Journal of High Energy Physics

, 2019:197 | Cite as

Emergent classical spacetime from microstates of an incipient black hole

  • Vijay Balasubramanian
  • David Berenstein
  • Aitor Lewkowycz
  • Alexandra Miller
  • Onkar ParrikarEmail author
  • Charles Rabideau
Open Access
Regular Article - Theoretical Physics


Black holes have an enormous underlying space of microstates, but universal macroscopic physics characterized by mass, charge and angular momentum as well as a causally disconnected interior. This leads to two related puzzles: (1) How does the effective factorization of interior and exterior degrees of freedom emerge in gravity?, and (2) How does the underlying degeneracy of states wind up having a geometric realization in the horizon area and in properties of the singularity? We explore these puzzles in the context of an incipient black hole in the AdS/CFT correspondence, the microstates of which are dual to half-BPS states of the \( \mathcal{N} \) = 4 super-Yang-Mills theory. First, we construct a code subspace for this black hole and show how to organize it as a tensor product of a universal macroscopic piece (describing the exterior), and a factor corresponding to the microscopic degrees of freedom (describing the interior). We then study the classical phase space and symplectic form for low-energy excitations around the black hole. On the AdS side, we find that the symplectic form has a new physical degree of freedom at the stretched horizon of the black hole, reminiscent of soft hair, which is absent in the microstates. We explicitly show how such a soft mode emerges from the microscopic phase space in the dual CFT via a canonical transformation and how it encodes partial information about the microscopic degrees of freedom of the black hole.


