Communications in Mathematical Physics

, Volume 28, Issue 1, pp 1-38

First online:

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

  • Arthur E. FischerAffiliated withDepartment of Mathematics, University of California
  • , Jerrold E. MarsdenAffiliated withDepartment of Mathematics, University of California

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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 ofR n 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.