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If a structure were sufficiently strong at a large enough radius from a black hole, could it completely enclose a black hole without collapsing in, since in some sense the material would pull on itself uniformly?

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The structure you are describing sounds similar to a Dyson sphere, which has been proposed as a method of extracting energy from a star by an extremely advanced civilisation. For distances much larger than the Schwarzschild radius, a black hole has the same gravitational field as a star of the same mass, so the same physics applies*. It is theoretically possible, but well beyond the realms of current materials science and engineering.

One problem with constructing such a structure is that it is unstable: small perturbations would cause it to drift off until the central mass collided with the wall. This is precisely because of the effect you mention: the gravitational forces cancel out leaving no net force on the shell (this is known as the shell theorem).

*Thanks to D. Halsey for suggesting this clarification.

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    $\begingroup$ The question asks about a structure surrounding a black hole, but your answer only discusses a structure around a star. I think your answer would be improved by noting that there could be essentially no difference in the two cases. The OP doesn't seem to realize that. $\endgroup$
    – D. Halsey
    Jun 28 '20 at 22:58
  • $\begingroup$ @D.Halsey thanks, I've added a clarifying sentence. $\endgroup$
    – DavidH
    Jun 29 '20 at 6:39

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