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Journal of High Energy Physics

, 2018:125 | Cite as

Symmetries and charges of general relativity at null boundaries

  • Venkatesa Chandrasekaran
  • Éanna É. Flanagan
  • Kartik Prabhu
Open Access
Regular Article - Theoretical Physics
  • 43 Downloads

Abstract

We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger.

Keywords

Black Holes Classical Theories of Gravity Space-Time Symmetries 

Notes

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.

References

  1. [1]
    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. A269 (1962) 21 [INSPIRE].
  2. [2]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A270 (1962) 103 [INSPIRE].
  3. [3]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    D. Kapec, M. Pate and A. Strominger, New Symmetries of QED, Adv. Theor. Math. Phys. 21 (2017) 1769 [arXiv:1506.02906] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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].
  7. [7]
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
  8. [8]
    A. Ashtekar and M. Streubel, Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity, Proc. Roy. Soc. Lond. A376 (1981) 585 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    T. Dray and M. Streubel, Angular momentum at null infinity, Class. Quant. Grav. 1 (1984) 15 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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
  11. [11]
    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].
  12. [12]
    C. Eling and Y. Oz, On the Membrane Paradigm and Spontaneous Breaking of Horizon BMS Symmetries, JHEP 07 (2016) 065 [arXiv:1605.00183] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    R.-G. Cai, S.-M. Ruan and Y.-L. Zhang, Horizon supertranslation and degenerate black hole solutions, JHEP 09 (2016) 163 [arXiv:1609.01056] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S.W. Hawking, The Information Paradox for Black Holes, 2015, arXiv:1509.01147 [INSPIRE].
  15. [15]
    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
  16. [16]
    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
  17. [17]
    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
  18. [18]
    M. Blau and M. O’Loughlin, Horizon Shells and BMS-like Soldering Transformations, JHEP 03 (2016) 029 [arXiv:1512.02858] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    R.F. Penna, Near-horizon BMS symmetries as fluid symmetries, JHEP 10 (2017) 049 [arXiv:1703.07382] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Grumiller and M.M. Sheikh-Jabbari, Membrane Paradigm from Near Horizon Soft Hair, arXiv:1805.11099 [INSPIRE].
  21. [21]
    J.-i. Koga, Asymptotic symmetries on Killing horizons, Phys. Rev. D 64 (2001) 124012 [gr-qc/0107096] [INSPIRE].
  22. [22]
    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].
  23. [23]
    R.F. Penna, BMS invariance and the membrane paradigm, JHEP 03 (2016) 023 [arXiv:1508.06577] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    S. Hollands, A. Ishibashi and R.M. Wald, BMS Supertranslations and Memory in Four and Higher Dimensions, Class. Quant. Grav. 34 (2017) 155005 [arXiv:1612.03290] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A.C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices, Phys. Rev. D 85 (2012) 104049 [Erratum ibid. D 87 (2013) 069904] [arXiv:1105.3445] [INSPIRE].
  28. [28]
    H. Casini, E. Teste and G. Torroba, Modular Hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
  29. [29]
    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
  30. [30]
    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
  31. [31]
    F. Hopfmüller and L. Freidel, Gravity Degrees of Freedom on a Null Surface, Phys. Rev. D 95 (2017) 104006 [arXiv:1611.03096] [INSPIRE].
  32. [32]
    W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    W. Wieland, Fock representation of gravitational boundary modes and the discreteness of the area spectrum, Annales Henri Poincaré 18 (2017) 3695 [arXiv:1706.00479] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, JHEP 02 (2018) 021 [arXiv:1706.05061] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    F. Hopfmüller and L. Freidel, Null Conservation Laws for Gravity, Phys. Rev. D 97 (2018) 124029 [arXiv:1802.06135] [INSPIRE].
