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Advances in High Energy Physics
Volume 2014, Article ID 907518, 14 pages
http://dx.doi.org/10.1155/2014/907518
Review Article

The Geometry of Black Hole Singularities

Horia Hulubei National Institute for Physics and Nuclear Engineering, 077125 Bucharest, Romania

Received 3 January 2014; Accepted 17 January 2014; Published 3 March 2014

Academic Editor: Christian Corda

Copyright © 2014 Ovidiu Cristinel Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Linked References

  1. V. K. Schwarzschild, “Über das gravitationsfeld einer kugel aus inkompressibler flüssigkeit nach der Einsteinschen theorie,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, vol. K1, pp. 424–434, 1916. View at Google Scholar
  2. V. K. Schwarzschild, “Über das gravitationsfeld eines massenpunktes nach der Einsteinschen theorie,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, vol. K1, pp. 189–196, 1916. View at Google Scholar
  3. A. Friedman, “Uber die krümmung des raumes,” Zeitschrift für Physik, vol. 10, no. 1, pp. 377–386, 1922. View at Publisher · View at Google Scholar
  4. A. Friedmann, “On the curvature of space,” General Relativity and Gravitation, vol. 31, no. 12, pp. 1991–2000, 1999. View at Publisher · View at Google Scholar
  5. A. Friedmann, “Über die möglichkeit einer welt mit konstanter negativer krümmung des raumes,” Zeitschrift für Physik, vol. 21, no. 1, pp. 326–332, 1924. View at Publisher · View at Google Scholar · View at Scopus
  6. G. Lemaître, “Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques,” Annales de la Société Scientifique de Bruxelles, vol. 47, pp. 49–59, 1927. View at Google Scholar
  7. H. P. Robertson, “Kinematics and world-structure,” The Astrophysical Journal, vol. 82, p. 284, 1935. View at Publisher · View at Google Scholar
  8. H. P. Robertson, “Kinematics and world-structure II,” The Astrophysical Journal, vol. 83, p. 187, 1936. View at Publisher · View at Google Scholar
  9. H. P. Robertson, “Kinematics and world-structure III,” The Astrophysical Journal, vol. 83, p. 257, 1936. View at Publisher · View at Google Scholar
  10. A. G. Walker, “On Milne's theory of world-structure,” Proceedings of the London Mathematical Society, vol. s2-42, no. 1, pp. 90–127, 1937. View at Publisher · View at Google Scholar
  11. R. Penrose, “Gravitational collapse and space-time singularities,” Physical Review Letters, vol. 14, no. 3, pp. 57–59, 1965. View at Publisher · View at Google Scholar · View at Scopus
  12. R. Penrose, “Gravitational collapse: the role of general relativity,” Revista del Nuovo Cimento, vol. 1, pp. 252–276, 1969. View at Google Scholar
  13. S. W. Hawking, “The occurrence of singularities in cosmology,” Proceedings of the Royal Society A, vol. 294, no. 1439, pp. 511–521, 1966. View at Google Scholar
  14. S. W. Hawking, “The occurrence of singularities in cosmology. II,” Proceedings of the Royal Society A, vol. 295, no. 1443, pp. 490–493, 1966. View at Publisher · View at Google Scholar
  15. S. W. Hawking, “The occurrence of singularities in cosmology. III. Causality and singularities,” Proceedings of the Royal Society A, vol. 300, no. 1461, pp. 187–201, 1967. View at Publisher · View at Google Scholar
  16. S. W. Hawking and R. W. Penrose, “The Singularities of gravitational collapse and cosmology,” Proceedings of the Royal Society A, vol. 314, no. 1519, pp. 529–548, 1970. View at Google Scholar · View at Scopus
  17. R. Penrose, “Singularities of spacetime,” in Theoretical Principles in Astrophysics and Relativity, N. R. Lebovitz, W. H. Reid, and P. O. Vandervoort, Eds., vol. 1, pp. 217–243, University of Chicago Press, Chicago, Ill, USA, 1978. View at Google Scholar
  18. R. Penrose, “Singularities and time-asymmetry,” in General Relativity: An Einstein Centenary Survey, vol. 1, pp. 581–638, Cambridge University Press, Cambridge, UK, 1979. View at Google Scholar
