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Extremality, Holography and Coarse Graining

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Supersymmetric Gravity and Black Holes

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 142))

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Abstract

I discuss some of the concepts at the crossroads of gravitational thermodynamics, holography and quantum mechanics. First, the origin of gravitational thermodynamics due to coarse graining of quantum information is exemplified using the half-BPS sector of \(\mathcal{N}\,=\,4\) SYM and its LLM description in type IIB supergravity.

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Notes

  1. 1.

    The language used in this argument may induce some readers to think of a transition between an open string (gauge theory) description to a closed string (gravitational) description. This may indeed be helpful, but the argument is more generic. If one assumes the existence of a fully quantum mechanical description of gravity, there is no guarantee that a typical pure state in such Hilbert space allows a reliable description in terms of a classical geometry when taking the classical limit.

  2. 2.

    There is more than one ensemble achieving this, see [71] for a discussion on this point.

  3. 3.

    The same methods were applied to the D1–D5 system in [135].

  4. 4.

    We are using the Husimi distribution Hu(q, p) given its nice properties in the classical limit. For further discussion, see [13]. For a review on phase space distributions, see [98].

  5. 5.

    The exact quantum state is an N-particle state. Thus, there is generically information loss when going from this to the one particle description. In the large N limit, this is typically expected to be a subleading effect not emerging in the classical gravitational description. See [16] for a discussion on this matter.

  6. 6.

    This result was independently obtained by Masaki Shigemori in an unpublished work by considering a gas of fermionic particles in phase space.

  7. 7.

    This observation may not be that surprising since we know of examples in quantum mechanics in which the saddle point approximation involves a complex configuration. It seems still meaningful to appreciate its conceptual consequences beyond its purely technical nature.

  8. 8.

    Cardy’s formula requires the temperature to be large. See [100] for a justification on the validity of Cardy’s regime for extremal Kerr.

  9. 9.

    The amount of literature here is immense. We refer the reader to a subset of reviews and references therein [97].

  10. 10.

    The action of this orbifold at the AdS3 boundary is like the one of a conical defect. It would be interesting to see whether the techniques developed in [119] to compute the worldsheet string perturbative spectrum can be extended to this case, and whether there is any interesting structure emerging in the large N limit.

  11. 11.

    The importance of this geometry for the physics of extremal black holes was already emphasised some time ago in [10, 148].

  12. 12.

    The precise definition of DLCQ in quantum field theory is rather subtle. As emphasised in [96], amplitudes computed in these theories diverge order by order in perturbation theory due to strong interactions among longitudinal zero modes. This quantization scheme was argued to be well defined non-perturbatively.

  13. 13.

    There are different ways of arguing the existence of this temperature. From the global version of the spacelike self-dual orbifold [10, 21] containing two disjoint causally connected boundaries, the finite temperature originates from entanglement entropy after integrating out part of the space leading to the single boundary metric (3.133), pretty much in the same way Rindler space has a finite temperature when viewed as a local patch of the full Minkowski spacetime.

  14. 14.

    Readers interested in the supersymmetric properties of this orbifold, see [63, 81]. In particular, [81] discusses the embedding of this orbifold in higher dimensional supergravities stressing the importance of the fermion chirality to assess the supersymmetry of this quotient.

  15. 15.

    The same structure appears in the near horizon of extremal black holes with vanishing horizon. See [4] for different examples of its appearance.

  16. 16.

    For a different emphasis on how to use the AdS3/CFT2 correspondence to learn how to formulate the AdS2/CFT1 correspondence, see [88].

  17. 17.

    Notice the transformation for the generator l 0 is due to the fact that we were working on the plane. Indeed, if we would have worked on the cylinder, the transformation is the expected one:

    $${l}_{n}^{\text{ cyl}} \equiv \frac{1} {K}{L}_{nK}^{\text{ cyl}}\,,\quad n\neq 0\,,\quad \quad {l}_{ 0}^{\text{ cyl}} \equiv \frac{1} {K}{L}_{0}^{\text{ cyl}}\,.$$

    We now see that the transformation quoted on the plane makes sure the above cylinder transformation brings us back to the plane.

