Could we build a spherical shell around a black hole singularity inside the event horizon? [duplicate]

Question

Let's say we have acquired the technology to build such colossal structures. If we built a spherical shell (supposedly made of a very strong material) with a black hole right at the center, would everything now be protected from said black hole? Also, would the black hole's gravity allow you to walk on the outside of the shell?

Answer 1

This is similar to the question on why we can't up a Dyson sphere around a star, or a planet for that matter. Newton's law of gravity $$ \vec F~=~-\frac{GMm\hat r}{r^2} $$ can be evaluated for a shell of matter of total mass $m$ around a large mass $M$. We break the mass up into units of mass $\mathrm dm$ around the thin shell as $$ \mathrm dm~=~\sigma ~\sin\theta~ \mathrm d\theta~\mathrm d\phi, $$ for $\sigma$ the mass per unit area of the shell. Now we consider integrating the magnitude of the force on the shell by $$ F~=~\int \mathrm dF\cdot\hat r~=~-\frac{GM}{r^2}\int_0^{\pi}\int_0^{2\pi} \sin\theta ~\mathrm d\theta~\mathrm d\phi~=~0. $$ The integration of the $\sin\theta$ around $0,~2\pi$ is what makes this vanish. As a result the net force on the mass shell is zero. The central gravitating mass will not be gravitationally bound to the shell. This in electromagnetism is why you have a constant electric potential in a conducting shell where $\vec E~=~-\nabla\Phi$ and the electric force $\vec F~=~q\vec E$ is zero. Therefore the electrostatic potential is constant inside the shell. Much the same hold for gravitation.

The result is similar for a ring, such as a ringworld of Niven and Pournelle or those in the game $\textrm{Halo}$. The star will attract the ring along the azimuthal direction, but not in the plane of the ring. As a result this is not a stable configuration over the long term. Any perturbation on the system can cause a Dyson sphere or ringworld to crash into the central mass or star.

There has been some hubbub over the prospect for a Dyson sphere around the so called Tabby star about $1400$ light years away. If so at least it is far enough away that maybe they will not try to plant a stellar empire flag here. However, I really doubt any intelligent life would invest the huge amount of energy on such a project.

A shell inside the event horizon would be inexorably crushed inwards. The tidal acceleration would not be resisted by any material.