso, R. Bekenstein bounds in de Sitter and flat space. JHEP
**2001**, 0111, 035:1–035:14. [Google Scholar] [CrossRef] - Padmanabhan, T. Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes. Class. Quant. Grav.
**2002**, 19, 5387–5408. [Google Scholar] [CrossRef] - Padmanabhan, T. The holography of gravity encoded in a relation between entropy, horizon area and the action for gravity. Gen. Rel. Grav.
**2002**, 34, 2029–2035. [Google Scholar] [CrossRef] - Choudhury, T.R.; Padmanabhan, T. Concept of temperature in multi-horizon spacetimes: Analysis of Schwarzschild-De Sitter metric. Gen. Rel. Grav.
**2007**, 39, 1789–1811. [Google Scholar] [CrossRef] - Cai, R.G. Cardy-Verlinde formula and asymptotically de Sitter spaces. Phys. Lett. B
**2002**, 525, 331–336. [Google Scholar] [CrossRef] - Teitelboim, C. Gravitational thermodynamics of Schwarzschild-de Sitter space. In Strings and Gravity. Tying the Forces Together. Proceedings of the 5th Francqui Colloquium on Strings and Gravity, Brussels, Belgium, 19–21 October 2001; De Boeck & Larcier: Brussels, Belgium, 2003. [Google Scholar]
- Gomberoff, A.; Teitelboim, C. De Sitter black holes with either of the two horizons as a boundary. Phys. Rev. D
**2003**, 67, 104024:1–104024:7. [Google Scholar] [CrossRef] - Aros, R. De Sitter thermodynamics: A glimpse into nonequilibrium. Phys. Rev. D
**2008**, 77, 104013:1–104013:7. [Google Scholar] [CrossRef] - Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiattia, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Iha, S.; Kirschner, R.P.; et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J.
**1998**, 116, 1009–1038. [Google Scholar] [CrossRef] - Riess, A.G.; Kirschner, R.P.; Schmidt, B.P.; Iha, S.; Challis, P.; Garnavich, P.M.; Esin, A.A.; Carpenter, C.; Grashins, R.; et al. BV RI light curves for 22 type Ia supernovae. Astron. J.
**1999**, 117, 707–724. [Google Scholar] [CrossRef] - Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Ω and Λ from 42 high-redshift supernovae. Astrophys. J.
**1999**, 517, 565–586. [Google Scholar] [CrossRef] - Bahcall, N.A.; Ostriker, J.P.; Perlmutter, S.; Steinhardt, P.J. The cosmic triangle: Revealing the state of the universe. Science
**1999**, 284, 1481–1488. [Google Scholar] [CrossRef] - Wang, L.; Caldwell, R.R.; Ostriker, J.P.; Steinhardt, P.J. Cosmic concordance and quintessence. Astrophys. J.
**2000**, 530, 17–35. [Google Scholar] [CrossRef] - Spergel, D.N.; Verde, L.; Peiris, H.V.; Komatsu, E.; Nolte, M.R.; Bennett, C.L.; Halpern, M.; Hinshaw, G.; Jarosik, N.; Kogut, A.; et al. First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters. Astrophys. J. Suppl. Ser.
**2003**, 148, 175–194. [Google Scholar] - Schubnell, M. Probing dark energy in the accelerating universe with SNAP. In Proceedings of 8th Conference on the Intersections of Particle and Nuclear Physics (CIPANP 2003), New York, NY, USA, 19–24 May 2003; Parsa, Z., Ed.; AIP: New York, NY, USA, 2004; pp. 323–327. [Google Scholar]
- Corasaniti, P.S.; Copeland, E.J. Constraining the quintessence equation of state with SnIa data and CMB peaks. Phys. Rev. D
**2002**, 65, 043004:1–043004:5. [Google Scholar] [CrossRef] - Corasaniti, P.S.; Kunz, M.; Parkinson, D.; Copeland, E.J.; Bassett, B.A. Foundations of observing dark energy dynamics with the Wilkinson Microwave Anisotropy Probe. Phys. Rev. D
**2004**, 70, 083006:1–083006:15. [Google Scholar] [CrossRef] - Hannestad, S.; Mortsell, E. Probing the dark side: Constraints on the dark energy equation of state from CMB, large scale structure, and type Ia supernovae. Phys. Rev. D
**2002**, 66, 063508:1–063508:5. [Google Scholar] [CrossRef] - Tonry, J.L.; Schmidt, B.P.; Barris, B.; Candia, P.; Challis, P.; Clocchiatti, A.; Cail, A.L.; Filippenko, A.V.; Garnavich, P.; Hogan, C.; et al. Cosmological results from high-z supernovae. Astrophys. J.
