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It says it all in the title, really - is there an equation to estimate the energy required to compress a given amount of matter into a black hole?

Is there a general formula somewhere, that I have not been able to find?

If you are able to provide the right equation, could you also give me a citation, or at least the name of the paper or physicist who derived the equation?

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    $\begingroup$ I’m voting to close this question because it is based on ChatGPT output $\endgroup$ Commented Sep 30, 2024 at 11:22
  • $\begingroup$ A one-ton black hole would be quantum-scale and probably impossible to achieve, so yes, ChatGPT was wrong, as expected. $\endgroup$ Commented Sep 30, 2024 at 12:18
  • $\begingroup$ @PM2Ring not really physics.meta.stackexchange.com/q/14281 and I for one will continue to downvote and close vote them until the technology becomes worth accepting as a valid source $\endgroup$ Commented Sep 30, 2024 at 12:45
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    $\begingroup$ Note that the ChatGPT material seems to have been removed from v4 of the question. Beware that a previous version, which included this output but obscured its source, would have broken our guidelines about plagiarism. @PM2Ring, my reading of the help page (which I didn't write, but which our community can edit) is consistent with the meta posts linked in these comments (which I mostly wrote, in both cases). If you think differently, let's talk on meta. $\endgroup$ Commented Sep 30, 2024 at 17:34
  • $\begingroup$ Ok, that's better. But you need to mention the things that you think are relevant to this calculation, and why you think they're relevant. $\endgroup$ Commented Sep 30, 2024 at 17:35

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There is no such formula because there is no fixed amount of compression needed for collapse.

As a scaling problem the Schwarzschild radius is directly proportional to the mass, while the volume enclosed before collapse is proportional to the cube of the radius. That means that the minimum density required before collapse decreases as the square of the mass.

You do not need to compress mass to a dense state if you simply have a lot of it. Specifically, the Schwarzschild radius is $$R=\frac{2GM}{c^2}$$ and the volume of a sphere (neglecting curvature) is $$V=\frac{4}{3}\pi R^3$$ so the critical density in flat spacetime is $$\rho=\frac{M}{V}=\frac{3}{32\pi}\frac{c^6}{G^3 M^2}$$ $$\rho M^2 = 7.287 \ 10^{79}\mathrm{\ kg^3 \ m^{-3}}$$

So if you have a mass $M$ of material of natural density $\rho$ then you do not need any compression at all for it to form a black hole. For instance, $2.70 \ 10^{38} \mathrm{\ kg}$ of ordinary liquid water would be sufficient to form a black hole without any compression

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  • $\begingroup$ You are using the Euclidean geometry not applicable in the Schwarzschild spacetime: “neglecting curvature” doesn’t work at the horizon since the Schwarzschild volume there is zero, so density is not uniquely defined: arxiv.org/abs/0801.1734 - “There is a zero volume inside the black hole in any Schwarzschild time slice of a Schwarzschild black hole spacetime”. $\endgroup$ Commented Oct 2, 2024 at 20:23

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