  19. C. J. Isham, R. Penrose, and D. W. Sciama, Quantum Gravity II, vol. 1, Oxford University Press, Oxford, UK, 1981.
  20. R. Penrose, “The question of cosmic censorship,” in Black Holes and Relativistic Stars, R. M. Wald, Ed., pp. 103–122, University of Chicago Press, Chicago, Ill, USA, 1998. View at Google Scholar
  21. S. Barve and T. P. Singh, “Are naked singularities forbidden by the second law of thermodynamics?” Modern Physics Letters A, vol. 12, no. 32, pp. 2415–2419, 1997. View at Google Scholar · View at Scopus
  22. S. Barve, T. P. Singh, C. Vaz, and L. Witten, “Particle creation in the marginally bound, self-similar collapse of inhomogeneous dust,” Nuclear Physics B, vol. 532, no. 1-2, pp. 361–375, 1998. View at Google Scholar · View at Scopus
  23. S. Barve, T. P. Singh, C. Vaz, and L. Witten, “Quantum stress tensor in self-similar spherical dust collapse,” Physical Review D, vol. 58, no. 10, Article ID 104018, 8 pages, 1998. View at Publisher · View at Google Scholar · View at Scopus
  24. S. Barve, T. P. Singh, and L. Witten, “Spherical gravitational collapse: tangential pressure and related equations of state,” General Relativity and Gravitation, vol. 32, no. 4, pp. 697–717, 2000. View at Google Scholar · View at Scopus
  25. P. S. Joshi, “Spacetime singularities,” http://arxiv.org/abs/1311.0449.
  26. H. A. Buchdahl, “Non-linear Lagrangians and cosmological theory,” Monthly Notices of the Royal Astronomical Society, vol. 150, article 1, 1970. View at Google Scholar
  27. A. A. Starobinsky, “A new type of isotropic cosmological models without singularity,” Physics Letters B, vol. 91, no. 1, pp. 99–102, 1980. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Borisov, B. Jain, and P. Zhang, “Spherical collapse in f(R) gravity,” Physical Review D, vol. 85, no. 6, Article ID 063518, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. S. H. Hendi and D. Momeni, “Black-hole solutions in F(R) gravity with conformal anomaly,” The European Physical Journal C, vol. 71, no. 12, article 1832, pp. 1–9, 2011. View at Publisher · View at Google Scholar
  30. G. J. Olmo and D. Rubiera-Garcia, “Palatini f(R) black holes in nonlinear electrodynamics,” Physical Review D, vol. 84, no. 12, Article ID 124059, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  31. C. Corda and H. J. M. Cuesta, “Removing black hole singularities with nonlinear electrodynamics,” Modern Physics Letters A, vol. 25, no. 28, pp. 2423–2429, 2010. View at Publisher · View at Google Scholar · View at Scopus
  32. G. 't Hooft and M. J. G. Veltman, “One loop divergencies in the theory of gravitation,” Annales de l'Institut Henri Poincaré A, vol. 20, no. 1, pp. 69–94, 1974. View at Google Scholar
  33. M. H. Goroff and A. Sagnotti, “The ultraviolet behavior of Einstein gravity,” Nuclear Physics B, vol. 266, no. 3-4, pp. 709–736, 1986. View at Google Scholar · View at Scopus
  34. M. Bojowald, “Absence of a singularity in loop quantum cosmology,” Physical Review Letters, vol. 86, no. 23, pp. 5227–5230, 2001. View at Publisher · View at Google Scholar · View at Scopus
  35. A. Ashtekar and P. Singh, “Loop quantum cosmology: a status report,” Classical and Quantum Gravity, vol. 28, no. 21, Article ID 213001, 2011. View at Publisher · View at Google Scholar
  36. M. Visinescu, “Bianchi type-I string cosmological model in the presence of a magnetic field: classical and quantum loop approach,” Romanian Reports on Physics, vol. 61, no. 3, pp. 427–435, 2009. View at Google Scholar · View at Scopus
  37. V. Rikhvitsky, B. Saha, and M. Visinescu, “Bianchi type-I string cosmological model in the presence of a magnetic field: classical versus loop quantum cosmology approaches,” Astrophysics and Space Science, vol. 339, no. 2, pp. 371–377, 2012. View at Publisher · View at Google Scholar · View at Scopus