References

  1. L.F. Abbott, S. Deser, Nucl. Phys. B 195, 76 (1982). V. Iyer, R.M. Wald, Phys. Rev. D 50, 846 (1994). I.M. Anderson, C.G. Torre, Phys. Rev. Lett. 77, 4109 (1996).C.G. Torre, Local cohomology in field theory with applications to the Einstein equations. arXiv:hep-th/9706092. G. Barnich, F. Brandt, M. Henneaux, Commun. Math. Phys. 174, 57 (1995). G. Barnich, F. Brandt, M. Henneaux, Phys. Rep. 338, 439 (2000)

    Google Scholar 

  2. A.J. Amsel, G.T. Horowitz, D. Marolf, M.M. Roberts, J. High Energy Phys. 0909, 044 (2009)

    Google Scholar 

  3. N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E. Trincherini, G. Villadoro, J. High Energy Phys. 0705, 055 (2007)

    Google Scholar 

  4. T. Azeyanagi, N. Ogawa, S. Terashima, Emergent AdS 3 in the zero entropy extremal black Holes. arXiv:1010.4291 (hep-th). Y. Matsuo, T. Nishioka, New near-horizon limit in Kerr/CFT. arXiv:1010.4549 (hep-th). T. Azeyanagi, N. Ogawa, S. Terashima, On non-chiral extension of Kerr/CFT. (arXiv:1102.3423 (hep-th))

    Google Scholar 

  5. H. Bacry, A. Grossmann, J. Zak, Phys. Rev. B 12, 1118 (1975)

    Google Scholar 

  6. V. Balasubramanian, B. Czech, Quantitative approaches to information recovery from black holes. (arXiv:1102.3566 (hep-th))

    Google Scholar 

  7. V. Balasubramanian, R. Gopakumar, F. Larsen, Nucl. Phys. B 526, 415 (1998)

    Google Scholar 

  8. V. Balasubramanian, P. Kraus, A.E. Lawrence, Phys. Rev. D 59, 046003 (1999)

    Google Scholar 

  9. V. Balasubramanian, M. Berkooz, A. Naqvi, M.J. Strassler, J. High Energy Phys. 0204, 034 (2002)

    Google Scholar 

  10. V. Balasubramanian, A. Naqvi, J. Simon, J. High Energy Phys. 0408, 023 (2004)

    Google Scholar 

  11. V. Balasubramanian, V. Jejjala, J. Simon, Int. J. Mod. Phys. D14, 2181–2186 (2005)

    Google Scholar 

  12. V. Balasubramanian, P. Kraus, M. Shigemori, Class. Quantum Gravity 22, 4803 (2005)

    Google Scholar 

  13. V. Balasubramanian, J. de Boer, V. Jejjala, J. Simon, J. High Energy Phys. 0512, 006 (2005)

    Google Scholar 

  14. V. Balasubramanian, B. Czech, K. Larjo, J. Simon, J. High Energy Phys. 0611, 001 (2006)

    Google Scholar 

  15. V. Balasubramanian, D. Marolf, M. Rozali, Gen. Relativ. Gravit. 38, 1529 (2006). (Int. J. Mod. Phys. D 15, 2285 (2006))

