New theoretical techniques in the study of gravity

Fischer, A. E. and J. E. Marsden

Gen. Relativity Gravitation, 4, (1973), 309-317

Abstract:

Using new methods based on first order techniques, it is shown how sharp theorems for existence, uniqueness, and continuous dependence on 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.