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On the geometric structure of the set of solutions of Einstein equations

Autorzy

Wiktor Szczyrba

Seria

Rozprawy Matematyczne tom/nr w serii: 150 wydano: 1977

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Pełne teksty:
rm150_01.pdf

Warianty tytułu

Abstrakty

EN

CONTENTS

1. Introduction .......................................................................................................................................................... 5
2. Notation and preliminary remarks............................................................................................................................ 7
3. A geometric approach to the calculus of variations............................................................................................... 9
4. Multisymplectic manifolds and a multiphase structure of a classical field theory........................................... 19
5. A multiphase structure of General Relativity............................................................................................................ 22
6. The Cauchy problem and ADMW coordinates in General Relativity................................................................... 26
7. A symplectic structure in the set of solutions of field equations.......................................................................... 29
8. A symplectic structure in the set of Einstein metrics.............................................................................................. 36
9. The gauge distribution and the action of the diffeomorphism group.................................................................. 39
10. Degrees of freedom and a superphase space -for General Relativity............................................................. 46
11. A pseudo-differential structure in the space ℋ. A Lie algebra of functionals on ℋ....................................... 48
12. A variational principle for General Relativity............................................................................................................ 55
13. The Hamilton-Jacobi equation in lagrangian field theories................................................................................ 57
14. The Hamilton-Jacobi equation in General Relativity............................................................................................. 62
15. Proofs............................................................................................................................................................................. 66
Appendix. Proof of the ellipticity of the operator AA*..................................................................................................... 79
References.......................................................................................................................................................................... 82

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 150

Liczba stron

83

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae CL

Daty

wydano
1977

Twórcy

autor
Wiktor Szczyrba

Bibliografia

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  • [4] M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, Journ. Differ. Geometry 3 (1969), pp. 379-392.
  • [5] I. Białynicki-Birula and Z. Białynicki-Birula, Quantum electrodynamics, PWN-Pergamon Press, Warsaw-Oxford 1976.
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  • [7] Y. Choquet-Bruhat, The Cauchy problem, in: L. Witten (Ed.), Gravitation-an introduction to current research, John Wiley, New York 1962.
  • [8] Y. Choquet-Bruhat and S. Deser, Stabilité initiale de l'espace de Minkowski, C. R. Acad. Sci. Paris 275 (1975), pp. 1019-1021.
  • [9] Y. Choquet-Bruhat and E. Geroch, Global aspects of the Cauchy problem in General Relativity, Comm. Math. Phys. 14 (1969), pp. 329-335.
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  • [11] B. De Witt, Quantum theory of gravity, I. The canonical theory, Phys. Rev. 160 (1967), pp. 1113-1148.
  • [12] P, A. M. Dirac, Generalized Hamiltonian dynamics, Proc. Roy. Soc. (London) A246 (1958), pp. 326-332; see also: The theory of gravitation in Hamiltonian form, ibid. A246 (1958), pp. 333-346.
  • [13] D. Ebin and J. Marsden, Group of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), pp. 102-163.
  • [14] L. Fadeev, Symplectic structure and quantisation of the Einstein gravitation theory, in: Actes du Congres Intern, d. Mathemat. Nice 1970, vol. 3, pp. 35-39.
  • [15] L. Fadeev and V. Popov, A covariant quantization of the gravitational field (in Russian), Uspechi Fiz. Nauk 111 (1973), pp. 427-450.
  • [16] A. Fischer, The theory of superspaces, in: M. Carmelli, S. Fiekler, L. Witten (Ed.), Relativity, Plenum Press, New York 1970.
  • [17] A. Fischer and J. Marsden, The Einstein equations of evolution. A geometric approach, Journal Math. Phys. 13 (1972), pp. 546-568; see also: General Relativity as a Hamiltonian system, in: Symposia Mathematica vol. XIV, published by Istituto Nazionale di Alta Matematica Roma, Academic Press, London-New York 1974.
  • [18] A. Fischer and J. Marsden, The Einstein evolution equations as a quasilinear first order symmetric hyperbolic system I, Comm. Math. Phys. 28 (1972), pp. 1-38.
  • [19] A. Fischer and J. Marsden, Linearization stability of the Einstein equations, Bull. Amer. Math. Soc. 79 (1973), pp. 997-1003.
  • [20] K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory, Reports on Math. Phys. 3 (1972), pp. 307-326.
  • [21] K. Gawędzki Fourier-like kernels in geometric quantization, Dissert. Mathematicae 128 (1976).
  • [22] H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier 23 (1973), pp. 203-267.
  • [23] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge 1973.
  • [24] J. Kijowski, A finite dimensional canonical formalism in the classical field theory, Comm. Math. Phys. 30 (1973), pp. 99-128.
  • [25] J. Kijowski, Multiphase spaces and gauge in the calculus of variations, Bull. Acad. Pol. Sci. 22 (1974), pp. 1219-1225.
  • [26] J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Comm. Math. Phys. 46 (1976), pp. 183-206, see also: Multisymplectic manifolds and the geometrical construction of the Poisson brackets in the classical field theory, Proceedings Coll. Intern. Géométrie symplectique et Physique Mathématique, Aix-en-Provence 1974, In: Colloques Internationaux CNRS n° 237 (1975), pp. 347-379.
  • [27] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 1, 1963, vol. 2, 1969, Interscience Publ., New York.
  • [28] B. Kostant, Quantizations and unitary representations, in: Lectures in Modern Analysis and Applications III, ed. C. T. Taam, Lecture Notes in Mathematics 170, Springer-Verlag, Berlin 1970; see also: Symplectic spinors, in: Symposia Mathematica vol. XIV, published by Istituto Nazionale di Alta Matematica Roma, Academic Press, London-New York 1974.
  • [29] K. Kuchar, A buble-time canonical formalism for geometrodynamics, Journ. Math. Phya. 13 (1972), pp. 768-781.
  • [30] W. Kundt, Canonical quantization of gauge invariant field theories, Springer tracts in modern physic 40, Springer-Verlag, Berlin-Heidelberg-New York 1966.
  • [31] S. Lang, Introduction to differentiable manifolds, Interscience, New York-London 1962.
  • [32] A. Lichnerowicz, Relativistic hydrodynamics and magnetohydrodynamics, Benjamin, New York 1967.
  • [33] Ch. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman and Company, San Francisco 1973.
  • [34] R. Narasimhan, Analysis on real and complex manifolds, Masson and Cie, Paris 1968.
  • [35] J. M. Souriau, Structure des systèmes dynamiques, Dunod, Paris 1969.
  • [36] S. Sternberg, Lectures on differential geometry, Inc. Englewood N. J., Prentice Hall, 1964.
  • [37] W. Szczyrba, Lagrangian formalism in the classical field theory, Ann. Pol. Math. 32 (1976), pp. 145-185.
  • [38] W. M. Tulczyjew, Seminar on phase spaces, Warsaw 1968 (unpublished); see also: Hamiltonian systems, Lagrangian systems and the Legendre transformation, in: Symposia Mathematica vol. XIV, published by Istituto Nazionale di Alta Matematica Roma, Academic Press, London-New York 1974.
  • [39] W. M. Tulczyjew, Canonical dynamics of relativistic charged particles, Ann. Inst. Poincaré A 15 (1971), pp. 177-187.
  • [40] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), pp. 329-346.
  • [41] J. A. Wheeler, Geometrodynamics and the issue of the final state, in: B. De Witt, C. De Witt (Ed.) Relativity, Groups and Topology, Gordon and Breach, New York 1964.
  • [42] K. Yano, Integral formulas in Riemannian Geometry, Marcel Dekker, New York 1970.

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