Holographic probes of collapsing black holes

  • Veronika E. HubenyEmail author
  • Henry Maxfield
Open Access


We continue the programme of exploring the means of holographically decoding the geometry of spacetime inside a black hole using the gauge/gravity correspondence. To this end, we study the behaviour of certain extremal surfaces (focusing on those relevant for equal-time correlators and entanglement entropy in the dual CFT) in a dynamically evolving asymptotically AdS spacetime, specifically examining how deep such probes reach. To highlight the novel effects of putting the system far out of equilibrium and at finite volume, we consider spherically symmetric Vaidya-AdS, describing black hole formation by gravitational collapse of a null shell, which provides a convenient toy model of a quantum quench in the field theory. Extremal surfaces anchored on the boundary exhibit rather rich behaviour, whose features depend on dimension of both the spacetime and the surface, as well as on the anchoring region. The main common feature is that they reach inside the horizon even in the post-collapse part of the geometry. In 3-dimensional spacetime, we find that for sub-AdS-sized black holes, the entire spacetime is accessible by the restricted class of geodesics whereas in larger black holes a small region near the imploding shell cannot be reached by any boundary-anchored geodesic. In higher dimensions, the deepest reach is attained by geodesics which (despite being asymmetric) connect equal time and antipodal boundary points soon after the collapse; these can attain spacetime regions of arbitrarily high curvature and simultaneously have smallest length. Higher-dimensional surfaces can penetrate the horizon while anchored on the boundary at arbitrarily late times, but are bounded away from the singularity. We also study the details of length or area growth during thermalization. While the area of extremal surfaces increases monotonically, geodesic length is neither monotonic nor continuous.