    Google Scholar 

  16. V. Balasubramanian, B. Czech, K. Larjo, D. Marolf, J. Simon, J. High Energy Phys. 0712, 067 (2007)

    Google Scholar 

  17. V. Balasubramanian, J. de Boer, S. El-Showk, I. Messamah, Class. Quantum Gravity 25, 214004 (2008)

    Google Scholar 

  18. V. Balasubramanian, B. Czech, V.E. Hubeny, K. Larjo, M. Rangamani, J. Simon, Gen. Relativ. Gravit. 40, 1863 (2008)

    Google Scholar 

  19. V. Balasubramanian, J. de Boer, V. Jejjala, J. Simon, J. High Energy Phys. 0805, 067 (2008)

    Google Scholar 

  20. V.  Balasubramanian, J. de Boer, M. Sheikh-Jabbari, J. Simón, J. High Energy Phys. 1002, 017 (2010)

    Google Scholar 

  21. V. Balasubramanian, J. Parsons, S.F. Ross, Class. Quantum Gravity 28, 045004 (2011)

    Google Scholar 

  22. M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992)

    Google Scholar 

  23. M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli, Phys. Rev. D 48, 1506 (1993)

    Google Scholar 

  24. T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys. Rev. D55, 5112–5128 (1997)

    Google Scholar 

  25. J.M. Bardeen, G.T. Horowitz, Phys. Rev. D 60, 104030 (1999)

    Google Scholar 

  26. J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phys. 31, 161 (1973)

    Google Scholar 

  27. V. Bargmann, P. Butera, L. Girardello, J.R. Klauder, Rep. Math. Phys. 2, 221 (1971)

    Google Scholar 

  28. G. Barnich, Class. Quantum Gravity 20, 3685 (2003)

    Google Scholar 

  29. G. Barnich, F. Brandt, Nucl. Phys. B 633, 3 (2002)

    Google Scholar 

  30. G. Barnich, G. Compere, J. Math. Phys. 49, 042901 (2008)

    Google Scholar 

  31. B. Bates, F. Denef, Exact solutions for supersymmetric stationary black hole composites. arXiv:hep-th/0304094

    Google Scholar 

  32. M. Becker, S. Cremonini, W. Schulgin, J. High Energy Phys. 1009, 022 (2010). M. Becker, S. Cremonini, W. Schulgin, J. High Energy Phys. 1102, 007 (2011)

    Google Scholar 

  33. K. Behrndt, A.H. Chamseddine, W.A. Sabra, Phys. Lett. B442, 97–101 (1998). K. Behrndt, M. Cvetic, W.A. Sabra, Nucl. Phys. B553, 317–332 (1999)

    Google Scholar 

  34. J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)

    Google Scholar 

  35. I. Bena, N.P. Warner, Phys. Rev. D 74, 066001 (2006)

    Google Scholar 

  36. I. Bena, N.P. Warner, Lect. Notes Phys. 755, 1 (2008)

    Google Scholar 

  37. I. Bena, C.-W. Wang, N.P. Warner, J. High Energy Phys. 0611, 042 (2006)

    Google Scholar 

  38. I. Bena, S. Giusto, C. Ruef, N.P. Warner, J. High Energy Phys. 0911, 089 (2009)

    Google Scholar 

  39. I. Bena, S. Giusto, C. Ruef, N.P. Warner, J. High Energy Phys. 1003, 047 (2010)

    Google Scholar 

  40. I. Bena, N. Bobev, S. Giusto, C. Ruef, N.P. Warner, J. High Energy Phys. 1103, 022 (2011)

    Google Scholar 

  41. D. Berenstein, J. High Energy Phys. 0407, 018 (2004)

    Google Scholar 

  42. D. Berenstein, J. High Energy Phys. 0601, 125 (2006)

    Google Scholar 

  43. P. Berglund, E.G. Gimon, T.S. Levi, J. High Energy Phys. 0606, 007 (2006)

    Google Scholar 

  44. L. Boltzmann, Kais. Akad. Wiss. Wien Math. Naturwiss. Classe 76, 373–435 (1877). L. Boltzmann, Kais. Akad. Wiss. Wien Math. Naturwiss. Classe 66, 275–370 (1872)

    Google Scholar 

  45. R. Bousso, J. High Energy Phys. 9907, 004 (1999)

    Google Scholar 

  46. R. Bousso, Rev. Mod. Phys. 74, 825–874 (2002)

    Google Scholar 

  47. D. Brecher, A. Chamblin, H.S. Reall, Nucl. Phys. B 607, 155 (2001)

    Google Scholar 

  48. I. Bredberg, T. Hartman, W. Song, A. Strominger, J. High Energy Phys. 1004, 019 (2010)

    Google Scholar 

  49. I. Bredberg, C. Keeler, V. Lysov, A. Strominger, Cargese lectures on the Kerr/CFT correspondence. arXiv:1103.2355 (hep-th)

