Conservation of asymptotic charges from past to future null infinity: supermomentum in general relativity

  • Kartik PrabhuEmail author
Open Access
Regular Article - Theoretical Physics


We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl tensor components on past null infinity are antipodally matched to those on future null infinity. The subalgebra of totally fluxless supertranslations near spatial infinity provides a natural isomorphism between the BMS-supertranslations on past and future null infinity. This proves that the flux of the supermomenta is conserved from past to future null infinity in a classical gravitational scattering process provided additional suitable conditions are satisfied at the timelike infinities.


Classical Theories of Gravity Gauge Symmetry Global Symmetries SpaceTime Symmetries 


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  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. A 269 (1962) 21.Google Scholar
  2. [2]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. A 270 (1962) 103.Google Scholar
  3. [3]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. Penrose, Zero rest-mass fields including gravitation: asymptotic behaviour, Proc. Roy. Soc. A 284 (1965) 159.Google Scholar
  5. [5]
    R.P. Geroch and J. Winicour, Linkages in general relativity, J. Math. Phys. 22 (1981) 803 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Ashtekar and M. Streubel, Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity, Proc. Roy. Soc. A 376 (1981) 585.Google Scholar
  7. [7]
    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].
  8. [8]
    R. Arnowitt, S. Deser and C.W. Misner, The Dynamics of General Relativity, in Gravitation: An Introduction to Current Research, L. Witten eds., Wiley, New York U.S.A. (1962).Google Scholar
  9. [9]
    R.P. Geroch, Structure of the gravitational field at spatial infinity, J. Math. Phys. 13 (1972) 956 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Corichi and J.D. Reyes, The gravitational Hamiltonian, first order action, Poincaré charges and surface terms, Class. Quant. Grav. 32 (2015) 195024 [arXiv:1505.01518] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Beig and B.G. Schmidt, Einstein’s equations near spatial infinity, Commun. Math. Phys. 87 (1982) 65.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Ashtekar and R.O. Hansen, A unified treatment of null and spatial infinity in general relativity. I — Universal structure, asymptotic symmetries and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].
  13. [13]
    P. Sommers, The geometry of the gravitational field at spacelike infinity, J. Math. Phys. 19 (1978) 549.ADSCrossRefGoogle Scholar
  14. [14]
    A. Ashtekar, Asymptotic Structure of the Gravitational Field at Spatial Infinity, in General Relativity and Gravitation. One Hundered Years After the Birth of Albert Einstein. Vol. 2, A. Held eds., Plenum Press, New York U.S.A. (1980) pg. 37.Google Scholar
  15. [15]
    A. Ashtekar and J.D. Romano, Spatial infinity as a boundary of space-time, Class. Quant. Grav. 9 (1992) 1069 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    H. Friedrich, Gravitational fields near space-like and null infinity, J. Geom. Phys. 24 (1998) 83.Google Scholar
  17. [17]
    A. Ashtekar, J. Engle and D. Sloan, Asymptotics and Hamiltonians in a First order formalism, Class. Quant. Grav. 25 (2008) 095020 [arXiv:0802.2527] [INSPIRE].
  18. [18]
    R. Geroch, Asymptotic structure of space-time, in Asymptotic structure of space-time, F.P. Esposito and L. Witten eds., Plenum Press, New York U.S.A. (1977).Google Scholar
  19. [19]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
  20. [20]
    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
  23. [23]
    S.W. Hawking, The Information Paradox for Black Holes, 2015, arXiv:1509.01147 [INSPIRE].
  24. [24]
    R. Bousso and M. Porrati, Soft Hair as a Soft Wig, Class. Quant. Grav. 34 (2017) 204001 [arXiv:1706.00436] [INSPIRE].
  25. [25]
    A. Ashtekar and A. Magnon-Ashtekar, Energy-Momentum in General Relativity, Phys. Rev. Lett. 43 (1979) 181 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. Ashtekar and M. Streubel, On angular momentum of stationary gravitating systems, J. Math. Phys. 20 (1979) 1362.ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    C. Troessaert, The BMS4 algebra at spatial infinity, Class. Quant. Grav. 35 (2018) 074003 [arXiv:1704.06223] [INSPIRE].
  28. [28]
    M. Herberthson and M. Ludvigsen, A relationship between future and past null infinity, Gen. Rel. Grav. 24 (1992) 1185 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton U.S.A. (1993).Google Scholar
  30. [30]
    A. Ashtekar, The BMS group, conservation laws, and soft gravitons, talk presented at the Perimeter Institute for Theoretical Physics, Waterloo Canada (2016). Available online at
  31. [31]
    K. Prabhu, Conservation of asymptotic charges from past to future null infinity: Maxwell fields, JHEP 10 (2018) 113 [arXiv:1808.07863] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    G. Compère and F. Dehouck, Relaxing the Parity Conditions of Asymptotically Flat Gravity, Class. Quant. Grav. 28 (2011) 245016 [Erratum ibid. 30 (2013) 039501] [arXiv:1106.4045] [INSPIRE].
  33. [33]
    S. Hawking and G. Ellis, The Large scale structure of space-time, Cambridge University Press, New York U.S.A. (1973).Google Scholar
