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Extended phase space thermodynamics of Bardeen black hole in massive gravity

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Abstract

We present an exact solution of Bardeen AdS black hole in massive gravity theory by taking into account the massive gravity parameter m and study its thermodynamic properties such as temperature, entropy and heat capacity at constant pressure. We also study the critical behaviour of Bardeen AdS massive black holes by taking the cosmological constant \(\Lambda \) as the thermodynamic pressure, and it is shown that thermodynamic volume is independent of the massive gravity parameter m. In addition to these, we investigate the dependence of critical parameters on the massive gravity parameter (m) and magnetic monopole charge (e). It is observed that the critical exponents (critical temperature and critical pressure) increase with massive gravity parameter but decrease in the presence of magnetic monopole charge. Also the numerical value of ratio \(P_\mathrm{c}r_\mathrm{c}/T_\mathrm{c}\) increases with increase in the values of the parameters m and e.

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References

  1. J.A. Wheeler, Our universe: the known and unknown. (1967). Public lecture

  2. J.D. Bekenstein, Black hole and entropy. Phys. Rev. D 7, 2333 (1973)

    ADS  MathSciNet  MATH  Google Scholar 

  3. J.D. Bekenstein, Black hole and the second law. Lett. al Nuovo Ciemnto 4, 15 (1972)

    Google Scholar 

  4. J.D. Bekenstein, Generialized second law of thermodynamics in black hole physics. Phys. Rev. D 9, 3292 (1974)

    ADS  Google Scholar 

  5. J.D. Bekenstein, Statistical black hole thermodynamics. Phys. Rev. D 12, 3077 (1975)

    ADS  MathSciNet  Google Scholar 

  6. S.W. Hawking, Black hole and thermodynamics. Phys. Rev. D 13, 191 (1976)

    ADS  Google Scholar 

  7. J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998)

    ADS  MathSciNet  Google Scholar 

  8. J. Bardeen Regular, in Proceedings of GR5 (Tiflis, U.S.S.R., 1968)

  9. E. Ayon-Beato, A. Garcia, Gen. Relativ. Gravit. 31, 629 (1999)

    ADS  Google Scholar 

  10. E. Ayon-Beato, A. Garcia, Gen. Relativ. Gravit. 37, 635 (2005)

    ADS  Google Scholar 

  11. E. Ayon-Beato, A. Garcia, Phys. Lett. B 493, 149 (2000)

    ADS  MathSciNet  Google Scholar 

  12. H. Culetu,. arXiv:1408.3334v1 [gr-qc]

  13. L. Balart, E.C. Vagenas, Phys. Lett. B 730, 14 (2014)

    ADS  MathSciNet  Google Scholar 

  14. L. Balart, E.C. Vagenas, Phys. Rev. D 90(12), 124045 (2014)

    ADS  Google Scholar 

  15. L. Xiang, Y. Ling, Y.G. Shen, Int. J. Mod. Phys. D 22, 1342016 (2013)

    ADS  Google Scholar 

  16. K.A. Bronnikov, Phys. Rev. D 63, 044005 (2001)

    ADS  MathSciNet  Google Scholar 

  17. O.B. Zaslavskii, Phys. Rev. D 80, 064034 (2009)

    ADS  MathSciNet  Google Scholar 

  18. J.P.S. Lemos, V.T. Zanchin, Phys. Rev. D 83, 124005 (2011)

    ADS  Google Scholar 

  19. D.V. Singh, S. Siwach. arXiv: 1909.11529 [hep-th]

  20. S. Ansoldi. arXiv:0802.0330 [gr-qc]

  21. S.G. Ghosh, Eur. Phys. J. C 75, 532 (2015)

    ADS  Google Scholar 

  22. D.V. Singh, N.K. Singh, Ann. Phys. 383, 600 (2017)

    ADS  Google Scholar 

  23. S. Fernando, Int. J. Mod. Phys. D 26, 1750071 (2017)

    ADS  Google Scholar 

  24. C. Bambi, L. Modesto, Phys. Lett. B 721, 329 (2013)

    ADS  MathSciNet  Google Scholar 

  25. M. Sharif, W. Javed, Can. J. Phys. 89, 1027 (2011)

    ADS  Google Scholar 

  26. Md Sabir Ali, S.G. Ghosh, Phys. Rev. D 98, 084025 (2018)

    ADS  MathSciNet  Google Scholar 

  27. D.V. Singh, S.G. Ghosh, S.D. Maharaj, Ann. Phys. 412, 168025 (2019)

    Google Scholar 

  28. A. Kumar, D.V. Singh, S.G. Ghosh, Eur. Phys. J. C 79, 275 (2019)

    ADS  Google Scholar 

  29. A. Kumar, D.V. Singh, S.G. Ghosh, Ann. Phys. 419, 168214 (2020)

    Google Scholar 

  30. D.V. Singh, S. Siwach, Phys. Lett. B 808, 135658 (2020)

    MathSciNet  Google Scholar 

  31. A. Kumar, R. Kumar. arXiv:2003.13104 [gr-qc]

  32. R.G. Cai, Y.P. Hu, Q.Y. Pan, Y.L. Zhang, Phys. Rev. D 91(2), 024032 (2015)

    ADS  MathSciNet  Google Scholar 

  33. E. Babichev, R. Brito, Class. Quantum Gravity 32, 154001 (2015)

    ADS  Google Scholar 

  34. B. Eslam Panah, S. Hendi, EPL 125(6), 60006 (2019)

    ADS  Google Scholar 

  35. B. Eslam Panah, S. Hendi, Y. Ong, Phys. Dark Univ. 27, 100452 (2020)

