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Nonsingular Black Holes in Palatini Extensions of General Relativity

  • Gonzalo J. Olmo
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 176)

Abstract

An introduction to extended theories of gravity formulated in metric-affine (or Palatini) spaces is presented. Focusing on spherically symmetric configurations with electric fields, we will see that in these theories the central singularity present in General Relativity is generically replaced by a wormhole structure. The resulting space-time becomes geodesically complete and, therefore, can be regarded as non-singular. We illustrate these properties considering two different models, namely, a quadratic f(R) theory and a Born-Infeld like gravity theory.

Keywords

Black Hole Field Equation Black Hole Solution Ricci Tensor Curvature Divergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The author is supported by a Ramon y Cajal contract, the Spanish grants FIS2014-57387-C3-1-P and FIS2011-29813-C02-02 from MINECO, the grants i-LINK0780 and i-COOPB20105 of the Spanish Research Council (CSIC), the Consolider Program CPANPHY- 1205388. This work has also been supported by the Severo Ochoa Grant SEV-2014-0398 (Spain) and the CNPq project No. 301137/2014-5 (Brazilian agency).

References

  1. 1.
    R. Penrose, Phys. Rev. Lett. 14, 57 (1965)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Penrose, Riv. Nuovo Cim. Numero Speciale 1, 252 (1969)ADSGoogle Scholar
  3. 3.
    R. Penrose, Gen. Relativ. Gravit. 34, 1141 (2002)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    S.W. Hawking, Phys. Rev. Lett. 17, 444 (1966)ADSCrossRefGoogle Scholar
  5. 5.
    B. Carter, Phys. Rev. Lett. 26, 331 (1971)ADSCrossRefGoogle Scholar
  6. 6.
    S.W. Hawking, Phys. Rev. D 14, 2460 (1976)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation and Cosmology (W.H. Freeman, London, 1973)Google Scholar
  8. 8.
    E. Curiel, P. Bokulich, Singularities and black holes, in The Stanford Encyclopedia of Philosophy ed. by E.N. Zalta. (http://plato.stanford.edu/archives/fall2012/entries/spacetime-singularities/-2012)
  9. 9.
    R.P. Geroch, Ann. Phys. 48, 526 (1968)ADSCrossRefGoogle Scholar
  10. 10.
    S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)zbMATHCrossRefGoogle Scholar
  11. 11.
    R.M. Wald, General Relativity (University Press, Chicago, 1984)zbMATHCrossRefGoogle Scholar
  12. 12.
    J.M.M. Senovilla, D. Garfinkle, Class. Quantum Gravit. 32(12), 124008 (2015). arXiv:1410.5226 [gr-qc]
  13. 13.
    V.F. Mukhanov, R.H. Brandenberger, Phys. Rev. Lett. 68, 1969 (1992)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    S. Ansoldi, arXiv:0802.0330 [gr-qc]
  15. 15.
    J.P.S. Lemos, V.T. Zanchin, Phys. Rev. D 83, 124005 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    E. Spallucci, A. Smailagic, P. Nicolini, Phys. Lett. B 670, 449 (2009)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    K.A. Bronnikov, J.C. Fabris, Phys. Rev. Lett. 96, 251101 (2006)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    S.A. Hayward, Phys. Rev. Lett. 96, 031103 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    G.J. Olmo, D. Rubiera-Garcia, A. Sanchez-Puente, Phys. Rev. D 92(4), 044047 (2015)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Visser, Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, New York, 1995)Google Scholar
  21. 21.
    F.S.N. Lobo, Classical and Quantum Gravity Research (Nova Science Publishers, New York, 2008), pp. 1–78. arXiv:0710.4474 [gr-qc]
  22. 22.
    J.A. Wheeler, Phys. Rev. 97, 511 (1955)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    F.S.N. Lobo, G.J. Olmo, D. Rubiera-Garcia, JCAP 1307, 011 (2013); Eur. Phys. J. C 74, 2924 (2014)ADSGoogle Scholar
  24. 24.
    F.S.N. Lobo, J. Martinez-Asencio, G.J. Olmo, D. Rubiera-Garcia, Phys. Lett. B 731, 163 (2014)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Ortin, Gravity and Strings (Cambridge Monographs on Mathematical Physics), 2nd edn. (2015)Google Scholar
  26. 26.