    Google Scholar 

  50. J.D. Brown, M. Henneaux, Commun. Math. Phys. 104, 207 (1986)

    Google Scholar 

  51. P. Calabrese, J.L. Cardy, J. Stat. Mech. 0406, P06002 (2004)

    Google Scholar 

  52. J.L. Cardy, Nucl. Phys. B 270, 186 (1986)

    Google Scholar 

  53. A. Castro, F. Larsen, J. High Energy Phys. 0912, 037 (2009)

    Google Scholar 

  54. A. Castro, D. Grumiller, F. Larsen, R. McNees, J. High Energy Phys. 0811, 052 (2008)

    Google Scholar 

  55. A. Castro, C. Keeler, F. Larsen, J. High Energy Phys. 1007, 033 (2010)

    Google Scholar 

  56. A. Castro, A. Maloney, A. Strominger, Phys. Rev. D 82, 024008 (2010)

    Google Scholar 

  57. D.D.K. Chow, M. Cvetic, H. Lu, C.N. Pope, Phys. Rev. D 79, 084018 (2009)

    Google Scholar 

  58. B.D. Chowdhury, S.D. Mathur, Class. Quantum Gravity 25, 135005 (2008). B.D. Chowdhury, S.D. Mathur, Class. Quantum Gravity 25, 225021 (2008). B.D. Chowdhury, S.D. Mathur, Class. Quantum Gravity 26, 035006 (2009)

    Google Scholar 

  59. G. Compere, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions. arXiv:0708.3153 (hep-th)

    Google Scholar 

  60. G. Compere, K. Murata, T. Nishioka, J. High Energy Phys. 0905, 077 (2009)

    Google Scholar 

  61. G. Compere, W. Song, A. Virmani, Microscopics of extremal Kerr from spinning M5 branes. (arXiv:1010.0685 (hep-th))

    Google Scholar 

  62. S. Corley, A. Jevicki, S. Ramgoolam, Adv. Theor. Math. Phys. 5, 809–839 (2002)

    Google Scholar 

  63. O. Coussaert, M. Henneaux, Self-dual solutions of 2 + 1 Einstein gravity with a negative cosmological constant. arXiv:hep-th/9407181

    Google Scholar 

  64. C. Crnkovic, E. Witten, Covariant description Of canonical formalism in geometrical theories, in Three Hundred Years of Gravitation, ed. by S.W. Hawking, W. Israel (Cambridge University Press, Cambridge/New York, 1987), pp. 676–684

    Google Scholar 

  65. M. Cvetic, F. Larsen, J. High Energy Phys. 0909, 088 (2009)

    Google Scholar 

  66. M. Cvetic, H. Lu, C.N. Pope, Nucl. Phys. B 545, 309 (1999)

    Google Scholar 

  67. G. Dall’Agata, S. Giusto, C. Ruef, J. High Energy Phys. 1102, 074 (2011)

    Google Scholar 

  68. J. de Boer, S. El-Showk, I. Messamah, D.V.d. Bleeken, Quantizing N  = 2 multicenter solutions. arXiv:0807.4556 (hep-th)

    Google Scholar 

  69. J. de Boer, S. El-Showk, I. Messamah, D.V.d. Bleeken, A bound on the entropy of supergravity? arXiv:0906.0011 (hep-th)

    Google Scholar 

  70. J. de Boer, M.M. Sheikh-Jabbari, J. Simon, Near horizon limits of massless BTZ and their CFT duals. arXiv:1011.1897 (hep-th)