  37. [37]
    P.R. Brady, S. Droz, W. Israel and S.M. Morsink, Covariant double null dynamics: (2+2) splitting of the Einstein equations, Class. Quant. Grav. 13 (1996) 2211 [gr-qc/9510040] [INSPIRE].
  38. [38]
    R.J. Epp, The Symplectic structure of general relativity in the double null (2+2) formalism, gr-qc/9511060 [INSPIRE].
  39. [39]
    M.P. Reisenberger, The Symplectic 2-form and Poisson bracket of null canonical gravity, gr-qc/0703134 [INSPIRE].
  40. [40]
    K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan, A Boundary Term for the Gravitational Action with Null Boundaries, Gen. Rel. Grav. 48 (2016) 94 [arXiv:1501.01053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    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., pp. 676-684. Cambridge University Press (1987) [INSPIRE].
  42. [42]
    A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Mechanics, Analysis and Geometry: 200 Years After Lagrange, M. Francaviglia, ed., North-Holland Delta Series, pp. 417-450, Elsevier, Amsterdam (1991) [INSPIRE].
  43. [43]
    J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
  45. [45]
    I. Khavkine, Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A 29 (2014) 1430009 [arXiv:1402.1282] [INSPIRE].
  46. [46]
    M. Geiller, Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity, Nucl. Phys. B 924 (2017) 312 [arXiv:1703.04748] [INSPIRE].
  47. [47]
    K. Prabhu, The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom, Class. Quant. Grav. 34 (2017) 035011 [arXiv:1511.00388] [INSPIRE].
  48. [48]
    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].
  49. [49]
    M.D. Seifert and R.M. Wald, A General variational principle for spherically symmetric perturbations in diffeomorphism covariant theories, Phys. Rev. D 75 (2007) 084029 [gr-qc/0612121] [INSPIRE].
  50. [50]
    R.M. Wald, General Relativity, The University of Chicago Press (1984) [INSPIRE].
  51. [51]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  52. [52]
    D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press (1993) [INSPIRE].
  53. [53]
    L.B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in General Relativity, Living Rev. Rel. 12 (2009) 4 [INSPIRE].CrossRefzbMATHGoogle Scholar
  54. [54]
    É. É. Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95 (2017) 044002 [arXiv:1510.03386] [INSPIRE].
  55. [55]
    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
  56. [56]
    R.M. Wald, On identically closed forms locally constructed from a field, J. Math. Phys. 31 (1990) 2378.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    A. Ashtekar, C. Beetle and J. Lewandowski, Geometry of generic isolated horizons, Class. Quant. Grav. 19 (2002) 1195 [gr-qc/0111067] [INSPIRE].
  58. [58]
    E. Gourgoulhon and J.L. Jaramillo, A 3+1 perspective on null hypersurfaces and isolated horizons, Phys. Rept. 423 (2006) 159 [gr-qc/0503113] [INSPIRE].
  59. [59]
    J.M. Bardeen, B. Carter and S.W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    A. Ashtekar, Geometry and Physics of Null Infinity, arXiv:1409.1800 [INSPIRE].
  61. [61]
    M. Hotta, K. Sasaki and T. Sasaki, Diffeomorphism on horizon as an asymptotic isometry of Schwarzschild black hole, Class. Quant. Grav. 18 (2001) 1823 [gr-qc/0011043] [INSPIRE].
  62. [62]
    D. Lüst, Supertranslations and Holography near the Horizon of Schwarzschild Black Holes, Fortsch. Phys. 66 (2018) 1800001 [arXiv:1711.04582] [INSPIRE].
  63. [63]
    S. Hou, Asymptotic Symmetries of the Null Infinity and the Isolated Horizon, arXiv:1704.05701 [INSPIRE].
  64. [64]
    E. Morales, On a Second Law of Black Hole Mechanics in a Higher Derivative Theory of Gravity, Ph.D. Thesis, University of Gottingen, (2008).Google Scholar
  65. [65]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  66. [66]