**2003**, 594, 1–24. [Google Scholar] [CrossRef] - Ellis, J. Dark matter and dark energy: Summary and future directions. Phil. Trans. A
**2003**, 361, 2607–2627. [Google Scholar] [CrossRef] [PubMed] - Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D
**2006**, 15, 1753–1935. [Google Scholar] [CrossRef] - Copeland, E.J. Models of dark energy. In Proceedings of the Invisible Universe International Conference, Paris, France, 29 June–3 July 2009; AIP: New York, NY, USA, 2010; pp. 132–138. [Google Scholar]
- Strominger, A. The dS/CFT correspondence. JHEP
**2001**, 0110, 034:1–034:18. [Google Scholar] [CrossRef] - Strominger, A. Inflation and the dS/CFT correspondence. JHEP
**2001**, 0111, 049:1–049:6. [Google Scholar] [CrossRef] - MacGibbon, J.H. Can Planck-mass relics of evaporating black holes close the Universe? Nature
**1987**, 329, 308–309. [Google Scholar] [CrossRef] - Rajagopal, K.; Turner, M.S.; Wilczek, F. Cosmological implications of axinos. Nucl. Phys. B
**1991**, 358, 447–470. [Google Scholar] [CrossRef] - Carr, B.J.; Gilbert, J.H.; Lidsey, J.E. Black hole relics and inflation: Limits on blue perturbation spectra. Phys. Rev. D
**1994**, 50, 4853–4867. [Google Scholar] [CrossRef] - Adler, R.J.; Chen, P.; Santiago, D. The generalized uncertainty principle and black hole remnants. Gen. Rel. Grav.
**2001**, 33, 2101–2108. [Google Scholar] [CrossRef] - Chen, P.; Adler, R.J. Black hole remnants and dark matter. Nucl. Phys. B
**2003**, 124, 103–106. [Google Scholar] [CrossRef] - Carr, B.J. Primordial black holes—Recent developments. In Presented at the 22nd Texas Symposium on Relativistic Astrophysics, Stanford, CA, USA, 13–17 December 2004. No. 0204.
- Nozari, K.; Mehdipour, S.H. Gravitational uncertainty and black hole remnants. Mod. Phys. Lett. A
**2005**, 20, 2937–2948. [Google Scholar] [CrossRef] - Koch, B.; Bleicher, M.; Hossenfelder, S. Black hole remnants at the LHC. JHEP
**2005**, 2005, 053:1–053:20. [Google Scholar] [CrossRef] - Nayak, G.C. Dark matter production at the LHC from black hole remnants. Physics of Particles and Nuclei Letters
**2011**, 4, 564–572. [Google Scholar] [CrossRef] - Susskind, L. The World as a hologram. J. Math. Phys.
**1995**, 36, 6377–6396. [Google Scholar] [CrossRef] - Lin, F.L. Black hole in de Sitter space. In Presented at the International Symposium on Particles, Strings and Cosmology PASCOS 98, Boston, MA, USA, 22–29 March 1998.
- Kin, F.K.; Soo, C. Quantum field theory with and without conical singularities: Black holes with a cosmological constant and the multi-horizon scenario. Class. Quant. Grav.
**1999**, 16, 551–562. [Google Scholar] - Bousso, R.; Hawking, S.W. (Anti-)evaporation of Schwarzschild-de Sitter black holes. Phys. Rev. D
**1998**, 57, 2436–2442. [Google Scholar] [CrossRef] - Huang, Q.G.; Ke, K.; Li, M. One conjecture and two observations on de Sitter space. JHEP
**2006**, 2006, 045:1–045:10. [Google Scholar] [CrossRef] - Dymnikova, I. Regular black hole remnants. In Proceedings of the Invisible Universe International Conference, Paris, France, 29 June–3 July 2009; AIP: New York, NY, USA, 2010. [Google Scholar]
- Sakharov, A.D. Expanding universe and the appearance of a nonuniform distribution of matter. Sov. Phys. JETP
**1966**, 22, 241–249. [Google Scholar] - Gliner, E.B. Algebraic properties of the energy-momentum tensor and vacuum-like states of matter. Sov. Phys. JETP
**1966**, 22, 378–383. [Google Scholar] - Poisson, E.; Israel, W. Structure of the black hole nucleus. Class. Quant. Grav.
