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Also could it reach parts of the universe that are receding faster than the speed of light from us?

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Theoretically, yes. Assuming that our physics for black holes is correct enough to tell us that the escape velocity of a black hole is just greater than the speed of light. If you think about going faster than the speed of light as going back in time, this also serves as a way of "escaping" the black hole, for you would go back in time to a point to where you were never in the black hole. Therefore, you "escaped."

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Thanks for the answer, just want to point out that Alcubierre drives can't go back in time, because the object inside the warp bubble is not moving faster than light locally - i.e. light inside the warp bubble would still move faster than the object. It just shrinks space ahead and expands it behind, so it travels a shorter path. –  sashoalm Oct 21 '11 at 15:39
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