Journal of High Energy Physics

, 2018:98 | Cite as

Black hole entropy and soft hair

  • Sasha HacoEmail author
  • Stephen W. Hawking
  • Malcolm J. Perry
  • Andrew Strominger
Open Access
Regular Article - Theoretical Physics


A set of infinitesimal Virasoro L ⊗ Virasoro R diffeomorphisms are presented which act non-trivially on the horizon of a generic Kerr black hole with spin J. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Integrability and associativity of the charge algebra are shown to require the inclusion of ‘Wald-Zoupas’ counterterms. A counterterm satisfying the known consistency requirement is constructed and yields central charges cL = cR = 12J. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the central charges reproduce the macroscopic area-entropy law for generic Kerr black holes.


Black Holes Gauge-gravity correspondence Space-Time Symmetries 


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]
    G.T. Horowitz and A. Strominger, Counting states of near extremal black holes, Phys. Rev. Lett. 77 (1996) 2368 [hep-th/9602051] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
  6. [6]
    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
  7. [7]
    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
  8. [8]
    E.E. Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95 (2017) 044002 [arXiv:1510.03386] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    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
  10. [10]
    G. Compère and J. Long, Vacua of the gravitational field, JHEP 07 (2016) 137 [arXiv:1601.04958] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M.M. Sheikh-Jabbari, Residual diffeomorphisms and symplectic soft hairs: The need to refine strict statement of equivalence principle, Int. J. Mod. Phys. D 25 (2016) 1644019 [arXiv:1603.07862] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  12. [12]
    J.E. Baxter, On the global existence of hairy black holes and solitons in Anti-de Sitter Einstein-Yang-Mills theories with compact semisimple gauge groups, Gen. Rel. Grav. 48 (2016) 133 [arXiv:1604.05012] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Compère, Bulk supertranslation memories: a concept reshaping the vacua and black holes of general relativity, Int. J. Mod. Phys. D 25 (2016) 1644006 [arXiv:1606.00377] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    P. Mao, X. Wu and H. Zhang, Soft hairs on isolated horizon implanted by electromagnetic fields, Class. Quant. Grav. 34 (2017) 055003 [arXiv:1606.03226] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    V. Cardoso and L. Gualtieri, Testing the black holeno-hairhypothesis, Class. Quant. Grav. 33 (2016) 174001 [arXiv:1607.03133] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    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
  18. [18]
    D. Grumiller et al., Higher spin black holes with soft hair, JHEP 10 (2016) 119 [arXiv:1607.05360] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    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
  20. [20]
    B. Gabai and A. Sever, Large gauge symmetries and asymptotic states in QED, JHEP 12 (2016) 095 [arXiv:1607.08599] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C. Gomez and M. Panchenko, Asymptotic dynamics, large gauge transformations and infrared symmetries, arXiv:1608.05630 [INSPIRE].
  22. [22]
    D. He and Q.-y. Cai, Gravitational correlation, black hole entropy and information conservation, Sci. China Phys. Mech. Astron. 60 (2017) 040011 [arXiv:1609.05825] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    F. Tamburini, M. De Laurentis, I. Licata and B. Thidé, Twisted soft photon hair implants on Black Holes, Entropy 19 (2017) 458 [arXiv:1702.04094] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Ammon et al.e, Higher-spin flat space cosmologies with soft hair, JHEP 05 (2017) 031 [arXiv:1703.02594] [INSPIRE].
  25. [25]
    P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, Soft gravitons and the memory effect for plane gravitational waves, Phys. Rev. D 96 (2017) 064013 [arXiv:1705.01378] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    R. Bousso and M. Porrati, Soft hair as a soft wig, Class. Quant. Grav. 34 (2017) 204001 [arXiv:1706.00436] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Strominger, Black hole information revisited, arXiv:1706.07143 [INSPIRE].
  28. [28]