    L.P. Eisenhart, Non-Riemannian Geometry. Am. Math. Soc. 184 (1927)Google Scholar
  27. 27.
    H. Feigl, M. Brodbeck (eds.), Readings in the Phylosophy of Science (Appleton-Century-Crofts, New York, 1953). (See page 193)Google Scholar
  28. 28.
    C.M. Will, Living Rev. Relativ. 17, 4 (2014). arXiv:1403.7377 [gr-qc]ADSCrossRefGoogle Scholar
  29. 29.
    E. Kröner, Z. angew. Math. Mech. 66, 5 (1986)Google Scholar
  30. 30.
    H. Kleinert, Gauged Fields in Condensed Matter, vol. II (World Scientific, Singapore, 1989)zbMATHCrossRefGoogle Scholar
  31. 31.
    E. Kröner, Int. J. Theor. Phys. 29, 1219 (1990)CrossRefGoogle Scholar
  32. 32.
    R. de Witt, Int. J. Eng. Sci. 19, 1475 (1981)CrossRefGoogle Scholar
  33. 33.
    E. Kröner, Trends in applications of pure mathematics to mechanics. Lect. Notes Phys. 249, 281 (1986)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    E. Kröner, Int. J. Solids Struct. 29, 1849 (1992)CrossRefGoogle Scholar
  35. 35.
    K. Kondo, Proceedings of the 2nd Japan Kat. Congr. of Appl. at Max-Planek-lnstitut fur Mctallforschung, Postfach 800665, D-7000 Stuttgart 80, BRD. Mechanics, p. 41 (1952)Google Scholar
  36. 36.
    B.A. Bilby, R. Bullough, E. Smith, Proc. Roy. Soc. Lond. Ser. A 231, 263 (1955)Google Scholar
  37. 37.
    F.S.N. Lobo, G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 91(12), 124001 (2015). arXiv:1412.4499 [hep-th]
  38. 38.
    G.J. Olmo, D. Rubiera-Garcia, Int. J. Mod. Phys. D 24(09), 1542013 (2015). arXiv:1507.07777 [hep-th]
  39. 39.
    G.J. Olmo, D. Rubiera-Garcia, J. Phys. Conf. Ser. 600(1), 012041 (2015). arXiv:1506.02139 [gr-qc]
  40. 40.
    M. Ferraris, M. Francaviglia, C. Reina, Gen. Relativ. Gravit. 14(3), 244 (1982)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 88, 084030 (2013). arXiv:1306.4210 [hep-th]ADSCrossRefGoogle Scholar
  42. 42.
    G.J. Olmo, Chapter of the book ‘Open Questions in Cosmology’, ed. by G.J. Olmo (InTech Publishing, Rijeka, Croatia, 2012) ISBN 978-953-51-0880-1. (Downloadable from http://www.intechopen.com/books/open-questions-in-cosmology). arXiv:1212.6393
  43. 43.
    G.J. Olmo, Int. J. Mod. Phys. D 20, 413 (2011). arXiv:1101.3864 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    L. Parker, D.J. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity (Cambridge University Press, Cambridge, 2009)zbMATHCrossRefGoogle Scholar
  45. 45.
    N.D. Birrel, P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982)CrossRefGoogle Scholar
  46. 46.
    J.A.R. Cembranos, Phys. Rev. Lett. 102, 141301 (2009). arXiv:0809.1653 [hep-ph]ADSCrossRefGoogle Scholar
  47. 47.
    R. Schimming, R.J. Schmidt, NTM Schriftenr. Gesch. Naturw. Tech. Med. 27, 41 (1990). arXiv:gr-qc/0412038
  48. 48.
    R. Utiyama, B.S. DeWitt, J. Math. Phys. 3, 608 (1962)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    K.S. Stelle, Phys. Rev. D 16(3), 953 (1977)Google Scholar
  50. 50.
    K.S. Stelle, Gen. Relativ. Gravit. 9, 353 (1978)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    H. Weyl, Ann. der Phys. 59, 101 (1919)ADSCrossRefGoogle Scholar
  52. 52.
    H. Weyl, Phys. Zeits. 22, 473 (1921)Google Scholar
  53. 53.
    W. Pauli, Phys. Zeits. 20, 457 (1919)Google Scholar
  54. 54.
