General Relativity and Gravitation

, Volume 7, Issue 12, pp 915–920

A new Hamiltonian structure for the dynamics of general relativity

  • Arthur E. Fischer
  • Jerrold E. Marsden
Research Articles


A new compact form of the dynamical equations of relativity is proposed. The new form clarifies the covariance of the equations under coordinate transformations of the space-time. On a deeper level, we obtain new insight into the infinite-dimensional symplectic geometry behind the dynamical equations, the decompositions of gravitational perturbations, and the space of gravitational degrees of freedom. Prospects for these results in studying fields coupled to gravity and the quantization of gravity are outlined.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, R. (1967).Foundations of Mechanics (W. A. Benjamin, New York).Google Scholar
  2. 2.
    Arms, J., Fischer, A., and Marsden, J. (1975).C. R. Acad. Sci. Paris,281, 517.Google Scholar
  3. 3.
    Arnowitt, R., Deser, S., and Misner, C. W. (1962).Gravitation: An Introduction to Current Research, ed. by Witten, L. (Wiley, New York).Google Scholar
  4. 4.
    Beiger, M., and Ebin, D. (1967).J. Diff. Geom.,3, 379.Google Scholar
  5. 5.
    Brill, D., and Deser, S. (1968).Ann. Phys. N. Y.,50, 548.Google Scholar
  6. 6.
    Choquet-Bruhat, Y. (1975).C. R. Acad. Sci. Paris,280, 169; see also Cantor, M., Fischer, A., Marsden, J., O'Murchadha, N., and York, J. W.Commun. Math. Phys.,49, 187.Google Scholar
  7. 7.
    Choquet-Bruhat, Y., and Deser, S. (1976).Ann. Phys. N. Y.,81, 165–178; see also C. R. Acad. Sci. Paris275 (1972), 1019–1021.Google Scholar
  8. 8.
    Choquet-Bruhat, Y., and Marsden, J. (1974). Solution of the local mass problem in general relativity (to appear in Comm. Math. Phys.); see also (1976).C. R. Acad. Sci.,282, 609.Google Scholar
  9. 9.
    Chernoff, P., and Marsden, J. (1974). “Properties of Infinite Dimensional Hamiltonian Systems”, Springer Lecture Notes No. 425.Google Scholar
  10. 10.
    Deser, S. (1967).Ann. Inst. H. Poincaré,7, 149.Google Scholar
  11. 11.
    Fischer, A., and Marsden, I. (1972).J. Math. Phys.,13, 546.Google Scholar
  12. 12.
    Fischer, A., and Marsden, J. (1973).Gen. Rel Grav.,5, 73.Google Scholar
  13. 13.
    Fischer, A., and Marsden, J. (1975).Proc. Symp. Pure Math. Vol. XXVII, part 2, 219–263; see also(1973).Bull. Am. Math. Soc.,79, 997; (1975).Duke Math. J.,42, 519.Google Scholar
  14. 14.
    Fischer, A., Marsden, J., and Moncrief, V. Linearization stability of the Einstein equations, in preparation.Google Scholar
  15. 15.
    Hughes, T., Kato, T., and Marsden, J. Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity (to appear inArch. Rat. Mech. and Analysis).Google Scholar
  16. 16.
    Marsden, J., and Weinstein, A. (1974).Rep. Math. Phys.,5, 121.Google Scholar
  17. 17.
    Misner, C. W. Thorne, K. S., and Wheeler, J. A. (1973).Gravitation (W. H. Freeman, San Francisco).Google Scholar
  18. 18.
    Moncrief, V. (1975).J. Math. Phys.,16, 493; and (1976).J. Math. Phys.,17, 1893.Google Scholar
  19. 19.
    Moncrief, V. (1975).J. Math. Phys.,16, 1556.Google Scholar
  20. 20.
    O'Murchadha, N., and York, J. W. (1974).Phys. Rev. D,10,428, 437.Google Scholar
  21. 21.
    Smale, S. (1970).Inventions Math.,10, 305;11, 45.Google Scholar
  22. 22.
    York, J. W. (1973).J. Math. Phys.,14, 456.Google Scholar
  23. 23.
    York, J. W. (1974).Ann. Inst. H. Poincaré,21, 319.Google Scholar
  24. 24.
    Dirac, P. A. M. (1965).Lectures on Quantum Mechanics (Academic Press, New York).Google Scholar
  25. 25.
    Kuchar, K. (1976).J. Math. Phys.,17, 777; 792; 801.Google Scholar
  26. 26.
    Kuchar, K. Tensor sources in geometrodynamics, to appear.Google Scholar

Copyright information

© Plenum Publishing Corp 1977

Authors and Affiliations

  • Arthur E. Fischer
    • 1
  • Jerrold E. Marsden
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSanta Cruz and Berkeley

Personalised recommendations