Black holes are often featured in advanced sci-fi technology. A significant challenge for such technology is the safe containment of a black hole. Some stories have electrically charged black holes. I'm wondering about another containment method. If one could encase a black hole in negative mass, either by casimir plates or exotic matter, would this be able to contain the black hole? In this context, I'm using "containment" to mean that 1) positive mass would need to break through the negative mass wall in order to fall into the black hole and 2) the positive mass wall (and the contained black hole) could be accelerated or otherwise moved to other locations.
Sadly, this will not work with the negative-mass exotic matter the OP suggests. There are several fatal problems:
First, you need the exotic matter. That does not seem to be available: it's recently been shown that antimatter has positive gravitational mass.
Then you need enough exotic matter. You presumably want enough that the black hole is thoroughly confined and you can subject the assembly to acceleration without the black hole falling out. That means you need a reasonable fraction of the mass of the black hole. Since you presumably want a heavy enough black hole that it doesn't quickly evaporate as Hawking radiation, you need quite a lot of exotic matter.
Then you have a problem with confinement. Newton's shell theorem tells us that the net gravitational effect of a spherical hollow shell on things inside it is zero, and it seems intuitively likely that this also applies to negative gravity.
There is a way to do move black holes around if you have a lot of exotic matter available. I call it the "stellar bulldozer," and it consists of a flat plate of exotic matter, of a Jupiter mass or more. It's relatively easy to fly matter to the edge of the plate if you come in sideways. Then you release it on the side opposite to the black hole, and it flies away under the negative gravity of the plate. This pushes the plate towards the black hole, and the negative gravity of the plate pushes the black hole away.
Yes, this breaks conservation of energy. Negative mass does that. That's one of the reasons it probably can't exist.
It really depends on what you mean by "Containment". Lets get some basics out of the way first.
A Black Hole is just a region of spacetime where the escape velocity is faster than the speed of light, hence why it is black. This is also what we know as the Event Horizon, and shadow of a black hole. At or around the center we have the Singularity, or Ringularity. There are 3 standard forms of black hole exteriors we can describe. The Schwarzschild (Stationary in time and space, no charge), Kerr (no Charge, Rotating) and Kerr-Newman (Rotating and Charged) Black Hole. For most all applications, Kerr is the model we use because there is no natural way for a black hole to obtain significant net charge.
All Black holes are described by their Mass, Angular Momentum and Charge. It appears as if by Containment you mean the ability to manipulate it. As not a single external quantity of a black hole can be contained. Neither the Gravitational Well of the mass, the Ergo Sphere of the Angular Momentum nor the increased Gravitational Well and weird spacetime of the Charge are things you can wall off. It does not matter what kind of stuff you throw around it, due to your Tags. Neither exotic nor negative mass are real things.
Besides, manipulating a black hole by using the repulsive force of some imaginary matter is about as effective as trying to use a Gravitational tug. So not at all. A Black hole, of any size, has for all intend near infinite inertia and will not be impressed by you hovering some mass over it. Let alone anything of useful Mass.
The way to move a black hole is to exploit the fact quantities are conserved, on a local scale, even for said black hole. If you throw something really hard into the Event Horizon, the black hole will move the other way.
You can move a near non Rotating black hole by firing a laser or particle beam into it. I am saying non Rotating because in the rotating Kerr case, light does not fall into the Horizon following a straight line. So you want as close to 0 spin as possible. Of course this is a very slow way of changing anything. But with a sufficiently powerful laser / particle beam you can do it.