AdS-CFT Correspondence Black Holes Spacetime Singularities 


Open Access

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  1. [1]
    J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
  2. [2]
    T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
  3. [3]
    J. Polchinski, S matrices from AdS space-time, hep-th/9901076 [INSPIRE].
  4. [4]
    L. Susskind, Holography in the flat space limit, hep-th/9901079 [INSPIRE].
  5. [5]
    S.B. Giddings, Flat space scattering and bulk locality in the AdS/CFT correspondence, Phys. Rev. D 61 (2000) 106008 [hep-th/9907129] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    G.T. Horowitz and V.E. Hubeny, CFT description of small objects in AdS, JHEP 10 (2000) 027 [hep-th/0009051] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    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].ADSMathSciNetGoogle Scholar
  8. [8]
    M. Gary and S.B. Giddings, The Flat space S-matrix from the AdS/CFT correspondence?, Phys. Rev. D 80 (2009) 046008 [arXiv:0904.3544] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    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].ADSGoogle Scholar
  12. [12]
    A.L. Fitzpatrick and J. Kaplan, Scattering States in AdS/CFT, arXiv:1104.2597 [INSPIRE].
  13. [13]
    I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    S. Raju, New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators, Phys. Rev. D 85 (2012) 126009 [arXiv:1201.6449] [INSPIRE].ADSGoogle Scholar
  15. [15]
    S. Raju, Four Point Functions of the Stress Tensor and Conserved Currents in AdS 4 /CFT 3, Phys. Rev. D 85 (2012) 126008 [arXiv:1201.6452] [INSPIRE].ADSGoogle Scholar
  16. [16]
    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    V.E. Hubeny, H. Liu and M. Rangamani, Bulk-cone singularities & signatures of horizon formation in AdS/CFT, JHEP 01 (2007) 009 [hep-th/0610041] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    J. Hammersley, Extracting the bulk metric from boundary information in asymptotically AdS spacetimes, JHEP 12 (2006) 047 [hep-th/0609202] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    S. Bilson, Extracting spacetimes using the AdS/CFT conjecture, JHEP 08 (2008) 073 [arXiv:0807.3695] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    V.E. Hubeny, Precursors see inside black holes, Int. J. Mod. Phys. D 12 (2003) 1693 [hep-th/0208047] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  27. [27]
    A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    J. Aparicio and E. Lopez, Evolution of Two-Point Functions from Holography, JHEP 12 (2011) 082 [arXiv:1109.3571] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    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] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].ADSGoogle Scholar
  33. [33]
    H. Liu and S.J. Suh, Entanglement Tsunami: Universal Scaling in Holographic Thermalization, Phys. Rev. Lett. 112 (2014) 011601 [arXiv:1305.7244] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    H. Liu and S.J. Suh, Entanglement growth during thermalization in holographic systems, arXiv:1311.1200 [INSPIRE].
  35. [35]
    D.A. Lowe and S. Roy, Holographic description of asymptotically AdS 2 collapse geometries, Phys. Rev. D 78 (2008) 124017 [arXiv:0810.1750] [INSPIRE].ADSMathSciNetGoogle Scholar
  36. [36]
    T. Albash and C.V. Johnson, Holographic Entanglement Entropy and Renormalization Group Flow, JHEP 02 (2012) 095 [arXiv:1110.1074] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    T. Albash and C.V. Johnson, Holographic Studies of Entanglement Entropy in Superconductors, JHEP 05 (2012) 079 [arXiv:1202.2605] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    W. Baron, D. Galante and M. Schvellinger, Dynamics of holographic thermalization, JHEP 03 (2013) 070 [arXiv:1212.5234] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    E. Caceres and A. Kundu, Holographic Thermalization with Chemical Potential, JHEP 09 (2012) 055 [arXiv:1205.2354] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    D. Galante and M. Schvellinger, Thermalization with a chemical potential from AdS spaces, JHEP 07 (2012) 096 [arXiv:1205.1548] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    W. Fischler, S. Kundu and J.F. Pedraza, Entanglement and out-of-equilibrium dynamics in holographic models of de Sitter QFTs, arXiv:1311.5519 [INSPIRE].
  42. [42]
    V.E. Hubeny and M. Rangamani, A Holographic view on physics out of equilibrium, Adv. High Energy Phys. 2010 (2010) 297916 [arXiv:1006.3675] [INSPIRE].CrossRefGoogle Scholar
  43. [43]
    V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  45. [45]
    P. Kraus, H. Ooguri and S. Shenker, Inside the horizon with AdS/CFT, Phys. Rev. D 67 (2003) 124022 [hep-th/0212277] [INSPIRE].ADSMathSciNetGoogle Scholar
  46. [46]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, arXiv:1306.0622 [INSPIRE].
  47. [47]
    V.E. Hubeny, M. Rangamani and E. Tonni, Thermalization of Causal Holographic Information, JHEP 05 (2013) 136 [arXiv:1302.0853] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    P. Bizoń and J. Jałmużna, Globally regular instability of AdS 3, Phys. Rev. Lett. 111 (2013) 041102 [arXiv:1306.0317] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    T. Takayanagi and T. Ugajin, Measuring Black Hole Formations by Entanglement Entropy via Coarse-Graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    B. Freivogel, V.E. Hubeny, A. Maloney, R.C. Myers, M. Rangamani et al., Inflation in AdS/CFT, JHEP 03 (2006) 007 [hep-th/0510046] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    M. Headrick, General properties of holographic entanglement entropy, arXiv:1312.6717 [INSPIRE].
  53. [53]
    V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [INSPIRE].ADSMathSciNetGoogle Scholar
  54. [54]
    J. Louko, D. Marolf and S.F. Ross, On geodesic propagators and black hole holography, Phys. Rev. D 62 (2000) 044041 [hep-th/0002111] [INSPIRE].ADSMathSciNetGoogle Scholar
  55. [55]
    J. Kaplan, Extracting data from behind horizons with the AdS/CFT correspondence, hep-th/0402066 [INSPIRE].
  56. [56]
    G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I., JHEP 04 (2006) 044 [hep-th/0506202] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  57. [57]
    S.H. Shenker and D. Stanford, Multiple Shocks, arXiv:1312.3296 [INSPIRE].
  58. [58]
    T. Andrade, S. Fischetti, D. Marolf, S.F. Ross and M. Rozali, Entanglement and Correlations near Extremality: CFTs dual to Reissner-Nordström AdS 5, arXiv:1312.2839 [INSPIRE].
  59. [59]
    E. Caceres, A. Kundu, J.F. Pedraza and W. Tangarife, Strong Subadditivity, Null Energy Condition and Charged Black Holes, JHEP 01 (2014) 084 [arXiv:1304.3398] [INSPIRE].CrossRefGoogle Scholar
  60. [60]
    N. Engelhardt and A.C. Wall, Extremal Surface Barriers, arXiv:1312.3699 [INSPIRE].
  61. [61]
    F. Nogueira, Extremal Surfaces in Asymptotically AdS Charged Boson Stars Backgrounds, Phys. Rev. D 87 (2013) 106006 [arXiv:1301.4316] [INSPIRE].ADSGoogle Scholar
  62. [62]
    S.A. Gentle and M. Rangamani, Holographic entanglement and causal information in coherent states, JHEP 01 (2014) 120 [arXiv:1311.0015] [INSPIRE].CrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Centre for Particle Theory & Department of Mathematical SciencesScience LaboratoriesDurhamU.K..

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