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Let's say a large object (think runaway planet) passed a much smaller one. The larger object's frame dragging distorts the space inhabited by the smaller object, so that it appears to briefly follow the larger one.

When the larger object has passed, does the smaller object keep following because the frame dragging has propelled it in that direction, or does the the object suddenly stop, because it was stationary in space, and the space was only temporarily dragged?

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Taking the larger object as generating a Schwarzschild background, we see that curvature of space decays with distance from the larger object. If you Lorentz boost the Schwarzschild metric, as to make it the metric of a speeding massive object, you will see the same thing. If we focus on the region surrounding your smaller object we then conclude that the curvature will peak at the point of closest approach, but it will fade away as the larger object gets away.

Regarding the question of whether "frame-dragging imparts momentum", it simple to answer it on the rest-frame of the larger object. From the point of view of the larger object, we see a smaller object approaching. So we can study the geodesics on a Schwarzschild metric. For larger impact parameter the smaller object will escape to infinity, while for smaller impact parameters it may orbit the bigger object or even fall into the event horizon if it is a black hole. At the end we can switch back to the original "lab frame" and our conclusions should remain the same.

So, yes, frame-dragging imparts momentum.

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