Black holes and the butterfly effect

Journal of High Energy Physics

Volumn 2014, Issue 3, 2014, Pages

  Shenker, Stephen H ^a,b   Stanford, Douglas ^a,b  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

AdS CFT Correspondence; Black Holes

Indexed keywords

EID: 84896449116     PISSN: 11266708     EISSN: 10298479     Source Type: Journal    
DOI: 10.1007/JHEP03(2014)067     Document Type: Article

References (69)

1. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [ hep-th/0510092 ].
2. M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405.
3. R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [ arXiv:1006.1263 ].
4. R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [ arXiv:1011.5819 ].
5. H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [ arXiv:1102.0440 ].
6. J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [ hep-th/0106112 ].
7. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [ hep-th/0603001 ].
8. S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [ hep-th/0605073 ]. (Pubitemid 44330349)
9. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [ arXiv:1304.4926 ].
10. V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [ arXiv:0705.0016 ].
11. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [ arXiv:0708.4025 ].
12. Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [ arXiv:0808.2096 ].
13. L. Susskind, Addendum to Fast Scramblers, arXiv:1101.6048.
14. N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [ arXiv:1111.6580 ].
15. A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [ arXiv:1207.3123 ].
16. S.L. Braunstein, S. Pirandola and K. Zyczkowski, Better Late than Never: Information Retrieval from Black Holes, Phys. Rev. Lett. 110 (2013) 101301 [ arXiv:0907.1190 ].
17. P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [ cond-mat/0503393 ]. (Pubitemid 40669590)
18. G. De Chiara, S. Montangero, P. Calabrese and R. Fazio, Entanglement entropy dynamics in Heisenberg chains, J. Stat. Mech. 0603 (2006) L03001 [ cond-mat/0512586 ]. (Pubitemid 43534995)
19. J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [ arXiv:1006.4090 ].
20. T. Albash and C.V. Johnson, Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches, New J. Phys. 13 (2011) 045017 [ arXiv:1008.3027 ].
21. V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [ arXiv:1012.4753 ].
22. V. Balasubramanian et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [ arXiv:1103.2683 ].
23. V. Balasubramanian, A. Bernamonti, N. Copland, B. Craps and F. Galli, Thermalization of mutual and tripartite information in strongly coupled two dimensional conformal field theories, Phys. Rev. D 84 (2011) 105017 [ arXiv:1110.0488 ].
24. A. Allais and E. Tonni, Holographic evolution of the mutual information, JHEP 01 (2012) 102 [ arXiv:1110.1607 ].
25. S.R. Das, Holographic Quantum Quench, J. Phys. Conf. Ser. 343 (2012) 012027 [ arXiv:1111.7275 ].
26. M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080 [ arXiv:1302.5703 ].
27. I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, arXiv:1211.2887.
28. T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [ arXiv:1303.1080 ].
29. T. Dray and G. 't Hooft, The Effect of Spherical Shells of Matter on the Schwarzschild Black Hole, Commun. Math. Phys. 99 (1985) 613.
30. M. Van Raamsdonk, Evaporating Firewalls, talk given at SITP, October 1 2012, summary of this talk was presented in arXiv:1307.1796.
31. B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [ arXiv:1206.1323 ].
32. E. Silverstein, private communication and unpublished notes, February 2013.
33. J. Maldacena and L. Susskind, Cool horizons for entangled black holes, to appear.
34. M. Bañuls, J. Cirac and M. Hastings, Strong and Weak Thermalization of Infinite Nonintegrable Quantum Systems, Phys. Rev. Lett. 106 (2011) 050405 [ arXiv:1007.3957 ].