  38. O. C. Stoica, Singular general relativity [Ph.D. thesis], 2013.
  39. O. C. Stoica, “On singular semi-Riemannian manifolds,” http://arxiv.org/abs/1105.0201.
  40. O. C. Stoica, “Warped products of singular semi-Riemannian manifolds,” http://arxiv.org/abs/1105.3404.
  41. O. C. Stoica, “Einstein equation at singularities,” Central European Journal of Physics, 2014. View at Publisher · View at Google Scholar
  42. O. C. Stoica, “Schwarzschild’s singularity is semi-regularizable,” The European Physical Journal Plus, vol. 127, no. 7, article 83, pp. 1–8, 2012. View at Publisher · View at Google Scholar
  43. O. C. Stoica, “Analytic Reissner-Nordström singularity,” Physica Scripta, vol. 85, no. 5, Article ID 055004, 2012. View at Publisher · View at Google Scholar
  44. O. C. Stoica, “Kerr-Newman solutions with analytic singularity and no closed timelike curves,” http://arxiv.org/abs/1111.7082.
  45. A. S. Eddington, “A comparison of Whitehead's and Einstein's formul?” Nature, vol. 113, no. 2832, p. 192, 1924. View at Publisher · View at Google Scholar · View at Scopus
  46. D. Finkelstein, “Past-future asymmetry of the gravitational field of a point particle,” Physical Review, vol. 110, no. 4, pp. 965–967, 1958. View at Publisher · View at Google Scholar · View at Scopus
  47. D. N. Kupeli, “Degenerate manifolds,” Geometriae Dedicata, vol. 23, no. 3, pp. 259–290, 1987. View at Publisher · View at Google Scholar · View at Scopus
  48. K. P. Tod, “Quasi-local mass and cosmological singularities,” Classical and Quantum Gravity, vol. 4, no. 5, pp. 1457–1468, 1987. View at Publisher · View at Google Scholar · View at Scopus
  49. K. P. Tod, “Isotropic singularities and the γ = 2 equation of state,” Classical and Quantum Gravity, vol. 7, no. 1, pp. L13–L16, 1990. View at Publisher · View at Google Scholar
  50. K. P. Tod, “Isotropic singularities and the polytropic equation of state,” Classical and Quantum Gravity, vol. 8, no. 4, pp. L77–L82, 1991. View at Publisher · View at Google Scholar
  51. K. P. Tod, “Isotropic singularities,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 50, no. 1, pp. 69–93, 1992. View at Google Scholar
  52. C. M. Claudel and K. P. Newman, “The Cauchy problem for quasi-linear hyperbolic evolution problems with a singularity in the time,” Proceedings of the Royal Society A, vol. 454, no. 1972, pp. 1073–1107, 1998. View at Google Scholar · View at Scopus
  53. K. Anguige and K. P. Tod, “Isotropic cosmological singularities: I. Polytropic perfect fluid spacetimes,” Annals of Physics, vol. 276, no. 2, pp. 257–293, 1999. View at Publisher · View at Google Scholar · View at Scopus
  54. K. Anguige and K. P. Tod, “Isotropic cosmological singularities: II. The Einstein-Vlasov system,” Annals of Physics, vol. 276, no. 2, pp. 294–320, 1999. View at Publisher · View at Google Scholar · View at Scopus
  55. K. P. Tod, “Isotropic cosmological singularities,” in The Conformal Structure of Space-Time, vol. 604 of Lecture Notes in Physics, pp. 123–134, Springer, Berlin, Germany, 2002. View at Publisher · View at Google Scholar
  56. K. P. Tod, “Isotropic cosmological singularities: other matter models,” Classical and Quantum Gravity, vol. 20, no. 3, pp. 521–534, 2003. View at Publisher · View at Google Scholar
  57. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1983.
