if it were possible to suspend a hemisphere of some exoric material around just half of a black hole so that it stayed tethered a certain distance away, would it act so as to force the opposite side of the black hole to push in the direction of the material?
You mean would it make the black hole ‘blow’ in that direction instead of ‘suck’?
The problem is that a black hole doesn’t really suck. A vacuum cleaner sucks because it has a motor in it that drives a fan that moves air. When you move air out of a space, other air and stuff (dust, cat hair, cracker crumbs, bugs) in the vicinity rush in to fill the space. ‘Nature abhors a vacuum’ is just a way of saying that if there’s no wall to separate regions of different air pressure, the pressure will equalise between those regions.
Gravity is not the same thing as sucking. A massive object pulls stuff towards it because of that curving of space, or so it’s thought. If you put a bowling ball on a trampoline, it curves the surface. Anything put on that surface will slide or roll towards the bowling ball. If you shoot a marble from one edge of the trampoline towards another edge, and it crosses the part that was curved by the bowling ball, its course will be changed. (That’s how you use a big planet like Jupiter to change the course of a spacecraft that you’re trying to send out of the solar system-- it’s a tried and true method.)
Now then, would the opposite be true also? If so, what would happen if it were true?
Opposite of what? You mean, can you make a mass more massive by curving space-time around it? It wouldn’t be ‘compressing’ the mass or making it denser. A larger mass makes a steeper curve, so you’re asking if an induced steeper curve would make the mass heavier.
If you hook a fishing hook into the trampoline directly under the bowling ball, and pull down on the fishing line, you will artificially curve that space. Any marble on the surface of the trampoline will see a greater curve than would have been there from an unassisted bowling ball. From the marble’s perspective, the bowling ball will look more massive than it is. The marble will have no way of knowing that the bowling ball’s mass hasn’t changed and that it has outside help. If the marble can observe the size of the bowling ball as well as how much it curves space, it might think the ball’s density has increased, since it’s mass has increased but its size has not.