35. M. Hotta and M. Tanaka, Shock wave geometry with nonvanishing cosmological constant, Class. Quant. Grav. 10 (1993) 307.
36. K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [ hep-th/9408169 ].
37. R.-G. Cai and J.B. Griffiths, Null particle solutions in three-dimensional (anti-)de Sitter spaces, J. Math. Phys. 40 (1999) 3465 [ gr-qc/9905011 ].
38. L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP 08 (2007) 019 [ hep-th/0611122 ].
39. T. Dray and G. 't Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173.
40. M.M. Wolf, F. Verstraete, M.B. Hastings and J.I. Cirac, Area Laws in Quantum Systems: Mutual Information and Correlations, Phys. Rev. Lett. 100 (2008) 070502 [ arXiv:0704.3906 ]. (Pubitemid 351393592)
41. V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [ hep-th/9906226 ].
42. J. Louko, D. Marolf and S.F. Ross, On geodesic propagators and black hole holography, Phys. Rev. D 62 (2000) 044041 [ hep-th/0002111 ].
43. P. Kraus, H. Ooguri and S. Shenker, Inside the horizon with AdS/CFT, Phys. Rev. D 67 (2003) 124022 [ hep-th/0212277 ].
44. L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [ hep-th/0306170 ].
45. J. Kaplan, Extracting data from behind horizons with the AdS/CFT correspondence, hep-th/0402066.
46. G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I., JHEP 04 (2006) 044 [ hep-th/0506202 ]. (Pubitemid 43684627)
47. J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Entanglement Entropy in Lovelock Gravities, JHEP 07 (2011) 109 [ arXiv:1101.5781 ].
48. L.-Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 04 (2011) 025 [ arXiv:1101.5813 ].
49. 3, Adv. Theor. Math. Phys. 2 (1998) 733 [ hep-th/9806194 ].
50. R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The Pomeron and gauge/string duality, JHEP 12 (2007) 005 [ hep-th/0603115 ].
51. D. Amati, M. Ciafaloni and G. Veneziano, Superstring Collisions at Planckian Energies, Phys. Lett. B 197 (1987) 81.
52. D. Amati, M. Ciafaloni and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3 (1988) 1615.
53. G.T. Horowitz and A.R. Steif, Space-Time Singularities in String Theory, Phys. Rev. Lett. 64 (1990) 260.
54. G.T. Horowitz and A.R. Steif, Strings in Strong Gravitational Fields, Phys. Rev. D 42 (1990) 1950.
55. G. Veneziano, String-theoretic unitary S-matrix at the threshold of black-hole production, JHEP 11 (2004) 001 [ hep-th/0410166 ].
56. S.B. Giddings, D.J. Gross and A. Maharana, Gravitational effects in ultrahigh-energy string scattering, Phys. Rev. D 77 (2008) 046001 [ arXiv:0705.1816 ].
57. D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [ arXiv:0803.1467 ].
58. L. Susskind, Strings, black holes and Lorentz contraction, Phys. Rev. D 49 (1994) 6606 [ hep-th/9308139 ].
59. J. Polchinski, String theory and black hole complementarity, hep-th/9507094.
60. A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [ arXiv:1304.6483 ].
61. G. Horowitz, A. Lawrence and E. Silverstein, Insightful D-branes, JHEP 07 (2009) 057 [ arXiv:0904.3922 ].
62. 5 and other spaces, Phys. Rev. D 62 (2000) 066007 [ hep-th/9905186 ].
63. A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A Boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [ hep-th/0506118 ].
64. I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [ arXiv:1201.3664 ].
65. R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus and C. Zukowski, Null Geodesics, Local CFT Operators and AdS/CFT for Subregions, Phys. Rev. D 88 (2013) 064057 [ arXiv:1209.4641 ].
66. S. Leichenauer and V. Rosenhaus, AdS black holes, the bulk-boundary dictionary and smearing functions, Phys. Rev. D 88 (2013) 026003 [ arXiv:1304.6821 ].
67. H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [ arXiv:1305.7244 ].
68. D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [ gr-qc/9305007 ].
69. M. Fannes, A continuity property of the entropy density for spin lattice systems, Commun. Math. Phys. 31 (1973) 291.