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A nonsingular, anisotropic universe in a black hole with torsion and particle production

Abstract

We consider a universe formed in a black hole in general relativity with spin and torsion. The interior of a Schwarzschild black hole can be represented by the Kantowski–Sachs metric that describes a closed anisotropic universe. We use this metric to derive the Einstein–Cartan field equations with a relativistic spin fluid as a source. We show that torsion may prevent a singularity and replace it with a nonsingular bounce if particle production dominates over shear. Particle production after the last bounce can generate a finite period of inflation, during which the universe expands and isotropizes to the current state. This scenario is only approximate: the Kantowski–Sachs metric is never reached and should be replaced with a more general metric that tends to that of a 3-sphere.

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References

  1. Di Valentino, E., Melchiorri, A., Silk, J.: Nat. Astron. 4, 196 (2020)

    ADS  Google Scholar 

  2. Novikov, I.D.: J. Exp. Theor. Phys. Lett. 3, 142 (1966)

    Google Scholar 

  3. Pathria, R.K.: Nature 240, 298 (1972)

    ADS  Google Scholar 

  4. Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Lett. B 216, 272 (1989)

    ADS  MathSciNet  Google Scholar 

  5. Frolov, V.P., Markov, M.A., Mukhanov, V.F.: Phys. Rev. D 41, 383 (1990)

    ADS  MathSciNet  Google Scholar 

  6. Smolin, L.: Class. Quantum Grav. 9, 173 (1992)

    ADS  Google Scholar 

  7. Hawking, S.: Black Holes and Baby Universes and other Essays. Bantam Dell, New York (1993)

    Google Scholar 

  8. Stuckey, W.M.: Am. J. Phys. 62, 788 (1994)

    ADS  Google Scholar 

  9. Easson, D.A., Brandenberger, R.H.: J. High Energy Phys. 06, 024 (2001)

    ADS  Google Scholar 

  10. Smoller, J., Temple, B.: Proc. Natl. Acad. Sci. USA 100, 11216 (2003)

    ADS  MathSciNet  Google Scholar 

  11. Popławski, N.J.: Phys. Lett. B 687, 110 (2010)

    ADS  MathSciNet  Google Scholar 

  12. Popławski, N.J.: Phys. Lett. B 694, 181 (2010)

    ADS  MathSciNet  Google Scholar 

  13. Popławski, N.J.: Phys. Lett. B 701, 672 (2011)

    ADS  Google Scholar 

  14. Popławski, N.: Astrophys. J. 832, 96 (2016)

    ADS  Google Scholar 

  15. Popławski, N.: Int. J. Mod. Phys. D 27, 1847020 (2018)

    ADS  MathSciNet  Google Scholar 

  16. Popławski, N.: arXiv:1910.10819; arXiv:1912.02173

  17. Lord, E.A.: Tensors, Relativity and Cosmology. McGraw-Hill, New York (1976)

    Google Scholar 

  18. Sciama, D.W.: Proc. Camb. Philos. Soc. 54, 72 (1958)

    ADS  MathSciNet  Google Scholar 

  19. Kibble, T.W.B.: J. Math. Phys. 2, 212 (1961)

    ADS  Google Scholar 

  20. Sciama, D.W.: Recent Developments in General Relativity, p. 415. Pergamon, Oxford (1962)

    Google Scholar 

  21. Sciama, D.W.: Rev. Mod. Phys. 36, 463 (1964)

    ADS  Google Scholar 

  22. Sciama, D.W.: Rev. Mod. Phys. 36, 1103 (1964)

    ADS  Google Scholar 

  23. Hehl, F.W., Datta, B.K.: J. Math. Phys. 12, 1334 (1971)

    ADS  Google Scholar 

  24. Hehl, F.W.: Gen. Relativ. Gravit. 4, 333 (1973)

    ADS  Google Scholar 

  25. Hehl, F.W.: Gen. Relativ. Gravit. 5, 491 (1974)

    ADS  Google Scholar 

  26. Hehl, F.W., von der Heyde, P., Kerlick, G.D., Nester, J.M.: Rev. Mod. Phys. 48, 393 (1976)

    ADS  Google Scholar 

  27. de Sabbata, V., Gasperini, M.: Introduction to Gravitation. World Scientific, Singapore (1985)

