Black hole perturbation theory, or more generally, perturbation theory on a Schwarzschild bockground, has been applied in several contexts, but usually under the simplifying assumption that the ADM momentum vanishes, namely, that the evolution is carried out and observed in the ``center of momentum frame. In this paper we consider some consequences of the inclusion of a non vanishing ADM momentum in the initial data. We first provide a justification for the validity of the transformation of the initial data to the ``center of momentum frame, and then analyze the effect of this transformation on the gravitational wave amplitude. The most significant result is the possibility of a type of gravitational memory effect that appears to have no simple relation with the well known Christodoulou effect.
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