AdS-CFT Correspondence Black Holes in String Theory Spacetime Singularities 


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]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, The library of Babel: on the origin of gravitational thermodynamics, JHEP 12 (2005) 006 [hep-th/0508023] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    R.C. Myers and O. Tafjord, Superstars and giant gravitons, JHEP 11 (2001) 009 [hep-th/0109127] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    O. Lunin and S.D. Mathur, Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, Phys. Rev. Lett. 88 (2002) 211303 [hep-th/0202072] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, Entropy of near-extremal black holes in AdS 5, JHEP 05 (2008) 067 [arXiv:0707.3601] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    L. Grant et al., Minisuperspace quantization of ‘bubbling AdS’ and free fermion droplets, JHEP 08 (2005) 025 [hep-th/0505079] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    L. Maoz and V.S. Rychkov, Geometry quantization from supergravity: The Case of ‘bubbling AdS’, JHEP 08 (2005) 096 [hep-th/0508059] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Ghosh, R.M. Soni and S.P. Trivedi, On the entanglement entropy for gauge theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94 (2016) 104053 [arXiv:1506.05792] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J.R. Fliss et al., Interface contributions to topological entanglement in abelian Chern-Simons theory, JHEP 09 (2017) 056 [arXiv:1705.09611] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    D. Harlow, The Ryu–Takayanagi formula from quantum error correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. Lin, Ryu-Takayanagi area as an entanglement edge term, arXiv:1704.07763 [INSPIRE].
  22. [22]
    J. Lin, Entanglement entropy in Jackiw-Teitelboim gravity, arXiv:1807.06575 [INSPIRE].
  23. [23]
    S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
  24. [24]
    L. Donnay, G. Giribet, H.A. Gonzalez and M. Pino, Supertranslations and superrotations at the black hole horizon, Phys. Rev. Lett. 116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    L. Donnay, G. Giribet, H.A. González and M. Pino, Extended symmetries at the black hole horizon, JHEP 09 (2016) 100 [arXiv:1607.05703] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    L. Donnay, G. Giribet, H.A. González and A. Puhm, Black hole memory effect, Phys. Rev. D 98 (2018) 124016 [arXiv:1809.07266] [INSPIRE].
  27. [27]
    S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, Black hole entropy and soft hair, JHEP 12 (2018) 098 [arXiv:1810.01847] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    D. Berenstein, A toy model for the AdS/CFT correspondence, JHEP 07 (2004) 018 [hep-th/0403110] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. Corley, A. Jevicki and S. Ramgoolam, Exact correlators of giant gravitons from dual N = 4 SYM theory, Adv. Theor. Math. Phys. 5 (2002) 809 [hep-th/0111222] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    A. Ghodsi, A.E. Mosaffa, O. Saremi and M.M. Sheikh-Jabbari, LLL vs. LLM: half BPS sector of N = 4 SYM equals to quantum Hall system, Nucl. Phys. B 729 (2005) 467 [hep-th/0505129] [INSPIRE].
  32. [32]
    D. Berenstein and A. Miller, Superposition induced topology changes in quantum gravity, JHEP 11 (2017) 121 [arXiv:1702.03011] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D. Berenstein and A. Miller, Code subspaces for LLM geometries, Class. Quant. Grav. 35 (2018) 065003 [arXiv:1708.00035] [INSPIRE].
  34. [34]
    A. Dhar, G. Mandal and S.R. Wadia, Classical Fermi fluid and geometric action for c = 1, Int. J. Mod. Phys. A 8 (1993) 325 [hep-th/9204028] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    A. Dhar, Two-dimensional string theory from the c = 1 matrix model, Nucl. Phys. Proc. Suppl. B 45 (1996) 234.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Dhar, G. Mandal and S.R. Wadia, W(∞) coherent states and path integral derivation of bosonization of nonrelativistic fermions in one-dimension, Mod. Phys. Lett. A 8 (1993) 3557 [hep-th/9309028] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    V. Balasubramanian et al., Quantum geometry and gravitational entropy, JHEP 12 (2007) 067 [arXiv:0705.4431] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    N.V. Suryanarayana, Half-BPS giants, free fermions and microstates of superstars, JHEP 01 (2006) 082 [hep-th/0411145] [INSPIRE].
  39. [39]
    J. Simon, Correlations vs connectivity in R-charge, JHEP 10 (2018) 048 [arXiv:1805.11279] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  40. [40]
    J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, JHEP 06 (2000) 008 [hep-th/0003075] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    V. Balasubramanian, M. Berkooz, A. Naqvi and M.J. Strassler, Giant gravitons in conformal field theory, JHEP 04 (2002) 034 [hep-th/0107119] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    V. Balasubramanian, B. Czech, K. Larjo and J. Simon, Integrability versus information loss: a simple example, JHEP 11 (2006) 001 [hep-th/0602263] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSCrossRefGoogle Scholar
  44. [44]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.ADSGoogle Scholar
  45. [45]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  46. [46]
    V. Balasubramanian et al., Typicality versus thermality: an analytic distinction, Gen. Rel. Grav. 40 (2008) 1863 [hep-th/0701122] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    P. Hayden and G. Penington, Learning the alpha-bits of black holes, arXiv:1807.06041 [INSPIRE].
  48. [48]
    T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP 06 (2018) 123 [arXiv:1712.03464] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    V. Balasubramanian et al., Emergent classical spacetime from microstates of an incipient black hole II, work in progress.Google Scholar
  50. [50]
    K. Papadodimas and S. Raju, Black hole interior in the holographic correspondence and the information paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].
  52. [52]
    A.E. Mosaffa and M.M. Sheikh-Jabbari, On classification of the bubbling geometries, JHEP 04 (2006) 045 [hep-th/0602270] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    K. Skenderis and M. Taylor, Anatomy of bubbling solutions, JHEP 09 (2007) 019 [arXiv:0706.0216] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    L.G. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys. 54 (1982) 407 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    E. Witten, Coadjoint orbits of the Virasoro group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    A. Belin, A. Lewkowycz and G. Sárosi, The boundary dual of the bulk symplectic form, Phys. Lett. B 789 (2019) 71 [arXiv:1806.10144] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    A. Caldeira and A. Leggett, Path integral approach to quantum brownian motion, Physica A 121 (1983) 587.Google Scholar
  58. [58]
    C. Agon, V. Balasubramanian, S. Kasko and A. Lawrence, Coarse grained quantum dynamics, Phys. Rev. D 98 (2018) 025019 [arXiv:1412.3148] [INSPIRE].ADSGoogle Scholar
  59. [59]
    V. Balasubramanian, J. de Boer, S. El-Showk and I. Messamah, Black holes as effective geometries, Class. Quant. Grav. 25 (2008) 214004 [arXiv:0811.0263] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    A. Dhar, G. Mandal and N.V. Suryanarayana, Exact operator bosonization of finite number of fermions in one space dimension, JHEP 01 (2006) 118 [hep-th/0509164] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2019

Authors and Affiliations

  • Vijay Balasubramanian
    • 1
    • 2
  • David Berenstein
    • 3
  • Aitor Lewkowycz
    • 4
  • Alexandra Miller
    • 5
  • Onkar Parrikar
    • 1
    Email author
  • Charles Rabideau
    • 1
    • 2
  1. 1.David Rittenhouse LaboratoryUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB), and International Solvay InstitutesBrusselsBelgium
  3. 3.Department of PhysicsUniversity of California at Santa BarbaraSanta BarbaraU.S.A.
  4. 4.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordU.S.A.
  5. 5.Department of PhysicsWellesley CollegeWellesleyU.S.A.

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