    Google Scholar 

  28. de Sabbata, V., Sivaram, C.: Spin and Torsion in Gravitation. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  29. Popławski, N.J.: Phys. Lett. B 690, 73 (2010)

    ADS  MathSciNet  Google Scholar 

  30. Popławski, N.J.: Phys. Lett. B 727, 575 (2013)

    ADS  Google Scholar 

  31. Popławski, N.: Classical Physics: Spacetime and Fields. arXiv:0911.0334

  32. Eisenhart, L.P.: Non-Riemannian Geometry. American Mathematical Society, Providence (1927)

    MATH  Google Scholar 

  33. Schrödinger, E.: Space–Time Structure. Cambridge University Press, Cambridge (1954)

    MATH  Google Scholar 

  34. Schouten, J.A.: Ricci-Calculus. Springer, Berlin (1954)

    MATH  Google Scholar 

  35. Hehl, F.W., McCrea, J.D.: Found. Phys. 16, 267 (1986)

    ADS  MathSciNet  Google Scholar 

  36. Popławski, N.: arXiv:1304.0047

  37. Nomura, K., Shirafuji, T., Hayashi, K.: Prog. Theor. Phys. 86, 1239 (1991)

    ADS  Google Scholar 

  38. Hehl, F.W.: Abh. Braunschw. Wiss. Ges. 18, 98 (1966)

    Google Scholar 

  39. Trautman, A.: Bull. Acad. Polon. Sci. Serie Sci. Math. Astr. Phys. 20, 185 (1972)

    Google Scholar 

  40. Trautman, A.: Symp. Math. 12, 139 (1973)

    MathSciNet  Google Scholar 

  41. Trautman, A.: Nat. Phys. Sci. 242, 7 (1973)

    ADS  Google Scholar 

  42. Kopczyński, W.: Phys. Lett. A 39, 219 (1972)

    ADS  MathSciNet  Google Scholar 

  43. Friedmann, A.: Z. Phys. A 10, 377 (1922)

    Google Scholar 

  44. Lemaître, G.: Ann. Soc. Sci. Bruxelles A 53, 51 (1933)

    Google Scholar 

  45. Robertson, H.P.: Astrophys. J. 82, 284 (1935)

    ADS  Google Scholar 

  46. Walker, A.G.: Proc. Lond. Math. Soc. 42, 90 (1937)

    Google Scholar 

  47. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon, Oxford (1975)

