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Baughn comments on Compartmentalization as a passive phenomenon - Less Wrong
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My rough calculation says that the density would need to be about a million times greater than Earth's, around 10^10 kg/m^3. This is too low to be a neutron star, but too high to be anything else I think. It may very well be impossible in this universe.
That's assuming uniform density though. Of course you could just have a microblackhole with a hard 1-meter-diameter shell encasing it. How you keep the shell centered is ... trickier.
Would it be possible to keep the black hole charged (use an electron gun), then manipulate electric fields to keep it centered? I don't know enough physics to tell.
Yes, this could work.