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Assume that all of sudden the Earth is turned into a black hole. And the moon revolves around the Earth (before turning into a black hole). What would happen to the Moon after earth changes to black hole will it be sucked to the black hole or continue to revolve the black hole?

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If the earth is changed into an earth-mass black hole, the only effect on the Moon's orbit will be a change in the back-reaction on the Moon of the tidal forces that the Moon exerts on the Earth, as well as subtle changes in the Moon's motion due to quadrupole corrections to the Earth's gravitational field. Neither of these effects would be visible to non-experts observing the Moon's orbit from, say, the ISS.

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Would the Earth's angular momentum be too much for a black hole? I would guess it would. The black hole would also want to get rid of its angular momentum, but I don't think it could transmit it to the moon because there's no mechanism like the quadrupole moment you mention. –  AlanSE Apr 17 '13 at 16:42
The extremal Kerr $J=GM^2/c\sim R_{bh} Mc$. Now, the Earth-mass black hole has radius 9 mm or so so we get about $10^{30}$ Js. The Earth's spin, actual angular momentum now, is indeed over $10^{40}$, ten orders of magnitude too much. This discrepancy is of course linked to Earth's too low density that makes the collapse de facto impossible. –  Luboš Motl Apr 17 '13 at 18:07

Simply, the moon remains in its orbit. It is a common misconception that black holes always "suck" which is quite wrong. They're just like any other celestial body - round, held by gravity, spinning, etc. Only because they're compressed to a much small size, they (in rough words) don't let go of the light (nor any other object) that hits the horizon. They're small in size. But, that doesn't mean that they've differed in their mass. So, the moon just faces its same side to the BH.

But, the observers in the moon or any other satellite rotating around the BH won't see its formation. They just see the Earth shrinking forever...

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I think that force of attraction between earth and moon is depend on their masses and the distance between them . Since any of these factors are not going to change if earth becomes a black hole. Hence it will continue to revolve in orbit

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