    MATH  Google Scholar 

  48. Fock, V.A.: The Theory of Space, Time and Gravitation. Macmillan, New York (1964)

    MATH  Google Scholar 

  49. Dirac, P.A.M.: General Theory of Relativity. Wiley, New York (1975)

    MATH  Google Scholar 

  50. Hehl, F.W., von der Heyde, P., Kerlick, G.D.: Phys. Rev. D 10, 1066 (1974)

    ADS  MathSciNet  Google Scholar 

  51. Nurgaliev, I.S., Ponomariev, W.N.: Phys. Lett. B 130, 378 (1983)

    ADS  Google Scholar 

  52. Unger, G., Popławski, N.: Astrophys. J. 870, 78 (2019)

    ADS  Google Scholar 

  53. Novello, M., Perez Bergliaffa, S.E.: Phys. Rep. 463, 127 (2008)

    ADS  MathSciNet  Google Scholar 

  54. Kuchowicz, B.: Gen. Relativ. Gravit. 9, 511 (1978)

    ADS  MathSciNet  Google Scholar 

  55. Gasperini, M.: Phys. Rev. Lett. 56, 2873 (1986)

    ADS  Google Scholar 

  56. Obukhov, Y.N., Korotky, V.A.: Class. Quantum Grav. 4, 1633 (1987)

    ADS  Google Scholar 

  57. Popławski, N.J.: Gen. Relativ. Grav. 44, 1007 (2012)

    ADS  Google Scholar 

  58. Popławski, N.J.: Class. Quantum Grav. 31, 065005 (2014)

    ADS  Google Scholar 

  59. Popławski, N.: Mod. Phys. Lett. A 33, 1850236 (2018)

    ADS  MathSciNet  Google Scholar 

  60. Desai, S., Popławski, N.J.: Phys. Lett. B 755, 183 (2016)

    ADS  Google Scholar 

  61. Popławski, N.: Phys. Rev. D 85, 107502 (2012)

    ADS  Google Scholar 

  62. Magueijo, J., Zlosnik, T.G., Kibble, T.W.B.: Phys. Rev. D 87, 063504 (2013)

    ADS  Google Scholar 

  63. Cubero, J.L., Popławski, N.J.: Class. Quantum Grav. 37, 025011 (2020)

    ADS  Google Scholar 

  64. Popławski, N.J.: Phys. Rev. D 83, 084033 (2011)

    ADS  Google Scholar 

  65. Popławski, N.: Gen. Relativ. Gravit. 46, 1625 (2014)

    ADS  MathSciNet  Google Scholar 

  66. Popławski, N.: Found. Phys., In press (2020)

  67. Brehme, R.W.: Am. J. Phys. 45, 423 (1977)

    ADS  Google Scholar 

  68. Doran, R., Lobo, F.S.N., Crawford, P.: Found. Phys. 38, 160 (2008)

    ADS  MathSciNet  Google Scholar 

  69. de Cesare, M., Seahra, S.S., Wilson-Ewing, E.: J. Cosmol. Astropart. Phys. 07, 018 (2020)

    Google Scholar 

  70. Kantowski, R., Sachs, R.K.: J. Math. Phys. 7, 443 (1966)

    ADS  Google Scholar 

  71. Collins, C.B.: J. Math. Phys. 18, 2116 (1977)

    ADS  Google Scholar 

  72. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of the Einstein’s Field Equations. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  73. Da̧browski, M.P.: J. Math. Phys. 36, 2985 (1995)

    ADS  MathSciNet  Google Scholar 

  74. Barrow, J.D., Da̧browski, M.P.: Mon. Not. Roy. Astron. Soc. 275, 850 (1995)

    ADS  Google Scholar 

  75. Barrow, J.D., Ganguly, C.: Int. J. Mod. Phys. D 26, 1743016 (2017)

    ADS  Google Scholar 

  76. Garcia de Andrade, L.C.: Nuovo Cim. B 116, 1107 (2001)

    ADS  Google Scholar 

  77. Weber, E.: J. Math. Phys. 25, 3279 (1984)

    ADS  MathSciNet  Google Scholar 

  78. Barrow, J.D.: Phys. Rev. D 51, 3113 (1995)

    ADS  Google Scholar 

  79. Da̧browski, M.P.: Astrophys. Space Sci. 240, 123 (1996)

    ADS  Google Scholar 

  80. Borgohain, P., Patgiri, M.: Indian J. Phys. A 72, 331 (1998)

    ADS  Google Scholar 

  81. Demiański, M.: Nature 307, 140 (1984)

    ADS  Google Scholar 

  82. Burd, A.B., Barrow, J.D.: Nucl. Phys. B 308, 929 (1988)

    ADS  Google Scholar 

  83. Burd, A.B., Barrow, J.D.: Nucl. Phys. B 324, 276 (1989)

    Google Scholar 

  84. de León, J.P.: J. Math. Phys. 31, 371 (1990)

