Communications in Mathematical Physics

, Volume 28, Issue 1, pp 1–38 | Cite as

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I

  • Arthur E. Fischer
  • Jerrold E. Marsden
Article

Abstract

A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented. A number of sharp regularity and smoothness properties of the solutions are obtained. The present paper is devoted to the case ofRn with suitable asymptotic conditions imposed. As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz. These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques. Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abraham, R.: Lectures on global analysis. Mimeographed, Princeton University.Google Scholar
  2. 2.
    Arnowitt, R., Deser, S., Misner, C. W.: The dynamics of general relativity. In: Witten, L. (Ed.): Gravitation; an introduction to current research. New York: John Wiley 1962.Google Scholar
  3. 3.
    Fourès-Bruhat, Y.: Théorèm d'existence pour certain systèmes d'équations aux dérivées partielles non linéaires. Acta Math.88, 141–225 (1952).Google Scholar
  4. 4.
    —— Cauchy problem. In: Witten, L. (Ed.): Gravitation; an introduction to current research. New York: John Wiley 1962.Google Scholar
  5. 5.
    Cantor, M.: Diffeomorphism groups over manifolds with non-compact base (preprint).Google Scholar
  6. 6.
    Choquet-Bruhat, Y.: Espaces-temps einsteiniens généraux, chocs gravitationels. Ann. Inst. Henri Poincaré8, 327–338 (1968).Google Scholar
  7. 7.
    —— SolutionsC d'équations hyperboliques non linéares. C. R. Acad. Sci. Paris272, 386–388 (1971).Google Scholar
  8. 8.
    —— Problèmes mathématiques en relativité. Actes Congres intern. Math. Tome3, 27–32 (1970).Google Scholar
  9. 9.
    C solutions of hyperbolic non-linear equations; applications in general relativity and gravitation. Gen. Rel. Grav.1 (1971).Google Scholar
  10. 10.
    — Stabilité de solutions d'équations hyperboliques non linéares. Application à l'espace-temps de Minkowski en relativité générale. C. R. Acad. Sci. Paris274, Ser. A. (843) (1972).Google Scholar
  11. 11.
    —— Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. math. Phys.14, 329–335 (1969).CrossRefGoogle Scholar
  12. 12.
    Chernoff, P.: Note on product formulas for operator semigroups. J. Funct. An.2, 238–242 (1968).CrossRefGoogle Scholar
  13. 13.
    Chernoff, P., Marsden, J.: On continuity and smoothness of group actions. Bull. Am. Math. Soc.76, 1044–1049 (1970).Google Scholar
  14. 14.
    — Hamiltonian systems and quantum mechanics (in preparation).Google Scholar
  15. 15.
    Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. II. New York: Interscience 1962.Google Scholar
  16. 16.
    Dionne, P.: Sur les problèmes de Cauchy bien posés. J. Anal. Math. Jerusalem10, 1–90 (1962/63).Google Scholar
  17. 17.
    Dunford, N., Schwartz, J.: Linear operators (I). New York: Interscience 1962.Google Scholar
  18. 18.
    Ebin, D.: On the space of Riemannian metrics. In: Proc. Symp. Pure Math., vol. 15. Providence, R. I.: Math. Soc. 1970.Google Scholar
  19. 19.
    —— Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math.92, 102–163 (1970).Google Scholar
  20. 20.
    Fischer, A., Marsden, J.: The Einstein equations of evolution — A geometric approach. J. Math. Phys.13, 546–568 (1972).CrossRefGoogle Scholar
  21. 21.
    —— —— General relativity, partial differential equations and dynamical systems. In: Proc. Symp. Pure Math., vol. 23. Providence, Rhode Island: Am. Math. Soc. 1972 (to appear).Google Scholar
  22. 22.
    Fock, V.: The theory of space, time and gravitation. New York: MacMillan 1964.Google Scholar
  23. 23.
    Friedrichs, K. O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math.7, 345–392 (1954).Google Scholar
  24. 24.
    — A limiting process leading to the equations of relativistic and nonrelativistic fluid dynamics. In: La Magnétohydrodynamique Classique et Relativiste, Lille 1969. Colloques Internat. CNRS, No. 184, Paris, 1970.Google Scholar
  25. 25.
    Hawking, S.: Stable and generic properties in general relativity. Gen. Rel. Grav.1, 393–400 (1971).CrossRefGoogle Scholar
  26. 26.
    Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966.Google Scholar
  27. 27.
    ——: Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan5, 208–234 (1953).Google Scholar
  28. 28.
    ——: Approximation theorems for evolution equations. In: Aziz, A. (Ed.): Lectures in Differential Equations. Mathematical Studies, Vol. 19. New York: Van Nostrand 1969.Google Scholar
  29. 29.
    ——: Linear evolution equations of “hyperbolic type”. J. Fac. Sci. Univ. Tokyo17, 241–258 (1970).Google Scholar
  30. 30.
    Lang, S.: Introduction to differentiable manifolds. New York: Interscience 1962.Google Scholar
  31. 31.
    Lax, P.: Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations. Commun. Pure Appl. Math.8, 615–633 (1955).Google Scholar
  32. 32.
    Leray, J.: Lectures on hyperbolic equations with variable coefficients. Princeton, Inst. for Adv. Stud. 1952.Google Scholar
  33. 33.
    Lichnerowicz, A.: Relativistic Hydrodynamics and Magnetohydrodynamics. New York: Benjamin 1967.Google Scholar
  34. 34.
    Marsden, J., Ebin, D., Fischer, A.: Diffeomorphism groups, hydrodynamics and relativity. In: Proc. Canadian Math. Congress XIII, Halifax (1971) (to appear).Google Scholar
  35. 35.
    Nirenberg, L.: On elliptic differential equations. Scuola Norm. Super. Pisa13, 115–162 (1959).Google Scholar
  36. 36.
    Palais, R.: Seminar on the Atiyah Singer index theorem. Ann. of Math. Studies, no. 57. Princeton: Princeton University Press 1965.Google Scholar
  37. 37.
    Phillips, R.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc.90, 193–254 (1959).Google Scholar
  38. 38.
    Schulenberger, J., Wilcox, C.: Completeness of the wave operators for perturbations of uniformly propagative systems. J. Funct. An.7, 447–474 (1971).CrossRefGoogle Scholar
  39. 39.
    Segal, I.: Non-linear semi-groups. Ann. Math.78, 339–364 (1963).Google Scholar
  40. 40.
    Sobolev, S. L.: Applications of functional analysis in mathematical physics. Providence, R. I.: Am. Math. Soc. 1963.Google Scholar
  41. 41.
    Synge, J. L.: Relativity: The general theory. Amsterdam: North-Holland Publ. Co. 1960.Google Scholar
  42. 42.
    Wilcox, C.: The domain of dependence inequality for symmetric hyperbolic systems. Bull. Am. Math. Soc.70, 149–154 (1964).Google Scholar
  43. 43.
    Yosida, K.: Functional Analysis. Berlin-Heidelberg-New York: Springer 1965.Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • Arthur E. Fischer
    • 1
  • Jerrold E. Marsden
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations