Journal of High Energy Physics

, 2017:151 | Cite as

Traversable wormholes via a double trace deformation

  • Ping Gao
  • Daniel Louis JafferisEmail author
  • Aron C. Wall
Open Access
Regular Article - Theoretical Physics


After turning on an interaction that couples the two boundaries of an eternal BTZ black hole, we find a quantum matter stress tensor with negative average null energy, whose gravitational backreaction renders the Einstein-Rosen bridge traversable. Such a traversable wormhole has an interesting interpretation in the context of ER=EPR, which we suggest might be related to quantum teleportation. However, it cannot be used to violate causality. We also discuss the implications for the energy and holographic entropy in the dual CFT description.


Black Holes Gauge-gravity correspondence 


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.


  1. [1]
    O. Aharony, M. Berkooz and B. Katz, Non-local effects of multi-trace deformations in the AdS/CFT correspondence, JHEP 10 (2005) 097 [hep-th/0504177] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  2. [2]
    D. Amati, M. Ciafaloni and G. Veneziano, Effective action and all order gravitational eikonal at Planckian energies, Nucl. Phys. B 403 (1993) 707 [INSPIRE].CrossRefADSGoogle Scholar
  3. [3]
    R.E. Arias, M. Botta Cantcheff and G.A. Silva, Lorentzian AdS, Wormholes and Holography, Phys. Rev. D 83 (2011) 066015 [arXiv:1012.4478] [INSPIRE].
  4. [4]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
  5. [5]
    M. Bañados, C. Teitelboim and J. Zanelli, Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. [6]
    C. Barcelo and M. Visser, Traversable wormholes from massless conformally coupled scalar fields, Phys. Lett. B 466 (1999) 127 [gr-qc/9908029] [INSPIRE].
  7. [7]
    T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  8. [8]
    M. Berkooz, A. Sever and A. Shomer, ’Double trace’deformations, boundary conditions and space-time singularities, JHEP 05 (2002) 034 [hep-th/0112264] [INSPIRE].
  9. [9]
    B. Bhawal and S. Kar, Lorentzian wormholes in Einstein-Gauss-Bonnet theory, Phys. Rev. D 46 (1992) 2464 [INSPIRE].MathSciNetADSGoogle Scholar
  10. [10]
    R. Bousso, A covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. [11]
    R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
  12. [12]
    P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  13. [13]
    W. Bunting, Z. Fu and D. Marolf, A coarse-grained generalized second law for holographic conformal field theories, Class. Quant. Grav. 33 (2016) 055008 [arXiv:1509.00074] [INSPIRE].
  14. [14]
    X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  15. [15]
    S. Carlip, The (2 + 1)-dimensional black hole, Class. Quant. Grav. 12 (1995) 2853 [gr-qc/9506079] [INSPIRE].
  16. [16]
    X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  17. [17]
    X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, arXiv:1705.08453 [INSPIRE].
  18. [18]
    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].CrossRefADSGoogle Scholar
  19. [19]
    T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  20. [20]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    E.E. Flanagan, D. Marolf and R.M. Wald, Proof of classical versions of the Bousso entropy bound and of the generalized second law, Phys. Rev. D 62 (2000) 084035 [hep-th/9908070] [INSPIRE].
  22. [22]
    N. Graham and K.D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76 (2007) 064001 [arXiv:0705.3193] [INSPIRE].
  23. [23]
    R. Haag, N.M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Commun. Math. Phys. 5 (1967) 215 [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  24. [24]
    P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  25. [25]
    D. Hochberg and M. Visser, Null energy condition in dynamic wormholes, Phys. Rev. Lett. 81 (1998) 746 [gr-qc/9802048] [INSPIRE].
  26. [26]
    D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  27. [27]
    D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  29. [29]
    I. Ichinose and Y. Satoh, Entropies of scalar fields on three-dimensional black holes, Nucl. Phys. B 447 (1995) 340 [hep-th/9412144] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  30. [30]
