Abstract
The canonical formalism of a spherically symmetric spacetime has been decomposition with respect to the radial coordinate r in the context of \(3 + 1\) split of spacetime. We have set up an effective Lagrangian that plays the role of independent variable couple with metric function. The surface gravity of Schwarzschild–de Sitter black hole has been quantized from the test particle moving around different energy states like Bohr’s atomic model. We have quantized the Hawking temperature and entropy of Schwarzschild–de Sitter black hole from quantization of surface gravity. We also have shown that the change of entropy reduced to zero when the boundary shrinks to very small size.
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References
S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space Time (Cambridge University Press, Combridge, 1973)
R.M. Wald, General Relativity (Chicago University Press, Chicago, 1984)
J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)
J.D. Bekenstein, Phys. Rev. D 9, 3292 (1974)
J.D. Bekenstein, Nuovo Cimento 11, 467 (1974)
J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phys. 31, 161 (1973)
S.W. Hawking, Commun. Math. Phys. 43, 199 (1975)
G.W. Gibbons, S.W. Hawking, Phys. Rev. D 15, 2752 (1977)
N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982)
T. Padmanabhan, Phys. Rep. 406, 49 (2005)
N. Seiberg, E. Witten, J. High Energy Phys. 9, 032 (1999)
G.T. Horowitz, New J. Phys. 7, 201 (2005)
A. Ashtekar, New J. Phys. 7, 198 (2005)
S. Mukherji and S.S. Pal, J. High Energy Phys. 205, 026 (2002). (arXiv: hep-th/ 0205164)
A. Ashtekar and K. Krasnov, in Black holes, Gravitational Radiation and the Universe, eds. B. Iyer and B. Bhawal (Kluwer Dodrecht, 1999) 149 (arXiv: gr-qc/9804039)
J. Louko, J. Makela, Phys. Rev. D 54, 4982 (1996)
J. Makela, Schroedinger Equation of the Schwarzschild Black Hole (arXiv: gr-qc/9602008)
J. Makela, P. Repo, Phys. Rev. D 57, 4899 (1998). (arXiv: gr-qc/9708029)
C. Kiefer, J. Marto and P.V. Moniz, Ann. Phys. (Berlin) 18, 722 (2009). (arXiv: 0812.2848 [grqc])
K. Nakamura, S. Konno, Y. Oshiro, A. Tomimatsu, Prog. Theor. Phys. 90, 861 (1993). (arXiv: gr-qc/9308029)
M. Kenmoku, H. Kubotani, E. Takasugi, Y. Yamazaki, Phys. Rev. D 59, 124004 (1999). (arXiv: gr-qc/9810042)
B.-B. Wang, Gen. Rel. Grav. 41, 1181 (2009)
J.C. Lopez-Domingues, O. Obregon, M. Sabido, Phys. Rev. D 74, 084024 (2006). (arXiv: hep-th/0607002)
O. Obregon, M. Sabido, V.I. Tkach, Gen. Rel. Grav. 33, 913 (2001). (arXiv: gr-qc/0003023)
M. Kenmoku, H. Kubotani, E. Takasugi, Y. Yamazaki, Phys. Rev. D 57, 4925 (1998). (arXiv: gr-qc/9711039)
M.A. Rahman, Commun. Theor. Phys. 71, 307 (2019)
A. Matte, Can. J. Math. 5, 1 (1953)
P. Teyssandier, Phys. Rev. D 16, 946 (1977)
L. Bel, Comput. Rend. 247, 1094 (1958)
P. Teyssandier, Phys. Rev. D 18, 1037 (1978)
V. Braginsky, C. Caves, K.S. Thorne, Phys. Rev. D 15, 2047 (1977)
M.A.G. Bonilla, J.M.M. Senovilla, Phys. Rev. Lett. 11, 783 (1977)
R. Jantzen, P. Carini, D. Bini, Ann. Phys. 215, 1 (1992)
R. Maartens, B.A. Bassett, Class. Quantum Gr. 15, 705 (1998)
S.J. Clark, R.W. Tucker, Class. Quantum Gr. 17, 4125 (2000)
J.M.M. Senovilla, Mod. Phys. Lett. A 15, 159 (2000)
J.M.M. Senovilla, Class. Quantum Gr. 17, 2799 (2000)
M.L. Ruggiero, A. Tartaglia, Nuovo Cimento B 117, 743 (2002)
L. Iorio, D.M. Lucchesi, Class. Quantum Gr. 20, 2477 (2003)
A. Zee, Phys. Rev. Lett. 55, 2379 (1985)
P.A.M. Dirac, R. Soc. London Ser. A 133, 60 (1931)
I. Ciufolini, A.J. Wheeler, Gravitation and Inertia (Princeton University Press, Princeton, 1995)
B. Mashhoon; arXiv:gr-qc/0311030 (1st chapter of The Measurement of Gravitomagnetism: A Challenging Enter- prise, edited by L. Iorio (Nova Science, New York, 2007), pp. 29-39)
M.A. Rahman, M.J. Hossain, M.I. Hossain, Astropart. Phys. 71, 71 (2015)
E. Simanek; preprint (arXiv:1209.3791v1)
W. Wilson, Phil. Mag. 29, 795 (1915)
A. Sommerfeld, Ann. Physik 51, 1 (1916)
I. Sakalli, M. Halilsoy, H. Pasaoglu, Astrophys. Space Sci. 340, 1555 (2012)
B. Zhang, Phys. Rev. D 92, 081501 (2015)
Y.C. Ong, Gen. Relativ. Gr. 47, 88 (2015)
B.R. Majhi, S. Samanta, Phys. Lett. B 770, 314 (2017)
X.G. He, B.Q. Ma, Mod. Phys. Lett. A 26, 2299 (2011)
E.P. Verlinde, JHEP 04, 029 (2011)
X.G. He, B.Q. Ma, Chin. Phys. Lett. 27, 070402 (2010)
Acknowledgements
The research is supported by Rajshahi University (No. A-1371/5/52/R.U./Sciece-32/1918-1919) from Faculty of Science, Rajshahi University, Bangladesh.
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Rahman, M.A. Entropy quantization of Schwarzschild–de Sitter black hole. Eur. Phys. J. Plus 135, 783 (2020). https://doi.org/10.1140/epjp/s13360-020-00827-5
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DOI: https://doi.org/10.1140/epjp/s13360-020-00827-5