    Google Scholar 

  71. L. D’Errico, W. Mueck, R. Pettorino, J. High Energy Phys. 0705, 063 (2007)

    Google Scholar 

  72. R. de Mello Koch, N. Ives, M. Stephanou, Phys. Rev. D79, 026004 (2009)

    Google Scholar 

  73. R. de Mello Koch, T.K. Dey, N. Ives, M. Stephanou, J. High Energy Phys. 0908, 083 (2009)

    Google Scholar 

  74. F. Denef, G.W. Moore, Split states, entropy enigmas, holes and halos. (hep-th/0702146 (HEP-TH))

    Google Scholar 

  75. A. Dhar, G. Mandal, N.V. Suryanarayana, J. High Energy Phys. 0601, 118 (2006)

    Google Scholar 

  76. O.J.C. Dias, H.S. Reall, J.E. Santos, J. High Energy Phys. 0908, 101 (2009)

    Google Scholar 

  77. R. Dijkgraaf, J.M. Maldacena, G.W. Moore, E.P. Verlinde, A black hole farey tail. arXiv: hep-th/0005003

    Google Scholar 

  78. A. Einstein, Ann. Phys. 49, 769 (1916). (Ann. Phys. 14, 517 (2005))

    Google Scholar 

  79. R. Fareghbal, C.N. Gowdigere, A.E. Mosaffa, M.M. Sheikh-Jabbari, J. High Energy Phys. 0808, 070 (2008)

    Google Scholar 

  80. R. Fareghbal, C.N. Gowdigere, A.E. Mosaffa, M.M. Sheikh-Jabbari, arXiv:0805.0203 (hep-th)

    Google Scholar 

  81. J.M. Figueroa-O’Farrill, J. Simon, Adv. Theor. Math. Phys. 8, 217 (2004)

    Google Scholar 

  82. V.P. Frolov, K.S. Thorne, Phys. Rev. D39, 2125 (1989)

    Google Scholar 

  83. W.D. Goldberger, J. High Energy Phys. 0903, 069 (2009)

    Google Scholar 

  84. L. Grant, L. Maoz, J. Marsano, K. Papadodimas, V.S. Rychkov, J. High Energy Phys. 0508, 025 (2005)

    Google Scholar 

  85. S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. B 428, 105 (1998)

    Google Scholar 

  86. M. Guica, A. Strominger, J. High Energy Phys. 1102, 010 (2011)

    Google Scholar 

  87. M. Guica, T. Hartman, W. Song, A. Strominger, Phys. Rev. D 80, 124008 (2009)

    Google Scholar 

  88. R.K. Gupta, A. Sen, J. High Energy Phys. 0904, 034 (2009)

    Google Scholar 

  89. T. Hartman, A. Strominger, J. High Energy Phys. 0904, 026 (2009)

    Google Scholar 

  90. T. Hartman, K. Murata, T. Nishioka, A. Strominger, J. High Energy Phys. 0904, 019 (2009)

    Google Scholar 

  91. S.W. Hawking, Commun. Math. Phys. 43, 199 (1975) (Erratum-ibid. 46, 206 (1976))

    Google Scholar 

  92. S.W. Hawking, Phys. Rev. D14, 2460–2473 (1976)

    Google Scholar 

  93. S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)

    Google Scholar 

  94. S.W. Hawking, R. Penrose, Proc. R. Soc. Lond. A314, 529–548 (1970)

    Google Scholar 

  95. I. Heemskerk, J. Polchinski, Holographic and Wilsonian renormalization groups. arXiv:1010.1264 (hep-th) T. Faulkner, H. Liu, M. Rangamani, Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm. arXiv:1010.4036 (hep-th)

    Google Scholar 

  96. S. Hellerman, J. Polchinski, Phys. Rev. D59, 125002 (1999)

    Google Scholar 

  97. C.P. Herzog, J. Phys. A 42, 343001 (2009). S.A. Hartnoll, Class. Quantum Gravity 26, 224002 (2009). G.T. Horowitz, Introduction to holographic superconductors. arXiv:1002.1722 (hep-th).J. McGreevy, Adv. High Energy Phys. 2010, 723105 (2010).T. Faulkner, H. Liu, J. McGreevy, D. Vegh, Emergent quantum criticality, Fermi surfaces, and AdS2. arXiv:0907.2694 (hep-th)

