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What happens to the fabric of space in the wake of a moving black hole? Is space permanently deformed by a moving black hole or does it rebound as the black hole passes?

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A black hole is only special and interesting when you are at or inside the event horizon. From far outside of the event horizon a black hole has the same effects gravitationally, as a normal star or planet of the same mass. – vsz Dec 4 '12 at 7:09
Moving relative to what? – Johannes Dec 4 '12 at 7:42

A black hole creates curvature in spacetime just like any other massive object, such as a star, which makes it attract other matter gravitationally. A black hole isn't any more exotic than a star in that regard.

The black hole's gravitational field is just a consequence of its mass existing in that location. If a black hole is in motion relative to you, then the curvature of spacetime that it creates will follow it.

You could, of course, also argue that the black hole is at rest, and you're the one who's in motion. That's why it wouldn't make sense for a black hole to "permanently" deform spacetime, since that would mean that there's a preferred frame of reference in which the black hole is in motion.

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Excellent answer. I'd actually be interested to know what the "blackhole wormhole" theories say about this - can an Einstein-Rosen bridge move, and (if so) does it leave distorted spacetime in its wake? – Polynomial Dec 4 '12 at 10:51
@Polynomial: As Dmitry says, you can't have a "permanent" distortion because that would mean there's a preferred frame of reference. – Mark Hurd Dec 4 '12 at 10:56
@Polynomial In order to leave a "wake", an object needs to travel through a medium in which a wake could be made (e.g. a meteor falling in the atmosphere). Space itself is not a "medium" of that kind. It is Lorentz-invariant, meaning that there are no preferred reference frames (no "absolute" motion through space, only relative motion). – Dmitry Brant Dec 4 '12 at 16:42
@DmitryBrant That makes sense. Nice one :) – Polynomial Dec 4 '12 at 16:45

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