    M. Hotta, Y. Nambu and K. Yamaguchi, Soft-hair-enhanced entanglement beyond page curves in a black-hole evaporation qubit model, Phys. Rev. Lett. 120 (2018) 181301 [arXiv:1706.07520] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    R.K. Mishra and R. Sundrum, Asymptotic symmetries, holography and topological hair, JHEP 01 (2018) 014 [arXiv:1706.09080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    C. Gomez and S. Zell, Black hole evaporation, quantum hair and supertranslations, Eur. Phys. J. C 78 (2018) 320 [arXiv:1707.08580] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    D. Grumiller, P. Hacker and W. Merbis, Soft hairy warped black hole entropy, JHEP 02 (2018) 010 [arXiv:1711.07975] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    A. Chatterjee and D.A. Lowe, BMS symmetry, soft particles and memory, Class. Quant. Grav. 35 (2018) 094001 [arXiv:1712.03211] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    C.-S. Chu and Y. Koyama, Soft hair of dynamical black hole and Hawking radiation, JHEP 04 (2018) 056 [arXiv:1801.03658] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    J. Kirklin, Localisation of soft charges and thermodynamics of softly hairy black holes, Class. Quant. Grav. 35 (2018) 175010 [arXiv:1802.08145] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    B. Cvetković and D. Simić, Near horizon OTT black hole asymptotic symmetries and soft hair, arXiv:1804.00484 [INSPIRE].
  36. [36]
    D. Grumiller and M.M. Sheikh-Jabbari, Membrane paradigm from near horizon soft hair, Int. J. Mod. Phys. D 27 (2018) 1847006 [arXiv:1805.11099] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    V. Chandrasekaran, E.E. Flanagan and K. Prabhu, Symmetries and charges of general relativity at null boundaries, JHEP 11 (2018) 125 [arXiv:1807.11499] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A. Averin, Schwarzschild/CFT from soft black hole hair?, arXiv:1808.09923 [INSPIRE].
  39. [39]
    S. Choi and R. Akhoury, Soft photon hair on Schwarzschild horizon from a Wilson line perspective, arXiv:1809.03467 [INSPIRE].
  40. [40]
    L. Donnay, G. Giribet, H.A. González and A. Puhm, Black hole memory effect, arXiv:1809.07266 [INSPIRE].
  41. [41]
    G. Compère and J. Long, Classical static final state of collapse with supertranslation memory, Class. Quant. Grav. 33 (2016) 195001 [arXiv:1602.05197] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    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
  43. [43]
    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].
  44. [44]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  45. [45]
    C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1989).Google Scholar
  46. [46]
    G.J. Zuckerman, Action principles and global geometry, Conf. Proc. C 8607214 (1986) 259 [INSPIRE].Google Scholar
  47. [47]
    J.D. Brown and J.W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
  48. [48]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    V. Iyer and R.M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].
  50. [50]
    R.M. Wald and A. Zoupas, A general definition ofconserved quantitiesin general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
  51. [51]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    G. Compère and A. Fiorucci, Advanced lectures in general relativity, arXiv:1801.07064 [INSPIRE].
  53. [53]
    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].ADSMathSciNetGoogle Scholar
  54. [54]
    C.-M. Chen and J.-R. Sun, Hidden conformal symmetry of the Reissner-Nordstrom black holes, JHEP 08 (2010) 034 [arXiv:1004.3963] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    Y.-Q. Wang and Y.-X. Liu, Hidden conformal symmetry of the Kerr-Newman black hole, JHEP 08 (2010) 087 [arXiv:1004.4661] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    B. Chen and J. Long, Real-time correlators and hidden conformal symmetry in Kerr/CFT correspondence, JHEP 06 (2010) 018 [arXiv:1004.5039] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    M. Becker, S. Cremonini and W. Schulgin, Correlation functions and hidden conformal symmetry of Kerr black holes, JHEP 09 (2010) 022 [arXiv:1005.3571] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    H. Wang, D. Chen, B. Mu and H. Wu, Hidden conformal symmetry of extreme and non-extreme Einstein-Maxwell-Dilaton-Axion black holes, JHEP 11 (2010) 002 [arXiv:1006.0439] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  59. [59]
    I. Agullo, J. Navarro-Salas, G.J. Olmo and L. Parker, Hawking radiation by Kerr black holes and conformal symmetry, Phys. Rev. Lett. 105 (2010) 211305 [arXiv:1006.4404] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    B. Chen and J. Long, Hidden conformal symmetry and quasi-normal modes, Phys. Rev. D 82 (2010) 126013 [arXiv:1009.1010] [INSPIRE].ADSGoogle Scholar
  61. [61]