    A.A. Starobinsky, Phys. Lett. 91B, 99 (1980)ADSCrossRefGoogle Scholar
  55. 55.
    P. Anderson, Phys. Rev. D 28, 271 (1983)Google Scholar
  56. 56.
    A. Economou, C.O. Lousto, Phys. Rev. D 49(1), 5278 (1994). arXiv:gr-qc/9310021
  57. 57.
    H. Lu, C. Pope, Phys. Rev. Lett. 106, 181302 (2011). arXiv:1101.1971 [hep-th]
  58. 58.
    I. Gullu, M. Gurses, T.C. Sisman, B. Tekin, Phys. Rev. D 83, 084015 (2011). arXiv:1102.1921 [hep-th]ADSCrossRefGoogle Scholar
  59. 59.
    H. Liu, H. Lu, M. Luo, Int. J. Mod. Phys. D 21, 1250020 (2012). arXiv:1104.2623 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    W. Berej, J. Matyjasek, D. Tryniecki, M. Woronowicz, Gen. Relativ. Gravit. 38, 885 (2006). arXiv:hep-th/0606185 ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 86, 044014 (2012)ADSCrossRefGoogle Scholar
  62. 62.
    G.J. Olmo, D. Rubiera-Garcia, Int. J. Mod. Phys. D 21, 1250067 (2012)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    G.J. Olmo, D. Rubiera-Garcia, Eur. Phys. J. C 72, 2098 (2012)ADSCrossRefGoogle Scholar
  64. 64.
    G.J. Olmo, H. Sanchis-Alepuz, S. Tripathi, Phys. Rev. D 80, 024013 (2009). arXiv:0907.2787 [gr-qc]
  65. 65.
    G.J. Olmo, Phys. Rev. D 72, 083505 (2005). arXiv:gr-qc/0505135
  66. 66.
    G.J. Olmo, Phys. Rev. Lett. 95, 261102 (2005). arXiv:gr-qc/0505101
  67. 67.
    J. Martinez-Asencio, G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 86, 104010 (2012). arXiv:1209.3371 [gr-qc]
  68. 68.
    S. Deser, G.W. Gibbons, Class. Quantum Gravit. 15, L35 (1998). arXiv:hep-th/9803049
  69. 69.
    D.N. Vollick, Phys. Rev. D 69, 064030 (2004). arXiv:gr-qc/0309101
  70. 70.
    X.-L. Du, K. Yang, X.-H. Meng, Y.X. Liu, arXiv:1403.0083 [gr-qc]
  71. 71.
    H.C. Kim, arXiv:1312.0703 [gr-qc]
  72. 72.
    S.I. Kruglov, Phys. Rev. D 89, 064004 (2014). arXiv:1310.6915 [gr-qc]ADSCrossRefGoogle Scholar
  73. 73.
    K. Yang, X.-L. Du, Y.-X. Liu, Phys. Rev. D 88, 124037 (2013). arXiv:1307.2969 [gr-qc]ADSCrossRefGoogle Scholar
  74. 74.
    P.P. Avelino, R.Z. Ferreira, Phys. Rev. D 86, 041501 (2012). arXiv:1205.6676 [astro-ph.CO]ADSCrossRefGoogle Scholar
  75. 75.
    A. De Felice, B. Gumjudpai, S. Jhingan, Phys. Rev. D 86, 043525 (2012). arXiv:1205.1168 [gr-qc]ADSCrossRefGoogle Scholar
  76. 76.
    C. Escamilla-Rivera, M. Bañados, P.G. Ferreira, Phys. Rev. D 85, 087302 (2012). arXiv:1204.1691 [gr-qc]ADSCrossRefGoogle Scholar
  77. 77.
    I. Cho, H.-C. Kim, T. Moon, Phys. Rev. D 86, 084018 (2012). arXiv:1208.2146 [gr-qc]ADSCrossRefGoogle Scholar
  78. 78.
    J.H.C. Scargill, M. Bañados, P.G. Ferreira, Phys. Rev. D 86, 103533 (2012). arXiv:1210.1521 [astro-ph.CO]ADSCrossRefGoogle Scholar
  79. 79.
    C. Escamilla-Rivera, M. Bañados, P.G. Ferreira, arXiv:1301.5264 [gr-qc]
  80. 80.