  58. O. C. Stoica, “Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime,” http://arxiv.org/abs/1112.4508.
  59. I. M. Singer and J. A. Thorpe, “The curvature of 4-dimensional Einstein spaces,” in Global Analysis (Papers in Honor of K. Kodaira), pp. 355–365, Princeton University Press, Princeton, NJ, USA, 1969. View at Google Scholar
  60. L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 10, Springer, Berlin, Germany, 3rd edition, 1987.
  61. S. Gallot, D. Hullin, and J. Lafontaine, Riemannian Geometry, Springer, Berlin, Germany, 3rd edition, 2004.
  62. O. C. Stoica, “Beyond the Friedmann-Lemaître-Robertson-Walker Big Bang singularity,” Communications in Theoretical Physics, vol. 58, no. 4, pp. 613–616, 2012. View at Publisher · View at Google Scholar
  63. O. C. Stoica, “On the Weyl curvature hypothesis,” Annals of Physics, vol. 338, pp. 186–194, 2013. View at Publisher · View at Google Scholar
  64. R. M. Wald, General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984.
  65. Y. Foures-Bruhat, “Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires,” Acta Mathematica, vol. 88, no. 1, pp. 141–225, 1952. View at Publisher · View at Google Scholar
  66. R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravitation: An Introduction to Current Research, pp. 227–264, John Wiley & Sons, New York, NY, USA, 1962. View at Google Scholar
  67. A. Anderson, Y. Choquet-Bruhat, and J. W. York, “Einstein's equations and equivalent hyperbolic dynamical systems,” in Mathematical and Quantum Aspects of Relativity and Cosmology, vol. 537 of Lecture Notes in Physics, pp. 30–54, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar
  68. Y. Choquet-Bruhat and J. York, “Constraints and evolution in cosmology,” in Cosmological Crossroads, vol. 592 of Lecture Notes in Physics, pp. 29–58, Springer, Berlin, Germany, 2002. View at Publisher · View at Google Scholar
  69. I. Rodnianski, “The cauchy problem in general relativity,” in Proceedings of the International Congress of Mathematicians, pp. 421–442, Madrid, Spain, August 2006.
  70. J. M. M. Senovilla, “Singularity theorems and their consequences,” General Relativity and Gravitation, vol. 30, no. 5, pp. 701–848, 1998. View at Google Scholar · View at Scopus
  71. O. C. Stoica, “Spacetimes with singularities,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 20, no. 2, pp. 213–238, 2013. View at Publisher · View at Google Scholar
  72. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space Time, Cambridge University Press, Cambridge, UK, 1995.
  73. J. D. Bekenstein, “Black holes and entropy,” Physical Review D, vol. 7, no. 8, pp. 2333–2346, 1973. View at Publisher · View at Google Scholar · View at Scopus
  74. J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics, vol. 31, no. 2, pp. 161–170, 1973. View at Publisher · View at Google Scholar · View at Scopus
  75. S. W. Hawking and R. W. Penrose, The Nature of Space and Time, Princeton University Press, Princeton, NJ, USA, 1996.
  76. A. Strominger, “Les Houches lectures on black holes,” http://arxiv.org/abs/hep–th/9501071.
  77. T. Jacobson, Introductory Lectures on Black Hole Thermodynamics, The University of Utrecht, Utrecht, The Netherlands, 1996, http://www.physics.umd.edu/grt/taj/776b/lectures.pdf.
