General Relativity and Gravitation

, Volume 4, Issue 4, pp 309–317

New theoretical techniques in the study of gravity

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

DOI: 10.1007/BF00759850

Cite this article as:
Fischer, A.E. & Marsden, J.E. Gen Relat Gravit (1973) 4: 309. doi:10.1007/BF00759850


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.

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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