On the stability of constraint propagation

ABSTRACT. The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to ‘constraint violating modes ’ propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. Our aim in this paper is to investigate one reason for this instability which is due to the choice of the time foliation. We demonstrate this for the Weyl system because this is the essential hyperbolic part in various formulations of the Einstein equations. 1