    M. Bañados, P.G. Ferreira, Phys. Rev. Lett. 105, 011101 (2010). arXiv:1006.1769 [astro-ph.CO]ADSMathSciNetCrossRefGoogle Scholar
  81. 81.
    T. Harko, F.S.N. Lobo, M.K. Mak, S.V. Sushkov, Mod. Phys. Lett. A 29, 1450049 (2014). arXiv:1305.0820 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  82. 82.
    P.P. Avelino, Phys. Rev. D 85, 104053 (2012). arXiv:1201.2544 [astro-ph.CO]ADSCrossRefGoogle Scholar
  83. 83.
    P.P. Avelino, JCAP 1211, 022 (2012). arXiv:1207.4730 [astro-ph.CO]ADSCrossRefGoogle Scholar
  84. 84.
    Y.-H. Sham, L.-M. Lin, P.T. Leung, Astrophys. J. 781, 66 (2014). arXiv:1312.1011 [gr-qc]ADSCrossRefGoogle Scholar
  85. 85.
    H.-C. Kim, arXiv:1312.0705 [gr-qc]
  86. 86.
    T. Harko, F.S.N. Lobo, M.K. Mak, S.V. Sushkov, Phys. Rev. D 88, 044032 (2013). arXiv:1305.6770 [gr-qc]ADSCrossRefGoogle Scholar
  87. 87.
    Y.H. Sham, P.T. Leung, L.M. Lin, Phys. Rev. D 87, 061503 (2013). arXiv:1304.0550 [gr-qc]ADSCrossRefGoogle Scholar
  88. 88.
    Y.-H. Sham, L.-M. Lin, P.T. Leung, Phys. Rev. D 86, 064015 (2012). arXiv:1208.1314 [gr-qc]ADSCrossRefGoogle Scholar
  89. 89.
    P. Pani, T.P. Sotiriou, Phys. Rev. Lett. 109, 251102 (2012). arXiv:1209.2972 [gr-qc]ADSCrossRefGoogle Scholar
  90. 90.
    P. Pani, V. Cardoso, T. Delsate, Phys. Rev. Lett. 107, 031101 (2011). arXiv:1106.3569 [gr-qc]ADSCrossRefGoogle Scholar
  91. 91.
    M. Bouhmadi-Lopez, C.Y. Chen, P. Chen, arXiv:1302.5013 [gr-qc]
  92. 92.
    R. Ferraro, F. Fiorini, J. Phys. Conf. Ser. 314, 012114 (2011). arXiv:1011.5196 [gr-qc]ADSCrossRefGoogle Scholar
  93. 93.
    G.J. Olmo, D. Rubiera-Garcia, A. Sanchez-Puente, Eur. Phys. J. C. 76(3), 143 (2016)Google Scholar
  94. 94.
    G.J. Olmo, D. Rubiera-Garcia, H. Sanchis-Alepuz, Eur. Phys. J. C 74, 2804 (2014). arXiv:1311.0815 [hep-th]ADSCrossRefGoogle Scholar
  95. 95.
    R. Shaikh, arXiv:1505.01314 [gr-qc]
  96. 96.
    C. Bambi, G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 91(10), 104010 (2015). arXiv:1504.01827 [gr-qc]
  97. 97.
    F.S.N. Lobo, G.J. Olmo, D. Rubiera-Garcia, Eur. Phys. J. C. 74(6), 2924 (2014)Google Scholar
  98. 98.
    T. Harko, F.S.N. Lobo, M.K. Mak, S.V. Sushkov, arXiv:1307.1883 [gr-qc]
  99. 99.
    S.D. Odintsov, G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 90(4), 044003 (2014). arXiv:1406.1205 [hep-th]
  100. 100.
    A.N. Makarenko, S. Odintsov, G.J. Olmo, arXiv:1403.7409 [hep-th]
  101. 101.
    A.N. Makarenko, S.D. Odintsov, G.J. Olmo, Phys. Lett. B 734, 36 (2014). arXiv:1404.2850 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar
  102. 102.
    D. Comelli, Phys. Rev. D 72, 064018 (2005)ADSMathSciNetCrossRefGoogle Scholar
  103. 103.
    D. Comelli, A. Dolgov, JHEP 0411(062), 062 (2004)ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    J.A. Feigenbaum, Phys. Rev. D 58, 124023 (1998)ADSMathSciNetCrossRefGoogle Scholar
  105. 105.