    W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  31. [31]
    W.R. Kelly and A.C. Wall, Holographic proof of the averaged null energy condition, Phys. Rev. D 90 (2014) 106003 [arXiv:1408.3566] [INSPIRE].ADSGoogle Scholar
  32. [32]
    J. Koeller and S. Leichenauer, Holographic Proof of the Quantum Null Energy Condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].
  33. [33]
    E.-A. Kontou and K.D. Olum, Averaged null energy condition in a classical curved background, Phys. Rev. D 87 (2013) 064009 [arXiv:1212.2290] [INSPIRE].
  34. [34]
    E.-A. Kontou and K.D. Olum, Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality, Phys. Rev. D 92 (2015) 124009 [arXiv:1507.00297] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  36. [36]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  37. [37]
    D. Marolf and A.C. Wall, Eternal black holes and superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
  38. [38]
    M.S. Morris and K.S. Thorne, Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56 (1988) 395 [INSPIRE].CrossRefzbMATHADSGoogle Scholar
  39. [39]
    M.S. Morris, K.S. Thorne and U. Yurtsever, Wormholes, time machines and the weak energy condition, Phys. Rev. Lett. 61 (1988) 1446 [INSPIRE].CrossRefADSGoogle Scholar
  40. [40]
    T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, EPR pairs, local projections and quantum teleportation in holography, JHEP 08 (2016) 077 [arXiv:1604.01772] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  41. [41]
    K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].CrossRefADSGoogle Scholar
  42. [42]
    L. Parker and D. Toms, Quantum field theory in curved spacetime: quantized field(s an)d gravity, Cambridge University Press, Cambridge U.K. (2009).Google Scholar
  43. [43]
    A.P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, Integrals and series, volume 3: More special functions, Gordon and Breach, New York U.A.A. (1992).Google Scholar
  44. [44]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  45. [45]
    M.J. Schlosser, Multiple hypergeometric series: Appell series and beyond, in Computer Algebra in Quantum Field Theory, Springer, Heidelberg Germany (2013), pg. 305.Google Scholar
  46. [46]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  47. [47]
    S.N. Solodukhin, Restoring unitarity in BTZ black hole, Phys. Rev. D 71 (2005) 064006 [hep-th/0501053] [INSPIRE].
  48. [48]
    A. Strominger and D.M. Thompson, A Quantum Bousso bound, Phys. Rev. D 70 (2004) 044007 [hep-th/0303067] [INSPIRE].
  49. [49]
    L. Susskind, ER=EPR, GHZ and the consistency of quantum measurements, Fortsch. Phys. 64 (2016) 72 [arXiv:1412.8483] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  50. [50]
    M. Thibeault, C. Simeone and E.F. Eiroa, Thin-shell wormholes in Einstein-Maxwell theory with a Gauss-Bonnet term, Gen. Rel. Grav. 38 (2006) 1593 [gr-qc/0512029] [INSPIRE].
  51. [51]
    W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].ADSGoogle Scholar
  52. [52]
    M. Visser, Lorentzian wormholes: From Einstein to Hawking, AIP Press, College Park U.S.A. (1996).Google Scholar
  53. [53]
    M. Visser, S. Kar and N. Dadhich, Traversable wormholes with arbitrarily small energy condition violations, Phys. Rev. Lett. 90 (2003) 201102 [gr-qc/0301003] [INSPIRE].
  54. [54]
    A.C. Wall, Ten proofs of the generalized second law, JHEP 06 (2009) 021 [arXiv:0901.3865] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  55. [55]
    A.C. Wall, Proving the Achronal Averaged Null Energy Condition from the Generalized Second Law, Phys. Rev. D 81 (2010) 024038 [arXiv:0910.5751] [INSPIRE].
  56. [56]
    A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [arXiv:1105.3445] [INSPIRE].
  57. [57]
    A.C. Wall, The generalized second law implies a quantum singularity theorem, Class. Quant. Grav. 30 (2013) 165003 [Erratum ibid. 30 (2013) 199501] [arXiv:1010.5513] [INSPIRE].
  58. [58]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeU.S.A.
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

Personalised recommendations