  78. S. W. Hawking, “Black hole explosions?” Nature, vol. 248, no. 5443, pp. 30–31, 1974. View at Publisher · View at Google Scholar · View at Scopus
  79. S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43, no. 3, pp. 199–220, 1975. View at Publisher · View at Google Scholar
  80. S. W. Hawking, “Breakdown of predictability in gravitational collapse,” Physical Review D, vol. 14, no. 10, pp. 2460–2473, 1976. View at Publisher · View at Google Scholar · View at Scopus
  81. L. Susskind, L. Thorlacius, and J. Uglum, “The stretched horizon and black hole complementarity,” Physical Review D, vol. 48, no. 8, pp. 3743–3761, 1993. View at Publisher · View at Google Scholar · View at Scopus
  82. S. W. Hawking, “Information loss in black holes,” Physical Review D, vol. 72, no. 8, Article ID 084013, 4 pages, 2005. View at Publisher · View at Google Scholar
  83. J. Preskill, “Do black holes destroy information?” in Proceedings of the International Symposium on Black Holes, Membranes, Wormholes and Superstrings,, vol. 1, pp. 22–39, World Scientific, River Edge, NJ, USA, 1993.
  84. D. N. Page, “Black hole information,” in Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics, vol. 1, pp. 1–41, 1994.
  85. T. Banks, “Lectures on black holes and information loss,” Nuclear Physics B, vol. 41, no. 1–3, pp. 21–65, 1995. View at Publisher · View at Google Scholar · View at Scopus
  86. T. P. Singh and C. Vaz, “The quantum gravitational black hole is neither black nor white,” International Journal of Modern Physics D, vol. 13, no. 10, pp. 2369–2373, 2004. View at Publisher · View at Google Scholar · View at Scopus
  87. P. D. Prester, “Curing black hole singularities with local scale invariance,” http://arxiv.org/abs/1309.1188.
  88. C. Corda, “Effective temperature, Hawking radiation and quasinormal modes,” International Journal of Modern Physics D, vol. 21, no. 11, 11 pages, 2012. View at Publisher · View at Google Scholar
  89. C. Corda, “Effective temperature for black holes,” Journal of High Energy Physics, vol. 2011, no. 8, article 101, pp. 1–10, 2011. View at Publisher · View at Google Scholar · View at Scopus
  90. C. Corda, “Black hole quantum spectrum,” The European Physical Journal C, vol. 73, no. 12, article 2665, pp. 1–12, 2013. View at Publisher · View at Google Scholar
  91. C. Corda, S. H. Hendi, R. Katebi, and N. O. Schmidt, “Effective state, Hawking radiation and quasi-normal modes for Kerr black holes,” Journal of High Energy Physics, vol. 2013, no. 6, article 8, pp. 1–12, 2013. View at Publisher · View at Google Scholar
  92. A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black holes: complementarity or firewalls?” Journal of High Energy Physics, vol. 2013, no. 2, article 62, pp. 1–20, 2013. View at Publisher · View at Google Scholar
  93. D.-I. Hwang, B.-H. Lee, and D.-H. Yeom, “Is the firewall consistent? Gedanken experiments on black hole complementarity and firewall proposal,” Journal of Cosmology and Astroparticle Physics, vol. 2013, no. 1, article 5, 2013. View at Publisher · View at Google Scholar
  94. D. Marolf and J. Polchinski, “Gauge-gravity duality and the black hole interior,” Physical Review Letters, vol. 111, no. 17, Article ID 171301, 5 pages, 2013. View at Publisher · View at Google Scholar
  95. N. Itzhaki, “Information loss in quantum gravity without black holes,” Classical and Quantum Gravity, vol. 12, no. 11, pp. 2747–2753, 1995. View at Publisher · View at Google Scholar · View at Scopus