    M.R. Setare and V. Kamali, Hidden conformal symmetry of extremal Kerr-Bolt spacetimes, JHEP 10 (2010) 074 [arXiv:1011.0809] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    M. Cvetič, G.W. Gibbons and C.N. Pope, Universal area product formulae for rotating and charged black holes in four and higher dimensions, Phys. Rev. Lett. 106 (2011) 121301 [arXiv:1011.0008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    D.A. Lowe, I. Messamah and A. Skanata, Scaling dimensions in hidden Kerr/CFT, Phys. Rev. D 84 (2011) 024030 [arXiv:1105.2035] [INSPIRE].ADSGoogle Scholar
  64. [64]
    M. Cvetič and F. Larsen, Conformal symmetry for general black holes, JHEP 02 (2012) 122 [arXiv:1106.3341] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    M. Cvetič and G.W. Gibbons, Conformal symmetry of a black hole as a scaling limit: a black hole in an asymptotically conical box, JHEP 07 (2012) 014 [arXiv:1201.0601] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    M. Cvetič and F. Larsen, Conformal symmetry for black holes in four dimensions, JHEP 09 (2012) 076 [arXiv:1112.4846] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    G. Compère, The Kerr/CFT correspondence and its extensions, Living Rev. Rel. 15 (2012) 11 [arXiv:1203.3561] [INSPIRE].CrossRefzbMATHGoogle Scholar
  68. [68]
    A. Virmani, Subtracted geometry from Harrison transformations, JHEP 07 (2012) 086 [arXiv:1203.5088] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    M.R. Setare and H. Adami, Near horizon symmetry and entropy formula for Kerr-Newman (A)dS black holes, JHEP 04 (2018) 133 [arXiv:1802.04665] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    H. Gonzalez, D. Grumiller, W. Merbis and R. Wutte, New entropy formula for Kerr black holes, EPJ Web Conf. 168 (2018) 01009 [arXiv:1709.09667] [INSPIRE].CrossRefGoogle Scholar
  71. [71]
    S. Carlip, Black hole entropy from horizon conformal field theory, Nucl. Phys. Proc. Suppl. 88 (2000) 10 [gr-qc/9912118] [INSPIRE].
  72. [72]
    S. Carlip, Symmetries, horizons and black hole entropy, Gen. Rel. Grav. 39 (2007) 1519 [Int. J. Mod. Phys. D 17 (2008) 659] [arXiv:0705.3024] [INSPIRE].
  73. [73]
    S. Carlip, Black hole entropy from Bondi-Metzner-Sachs symmetry at the horizon, Phys. Rev. Lett. 120 (2018) 101301 [arXiv:1702.04439] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].ADSMathSciNetGoogle Scholar
  75. [75]
    K. Hajian, M. Sheikh-Jabbari and H. Yavartanoo, Extreme Kerr black hole microstates with horizon fluff, Phys. Rev. D 98 (2018) 026025 [arXiv:1708.06378] [INSPIRE].ADSMathSciNetGoogle Scholar
  76. [76]
    P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022 [hep-th/0508218] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  77. [77]
    S. Detournay, T. Hartman and D.M. Hofman, Warped conformal field theory, Phys. Rev. D 86 (2012) 124018 [arXiv:1210.0539] [INSPIRE].ADSGoogle Scholar
  78. [78]
    O. Aharon et al., Modular covariance and uniqueness of \( J\overline{T} \) deformed CFTs, arXiv:1808.08978 [INSPIRE].
  79. [79]
    A. Bzowski and M. Guica, The holographic interpretation of \( J\overline{T} \) -deformed CFTs, arXiv:1803.09753 [INSPIRE].
  80. [80]
    M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  82. [82]
    E. Gourgoulhon and J.L. Jaramillo, New theoretical approaches to black holes, New Astron. Rev. 51 (2008) 791 [arXiv:0803.2944] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    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
  86. [86]
    A. Blommaert, T.G. Mertens, H. Verschelde and V.I. Zakharov, Edge state quantization: vector fields in Rindler, JHEP 08 (2018) 196 [arXiv:1801.09910] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    D. Harlow, Wormholes, emergent gauge fields and the weak gravity conjecture, JHEP 01 (2016) 122 [arXiv:1510.07911] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  88. [88]
    J.H. Schwarz, Can string theory overcome deep problems in quantum gravity?, Phys. Lett. B 272 (1991) 239 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  89. [89]
    A. Strominger, Statistical hair on black holes, Phys. Rev. Lett. 77 (1996) 3498 [hep-th/9606016] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Sasha Haco
    • 1
    • 2
    Email author
  • Stephen W. Hawking
    • 1
  • Malcolm J. Perry
    • 1
    • 2
    • 3
  • Andrew Strominger
    • 2
  1. 1.DAMTP, Cambridge University, Centre for Mathematical SciencesCambridgeU.K.
  2. 2.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeUSA
  3. 3.Radcliffe Institute for Advanced StudyCambridgeUSA

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