    A. Strominger, Black Hole Information Revisited, arXiv:1706.07143 [INSPIRE].
  67. [67]
    A. Ashtekar and A. Magnon-Ashtekar, Energy-Momentum in General Relativity, Phys. Rev. Lett. 43 (1979) 181 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    M. Campiglia, Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime, JHEP 11 (2015) 160 [arXiv:1509.01408] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    M. Campiglia and R. Eyheralde, Asymptotic U(1) charges at spatial infinity, JHEP 11 (2017) 168 [arXiv:1703.07884] [INSPIRE].
  70. [70]
    C. Troessaert, The BMS4 algebra at spatial infinity, Class. Quant. Grav. 35 (2018) 074003 [arXiv:1704.06223] [INSPIRE].
  71. [71]
    K. Prabhu, Conservation of asymptotic charges from past to future null infinity: Maxwell fields, JHEP 10 (2018) 113 [arXiv:1808.07863] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    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
  73. [73]
    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].
  74. [74]
    G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].
  75. [75]
    V.I. Arnol’d, Mathematical methods of classical mechanics, Graduate Texts in Mathematicals, Springer, New York, NY (1978).Google Scholar
  76. [76]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 (2010) [arXiv:1102.4632] [INSPIRE].
  77. [77]
    H. Goldstein, C. Poole and J. Safko, Classical mechanics, Addison-Wesley (2002).Google Scholar
  78. [78]
    J.D. Brown and M. Henneaux, On the Poisson Brackets of Differentiable Generators in Classical Field Theory, J. Math. Phys. 27 (1986) 489 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  79. [79]
    A. Seraj, Conserved charges, surface degrees of freedom and black hole entropy, Ph.D. Thesis, IPM, Tehran (2016) [arXiv:1603.02442] [INSPIRE].
  80. [80]
    H.-Y. Guo, C.-G. Huang and X.-n. Wu, Noether charge realization of diffeomorphism algebra, Phys. Rev. D 67 (2003) 024031 [gr-qc/0208067] [INSPIRE].
  81. [81]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    A. Strominger, Progress and Questions in Soft Physics, presentation at Strings 2018, https://indico.oist.jp/indico/event/5/picture/106.pdf.
  83. [83]
    S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, Black Hole Entropy and Soft Hair, arXiv:1810.01847 [INSPIRE].
  84. [84]
    A. Ashtekar, J. Engle, T. Pawlowski and C. Van Den Broeck, Multipole moments of isolated horizons, Class. Quant. Grav. 21 (2004) 2549 [gr-qc/0401114] [INSPIRE].
  85. [85]
    V. Chandrasekaran, É. É. Flanagan and K. Prabhu, in preparation.Google Scholar
  86. [86]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Higher-Dimensional Supertranslations and Weinberg’s Soft Graviton Theorem, arXiv:1502.07644 [INSPIRE].
  87. [87]
    M. Pate, A.-M. Raclariu and A. Strominger, Gravitational Memory in Higher Dimensions, JHEP 06 (2018) 138 [arXiv:1712.01204] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  88. [88]
    S. Hollands and A. Ishibashi, Asymptotic flatness and Bondi energy in higher dimensional gravity, J. Math. Phys. 46 (2005) 022503 [gr-qc/0304054] [INSPIRE].
  89. [89]
    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
  90. [90]
    M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, arXiv:1810.00377 [INSPIRE].
  92. [92]
    S.A. Hayward, The general solution to the Einstein equations on a null surface, Class. Quant. Grav. 10 (1993) 773.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    L. Barack, Late time decay of scalar, electromagnetic and gravitational perturbations outside rotating black holes, Phys. Rev. D 61 (2000) 024026 [gr-qc/9908005] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Venkatesa Chandrasekaran
    • 1
  • Éanna É. Flanagan
    • 2
    • 3
  • Kartik Prabhu
    • 3
  1. 1.Center for Theoretical Physics and Department of PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.Department of PhysicsCornell UniversityIthacaU.S.A.
  3. 3.Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE)Cornell UniversityIthacaU.S.A.

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