Abstract
In the eternal AdS black hole geometry, we consider two signals sent from the boundaries into the black hole interior shared between the two asymptotic regions. We compute three different out-of-time-order six-point functions to quantify various properties of the collision of these signals behind the horizons: (i) We diagnose the strength of the collision by probing the two-signal state on a late time slice with boundary operators. (ii) We quantify two-sided operator growth, which provides a dual description of the signals meeting in the black hole interior, in terms of the quantum butterfly effect and quantum circuits. (iii) We consider an explicit coupling between the left and right CFTs to make the wormhole traversable and extract information about the collision product from behind the horizon. At a technical level, our results rely on the method of eikonal resummation to obtain the relevant gravitational contributions to Lorentzian six-point functions at all orders in the GN-expansion. We observe that such correlation functions display an intriguing factorization property. We corroborate these results with geodesic computations of six-point functions in two- and three-dimensional gravity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
D. Marolf and A.C. Wall, Eternal Black Holes and Superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [hep-th/9408169] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
L. Susskind, Why do Things Fall?, arXiv:1802.01198 [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
X.-L. Qi and A. Streicher, Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK, JHEP 08 (2019) 012 [arXiv:1810.11958] [INSPIRE].
H.W. Lin, J. Maldacena and Y. Zhao, Symmetries Near the Horizon, JHEP 08 (2019) 049 [arXiv:1904.12820] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
Y. Zhao, Collision in the interior of wormhole, JHEP 03 (2021) 144 [arXiv:2011.06016] [INSPIRE].
F.M. Haehl and Y. Zhao, Size and momentum of an infalling particle in the black hole interior, JHEP 06 (2021) 056 [arXiv:2102.05697] [INSPIRE].
F.M. Haehl and Y. Zhao, Diagnosing collisions in the interior of a wormhole, Phys. Rev. D 104 (2021) L021901 [arXiv:2104.02736] [INSPIRE].
P. Gao, D.L. Jafferis and A.C. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
F.M. Haehl and M. Rozali, Fine Grained Chaos in AdS2 Gravity, Phys. Rev. Lett. 120 (2018) 121601 [arXiv:1712.04963] [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200.
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].
P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127 [arXiv:1601.06794] [INSPIRE].
T. Anous and J. Sonner, Phases of scrambling in eigenstates, SciPost Phys. 7 (2019) 003 [arXiv:1903.03143] [INSPIRE].
F.M. Haehl, W. Reeves and M. Rozali, Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs, JHEP 11 (2019) 102 [arXiv:1909.05847] [INSPIRE].
T. Anous and F.M. Haehl, On the Virasoro six-point identity block and chaos, JHEP 08 (2020) 002 [arXiv:2005.06440] [INSPIRE].
B. Chen, J.-q. Wu and J.-j. Zhang, Holographic Description of 2D Conformal Block in Semi-classical Limit, JHEP 10 (2016) 110 [arXiv:1609.00801] [INSPIRE].
T. Anous, T. Hartman, A. Rovai and J. Sonner, From Conformal Blocks to Path Integrals in the Vaidya Geometry, JHEP 09 (2017) 009 [arXiv:1706.02668] [INSPIRE].
V. Rosenhaus, Multipoint Conformal Blocks in the Comb Channel, JHEP 02 (2019) 142 [arXiv:1810.03244] [INSPIRE].
C.B. Jepsen and S. Parikh, Propagator identities, holographic conformal blocks, and higher-point AdS diagrams, JHEP 10 (2019) 268 [arXiv:1906.08405] [INSPIRE].
K. Jensen, Scrambling in nearly thermalized states at large central charge, arXiv:1906.05852 [INSPIRE].
Y. Kusuki and M. Miyaji, Entanglement Entropy, OTOC and Bootstrap in 2D CFTs from Regge and Light Cone Limits of Multi-point Conformal Block, JHEP 08 (2019) 063 [arXiv:1905.02191] [INSPIRE].
K. Alkalaev and M. Pavlov, Holographic variables for CFT2 conformal blocks with heavy operators, Nucl. Phys. B 956 (2020) 115018 [arXiv:2001.02604] [INSPIRE].
J.-F. Fortin, W.-J. Ma, V. Prilepina and W. Skiba, Efficient Rules for All Conformal Blocks, arXiv:2002.09007 [INSPIRE].
S. Hoback and S. Parikh, Towards Feynman rules for conformal blocks, JHEP 01 (2021) 005 [arXiv:2006.14736] [INSPIRE].
J.-F. Fortin, W.-J. Ma and W. Skiba, Six-point conformal blocks in the snowflake channel, JHEP 11 (2020) 147 [arXiv:2004.02824] [INSPIRE].
A. Kundu, A.K. Patra and R.R. Poojary, Reparametrization mode Ward Identities and chaos in higher-pt. correlators in CFT2, arXiv:2103.00824 [INSPIRE].
G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198 (1987) 61 [INSPIRE].
H.L. Verlinde and E.P. Verlinde, Scattering at Planckian energies, Nucl. Phys. B 371 (1992) 246 [hep-th/9110017] [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [INSPIRE].
L. Cornalba, M.S. Costa and J. Penedones, Eikonal approximation in AdS/CFT: Resumming the gravitational loop expansion, JHEP 09 (2007) 037 [arXiv:0707.0120] [INSPIRE].
R.C. Brower, M.J. Strassler and C.-I. Tan, On the eikonal approximation in AdS space, JHEP 03 (2009) 050 [arXiv:0707.2408] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
F.M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].
D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
A. Goel, H.T. Lam, G.J. Turiaci and H. Verlinde, Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK, JHEP 02 (2019) 156 [arXiv:1807.03916] [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan and M. Rangamani, Classification of out-of-time-order correlators, SciPost Phys. 6 (2019) 001 [arXiv:1701.02820] [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan, A.A. Nizami and M. Rangamani, Thermal out-of-time-order correlators, KMS relations, and spectral functions, JHEP 12 (2017) 154 [arXiv:1706.08956] [INSPIRE].
Y.D. Lensky and X.-L. Qi, Rescuing a black hole in the large-q coupled SYK model, JHEP 04 (2021) 116 [arXiv:2012.15798] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, M.T. Walters and J. Wang, Eikonalization of Conformal Blocks, JHEP 09 (2015) 019 [arXiv:1504.01737] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Conformal Blocks Beyond the Semi-Classical Limit, JHEP 05 (2016) 075 [arXiv:1512.03052] [INSPIRE].
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian Quantum Mechanics, JHEP 11 (2018) 182 [arXiv:1804.09834] [INSPIRE].
H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate Operators and the 1/c Expansion: Lorentzian Resummations, High Order Computations, and Super-Virasoro Blocks, JHEP 03 (2017) 167 [arXiv:1606.02659] [INSPIRE].
D. Langlois, K.-i. Maeda and D. Wands, Conservation laws for collisions of branes (or shells) in general relativity, Phys. Rev. Lett. 88 (2002) 181301 [gr-qc/0111013] [INSPIRE].
Y.-H. Qi, S.-J. Sin and J. Yoon, Quantum Correction to Chaos in Schwarzian Theory, JHEP 11 (2019) 035 [arXiv:1906.00996] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2105.12755
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Haehl, F.M., Streicher, A. & Zhao, Y. Six-point functions and collisions in the black hole interior. J. High Energ. Phys. 2021, 134 (2021). https://doi.org/10.1007/JHEP08(2021)134
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2021)134