A question that has always intrigued me is: "Imagine a star moving as it evolves into a black hole, Ignore the effect of debris from the supernova. Assume also that before the collapse, the star was moving significantly. How precisely does the information that the star is moving propagate out? Especially the information on the change in the gravitational field."
Now of course the first point is that once the mass collapses beyond the event horizon, we can no longer know that the mass is moving. However we can observe that the event horizon, and surface phenomena are behaving as though the contained mass is moving. Or maybe not, it could be that the event horizon becomes a static sphere, but that just doesn't feel right intuitively.
In any event, the starting point of such an analysis would be to preform a Lorentz, transform on the Schwarzschild solution, and then perhaps study the normal-modes (?) of the resulting solution. Before I begin to do that, does anyone know if that has been done and "published"?