Lets say we have a black hole that has a diameter > 1 light year and we can somehow push and pull this black hole {though the push and pull itself can never exceed lightspeed of course}.Will the other side of the black hole move in less than a year? Why or why not?
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A black hole is not a rigid object and you can't push it. The event horizon is not a physical shell surrounding the black hole, it's just a place in space.
What actually happens if you try to push a black hole, e.g. with a long pole, is surprisingly complicated due to the time dilation and length contraction that happen near the event horizon. See Thought Experiment - Poking a stick across a Black Hole's Event Horizon for more on this. However the bottom line is that you can't push the black hole by pushing it with a pole (or anything else).
Hurling heavy objects into the black hole would cause oscillations in the event horizon, but those oscillations propagate at the speed of light so you couldn't use this for FTL communication either.
John Rennie
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