Abstract
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.
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References
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).
Brill, D. (1972). ‘Isolated Solutions in General Relativity’, inGravitation: Problems and Prospects, (Petrov Jubilee Volume), (Naukova Dumka, Kiev).
Brill, D. and Deser, S. (1968). ’Variational Methods and Positive Energy in Relativity’,Ann. Phys.,50, 548–570.
Fourès-Bruhat, Y. (1962). ‘Cauchy Problem’, inGravitation: An Introduotion to Current Research, (ed. Witten, L.), (Wiley, New York).
Choquet-Bruhat, Y. and Deser, S. (1972). ‘Stabilité initiale de l'espace temps de Minkowski’,C. R. Acad. Sci., Paris,275, 1019–1021.
Ebin, D. and Marsden, J. (1970). ‘Groups of Diffeomorphisms and the Motion of an Incompressible Fluid’,Ann. Math.,92, 102–163.
Fischer, A. (1967). ‘The Theory of Superspace’, inRelativity, (eds. Carmeli, M., Fickler, S. and Witten, L.), (Plenum Press, New York).
Fischer, A. and Marsden, J. (1972). ‘The Einstein Equations of Evolution — A Geometric Approach’,J. Math. Phys.,13, 546–568.
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.
Fischer, A. and Marsden, J.(1973). ‘Linearization Stability of the Einstein Equations’,Bull. Amer. Math. Soc.,79, 995–1001.
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).
Lang, S. (1972).Differential Manifolds, (Addison Wesley).
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.
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.
Moncrieff, V. and Taub, A.Second Variation and Stability of Relativistic, Nonrotating Stars, (in preparation).
Smale, S. (1964). ‘Morse Theory and a Non-Linear Generalization of the Dirichlet Problem’,Ann. Math.,80, 382–396.
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).
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Editor's Note: This was the prize winning submission in this year's Gravity Research Foundation essay contest.
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Fischer, A.E., Marsden, J.E. Global analysis and general relativity. Gen Relat Gravit 5, 73–77 (1974). https://doi.org/10.1007/BF00758076
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DOI: https://doi.org/10.1007/BF00758076