    J.A. Feigenbaum, P.G.O. Freund, M. Pigli, Phys. Rev. D 57, 4738 (1998)ADSMathSciNetCrossRefGoogle Scholar
  106. 106.
    R. Ferraro, F. Fiorini, Phys. Lett. B 692, 206 (2010)ADSMathSciNetCrossRefGoogle Scholar
  107. 107.
    F. Fiorini, Phys. Rev. Lett. 111, 041104 (2013)ADSCrossRefGoogle Scholar
  108. 108.
    I. Gullu, T.C. Sisman, B. Tekin, Class. Quantum Gravit. 27, 162001 (2010)ADSMathSciNetCrossRefGoogle Scholar
  109. 109.
    I. Gullu, T.C. Sisman, B. Tekin, Phys. Rev. D 81, 104018 (2010)ADSMathSciNetCrossRefGoogle Scholar
  110. 110.
    J.A. Nieto, Phys. Rev. D 70, 044042 (2004)ADSMathSciNetCrossRefGoogle Scholar
  111. 111.
    M.N.R. Wohlfarth, Class. Quantum Gravit. 21, 1927 (2004); Corrigendum, Class. Quantum Gravit. 21, 5297 (2004)Google Scholar
  112. 112.
    J.B. Jiménez, L. Heisenberg, G.J. Olmo, JCAP 1411, 004 (2014). arXiv:1409.0233 [hep-th]CrossRefGoogle Scholar
  113. 113.
    J.B. Jiménez, L. Heisenberg, G.J. Olmo, JCAP 1506, 026 (2015). arXiv:1504.00295 [gr-qc]CrossRefGoogle Scholar
  114. 114.
    J.B. Jiménez, L. Heisenberg, G.J. Olmo, C. Ringeval, arXiv:1509.01188 [gr-qc]
  115. 115.
    D. Bazeia, L. Losano, R. Menezes, G.J. Olmo, D. Rubiera-Garcia, Class. Quantum Gravit. 32(21), 215011 (2015). arXiv:1509.04895 [hep-th]
  116. 116.
    D. Bazeia, L. Losano, G.J. Olmo, D. Rubiera-Garcia, A. Sanchez-Puente, Phys. Rev. D 92(4), 044018 (2015). arXiv:1507.07763 [hep-th]
  117. 117.
    C. Barragan, G.J. Olmo, Phys. Rev. D 82, 084015 (2010). arXiv:1005.4136 [gr-qc]ADSCrossRefGoogle Scholar
  118. 118.
    C.W. Misner, J.A. Wheeler, Ann. Phys. 2, 525 (1957)ADSMathSciNetCrossRefGoogle Scholar
  119. 119.
    M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP, Woodbury, 1995)Google Scholar
  120. 120.
    F.S.N. Lobo, G.J. Olmo, D. Rubiera-Garcia, JCAP 1307, 011 (2013). arXiv:1306.2504 [hep-th]ADSCrossRefGoogle Scholar
  121. 121.
    H. Stephani, E. Herlt, M. MacCullum, C. Hoenselaers, D. Kramer, Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge, 2003), p. 726zbMATHCrossRefGoogle Scholar
  122. 122.
    A. Einstein, N. Rosen, Phys. Rev. 48, 73 (1935)ADSCrossRefGoogle Scholar
  123. 123.
    A.V.B. Arellano, F.S.N. Lobo, Class. Quantum Gravit. 23, 7229 (2006). arXiv:gr-qc/0604095 ADSMathSciNetCrossRefGoogle Scholar
  124. 124.
    A. Fabbri, J. Navarro-Salas, Modeling Black Hole Evaporation (Imperial College Press, London, 2005)zbMATHCrossRefGoogle Scholar
  125. 125.
    S. Hawking, Mon. Not. Roy. Astron. Soc. 152, 75 (1971)Google Scholar
  126. 126.
    G.J. Olmo, D. Rubiera-Garcia, Universe 2015, 1(2), 173–185 (2015). arXiv:1509.02430 [hep-th]
  127. 127.
    C. Bejarano, G.J. Olmo, and D. Rubiera-Garcia, Work in ProgressGoogle Scholar
  128. 128.
    G.J. Olmo, D. Rubiera-Garcia, Phys. Rev. D 84, 124059 (2011). arXiv:1110.0850 [gr-qc]ADSCrossRefGoogle Scholar
  129. 129.
