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The frame-dragging-effect implies the rotation (in the direction of black hole angular momentum) of every object in the ergosphere. But what about the conservation of angular momentum if the object has a angular momentum in the other direction as the black hole? Would the difference be taken by the black hole?

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If we consider the total angular momentum of the whole system, i.e. the Kerr black hole plus the satellite, then the total angular momentum must be conserved. So yes, any change in the angular momentum of the satellite will be matched by an equal and opposite change in the angular momentum of the black hole. This is the principle behind the Penrose process for extracting energy from a Kerr black hole.

However when analysing motion around a black hole we usually use the approximation that the satellite is so much less massive than the black hole that the backreaction of the satellite on the black hole can be ignored. When analysed this way angular momentum is not conserved, but then we wouldn't expect it to be.

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