General Relativity and Gravitation

, Volume 5, Issue 1, pp 73–77 | Cite as

Global analysis and general relativity

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


An outline of recent applications of modern infinitedimensional manifold techniques to general relativity is presented. The uses, scope, and future of such methods are delineated. It is argued that the mixing of the two active fields of general relativity and global analysis provides stimulation for both fields as well as producing good theorems. The authors' work on linearization stability of the Einstein equations is sketched out to substantiate the arguments.


Manifold General Relativity Differential Geometry Einstein Equation Linearization Stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnowitt, R., Deser, S. and Misner, C. W. (1962). ‘The Dynamics of General Relativity’, inGravitation: An Introduction to Current Research, (ed. Wittert, L.), (Wiley, New York).Google Scholar
  2. 2.
    Brill, D. (1972). ‘Isolated Solutions in General Relativity’, inGravitation: Problems and Prospects, (Petrov Jubilee Volume), (Naukova Dumka, Kiev).Google Scholar
  3. 3.
    Brill, D. and Deser, S. (1968). ’Variational Methods and Positive Energy in Relativity’,Ann. Phys.,50, 548–570.Google Scholar
  4. 4.
    Fourès-Bruhat, Y. (1962). ‘Cauchy Problem’, inGravitation: An Introduotion to Current Research, (ed. Witten, L.), (Wiley, New York).Google Scholar
  5. 5.
    Choquet-Bruhat, Y. and Deser, S. (1972). ‘Stabilité initiale de l'espace temps de Minkowski’,C. R. Acad. Sci., Paris,275, 1019–1021.Google Scholar
  6. 6.
    Ebin, D. and Marsden, J. (1970). ‘Groups of Diffeomorphisms and the Motion of an Incompressible Fluid’,Ann. Math.,92, 102–163.Google Scholar
  7. 7.
    Fischer, A. (1967). ‘The Theory of Superspace’, inRelativity, (eds. Carmeli, M., Fickler, S. and Witten, L.), (Plenum Press, New York).Google Scholar
  8. 8.
    Fischer, A. and Marsden, J. (1972). ‘The Einstein Equations of Evolution — A Geometric Approach’,J. Math. Phys.,13, 546–568.Google Scholar
  9. 9.
    Fischer, A. and Marsden, J. (1972). ‘The Einstein Evolution Equations as a First Order Quasi-Linear Symmetric Hyperbolic System I’,Commun. Math. Phys.,28, 1–38.Google Scholar
  10. 10.
    Fischer, A. and Marsden, J.(1973). ‘Linearization Stability of the Einstein Equations’,Bull. Amer. Math. Soc.,79, 995–1001.Google Scholar
  11. 11.
    Kundt, W. (1972). ‘Global Theory of Spacetime’, inProceedings of The Thirteenth Biennial Seminar of the Canadian Mathematical Congress, (ed. Vanstone, J.R.), (Canadian Mathematical Congress, Montreal).Google Scholar
  12. 12.
    Lang, S. (1972).Differential Manifolds, (Addison Wesley).Google Scholar
  13. 13.
    Marsden, J. and Fischer, A. (1972).On The Existence of Complete Asymptotically Flat Spacetimes, (Departement des Mathématiques, université de Lyon), Vol. 4, No. 2, 182–193.Google Scholar
  14. 14.
    Marsden, J., Ebin, D. and Fischer, A. (1972). ‘Diffeomorphism Groups, Hydrodynamics and Relativity’, inProceedings of The Thirteenth Biennial Seminar of the Canadian Mathematical Congress, (ed. Vanstone, J. R.), (Canadian Mathematical Congress, Montreal), 135–272.Google Scholar
  15. 15.
    Moncrieff, V. and Taub, A.Second Variation and Stability of Relativistic, Nonrotating Stars, (in preparation).Google Scholar
  16. 16.
    Smale, S. (1964). ‘Morse Theory and a Non-Linear Generalization of the Dirichlet Problem’,Ann. Math.,80, 382–396.Google Scholar
  17. 17.
    Wheeler, J. A. (1964). ‘Geometrodynamics and the Issue of the Final State’, inRelativity, Groups and Topology, (eds. De Witt and De Witt), (Gordon and Breach Science Publishers, New York).Google Scholar

Copyright information

© Plenum Publishing Company Limited 1974

Authors and Affiliations

  • Arthur E. Fischer
    • 1
    • 2
  • Jerrold E. Marsden
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaSanta Cruz
  2. 2.Berkeley

Personalised recommendations