    A. García, H. Salazar, J.F. Plebański, Nuovo Cimento Soc. Ital. Fis. 84B, 65 (1984)ADSCrossRefGoogle Scholar
  130. 130.
    M. Demianski, Found. Phys. 16, 187 (1986)ADSMathSciNetCrossRefGoogle Scholar
  131. 131.
    H.P. de Oliveira, Class. Quantum Gravit. 11, 1469 (1994)ADSCrossRefGoogle Scholar
  132. 132.
    G.W. Gibbons, D.A. Rasheed, Nucl. Phys. B 454, 185 (1995)ADSMathSciNetCrossRefGoogle Scholar
  133. 133.
    E. Ayón-Beato, A. García, Phys. Rev. Lett. 80, 5056–5059 (1998)ADSCrossRefGoogle Scholar
  134. 134.
    E. Ayón-Beato, A. García, Gen. Relativ. Gravit. 31, 629–633 (1999)ADSCrossRefGoogle Scholar
  135. 135.
    E. Ayón-Beato, A. García, Phys. Lett. B 464, 25 (1999)ADSMathSciNetCrossRefGoogle Scholar
  136. 136.
    N. Breton, Phys. Rev. D 67, 124004 (2003)ADSMathSciNetCrossRefGoogle Scholar
  137. 137.
    M. Hassaine, C. Martinez, Phys. Rev. D 75, 027502 (2007)ADSMathSciNetCrossRefGoogle Scholar
  138. 138.
    M. Hassaine, C. Martinez, Class. Quantum Gravit. 25, 195023 (2008)ADSMathSciNetCrossRefGoogle Scholar
  139. 139.
    J. Diaz-Alonso, D. Rubiera-Garcia, Phys. Rev. D 81, 064021 (2010)ADSCrossRefGoogle Scholar
  140. 140.
    J. Diaz-Alonso, D. Rubiera-Garcia, Phys. Rev. D 82, 085024 (2010)ADSCrossRefGoogle Scholar
  141. 141.
    J. Diaz-Alonso, D. Rubiera-Garcia, Gen. Relativ. Gravit. 45, 1901 (2013)ADSMathSciNetCrossRefGoogle Scholar
  142. 142.
    J.A.R. Cembranos, A. de la Cruz-Dombriz, J. Jarillo, JCAP 1502(02), 042 (2015)ADSCrossRefGoogle Scholar
  143. 143.
    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1992)zbMATHGoogle Scholar
  144. 144.
    G.J. Olmo, AIP Conf. Proc. 1458, 222–237 (2011). arXiv:1207.4815v1 [gr-qc]
  145. 145.
    G.F.R. Ellis, B.G. Schmidt, Gen. Relativ. Gravit. 8, 915 (1977)ADSMathSciNetCrossRefGoogle Scholar
  146. 146.
    F.J. Tipler, Phys. Rev. D 15, 942 (1977)ADSMathSciNetCrossRefGoogle Scholar
  147. 147.
    C.J.S. Clarke, A. Krolak, J. Geom. Phys. 2, 127 (1985)ADSMathSciNetCrossRefGoogle Scholar
  148. 148.
    B.C. Nolan, Phys. Rev. D 60, 024014 (1999)ADSMathSciNetCrossRefGoogle Scholar
  149. 149.
    A. Ori, Phys. Rev. D 61, 064016 (2000)ADSMathSciNetCrossRefGoogle Scholar
  150. 150.
    F.J. Tipler, Phys. Lett. A 64, 8 (1977)ADSMathSciNetCrossRefGoogle Scholar
  151. 151.
    B.C. Nolan, Phys. Rev. D 62, 044015 (2000)ADSMathSciNetCrossRefGoogle Scholar
  152. 152.
    A. Sanchez-Puente, G.J. Olmo, D. Rubiera-Garcia, Impact of Curvature Divergences on Physical Observers in a Wormhole Space-time with Horizons. Class. Quant. Grav., in press (2016). arXiv:1602.01798 [hep-th]

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Departamento de Física Teórica and IFICCentro Mixto Universidad de Valencia - CSICPaternaSpain
  2. 2.Universidad de ValenciaValenciaSpain
  3. 3.Departamento de FísicaUniversidade Federal da ParaíbaJoão PessoaBrazil

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