General Relativity and Gravitation

, Volume 4, Issue 4, pp 309–317 | Cite as

New theoretical techniques in the study of gravity

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

Abstract

Using new methods based on first order techniques, it is shown how sharp theorems for existence, uniqueness, and continuous dependence on the Cauchy data for the exterior Einstein equations can be proved simply and directly. Our main tools are obtained from the theory of quasilinear first order symmetric hyperbolic systems of partial differential equations. Einstein's equations in harmonic coordinates are cast into this form, thus achieving a certain uniformity of the description of gravity with other systems of partial differential equations occurring frequently in mathematical physics. In this symmetric hyperbolic form, the Cauchy problem for the exterior equations is easily resolved. Similarly, using first order techniques, a uniqueness theorem can be proved which increases by one the degree of differentiability of the coordinate-transformation between two solutions of Einstein's equations with the same Cauchy data. Finally it is shown how the theory of first order symmetric hyperbolic systems admits a global intrinsic treatment on manifolds.

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References

  1. 1.
    Choquet-Bruhat, Y. (1968). Espaces-temps einsteiniens généraux chocs gravitationels,Ann. Inst. Henri Poincaré,8, 327–338.MATHMathSciNetGoogle Scholar
  2. 2.
    Choquet-Bruhat, Y. (1971). Solutions C d'équations hyperboliques non linéaires,C.R. Acad. Sci. Paris,272, 386–388.MATHMathSciNetGoogle Scholar
  3. 3.
    Courant, R. and Hilbert, D. (1962).Methods of Mathematical Physics, Vol.II, (Interscience, New York).MATHGoogle Scholar
  4. 4.
    Dionne, P. (1962/3). Sur les problèmes de Cauchy bien posés,J. Anal. Math.,10, 1–90.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fischer, A. and Marsden, J. (1972). General Relativity, Partial Differential Equations, and Dynamical Systems, inProc. Symp. Pure Math.,23, (A.M.S., Providence, Rhode Island), (to appear).Google Scholar
  6. 6.
    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.MATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Fischer, A. and Marsden, J. Quasi-Linear Symmetric Hyperbolic Systems on Manifolds and Applications to General Relativity, (to appear).Google Scholar
  8. 8.
    Fourès-Bruhat, Y. (1952). Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non-linéaires,Act Math.,88, 141–225.MATHCrossRefGoogle Scholar
  9. 9.
    Friedrichs, K.O. (1954). Symmetric Hyperbolic Linear Differential Equations,Commun. Pure Appl. Math.,8, 345–392.MathSciNetGoogle Scholar
  10. 10.
    Lax, P. (1955). Cauchy's Problem for Hyperbolic Equations and the Differentiability of Solutions of Elliptic Equations,Commun. Pure Appl. Math.,8, 615–633.MATHMathSciNetGoogle Scholar
  11. 11.
    Leray, J. (1952).Lectures on Hyperbolic Equations with Variable Coefficients, (Institute for Advanced Study, Princeton, New Jersey).Google Scholar
  12. 12.
    Lichnerowicz, A. (1967).Relativistic Hydrodynamics and Magnetohydrodynamics, (W.A. Benjamin, New York).MATHGoogle Scholar
  13. 13.
    Marsden, J., Ebin, D. and Fischer, A. (1972). Diffeomorphism Groups, Hydrodynamics, and General Relativity, inProceedings of the Canadian Mathematical Congress, XIII, Halifax, pp. 135–279.MathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Company Limited 1973

Authors and Affiliations

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

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