# Mass Evolution of Schwarzschild Black Holes

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## Abstract

In the classical theory of general relativity, black holes can only absorb and not emit particles. When quantum mechanical effects are taken into account, then the black holes emit particles as hot bodies with temperature proportional to *κ*, its surface gravity. This thermal emission can lead to a slow decrease in the mass of the black hole, and eventually to its disappearance, also called black hole evaporation. This characteristic allows us to analyze what happens with the mass of the black hole when its temperature is increased or decreased and how the energy is exchanged with the external environment. This paper has the aim to make a discussion about the mass evolution, due to Hawking radiation emission, of Schwarzschild black holes with different initial masses and external conditions as the empty space, the cosmic microwave background with constant temperature, and with temperature varying in accordance with the eras of the universe. Our main contributions are to take into account these variations in the temperature of the cosmic background radiation and use them into the black holes dynamics and also to analyze the asymptotic behavior of the solutions. As a result, we have the complete evaporation of the black holes in most cases, although their masses can increase in some cases, and even diverge for specific conditions.

## Keywords

Black hole thermodynamics Black hole evaporation Mass evolution of black holes Schwarzschild black hole## Notes

### Acknowledgements

We would like to thank Prof. Dr. Alberto Vazquez Saa for overall support and comments. We are also grateful to “Conselho Nacional de Desenvolvimento Cientifico e Tecnologico” (CNPq) for financial support.

