Schwarzschild/CFT from soft black hole hair?

  • Artem AverinEmail author
Open Access
Regular Article - Theoretical Physics


Recent studies of asymptotic symmetries suggest, that a Hamiltonian phase space analysis in gravitational theories might be able to account for black hole microstates. In this context we explain, why the use of conventional Bondi fall-off conditions for the gravitational field is too restrictive in the presence of an event horizon. This implies an enhancement of physical degrees of freedom (\( \mathcal{A} \)-modes). They provide new gravitational hair and are responsible for black hole microstates. Using covariant phase space methods, for the example of a Schwarzschild black hole, we give a proposal for the surface degrees of freedom and their surface charge algebra. The obtained two-dimensional dual theory is conjectured to be conformally invariant as motivated from the criticality of the black hole. Carlip’s approach to entropy counting reemerges as a Sugawara-construction of a 2D stress-tensor.


Black Holes Classical Theories of Gravity Gauge Symmetry 


Open Access

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  1. [1]
    S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
  3. [3]
    S. Carlip, Black Hole Thermodynamics, Int. J. Mod. Phys. D 23 (2014) 1430023 [arXiv:1410.1486] [INSPIRE].
  4. [4]
    G. Dvali and C. Gomez, Black Hole’s Quantum N-Portrait, Fortsch. Phys. 61 (2013) 742 [arXiv:1112.3359] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    G. Dvali and C. Gomez, Black Hole’s 1/N Hair, Phys. Lett. B 719 (2013) 419 [arXiv:1203.6575] [INSPIRE].
  6. [6]
    G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation, JCAP 01 (2014) 023 [arXiv:1312.4795] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    G. Dvali and C. Gomez, Black Hole Macro-Quantumness, arXiv:1212.0765 [INSPIRE].
  8. [8]
    D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
  10. [10]
    G. Dvali and C. Gomez, Black Holes as Critical Point of Quantum Phase Transition, Eur. Phys. J. C 74 (2014) 2752 [arXiv:1207.4059] [INSPIRE].
  11. [11]
    D. Flassig, A. Pritzel and N. Wintergerst, Black holes and quantumness on macroscopic scales, Phys. Rev. D 87 (2013) 084007 [arXiv:1212.3344] [INSPIRE].
  12. [12]
    G. Dvali, D. Flassig, C. Gomez, A. Pritzel and N. Wintergerst, Scrambling in the Black Hole Portrait, Phys. Rev. D 88 (2013) 124041 [arXiv:1307.3458] [INSPIRE].
  13. [13]
    M. Heusler, Black Hole Uniqueness Theorems, Cambridge University Press, Cambridge, New York (1996).Google Scholar
  14. [14]
    A. Castro, A. Maloney and A. Strominger, Hidden Conformal Symmetry of the Kerr Black Hole, Phys. Rev. D 82 (2010) 024008 [arXiv:1004.0996] [INSPIRE].
  15. [15]
    M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
  16. [16]
    S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82 (1999) 2828 [hep-th/9812013] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
  20. [20]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  21. [21]
    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    R.M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
  25. [25]
    A. Seraj, Conserved charges, surface degrees of freedom and black hole entropy, Ph.D. Thesis, IPM, Tehran (2016) [arXiv:1603.02442] [INSPIRE].
  26. [26]
    C. Troessaert, Canonical Structure of Field Theories with Boundaries and Applications to Gauge Theories, arXiv:1312.6427 [INSPIRE].
  27. [27]
    S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Averin, G. Dvali, C. Gomez and D. Lüst, Gravitational Black Hole Hair from Event Horizon Supertranslations, JHEP 06 (2016) 088 [arXiv:1601.03725] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Averin, G. Dvali, C. Gomez and D. Lüst, Goldstone origin of black hole hair from supertranslations and criticality, Mod. Phys. Lett. A 31 (2016) 1630045 [arXiv:1606.06260] [INSPIRE].
  31. [31]
    M. Mirbabayi and M. Porrati, Dressed Hard States and Black Hole Soft Hair, Phys. Rev. Lett. 117 (2016) 211301 [arXiv:1607.03120] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    R. Bousso and M. Porrati, Soft Hair as a Soft Wig, Class. Quant. Grav. 34 (2017) 204001 [arXiv:1706.00436] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    R. Bousso and M. Porrati, Observable Supertranslations, Phys. Rev. D 96 (2017) 086016 [arXiv:1706.09280] [INSPIRE].
  34. [34]
    B. Gabai and A. Sever, Large gauge symmetries and asymptotic states in QED, JHEP 12 (2016) 095 [arXiv:1607.08599] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    L. Donnay, G. Giribet, H.A. González and A. Puhm, Black hole memory effect, Phys. Rev. D 98 (2018) 124016 [arXiv:1809.07266] [INSPIRE].
  36. [36]
    S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [INSPIRE].
  37. [37]
    J.-i. Koga, Asymptotic symmetries on Killing horizons, Phys. Rev. D 64 (2001) 124012 [gr-qc/0107096] [INSPIRE].
  38. [38]
    M. Perry, Black Hole Entropy from Soft Hair, talk delivered at Second Annual BHI Conference on Black Holes, Cambridge, MA, U.S.A. (2018).Google Scholar
  39. [39]
    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
  40. [40]
    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].
  41. [41]
    G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
  42. [42]
    L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
  45. [45]
    J.D. Brown, Lower Dimensional Gravity, World Scientific, Singapore, Singapore (1988).Google Scholar
  46. [46]
    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
  47. [47]
    I. Bredberg, C. Keeler, V. Lysov and A. Strominger, Cargese Lectures on the Kerr/CFT Correspondence, Nucl. Phys. Proc. Suppl. 216 (2011) 194 [arXiv:1103.2355] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    M.-I. Park, Hamiltonian dynamics of bounded space-time and black hole entropy: Canonical method, Nucl. Phys. B 634 (2002) 339 [hep-th/0111224] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Arnold-Sommerfeld-Center for Theoretical PhysicsLudwig-Maximilians-UniversitätMünchenGermany
  2. 2.Max-Planck-Institut für Physik, Werner-Heisenberg-InstitutMünchenGermany

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