Boundary conditions for metric fluctuations in Lifshitz

We consider the quantization of linearized fluctuations of the metric and matter fields about a Lifshitz background, exploring the possibility of alternative boundary conditions, allowing the slow fall-off modes to fluctuate. We find that for z > 2, slow fall-off modes for some of the linearized fluctuations are normalizable, which opens up the possibility of considering alternative boundary conditions. Analysing stability, we find that alternative boundary conditions for the momentum density are allowed, but alternative boundary conditions for the energy density lead to an instability of the type we recently discovered in a similar analysis for scalar fields on a fixed Lifshitz background. Our investigation is in the context of the simple massive vector model, but we would expect the conclusions to be more general.