    Google Scholar 

  98. M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Phys. Rep. 106(3), 121–167 (1984)

    Google Scholar 

  99. C.M. Hull, P.K. Townsend, Nucl. Phys. B 438, 109 (1995)

    Google Scholar 

  100. V. Jejjala, S. Nampuri, J. High Energy Phys. 1002, 088 (2010)

    Google Scholar 

  101. V. Jejjala, O. Madden, S.F. Ross, G. Titchener, Phys. Rev. D71, 124030 (2005)

    Google Scholar 

  102. J. Kinney, J.M. Maldacena, S. Minwalla, S. Raju, Commun. Math. Phys. 275, 209 (2007)

    Google Scholar 

  103. H.K. Kunduri, J. Lucietti, H.S. Reall, Class. Quantum Gravity 24, 4169 (2007)

    Google Scholar 

  104. P. Kraus, H. Ooguri, S. Shenker, Phys. Rev. D 67, 124022 (2003). L. Fidkowski, V. Hubeny, M. Kleban, S. Shenker, J. High Energy Phys. 0402, 014 (2004). T.S. Levi, S.F. Ross, Phys. Rev. D 68, 044005 (2003).V. Balasubramanian, T.S. Levi, Phys. Rev. D 70, 106005 (2004).D. Brecher, J. He, M. Rozali, J. High Energy Phys. 0504, 004 (2005).G. Festuccia, H. Liu, J. High Energy Phys. 0604, 044 (2006).B. Freivogel, V.E. Hubeny, A. Maloney, R. Myers, M. Rangamani, S. Shenker, J. High Energy Phys. 0603, 007 (2006).K. Maeda, M. Natsuume, T. Okamura, Phys. Rev. D 74, 046010 (2006).A. Hamilton, D. Kabat, G. Lifschytz, D.A. Lowe. arXiv:hep-th/0612053

    Google Scholar 

  105. H. Lin, O. Lunin, J. Maldacena, J. High Energy Phys. 0410, 025 (2004)

    Google Scholar 

  106. F. Loran, H. Soltanpanahi, Class. Quantum Gravity 26, 155019 (2009)

    Google Scholar 

  107. H. Lu, J. Mei, C. N. Pope, J. High Energy Phys. 0904, 054 (2009). T. Azeyanagi, N. Ogawa, S. Terashima, J. High Energy Phys. 0904, 061 (2009)

    Google Scholar 

  108. O. Lunin, S.D. Mathur, Nucl. Phys. B 623, 342 (2002)

    Google Scholar 

  109. O. Lunin, S.D. Mathur, Phys. Rev. Lett. 88, 211303 (2002)

    Google Scholar 

  110. O. Lunin, J.M. Maldacena, L. Maoz, Gravity solutions for the D1-D5 system with angular momentum. arXiv:hep-th/0212210

    Google Scholar 

  111. J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). (Int. J. Theor. Phys. 38, 1113 (1999))

    Google Scholar 

  112. J.M. Maldacena, J. High Energy Phys. 0304, 021 (2003). L. Dyson, M. Kleban, L. Susskind, J. High Energy Phys. 0210, 011 (2002). L. Dyson, J. Lindesay, L. Susskind, J. High Energy Phys. 0208, 045 (2002). N. Goheer, M. Kleban, L. Susskind, J. High Energy Phys. 0307, 056 (2003). D. Birmingham, I. Sachs, S.N. Solodukhin, Phys. Rev. D 67, 104026 (2003). J.L.F. Barbon, E. Rabinovici, J. High Energy Phys. 0311, 047 (2003). J.L.F. Barbon, E. Rabinovici, Fortschr. Phys. 52, 642–649 (2004). J.L.F. Barbon, E. Rabinovici, Topology change and unitarity in quantum black hole dynamics. arXiv:hep-th/0503144. M. Kleban, M. Porrati, R. Rabadan, J. High Energy Phys. 0410, 030 (2004)

    Google Scholar 

  113. J.M. Maldacena, H. Ooguri, J. Math. Phys. 42, 2929–2960 (2001)

    Google Scholar 

  114. J.M. Maldacena, J. Michelson, A. Strominger, J. High Energy Phys. 9902, 011 (1999)

    Google Scholar 

  115. J.M. Maldacena, H. Ooguri, J. Son, J. Math. Phys. 42, 2961–2977 (2001). S. Hemming, E. Keski-Vakkuri, P. Kraus, J. High Energy Phys. 0210, 006 (2002)

