Abstract
We consider bulk quantum fields in AdS/CFT in the background of an eternal black hole. We show that for black holes with finite entropy, correlation functions of semiclassical bulk operators close to the horizon deviate from their semiclassical value and are ill-defined inside the horizon. This is due to the large-time behavior of correlators in a unitary CFT, and means the region near and inside the horizon receives corrections. We give a prescription for modifying the definition of a bulk field in a black hole background, such that one can still define operators that mimic the inside of the horizon, but at the price of violating microcausality. For supergravity fields we find that commutators at spacelike separation generically ~ e −S/2. Similar results hold for stable black holes that form in collapse. The general lesson may be that a small amount of non-locality, even over arbitrarily large spacelike distances, is an essential aspect of non-perturbative quantum gravity.
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References
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
V.K. Dobrev, Intertwining operator realization of the AdS/CFT correspondence, Nucl. Phys. B 553 (1999) 559 [hep-th/9812194] [INSPIRE].
I. Bena, On the construction of local fields in the bulk of AdS 5 and other spaces, Phys. Rev. D 62 (2000) 066007 [hep-th/9905186] [INSPIRE].
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] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: a holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum ibid. D 75 (2007) 129902] [hep-th/0612053] [INSPIRE].
I. Heemskerk, Construction of bulk fields with gauge redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].
D. Kabat, G. Lifschytz, S. Roy and D. Sarkar, Holographic representation of bulk fields with spin in AdS/CFT, Phys. Rev. D 86 (2012) 026004 [arXiv:1204.0126] [INSPIRE].
D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and transhorizon measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge fields in AdS, Phys. Rev. D 87 (2013) 086004 [arXiv:1212.3788] [INSPIRE].
D. Kabat and G. Lifschytz, Decoding the hologram: scalar fields interacting with gravity, Phys. Rev. D 89 (2014) 066010 [arXiv:1311.3020] [INSPIRE].
R. Bousso, A covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].
L. Susskind and E. Witten, The holographic bound in anti-de Sitter space, hep-th/9805114 [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
S.D. Mathur, Fuzzballs and the information paradox: a summary and conjectures, arXiv:0810.4525 [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
S.L. Braunstein, S. Pirandola and K. Życzkowski, Better late than never: information retrieval from black holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
S.B. Giddings, Models for unitary black hole disintegration, Phys. Rev. D 85 (2012) 044038 [arXiv:1108.2015] [INSPIRE].
S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070] [INSPIRE].
S.B. Giddings and Y. Shi, Effective field theory models for nonviolent information transfer from black holes, Phys. Rev. D 89 (2014) 124032 [arXiv:1310.5700] [INSPIRE].
A. Jevicki and S. Ramgoolam, Noncommutative gravity from the AdS/CFT correspondence, JHEP 04 (1999) 032 [hep-th/9902059] [INSPIRE].
D. Garner, S. Ramgoolam and C. Wen, Thresholds of large-N factorization in CFT 4 : exploring bulk locality in AdS 5, arXiv:1403.5281 [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT and the fate of the BTZ singularity, arXiv:0710.4334 [INSPIRE].
D.A. Lowe, Black hole complementarity from AdS/CFT, Phys. Rev. D 79 (2009) 106008 [arXiv:0903.1063] [INSPIRE].
S.-J. Rey and V. Rosenhaus, Scanning tunneling macroscopy, black holes and AdS/CFT bulk locality, JHEP 07 (2014) 050 [arXiv:1403.3943] [INSPIRE].
I.A. Morrison, Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [INSPIRE].
K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, arXiv:1402.5674 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
L. Dyson, J. Lindesay and L. Susskind, Is there really a de Sitter/CFT duality?, JHEP 08 (2002) 045 [hep-th/0202163] [INSPIRE].
J.L.F. Barbon and E. Rabinovici, Very long time scales and black hole thermal equilibrium, JHEP 11 (2003) 047 [hep-th/0308063] [INSPIRE].
J.L.F. Barbon and E. Rabinovici, Geometry and quantum noise, arXiv:1404.7085 [INSPIRE].
D. Marolf and J. Polchinski, Gauge/gravity duality and the black hole interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
S.D. Mathur and C.J. Plumberg, Correlations in Hawking radiation and the infall problem, JHEP 09 (2011) 093 [arXiv:1101.4899] [INSPIRE].
D. Birmingham, I. Sachs and S.N. Solodukhin, Relaxation in conformal field theory, Hawking-Page transition and quasinormal normal modes, Phys. Rev. D 67 (2003) 104026 [hep-th/0212308] [INSPIRE].
R.B. Mann, Topological black holes: outside looking in, gr-qc/9709039 [INSPIRE].
D. Birmingham, Topological black holes in anti-de Sitter space, Class. Quant. Grav. 16 (1999) 1197 [hep-th/9808032] [INSPIRE].
I. Ichinose and Y. Satoh, Entropies of scalar fields on three-dimensional black holes, Nucl. Phys. B 447 (1995) 340 [hep-th/9412144] [INSPIRE].
G. Lifschytz and M. Ortiz, Scalar field quantization on the (2 + 1)-dimensional black hole background, Phys. Rev. D 49 (1994) 1929 [gr-qc/9310008] [INSPIRE].
K. Papadodimas and S. Raju, State-dependent bulk-boundary maps and black hole complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
D.A. Lowe and L. Thorlacius, Comments on the black hole information problem, Phys. Rev. D 73 (2006) 104027 [hep-th/0601059] [INSPIRE].
K. Papadodimas and S. Raju, The unreasonable effectiveness of exponentially suppressed corrections in preserving information, Int. J. Mod. Phys. D 22 (2013) 1342030 [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
L. Susskind and L. Thorlacius, Gedanken experiments involving black holes, Phys. Rev. D 49 (1994) 966 [hep-th/9308100] [INSPIRE].
N. Iizuka and D. Kabat, Mutual information in Hawking radiation, Phys. Rev. D 88 (2013) 084010 [arXiv:1308.2386] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
M. Parikh and P. Samantray, Rindler-AdS/CFT, arXiv:1211.7370 [INSPIRE].
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Kabat, D., Lifschytz, G. Finite N and the failure of bulk locality: black holes in AdS/CFT. J. High Energ. Phys. 2014, 77 (2014). https://doi.org/10.1007/JHEP09(2014)077
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DOI: https://doi.org/10.1007/JHEP09(2014)077