  96. O. C. Stoica, “Quantum gravity from metric dimensional reduction at singularities,” http://arxiv.org/abs/1205.2586.
  97. G. Calcagni, “Quantum field theory, gravity and cosmology in a fractal universe,” Journal of High Energy Physics, vol. 2010, no. 3, article 120, pp. 1–38, 2010. View at Publisher · View at Google Scholar
  98. G. Calcagni, “Fractal universe and quantum gravity,” Physical Review Letters, vol. 104, no. 25, Article ID 251301, 4 pages, 2010. View at Publisher · View at Google Scholar
  99. G. Calcagni, “Geometry of fractional spaces,” Advances in Theoretical and Mathematical Physics, vol. 16, no. 2, pp. 549–644, 2012. View at Publisher · View at Google Scholar
  100. R.A. El-Nabulsi, “A fractional action-like variational approach of some classical, quantum and geometrical dynamics,” International Journal of Applied Mathematics, vol. 17, no. 3, pp. 299–318, 2005. View at Google Scholar
  101. R.A. El-Nabulsi, “Differential geometry and modern cosmology with fractionaly differentiated lagrangian function and fractional decaying force term,” Romanian Journal of Physics, vol. 52, no. 3-4, pp. 467–481, 2007. View at Google Scholar
  102. R.A. El-Nabulsi, “Some fractional geometrical aspects of weak field approximation and Schwarzschild spacetime,” Romanian Journal of Physics, vol. 52, no. 5–7, pp. 705–715, 2007. View at Google Scholar
  103. R.A. El-Nabulsi, “Cosmology with a fractional action principle,” Romanian Reports in Physics, vol. 59, no. 3, pp. 759–765, 2007. View at Google Scholar
  104. R. A. El-Nabulsi and D. F. M. Torres, “Fractional actionlike variational problems,” Journal of Mathematical Physics, vol. 49, no. 5, Article ID 053521, 2008. View at Publisher · View at Google Scholar · View at Scopus
  105. C. Udriste and D. Opris, “Euler-Lagrange-Hamilton dynamics with fractional action,” WSEAS Transactions on Mathematics, vol. 7, no. 1, pp. 19–30, 2008. View at Google Scholar · View at Scopus
  106. R. A. El-Nabulsi, “Modifications at large distances from fractional and fractal arguments,” Fractals, vol. 18, no. 2, pp. 185–190, 2010. View at Publisher · View at Google Scholar · View at Scopus
  107. R. A. El-Nabulsi and C. G. Wu, “Fractional complexified field theory from Saxena-Kumbhat fractional integral, fractional derivative of order (α,β) and dynamical fractional integral exponent,” African Diaspora Journal of Mathematics, vol. 13, no. 2, pp. 45–61, 2012. View at Google Scholar
  108. R. A. El-Nabulsi, “Gravitons in fractional action cosmology,” International Journal of Theoretical Physics, vol. 51, no. 12, pp. 3978–3992, 2012. View at Publisher · View at Google Scholar
  109. R. A. El-Nabulsi, “Fractional derivatives generalization of einstein's field equations,” Indian Journal of Physics, vol. 87, no. 2, pp. 195–200, 2013. View at Publisher · View at Google Scholar
  110. D. V. Shirkov, “Coupling running through the looking-glass of dimensional reduction,” Physics of Particles and Nuclei Letters, vol. 7, no. 6, pp. 379–383, 2010. View at Publisher · View at Google Scholar
  111. P. P. Fiziev and D. V. Shirkov, “Solutions of the Klein-Gordon equation on manifolds with variable geometry including dimensional reduction,” Theoretical and Mathematical Physics, vol. 167, no. 2, pp. 680–691, 2011. View at Publisher · View at Google Scholar · View at Scopus
  112. P. P. Fiziev, “Riemannian (1+d)-dim space-time manifolds with nonstandard topology which admit dimensional reduction to any lower dimension and transformation of the Klein-Gordon equation to the 1-dim Schrödinger like equation,” http://arxiv.org/abs/1012.3520.
