Einstein-Bianchi hyperbolic system for general relativity

Abstract

We consider the evolution part of the Cauchy problem in General Relativity [8] as the time history of the two fundamental forms of a spacelike hypersurface: its metric g and its extrinsic curvature K. On such a hypersurface, for example an “initial” one, these two quadratic forms must satisfy four initial value or constraint equations. These constraints can be posed and solved as an elliptic system by known methods that will not be discussed here. (See, for example, [8].) The Ricci tensor of the spacetime metric can be displayed in a straightforward 3 + 1 decomposition giving the time derivatives of g and K in terms of the space derivatives of these quantities. These expressions contain also the lapse and shift functions characterizing the threading of the spacelike hypersurfaces by time lines. However, proof of the existence of a causal evolution in local Sobolev spaces into an Einsteinian spacetime does not result directly from these equations, which do not form a hyperbolic system for arbitrary lapse and shift, despite the fact that their characteristics are only the light cone and the normal to the time slices [13].