Null Surface Canonical Formalism

  • J. N. Goldberg
  • D. C. Robinson
  • C. Soteriou
Part of the Ettore Majorana International Science Series book series (EMISS, volume 56)


More than 20 years ago, Peter Bergmann together with one of us (JNG) considered the problem of constructing the canonical formalism for general relativity on a null cone. The motivation for doing so came from the difficulty in constructing the observables which could then become the basic operators in a quantum theory of gravity. The analysis of Bondi1, Sachs2, and Newman-Unti3 of the Einstein equa-tions in the vicinity of null infinity indicated that the constraint equations were easy to integrate and that the data to be specified freely was easily recognizable. On the outgoing null cone, the geometry of the 2-surface foliation is given and at null infinity one gives the news function at all times and the mass aspect and the dipole aspect at one time. This result gave hope that in the canonical formalism one would be able to recognize the appropriate variables for the quantum theory.


Poisson Bracket Class Constraint Dirac Bracket Null Cone Secondary Constraint 
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  1. 1.
    H. Bondi, M.G.J. van der Burg, and A.W.K. Metzner, Gravitational waves in general relativity: VII Waves from axi-symmetric isolated systems, Proc. Rov. Soc. A269, 21 (1962).ADSzbMATHCrossRefGoogle Scholar
  2. 2.
    R.K. Sachs, Gravitational waves in general relativity: VIII Waves in asymptotically flat space-time, Proc. Rov. Soc. A270, 103 (1962).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    E.T. Newman and T.W.J. Unti, Note on the dynamics of gravitational sources, J. Math. Phys. 6, 1806 (1965).ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    J.N. Goldberg, The Hamiltonian of general relativity on a null surface, Found. of Phys. 14, 1211 (1984).ADSCrossRefGoogle Scholar
  5. 5.
    J.N. Goldberg, Dirac brackets for general relativity on a null cone, Found. of Phys. 15, 439 (1985).ADSCrossRefGoogle Scholar
  6. 6.
    C.G. Torre, Null surface geometrodynamics, Class. Quantum Gray. 3, 773 (1986).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R.A. d’Inverno and J. Stachel, Conformal two-structure as the gravitational degrees of freedom in general relativity, J. Math. Phys. 19, 2447 (1978).ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    R.A. d’Inverno and J. Smallwood, Covariant 2+2 formulation of the initial value problem in general relativity, Phys. Rev. D22, 1233 (1980).CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57, 2244 (1986).ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Samuel, A Lagrangian basis for Ashtekar’s reformulation of canonical gravity, Pramana J. Phvs. 28, L429 (1987).ADSCrossRefGoogle Scholar
  11. 11.
    T. Jacobson and L. Smolin, Covariant Action for Ashtekar’s form of canonical gravity, Class. Quantum Gray. 5, 583 (1988).ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • J. N. Goldberg
    • 1
  • D. C. Robinson
    • 2
  • C. Soteriou
    • 2
  1. 1.Department of PhysicsSyracuse UniversitySyracuseUSA
  2. 2.Department of MathematicsKing’s College StrandLondonUK

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