  In the Cauchy problem of general relativity one considers initial data that
satisfies certain constraints. The evolution equations guarantee that the
evolved variables will satisfy the constraints at later instants of time. This
is only true within the domain of dependence of the initial data. If one wishes
to consider situations where the evolutions are studied for longer intervals
than the size of the domain of dependence, as is usually the case in three
dimensional numerical relativity, one needs to give boundary data. The boundary
data should be specified in such a way that the constraints are satisfied
everywhere, at all times. In this paper we address this problem for the case of
general relativity linearized around Minkowski space using the generalized
Einstein-Christoffel symmetric hyperbolic system of evolution equations. We
study the evolution equations for the constraints, specify boundary conditions
for them that make them well posed and further choose these boundary conditions
in such a way that the evolution equations for the metric variables are also
well posed. We also consider the case of a manifold with a non-smooth boundary,
as is the usual case of the cubic boxes commonly used in numerical relativity.
The techniques discussed should be applicable to more general cases, as
linearizations around more complicated backgrounds, and may be used to
establish well posedness in the full non-linear case.
