A NONSTRICTLY HYPERBOLIC SYSTEM FOR THE EINSTEIN EQUATIONS WITH ARBITRARY LAPSE AND SHIFT

We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict in the sense of Leray a nd Ohya and is equivalent to the Einstein equations. Its characteristi cs are the light cone and the normal to the slice for any choice of la pse and shift functions, and it admits a well-posed causal Cauchy prob lem in a Gevrey class of index alpha = 2. The system becomes quasidiag onal hyperbolic if we posit a certain wave equation for the lapse func tion, and we can then relate the results to our previously obtained fi rst order symmetric hyperbolic system for general relativity.