**1988**, 5, L201–L205. [Google Scholar] [CrossRef] - Dymnikova, I. Nonsingular spherically symmetric black hole. Centrum Astronomiczne im. Mikolaja Kopernika
**1990**, CAMK preprint 216, 1–9. [Google Scholar] - Dymnikova, I. Vacuum nonsingular black hole. Gen. Rel. Grav.
**1992**, 24, 235–242. [Google Scholar] [CrossRef] - Perez, A. Spin foam models for quantum gravity. Class. Quant. Grav.
**2003**, 20, R43–R104. [Google Scholar] [CrossRef] - Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Modesto, L. Loop quantum gravity and black hole singularity. In Presented at the 17th SIGRAV Conference on General Relativity and Gravitational Physics, Turin, Italy, 4–7 September 2006.
- Bonanno, A.; Reuter, M. Renormalization group improved black hole spacetimes. Phys. Rev. D
**2000**, 62, 043008–043032. [Google Scholar] [CrossRef] - Bonanno, A.; Reuter, M. Spacetime structure of an evaporating black hole in quantum gravity. Phys. Rev. D
**2006**, 73, 083005–083017. [Google Scholar] [CrossRef] - Nicolini, P. Noncommutative black holes, the final appeal to quantum gravity: A review. Int. J. Mod. Phys. A
**2009**, 24, 1229–1308. [Google Scholar] [CrossRef] - Banerjee, R.; Gangopadhyay, S.; Modak, S.K. Voros product, noncommutative black hole and corrected area law. Phys. Lett. B
**2010**, 686, 181–187. [Google Scholar] [CrossRef] - Nicolini, P.; Smailagic, A.; Spalucci, E. Noncommutative geometry inspired Schwarzschild black hole. Phys. Lett. B
**2006**, 632, 547–551. [Google Scholar] [CrossRef] - Mann, R.B.; Nicolini, P. Cosmological production of noncommutative black hole. arXiv
**2011**. arXiv: 1102.5096 [gr-qc]. [Google Scholar] - Dymnikova, I. The algebraic structure of a cosmological term in spherically symmetric solutions. Phys. Lett. B
**2000**, 472, 33–38. [Google Scholar] [CrossRef] - Dymnikova, I. The cosmological term as a source of mass. Class. Quant. Grav.
**2002**, 19, 725–740. [Google Scholar] [CrossRef] - Dymnikova, I. Spherically symmetric space-time with regular de Sitter center. Int. J. Mod. Phys. D
**2003**, 12, 1015–1034. [Google Scholar] [CrossRef] - Dymnikova, I.; Galaktionov, E. Vacuum dark fluid. Phys. Lett. B
**2007**, 645, 358–364. [Google Scholar] [CrossRef] - Dymnikova, I.; Soltysek, B. Spherically symmetric space-time with two cosmological constants. Gen. Rel. Grav.
**1998**, 30, 1775–1793. [Google Scholar] [CrossRef] - Dymnikova, I.; Soltysek, B. Nonsingular cosmological black hole. In Particles, Fields and Gravitation; Rembielinsky, J., Ed.; AIP: New York, NY, USA, 1998; pp. 460–471. [Google Scholar]
- Dymnikova, I. Possibilities and surprises of vacuum dark fluid. Gravitation and Cosmology
**2011**, 17, 185–189. [Google Scholar] [CrossRef] - Dymnikova, I.; Galaktionov, E. Dark ingredients in one drop. Cent. Eur. J. Phys.
**2011**, 9, 644–653. [Google Scholar] [CrossRef] - It is invariant under radial Lorentz boosts which makes impossible to single out a preferred comoving reference frame and thus to fix the velocity with respect to a medium specified by ${T}_{t}^{t}={T}_{r}^{r}$— which is the intrinsic property of a vacuum, according to general euristic definition of a vacuum given in [92].
- Dymnikova, I. Variable cosmological term—Geometry and physics. In Woprosy Matematicheskoj Fiziki i Prikladnoj Matematiki; Tropp, A., Ed.; A.F. Ioffe Physico-Technical Institute: St. Petersburg, Russia, 2000; pp. 29–71, gr-qc/0010016. [Google Scholar]
- Dymnikova, I. From vacuum nonsingular black hole to variable cosmological constant. Gravitation and Cosmology Supplement
**2002**, 8, 131–147. [Google Scholar] - Dymnikova, I. Self-gravitating spherically symmetric vacuum. In General Relativity, Cosmology and Gravitational Lensing; Marmo, G., Rubano, C., Scudellaro, P., Eds.; Bibliopolis: Napoli, Italy, 2002; pp. 95–129. [Google Scholar]
- Dymnikova, I. Variable cosmological term. In Cosmology and Gravitation; Novello, M., Bergliaffa, S.E.P., Eds.; AIP: Melville, NY, USA, 2003; pp. 204–225. [Google Scholar]
- Dymnikova, I. Cosmological term, mass and space-time symmetries. In Beyond the Desert 2003. Proceedings of the Fourth Tegernsee International Conference on Particle Physics Beyond the Standard BEYOND 2003, Castle Ringberg, Tegernsee, Germany, 9–14 June 2003; Klapdor-Kleinhaus, H.V., Ed.; Springer Verlag: Berlin, Germany, 2004; pp. 485–502, hep-th/0310047. [Google Scholar]
- Dymnikova, I.; Galaktionov, E. Stability of a vacuum non-singular black hole. Class. Quant. Grav.
