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A persistent challenge in numerical relativity is the correct specificationof boundary conditions. In this work we consider a many parameter family ofsymmetric hyperbolic initial-boundary value formulations for the linearizedEinstein equations and analyze its well posedness using the Laplace-Fouriertechnique. By using this technique ill posed modes can be detected and thus anecessary condition for well posedness is provided. We focus on the followingtypes of boundary conditions: i) Boundary conditions that have been shown topreserve the constraints, ii) boundary conditions that result from setting theingoing constraint characteristic fields to zero and iii) boundary conditionsthat result from considering the projection of Einstein's equations along thenormal to the boundary surface. While we show that in case i) there are no illposed modes, our analysis reveals that, unless the parameters in theformulation are chosen with care, there exist ill posed constraint violatingmodes in the remaining cases.Comment: 10 pages, no figures. Revised version. Added more details on the  evolution of the constraint variable

Detecting ill posed boundary conditions in general relativity