    Google Scholar 

  36. M. Cvetic, G.W. Gibbons, D. Kubiznak, C.N. Pope, Phys. Rev. D 84, 024037 (2011)

    ADS  Google Scholar 

  37. B.P. Dolan, D. Kastor, D. Kubiznak, R.B. Mann, J. Traschen, Phys. Rev. D 87, 104017 (2013)

    ADS  Google Scholar 

  38. S. Hawking, D. Page, Commun. Math. Phys. 87, 577 (1983)

    ADS  Google Scholar 

  39. E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)

    MathSciNet  Google Scholar 

  40. A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Phys. Rev. D 60, 104026 (1999)

    ADS  MathSciNet  Google Scholar 

  41. A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Phys. Rev. D 60, 06401 (1999)

    Google Scholar 

  42. D. Kubiznák, R.B. Mann, JHEP 1207, 033 (2012)

    ADS  Google Scholar 

  43. S.H. Hendi, M.H. Vahidinia, Phys. Rev. D 88, 084045 (2013)

    ADS  Google Scholar 

  44. R.-G. Cai, L.-M. Cao, L. Li, R.-Q. Yang, JHEP 1309, 005 (2013)

    ADS  Google Scholar 

  45. J.-X. Mo, W.-B. Liu, Phys. Lett. B 727, 336 (2013)

    ADS  Google Scholar 

  46. J.-X. Mo, W.-B. Liu, Eur. Phys. J. C 74, 2836 (2014)

    ADS  Google Scholar 

  47. J.-X. Mo, G.-Q. Li, W.-B. Liu, Phys. Lett. B 730, 111 (2014)

    ADS  Google Scholar 

  48. G.-Q. Li, Phys. Lett. B 735, 256 (2014)

    ADS  Google Scholar 

  49. H.-H. Zhao, L.-C. Zhang, M.-S. Ma, R. Zhao, Phys. Rev. D 90, 064018 (2014)

    ADS  Google Scholar 

  50. M.H. Dehghani, S. Kamrani, A. Sheykhi, Phys. Rev. D 90, 104020 (2014)

    ADS  Google Scholar 

  51. R.A. Hennigar, W.G. Brenna, R.B. Mann, JHEP 1507, 077 (2015)

    ADS  Google Scholar 

  52. J. Xu, L.M. Cao, Y.P. Hu, Phys. Rev. D 91, 124033 (2015)

    ADS  Google Scholar 

  53. S.H. Hendi, R.M. Tad, Z. Armanfard, M.S. Talezadeh, Eur. Phys. J. C 76, 263 (2016)

    ADS  Google Scholar 

  54. J. Sadeghi, Int. J. Theor. Phys. 55, 2455 (2016)

    Google Scholar 

  55. J. Liang, C.-B. Sun, H.-T. Feng, Europhys. Lett. 113, 30008 (2016)

    ADS  Google Scholar 

  56. S. Fernando, Phys. Rev. D 94, 124049 (2016)

    ADS  MathSciNet  Google Scholar 

  57. Z.-Y. Fan, Eur. Phys. J. C 77, 266 (2016)

    ADS  Google Scholar 

  58. J. Sadeghi, B. Pourhassan, M. Rostami, Phys. Rev. D 94, 064006 (2016)

    ADS  MathSciNet  Google Scholar 

  59. D. Hansen, D. Kubiznak, R.B. Mann, JHEP 1701, 047 (2017)

    ADS  Google Scholar 

  60. B.R. Majhi, S. Samanta, Phys. Lett. B 773, 203 (2017)

    ADS  Google Scholar 

  61. S.H. Hendi, B.E. Panah, S. Panahiyan, M.S. Talezadeh, Eur. Phys. J. C 77, 133 (2017)

    ADS  Google Scholar 

  62. S. Upadhyay, B. Pourhassan, H. Farahani, Phys. Rev. D 95, 106014 (2017)

    ADS  MathSciNet  Google Scholar 

  63. C.H. Nam, Eur. Phys. J. C 78, 581 (2018)

    ADS  Google Scholar 

  64. P. Pradhan, Mod. Phys. Lett. A 32, 1850030 (2018)

    Google Scholar 

  65. D. Kastor, S. Ray, Class. Quantum Gravity 26, 195011 (2009)

    ADS  Google Scholar 

  66. Cao H. Nam, Eur. Phys. J. C 78, 1016 (2018)

    ADS  Google Scholar 

  67. D. Vegh, Holography without translational symmetry. arXiv: 1301.0537

  68. Rong-Gen Cai, Hu Ya-Peng, Qi-Yuan Pan, Yun-Long Zhang, Phys. Rev. D 92, 024032 (2015)

    ADS  MathSciNet  Google Scholar 

  69. A.G. Tzikas, Phys. Lett. B 788, 219–224 (2019)

    ADS  Google Scholar 

  70. S.H. Hendi, B.Eslam Panah, S. Panahiyan, H. Liu, X.-H. Meng, Phys. Lett. B 781, 40 (2018)

    ADS  Google Scholar 

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Acknowledgements

Authors would like to thank to Dr. Sanjay Siwach for useful discussions and remarks.

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Authors and Affiliations

  1. Department of Physics, Institute of Applied Science and Humanities, GLA University, Mathura, 281406, India

    Benoy Kumar Singh, Raj Pal Singh & Dharm Veer Singh

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  1. Benoy Kumar Singh
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  2. Raj Pal Singh
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  3. Dharm Veer Singh
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Correspondence to Dharm Veer Singh.

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Singh, B.K., Singh, R.P. & Singh, D.V. Extended phase space thermodynamics of Bardeen black hole in massive gravity. Eur. Phys. J. Plus 135, 862 (2020). https://doi.org/10.1140/epjp/s13360-020-00880-0

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