**2005**, 22, 2331–2358. [Google Scholar] [CrossRef] - Dymnikova, I. Space-time symmetry and mass of a lepton. Presented at the 5th International Symposium on Quantum Theory and Symmetries, Valladolid, Spain, 22–28 July 2007. J. Phys. A
**2008**, 41, 304033–304052. [Google Scholar] [CrossRef] - Dymnikova, I. De Sitter-Schwarzschild black hole: Its particlelike core and thermodynamical properties. Int. J. Mod. Phys. D
**1996**, 5, 529–540. [Google Scholar] [CrossRef] - Dymnikova, I. Internal structure of nonsingular spherical black holes. In Internal Structure of Black Holes and Spacetime Singularities; Burko, L.M., Ori, A., Eds.; IOP: London, UK, 1997; pp. 422–438. [Google Scholar]
- Myung, Y.S.; Kim, Y.W.; Park, Y.J. Black hole thermodynamics with generalized uncertainty principle. Phys. Lett. B
**2007**, 645, 393–397. [Google Scholar] [CrossRef] - Hayward, S.A. Formation and evaporation of nonsingular black holes. Phys. Rev. Lett.
**2006**, 96, 031103:1–031103:4. [Google Scholar] [CrossRef] - Dymnikova, I.; Korpusik, M. Regular black hole remnants in de Sitter space. Phys. Lett. B
**2010**, 685, 12–18. [Google Scholar] [CrossRef] - Bronnikov, K.A.; Dobosz, A.; Dymnikova, I. Nonsingular vacuum cosmologies with a variable cosmological term. Class. Quant. Grav.
**2003**, 20, 3797–3814. [Google Scholar] [CrossRef] - Novikov, I.D.; Frolov, V.P. Physics of Black Holes; Kluwer Acad. Publ.: Dordrecht, The Netherlands, 1989; Ch.9. [Google Scholar]
- Frolov, V.P.; Markov, M.A.; Mukhanov, V.F. Black holes as possible sources of closed and semiclosed worlds. Phys. Rev. D
**1990**, 41, 383–394. [Google Scholar] [CrossRef] - Bonanno, A.; Reuter, M. Quantum gravity effects near the null black hole singularity. Phys. Rev. D
**1999**, 60, 084011–084018. [Google Scholar] [CrossRef] - Dymnikova, I. ${\mathrm{\Lambda}}_{\nu}^{\mu}$ geometries from the point of view of different observers. In Beyond the Desert 2003. Proceedings of the Fourth Tegernsee International Conference on Particle Physics Beyond the Standard BEYOND 2003, Castle Ringberg, Tegernsee, Germany, 9–14 June 2003; Klapdor-Kleinhaus, H.V., Ed.; Springer Verlag: Berlin, Germany, 2004; pp. 521–539, gr-qc/03100314. [Google Scholar]
- Bronnikov, K.; Dymnikova, I. Regular homogeneous T-models with vacuum dark fluid. Class. Quant. Grav.
**2007**, 24, 5803–5816. [Google Scholar] [CrossRef] - Frampton, P.H. High longevity microlensing events and dark matter black holes. In Presented at the 11th confererence on cosmology COSMO 08, Madison, WI, USA, 25–29 August 2008.
- Landau, L.D.; Lifshitz, E.M. Classical Theory of Fields, 4th ed.; Butterworth-Heinemann: Oxford, UK, 1975. [Google Scholar]

© 2011 by the authors; 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**

Dymnikova, I.; Korpusik, M.
Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior. *Entropy* **2011**, *13*, 1967-1991.
https://doi.org/10.3390/e13121967

**AMA Style**

Dymnikova I, Korpusik M.
Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior. *Entropy*. 2011; 13(12):1967-1991.
https://doi.org/10.3390/e13121967

**Chicago/Turabian Style**

Dymnikova, Irina, and Michał Korpusik.
2011. "Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior" *Entropy* 13, no. 12: 1967-1991.
https://doi.org/10.3390/e13121967