  113. P. P. Fiziev and D. V. Shirkov, “The (2+1)-dimensional axial universes—solutions to the Einstein equations, dimensional reduction points and Klein—Fock—Gordon waves,” Journal of Physics A, vol. 45, no. 5, Article ID 055205, 2012. View at Publisher · View at Google Scholar
  114. D. V. Shirkov, “Dreamland with classic higgs field, dimensional reduction and all that,” Proceedings of the Steklov Institute of Mathematics, vol. 272, no. 1, pp. 216–222, 2011. View at Publisher · View at Google Scholar · View at Scopus
  115. L. Anchordoqui, D. C. Dai, M. Fairbairn, G. Landsberg, and D. Stojkovic, “Vanishing dimensions and planar events at the LHC,” Modern Physics Letters A, vol. 27, no. 4, Article ID 1250021, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  116. S. Carlip, “Lectures on (2+1)-dimensional gravity,” Journal of the Korean Physical Society, vol. 28, no. 4, pp. S447–S467, 1995. View at Google Scholar · View at Scopus
  117. S. Carlip, “Spontaneous dimensional reduction in short-distance quantum gravity?” in Proceedings of the 25th Max Born Symposium on the Planck Scale, J. Kowalski-Glikman, R. Durka, and M. Szczachor, Eds., vol. 1196 of AIP Conference Proceedings, pp. 72–80, Wroclaw, Poland, July 2009. View at Publisher · View at Google Scholar · View at Scopus
  118. S. Carlip, “The small scale structure of spacetime,” http://arxiv.org/abs/1009.1136.
  119. S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation,” in General Relativity: An Einstein Centenary Survey, vol. 1, pp. 790–831, Cambridge University Press, Cambridge, UK, 1979. View at Google Scholar
  120. P. Hořava, “Quantum gravity at a Lifshitz point,” Physical Review D, vol. 79, no. 8, Article ID 084008, 15 pages, 2009. View at Publisher · View at Google Scholar
  121. I. Oda, “Quantum instability of black hole singularity in three-dimensions,” http://arxiv.org/abs/gr-qc/9703056.
  122. K. Umetsu, “Tunneling mechanism in Kerr-Newman black hole and dimensional reduction near the horizon,” Physics Letters B, vol. 692, no. 1, pp. 61–63, 2010. View at Publisher · View at Google Scholar · View at Scopus
  123. J. W. Moffat, “Lorentz violation of quantum gravity,” Classical and Quantum Gravity, vol. 27, no. 13, Article ID 135016, 2010. View at Publisher · View at Google Scholar · View at Scopus
  124. J. Mureika and P. Nicolini, “Self-completeness and spontaneous dimensional reduction,” The European Physical Journal Plus, vol. 128, no. 7, article 78, 2013. View at Publisher · View at Google Scholar
  125. J. R. Mureika, “Primordial black hole evaporation and spontaneous dimensional reduction,” Physics Letters B, vol. 716, no. 1, pp. 171–175, 2012. View at Publisher · View at Google Scholar
  126. C. Charmousis, B. Goutéraux, and E. Kiritsis, “Higher-derivative scalar-vector-tensor theories: black holes, Galileons, singularity cloaking and holography,” Journal of High Energy Physics, vol. 2012, no. 9, article 11, pp. 1–44, 2012. View at Publisher · View at Google Scholar
  127. G. Calcagni, “Gravity on a multifractal,” Physics Letters B, vol. 697, no. 3, pp. 251–253, 2011. View at Publisher · View at Google Scholar
  128. J. D. Brown, Lower Dimensional Gravity, World Scientific, New York, NY, USA, 1988.
  129. R. Emparan and H. S. Reall, “Black holes in higher dimensions,” Living Reviews in Relativity, vol. 11, no. 6, 2008. View at Publisher · View at Google Scholar · View at Scopus
  130. A. Watcharangkool, The algebraic properties of black holes in higher dimension [M.S. Dissertations], 2012,, https://workspace.imperial.ac.uk/theoreticalphysics/Public/MSc/Dissertations/2012/Apimook%20Watcharangkool%20Dissertation.pdf.