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On a symplectic structure of general relativity

Chapter V. Riemannian Spaces — General Relativity

Part of the Lecture Notes in Mathematics book series (LNM,volume 570)

Keywords

  • Vector Field
  • Einstein Equation
  • Einstein Metrics
  • Tensor Density
  • Classical Field Theory

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References

  1. Abraham, R. Foundations of mechanics, New York: Benjamin 1967.

    Google Scholar 

  2. Adler, R, Bazin, M, Sohiffer, M. Introduction to General Relativity. New York: Mc Graw Hill 1965.

    MATH  Google Scholar 

  3. Arnowitt, R, Deser, S, Misner, C.W. The dynamics of General Relativity. In: Witten, L (ed), Gravitation-on introduction to current research, New York: John Wiley 1962.

    Google Scholar 

  4. Berger, M, Ebin, D, Some decompositions of the space of symmetric tensors on a Riemannian manifold, Journ. of Differential Geometry 3 (1969) 379–392.

    MathSciNet  MATH  Google Scholar 

  5. Bergmann, P.G, Komar, A.B, Status report on the quantization of the gravitational field. In: Recent Developments in General Relativity. London-Warsaw. Pergamon Press-PWN: London-Warsaw 1962.

    Google Scholar 

  6. Choquet-Bruhat, Y, The Cauchy problem. In: Witten, L (ed) Gravitation — an introduction to current research. New York: J.Wiley 1962.

    Google Scholar 

  7. Choquet-Bruhat, Y, Geroch, R, Global aspects of the Cauchy problem in G.R. Comm. Math. Phys. 14 (1969) 329–335.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Dedecker,P, Calcul des variations, formes differentieles et champs geodesiques. In: Coll. Intern. Geometrie Differ. Strasbourg 1953.

    Google Scholar 

  9. De Witt, B, Quantum theory of gravity I. The canonical theory. Phys. Rev. 160 (1967) 1113–1148.

    Google Scholar 

  10. Dirac, P.A.M. Generalized Hamiltonian dynamics, Proc.Roy.Soc.(London) A246 (1958) 326–332; The theory of gravitation in Hamiltonian form, Proc. Roy.Soc. A246 333–346.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Ebin, D, Marsden, J, Group of diffeomorphisms and the motion of an incompressible fluid. Ann of Mathematics 92 (1970) 102–163.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Fadeev, L, Symplectic structure and quantization of the Einstein gravitation theory. In: Actes du Congres Intern. a. Math. Nice 1970 35–39. (vol.3)

    Google Scholar 

  13. Fadeev, L, Popov, V, A covariant quantization of the gravitational field, Uspechi fiz. Nauk 111 (1973) 427–450 (in Russian).

    CrossRef  Google Scholar 

  14. Fischer, A, The theory of superspaces. In: Carmell, M, Fickler, S, Witten, L (ed) Relativity. New York: Plenum Press 1970.

    Google Scholar 

  15. Fischer, A, Marsden, J, The Einstein equations of evolution. A geometric approach, Journ. Mat.Phys. 13 (1972) 546–568.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Fischer, A, Marsden, J, The Einstein evolution equations as a quasilinear first order symmetric hyperbolic system I. Comm. Math.Phys. 28 (1972) 1–38.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Fischer, A, Marsden, J, Linearization stability of the Einstein equations, Bull. Am.Math.Soc. 79 (1973) 997–1003.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Fischer, A, Marsden, J, General Relativity as a Hamiltonian system. Symposia Mathematica XIV (published by Istituto di Alta Matematica Roma) London-New York Academic press 1974, 193–205.

    MATH  Google Scholar 

  19. Gawȩdzki, K, On the geometrization of the canonical formalism in the classical field theory, Reports on Math. Phys. 3 (1972) 307–326.

    CrossRef  MathSciNet  Google Scholar 

  20. Goldschmidt, H, Sternberg, S The Hamilton-Cartan formalism in the calculus of variations, Ann.Inst.Fourier 23 (1973) 203–267.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Hawking,S.W. Ellis, G,F.R, The large scale structure of space-time, Cambridge University Press 1973.

    Google Scholar 

  22. Kijowski, J, A finite dimensional canonical formalism in the classical field theory, Comm. Math.Phys. 30 (1973) 99–128.

    CrossRef  MathSciNet  Google Scholar 

  23. Kijowski,J, Szczyrba,W,A canonical structure of classical field theories (to appear in Comm. Math.Phys.).

    Google Scholar 

  24. Kobayashi, S, Nomozu, K, Foundations of differential geometry, New York: Interscience Publ. vol. 1 1963,vol.2 1969.

    Google Scholar 

  25. Kostant, B Quantization and unitary representations, in: Lecture Notes in Mathematics 170, Berlin: Springer-Verlag 1970.

    MATH  Google Scholar 

  26. _____, Symplectic spinors, in Symposia Mathematica vol.XIV London-New York: Academic Press 1974.

    Google Scholar 

  27. Kuchar, K A buble-time canonical formalism for geometrodynamics, Journ. Math. Phys. 13 (1972) 768–781.

    CrossRef  MathSciNet  Google Scholar 

  28. Kundt, W, Canonical quantization of gauge invariant field theories, Springer tracts in modern physics 40. Berlin-Heidelberg-New York: Springer-Verlag 1966.

    Google Scholar 

  29. Lichnerowicz, A Relativistic hydrodynamics and magmotohydrodynamics, New York: Ben jamin 1967.

    Google Scholar 

  30. Misner, Ch.W, Thorne, K.S, Wheeler, J.A, Gravitation, San fransioso: W.H.Freeman and Co 1973.

    Google Scholar 

  31. Narasimhan, R Analysis on real and complex manifolds, Paris:Masson ans Cie 1968.

    MATH  Google Scholar 

  32. Souriau, J.M, Structure des systomes dynamiques, Paris:Dunod 1969.

    Google Scholar 

  33. Szozyrba, Lagrangian formalism in the classical field theory, Ann, Pol.Math. 32 (1975) (to appear).

    Google Scholar 

  34. Wheeler, J.A, Geometrodynamics and the issue of the final state. In: DeWitt B, DeWitt C (ed), Relativity, Groups and Topology, New York:Gordon and Breach 1964.

    Google Scholar 

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Szczyrba, W. (1977). On a symplectic structure of general relativity. In: Bleuler, K., Reetz, A. (eds) Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087798

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  • DOI: https://doi.org/10.1007/BFb0087798

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