## References

- 1.J.K. Martin, A.J. Kox, R. Schulman, Vol. 6.
*The Collected Papers of Albert Einstein*(Princeton University Press, New Jersey, 1996)Google Scholar - 2.S.M. Carroll.
*Spacetime and Geometry*(Addison Wesley, San Francisco, 2004)zbMATHGoogle Scholar - 3.R.M. Wald.
*General Relativity*(University of Chicago Press, Chicago, 1984)zbMATHGoogle Scholar - 4.J.B. Hartle.
*Gravity: an Introduction to General Relativity*(Addison-Wesley, San Francisco, 2003)Google Scholar - 5.B.F. Schutz.
*A First Course in General Relativity*(Cambridge University Press, New York, 2009)zbMATHGoogle Scholar - 6.R. d’Inverno.
*Introducing Einstein’s Relativity*(Claredon Press, Oxford New York, 1998)zbMATHGoogle Scholar - 7.C.W. Misner, K.S. Thorne, J.A. Wheeler.
*Gravitation*(Freemand and Company, San Francisco, 1973)Google Scholar - 8.S. Chandrasekhar, . Mont. Not. R. Astron. Soc.
**95**, 207–225 (1935)ADSGoogle Scholar - 9.R. Penrose, . Gen. Relat. Gravit.
**34**, 1141–1165 (2002)ADSGoogle Scholar - 10.J.R. Oppenheimer, H. Snyder, . Phys. Rev.
**56**, 455–459 (1939)ADSMathSciNetGoogle Scholar - 11.K. Schwarzschild, . Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.Phys).
**3**, 189–196 (1999). Tradução por Antoci, S. e Loinger. AGoogle Scholar - 12.R.P. Kerr, . Phys. Rev. Lett.
**11**, 237–238 (1963)ADSMathSciNetGoogle Scholar - 13.R.W. Brehme, . Am. J. Phys.
**45**, 423–428 (1977)ADSGoogle Scholar - 14.R. Ruffini, J.A. Wheeler, . Phys. Today.
**24**, 30–41 (1971)ADSGoogle Scholar - 15.R. Blandford, N. Gehrels, . Phys. Today.
**52**, 40–46 (1999)Google Scholar - 16.L.I. Schiff, . Am. J. Phys.
**28**, 340–343 (1960)ADSGoogle Scholar - 17.C.M. Will, . Living Rev. Relativ.
**9**, 1–100 (2001)Google Scholar - 18.S. Deser, . Rev. Mod. Phys.
**29**, 417–423 (1957)ADSMathSciNetGoogle Scholar - 19.A Shomer, arXiv:http://arxiv.org/abs/0709.3555v2 (2007)
- 20.A. Macias, A. Camacho, . Phys. Lett. B.
**663**, 99–102 (2008)ADSMathSciNetGoogle Scholar - 21.S.W. Hawking, R. Penrose, . Proc. Roy. Soc. Lond. A.
**314**, 529–548 (1970)ADSGoogle Scholar - 22.V. Mukhanov, S. Winitzki.
*Introduction to Quantum Effects in Gravity*(Cambridge University Press, Cambridge, 2007)zbMATHGoogle Scholar - 23.L.E. Parker, D.J. Toms.
*Quantum Field Theory in Curved Spacetime*(Cambridge University Press, United Kingdom, 2009)zbMATHGoogle Scholar - 24.N.D. Birrel, P.C.W. Davies.
*Quantum Fields in Curved Space*(Cambridge University Press, New York, 1982)Google Scholar - 25.S.A. Fulling.
*Aspects of Quantum Field Theory in Curved Spacetime*(Cambridge University Press, New York, 1982)Google Scholar - 26.R.M. Wald.
*Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics*(University of Chicago Press, Chicago/London, 1994)zbMATHGoogle Scholar - 27.I. Klebanov, J. Maldecena, . Phys. Today.
**62**, 28 (2009)Google Scholar - 28.S.W. Hawking, . Comm. Math Phys.
**43**, 199–220 (1975)ADSMathSciNetGoogle Scholar - 29.J.D. Bekenstein, . Phys. Rev. D.
**7**, 2333–2346 (1973)ADSMathSciNetGoogle Scholar - 30.S.W. Hawking, . Commun. Math Phys.
**25**, 152–166 (1972)ADSGoogle Scholar - 31.M.C. LoPresto, . Phys. Teacher.
**41**, 299–301 (2003)ADSGoogle Scholar - 32.B.R. Parker, R.J. McLeod, . Am. J. Phys.
**48**, 1066–1070 (1980)ADSGoogle Scholar - 33.J.D. Bekenstein, . Phys. Today.
**33**, 24–31 (1980)ADSGoogle Scholar - 34.J.D. Bekenstein, . Phys. Rev. D.
**12**, 3077–3085 (1975)ADSGoogle Scholar - 35.S.W. Hawking, . Phys. Rev. D.
**13**, 191–197 (1976)ADSGoogle Scholar - 36.P.-H. Lambert, arXiv:http://arxiv.org/abs/1310.8312 (2013)
- 37.F. Belgiorno, M. Martellini, . Int. J. Mod. Phys. D.
**13**, 739–770 (2004)ADSGoogle Scholar - 38.J.H. Traschen, arXiv:http://arxiv.org/abs/0010055 (2000)
- 39.N.S.M. de Santi, R. Santarelli, . Rev. Bras. Ens. Fis.
**41**, e20180312 (2019)Google Scholar - 40.W.G. Unruh, R.M. Wald, . Rep. Prog. Phys.
**80**, 092002 (2017)ADSGoogle Scholar - 41.K. Thorne, R. Blanford.
*Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity and Statistical Physics*(Princeton University Press, New Jersey, 2017)Google Scholar - 42.F. Reif.
*Fundamentals of Statistical and Thermal Physics*(McGray-Hill, New York, 1965)Google Scholar - 43.H.B. Callen.
*Thermodynamics and an Introduction to Thermostatistics*(Wiley, New York, 1985)zbMATHGoogle Scholar - 44.R.A. Jacobson, P.G.A. Treasian, J.J. Bordi, K.E. Criddle, R. Ionasescu, J.B. Jones, R.A. Mackenzie, M.C. Meek, D. Parcher, F.J. Pelletier, J.W.M. Owen, D.C. Roth, I.M. Roundhill, J.R. Stauuch, . AJ.
**132**(6), 2520–2526 (2006)ADSGoogle Scholar - 45.K.A. Olive, et al., Review of particle physics. Chin. Phys.
**C38**, 090001 (2014)ADSGoogle Scholar - 46.A.M. Ghez, S. Salim, N.N. Weinberg, J.R. Lu, T. Do, J.K. Dunn, K. Matthews, M. Morris, S. Yelda, E.E. Becklin, et al., . Astrophys. J.
**689**, 1044–1062 (2008)ADSGoogle Scholar - 47.D.F. Page, . Phys. Rev. D.
**13**, 198–206 (1976)ADSGoogle Scholar - 48.V. Faraoni, . Am. J. Phys.
**85**, 865 (2017)ADSGoogle Scholar - 49.P. Rioseco, O. Sarbach, . Class. Quantum Grav.
**34**, 095007 (2017)ADSGoogle Scholar - 50.E. Bianchi, M. Christodoulou, F. D’Ambrosio, H.M. Haggard, C. Rovelli, . Class. Quantum Grav.
**35**, 225003 (2018)ADSGoogle Scholar - 51.J.E. Horvath, P.S. Custódio, Braz. J. Phys. 35 (2005)ADSGoogle Scholar
- 52.E. Abdalla, C.B.M.H. Chirenti, A. Saa, . J. High Energy Phys.
**2007**, 086 (2007)Google Scholar - 53.Y. Nomura, . Phys. Rev. D.
**99**, 086004 (2019)ADSMathSciNetGoogle Scholar - 54.N.S.M. de Santi.
*Termodinâmica De Buracos Negros De Schwarzschild*(Universidade Federal de São Carlos, Dissertação de mestrado, 2018)Google Scholar