Imagine we have a small black hole, with a mass somewhere around a mountain or small planet. Since it is a single point, it should have an even amount of pull across all matter at the same distance, I think.

If we took this black hole and surrounded it in a thin sphere of ordinary, breakable metal with the exact same strength and put it in a hypothetical 100% empty vacuum, would it damage the sphere?

Because it's a perfect sphere, there should be equal force across its entirety, so it shouldn't bend or break in any one part, I think.

If the black hole were to try to absorb the matter in the sphere, it couldn't pull the entire thing into the black hole without bending or breaking it in some way. But wouldn't one part of it have to be weaker?

This is of course hypothetical, since the existence of a perfect sphere of identical metal is unlikely/impossible and you can't have a perfect vacuum. But would this sphere survive the black hole?

  • $\begingroup$ static Spherically symmetric solution exists and unique. But it doesn't rotate. If it rotates it is not it anymore :) $\endgroup$ – user192234 Nov 27 '19 at 20:08
  • $\begingroup$ The black hole could still compress the metal sphere symmetrically... $\endgroup$ – mmeent Nov 27 '19 at 22:47

The somewhat surprising answer is that yes, for certain sizes of the sphere it would implode even if it was perfect.

The reason is that while the forces are evenly distributed everywhere, they put the sphere under a lot of compressive stress. Beyond a certain limit it will fail and the sphere will move inward since the energy released by moving closer to the black hole is greater than the energy stored in compressing the shell more. This is not that odd for normal materials: clearly they fail when forces get too extreme.

However, this can be shown for any kind of matter that obeys the relativistic energy conditions (positive density, compressive energy has less mass than the matter) the shell will fail when it has a radius less than $$R_{crit}=\frac{25G}{24c^2}\left[M+m+\sqrt{M^2+m^2-\frac{46}{25}Mm}\right]$$ where $m$ is the mass of the black hole and $M$ is the total mass of the system (not quite the mass of the hole plus shell, since there is a fair bit of gravitational binding energy involved). Inside this radius no material shell can persist.

Frauendiener, J., Hoenselaers, C., & Konrad, W. (1990). A shell around a black hole. Classical and Quantum Gravity, 7(4), 585.


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