    Google Scholar 

  116. J. Maldacena, D. Martelli, Y. Tachikawa, J. High Energy Phys. 0810, 072 (2008)

    Google Scholar 

  117. G. Mandal, J. High Energy Phys. 0508, 052 (2005)

    Google Scholar 

  118. L. Maoz, V.S. Rychkov, J. High Energy Phys. 0508, 096 (2005)

    Google Scholar 

  119. E.J. Martinec, W. McElgin, J. High Energy Phys. 0204, 029 (2002)

    Google Scholar 

  120. S.D. Mathur, Fortschr. Phys. 53, 793 (2005)

    Google Scholar 

  121. S.D. Mathur, Class. Quantum Gravity 23, R115 (2006)

    Google Scholar 

  122. S.D. Mathur, Lect. Notes Phys. 769, 3–48 (2009)

    Google Scholar 

  123. S. D. Mathur, Class. Quantum Gravity 26, 224001 (2009)

    Google Scholar 

  124. S.D. Mathur, Fuzzballs and the information paradox: a summary and conjectures. arXiv:0810.4525 (hep-th)

    Google Scholar 

  125. S.D. Mathur, The information paradox and the infall problem. (arXiv:1012.2101 (hep-th))

    Google Scholar 

  126. S.D. Mathur, C.J. Plumberg, Correlations in Hawking radiation and the infall problem. (arXiv:1101.4899 (hep-th))

    Google Scholar 

  127. J. McGreevy, L. Susskind, N. Toumbas, J. High Energy Phys. 0006, 008 (2000)

    Google Scholar 

  128. R.C. Myers, J. High Energy Phys. 9912, 022 (1999)

    Google Scholar 

  129. R.C. Myers, O. Tafjord, J. High Energy Phys. 0111, 009 (2001)

    Google Scholar 

  130. J. Navarro-Salas, P. Navarro, Nucl. Phys. B579, 250–266 (2000). M. Cadoni, S. Mignemi, Phys. Lett. B490, 131–135 (2000)

    Google Scholar 

  131. D.N. Page, Phys. Rev. Lett. 71, 3743 (1993)

    Google Scholar 

  132. R. Penrose, Phys. Rev. Lett. 14, 57–59 (1965)

    Google Scholar 

  133. A.M. Perelomov, Teor. Mat. Fiz. 6, 213 (1971)

    Google Scholar 

  134. J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995)

    Google Scholar 

  135. V.S. Rychkov, J. High Energy Phys. 0601, 063 (2006)

    Google Scholar 

  136. S. Ryu, T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006). S. Ryu, T. Takayanagi, J. High Energy Phys. 0608, 045 (2006). S. Ryu, T. Takayanagi, J. Phys. A A42, 504008(2009)

    Google Scholar 

  137. N. Seiberg, Phys. Rev. Lett. 79, 3577 (1997)

    Google Scholar 

  138. A. Sen, J. High Energy Phys. 0509, 038 (2005)

    Google Scholar 

  139. A. Sen, Gen. Relativ. Gravit. 40, 2249 (2008)

    Google Scholar 

  140. A. Sen, J. High Energy Phys. 0811, 075 (2008)

    Google Scholar 

  141. A. Sen, Int. J. Mod. Phys. A 24, 4225 (2009)

    Google Scholar 

  142. A. Sen, J. High Energy Phys. 0908, 068 (2009)

    Google Scholar 

  143. A. Sen, J. High Energy Phys. 1005, 097 (2010)

    Google Scholar 

  144. C.E. Shannon, Bell Syst. Tech. J. 27, 379–423 (1948)

    Google Scholar 

  145. J. Simon, Phys. Rev. D 81, 024003 (2010)

    Google Scholar 

  146. K. Skenderis, M. Taylor, Phys. Rep. 467, 117 (2008)

    Google Scholar 

  147. A. Strominger, J. High Energy Phys. 9802, 009 (1998)

    Google Scholar 

  148. A. Strominger, J. High Energy Phys. 9901, 007 (1999)

    Google Scholar 

  149. A. Strominger, C. Vafa, Phys. Lett. B 379, 99 (1996)

    Google Scholar 

  150. L. Susskind, J. Math. Phys. 36, 6377 (1995)

    Google Scholar 

  151. Y. Takayama, A. Tsuchiya, J. High Energy Phys. 0510, 004 (2005)

    Google Scholar 

  152. G. ’t Hooft, Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026

    Google Scholar 

  153. J.A. Wheeler, Phys. Rev. 97, 511 (1955). J.A. Wheeler, Ann. Phys. 2, 604 (1957)

    Google Scholar 

  154. E. Witten, Nucl. Phys. B 443, 85 (1995)

    Google Scholar 

  155. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)

    Google Scholar 

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  1. School of Mathematics, University of Edinburgh, Mayfield Road, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Joan Simón

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    Stefano Bellucci

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Simón, J. (2013). Extremality, Holography and Coarse Graining. In: Bellucci, S. (eds) Supersymmetric Gravity and Black Holes. Springer Proceedings in Physics, vol 142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31380-6_3

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