  A persistent challenge in numerical relativity is the correct specification
of boundary conditions. In this work we consider a many parameter family of
symmetric hyperbolic initial-boundary value formulations for the linearized
Einstein equations and analyze its well posedness using the Laplace-Fourier
technique. By using this technique ill posed modes can be detected and thus a
necessary condition for well posedness is provided. We focus on the following
types of boundary conditions: i) Boundary conditions that have been shown to
preserve the constraints, ii) boundary conditions that result from setting the
ingoing constraint characteristic fields to zero and iii) boundary conditions
that result from considering the projection of Einstein's equations along the
normal to the boundary surface. While we show that in case i) there are no ill
posed modes, our analysis reveals that, unless the parameters in the
formulation are chosen with care, there exist ill posed constraint violating
modes in the remaining cases.