  34. [34]
    P.G. Bergmann, ’Gauge-Invariant’ Variables in General Relativity, Phys. Rev. 124 (1961) 274 [INSPIRE].
  35. [35]
    A. Ashtekar, Logarithmic ambiguities in the description of spatial infinity, Found. Phys. 15 (1985) 419.Google Scholar
  36. [36]
    P.T. Chrusciel, On the Structure of Spatial Infinity. 2. Geodesically Regular Ashtekar-hansen Structures, J. Math. Phys. 30 (1989) 2094 [INSPIRE].
  37. [37]
    R.M. Wald, General Relativity, The University of Chicago Press, Chicago U.S.A. (1984).Google Scholar
  38. [38]
    J. Harris, Graduate Texts in Mathematics. Vol. 133: Algebraic Geometry: A First Course, first edition, Springer-Verlag, New York U.S.A. (1992).Google Scholar
  39. [39]
    R. Penrose and W. Rindler, Spinors and Space-Time Vol. 2: Spinor and Twistor Methods in Space-Time Geometry, Cambridge University Press, Cambridge U.K. (1988).Google Scholar
  40. [40]
    A. Kesavan, Asymptotic structure of space-time with a positive cosmological constant, Ph.D. Thesis, The Pennsylvania State University, State College U.S.A. (2016).Google Scholar
  41. [41]
    A. Ashtekar and A. Magnon, From i 0 to the 3 + 1 description of spatial infinity, J. Math. Phys. 25 (1984) 2682.Google Scholar
  42. [42]
    A. Ashtekar, M. Campiglia and A. Laddha, Null infinity, the BMS group and infrared issues, Gen. Rel. Grav. 50 (2018) 140 [arXiv:1808.07093] [INSPIRE].
  43. [43]
    R. Penrose and W. Rindler, Spinors and Space-Time. Vol. 1: Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, Cambridge U.K. (1988).Google Scholar
  44. [44]
    J. Porrill, The structure of timelike infinity for isolated systems, Proc. Roy. Soc. A 381 (1982) 323.Google Scholar
  45. [45]
    C. Cutler, Properties of spacetimes that are asymptotically flat at timelike infinity, Class. Quant. Grav. 6 (1989) 1075.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    D. Christodoulou, The Formation of Black Holes in General Relativity, in On recent developments in theoretical and experimental general relativity, astrophysics and relativistic field theories. Proceedings of 12th Marcel Grossmann Meeting on General Relativity, Paris France (2009), vol. 1-3, pg. 24 [arXiv:0805.3880] [INSPIRE].
  47. [47]
    M. Herberthson and M. Ludvigsen, Time-like infinity and direction-dependent metrics, Class. Quant. Grav. 11 (1994) 187.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    V. Chandrasekaran, É.É. Flanagan and K. Prabhu, Symmetries and charges of general relativity at null boundaries, JHEP 11 (2018) 125 [arXiv:1807.11499] [INSPIRE].
  49. [49]
    P.T. Chrusciel, M.A.H. MacCallum and D.B. Singleton, Gravitational waves in general relativity. XIV: Bondi expansions and the polyhomogeneity of Scri, gr-qc/9305021 [INSPIRE].
  50. [50]
    M. Herberthson, On the differentiability conditions at space-like infinity, Class. Quant. Grav. 15 (1998) 3873 [gr-qc/9712058] [INSPIRE].
  51. [51]
    M. Herberthson, A C >1 Completion of the Kerr Space-Time at Spacelike Infinity Including I + and I ,Gen. Rel. Grav. 33 (2001) 1197.Google Scholar
  52. [52]
    L. Bieri, An Extension of the Stability Theorem of the Minkowski Space in General Relativity, Ph.D. Thesis, ETH Zurich, Zurich Switzerland (2007).Google Scholar
  53. [53]
    L. Bieri, Solutions of the Einstein Vacuum Equations, in AMS/IP Studies in Advanced Mathematics. Vol. 45: Extensions of the Stability Theorem of the Minkowski Space in General Relativity, AMS Press, Providence U.S.A. (2009).Google Scholar
  54. [54]
    L. Bieri, An Extension of the Stability Theorem of the Minkowski Space in General Relativity, J. Diff. Geom. 86 (2010) 17 [arXiv:0904.0620] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    L. Bieri and P.T. Chruściel, Future-complete null hypersurfaces, interior gluings and the Trautman-Bondi mass, in Harvard CMSA Series in Mathematics. Vol. 1: Nonlinear Analysis in Geometry and Applied Mathematics, L. Bieri, P.T. Chruściel and S.-T. Yau eds., International Press of Boston, Inc., Boston U.S.A. (2017) [arXiv:1612.04359] [INSPIRE].
  56. [56]
    L. Bieri, Gravitational radiation and asymptotic flatness, preprint.Google Scholar
  57. [57]
    L. Bieri, Answering the Parity Question for Gravitational Wave Memory, Phys. Rev. D 98 (2018) 124038 [arXiv:1811.09907] [INSPIRE].
  58. [58]
    E.T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs group, J. Math. Phys. 7 (1966) 863 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  59. [59]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    E. Newman and R. Penrose, An Approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    R.P. Geroch, A. Held and R. Penrose, A space-time calculus based on pairs of null directions, J. Math. Phys. 14 (1973) 874 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions of Einstein’s Field Equations, second edition, Cambridge University Press, New York U.S.A. (2009).Google Scholar
  63. [63]
    A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill Book Company, Inc., New York U.S.A. (1953).Google Scholar
  64. [64]
    F.W.J. Olver et al. eds., NIST Digital Library of Mathematical Functions,, Release 1.0.18 of 2018-03-27.

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE)Cornell UniversityIthacaU.S.A.

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