    ADS  MathSciNet  Google Scholar 

  85. Mendes, L.E., Henriques, A.B.: Phys. Lett. B 254, 44 (1991)

    ADS  Google Scholar 

  86. Byland, S., Scialom, D.: Phys. Rev. D 57, 6065 (1998)

    ADS  MathSciNet  Google Scholar 

  87. Katore, S.D., Rane, R.S.: Astrophys. Space Sci. 323, 293 (2009)

    ADS  Google Scholar 

  88. Li, X.-Z., Hao, J.-G.: Phys. Rev. D 68, 083512 (2003)

    ADS  MathSciNet  Google Scholar 

  89. Barrow, J.D., Tsagas, C.G.: Class. Quantum Grav. 22, 1563 (2005)

    ADS  Google Scholar 

  90. Solomons, D., Dunsby, P.K.S., Ellis, G.F.R.: Class. Quantum Grav. 23, 6585 (2006)

    Google Scholar 

  91. Kopczyński, W.: Phys. Lett. A 43, 63 (1973)

    ADS  Google Scholar 

  92. Kuchowicz, B.: J. Phys. A: Math. Gen. 8, L29 (1975)

    ADS  Google Scholar 

  93. Tafel, J.: Phys. Lett. A 45, 341 (1973)

    ADS  Google Scholar 

  94. Tsoubelis, D.: Phys. Rev. D 20, 3004 (1979)

    ADS  Google Scholar 

  95. Batakis, N.A., Tsoubelis, D.: Phys. Rev. D 26, 2611 (1982)

    ADS  MathSciNet  Google Scholar 

  96. Lorenz, D.: Gen. Relativ. Gravit. 14, 691 (1982)

    ADS  Google Scholar 

  97. Kuchowicz, B.: Phys. Lett. A 54, 13 (1975)

    ADS  MathSciNet  Google Scholar 

  98. Kuchowicz, B.: Astrophys. Space Sci. 39, 157 (1976)

    ADS  Google Scholar 

  99. Kuchowicz, B.: Astrophys. Space Sci. 40, 167 (1976)

    ADS  Google Scholar 

  100. Kuchowicz, B.: Acta Phys. Pol. B 7, 81 (1976)

    ADS  Google Scholar 

  101. Lorenz, D.: J. Phys. A: Math. Gen. 15, 2809 (1982)

    ADS  Google Scholar 

  102. Parker, L.: Phys. Rev. Lett. 21, 562 (1968)

    ADS  Google Scholar 

  103. Parker, L.: Phys. Rev. 183, 1057 (1969)

    ADS  Google Scholar 

  104. Zel’dovich, Y.B.: J. Exp. Theor. Phys. Lett. 12, 307 (1970)

    Google Scholar 

  105. Parker, L.: Phys. Rev. D 3, 346 (1971)

    ADS  Google Scholar 

  106. Parker, L.: Phys. Rev. D 3, 2546 (1971)

    ADS  Google Scholar 

  107. Zel’dovich, Y.B., Starobinskii, A.A.: J. Exp. Theor. Phys. Lett. 26, 252 (1977)

    Google Scholar 

  108. Beilin, V.A., Vereshkov, G.M., Grishkan, Y.S., Ivanov, N.M., Nesterenko, V.A., Poltavtsev, A.N.: J. Exp. Theor. Phys. 51, 1045 (1980)

    ADS  Google Scholar 

  109. Kerr, R.P.: Phys. Rev. Lett. 11, 237 (1963)

    ADS  MathSciNet  Google Scholar 

  110. Saadeh, D., Feeney, S.M., Pontzen, A., Peiris, H.V., McEwen, J.D.: Phys. Rev. Lett. 117, 131302 (2016)

    ADS  Google Scholar 

  111. Colin, J., Mohayaee, R., Rameez, M., Sarkar, S.: Astron. Astrophys. Lett. 631, L13 (2019)

    ADS  Google Scholar 

  112. Migkas, K., Schellenberger, G., Reiprich, T.H., Pacaud, F., Ramos-Ceja, M.E., Lovisari, L.: Astron. Astrophys. 636, A15 (2020)

    ADS  Google Scholar 

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Acknowledgements

We thank John Barrow for pointing out the role of shear in the early universe. This work was funded by the University Research Scholar program at the University of New Haven.

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Correspondence to Nikodem J. Popławski.

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Popławski, N.J. A nonsingular, anisotropic universe in a black hole with torsion and particle production. Gen Relativ Gravit 53, 18 (2021). https://doi.org/10.1007/s10714-021-02790-7

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  • DOI: https://doi.org/10.1007/s10714-021-02790-7

Keywords

  • Einstein–Cartan theory
  • Torsion
  • Spin fluid
  • Black hole
  • Kantowski–Sachs metric
  • Anisotropy
  • Shear
  • Particle production
  • Bounce
  • Nonsingular universe