This was a humorous thought experiment that occurred while chatting about black holes. The person that I was talking to assumed that a black hole required a specific density to be achieved. I pointed to the formula for the Schwarzschild radius. This suggests that low density black holes are possible if they are large enough. I'll assume simple spherical, non-rotating, uncharged black holes.
$$
r = \frac{2 G M}{c ^ 2}
$$
Using $\rho$ as the the average density.
$$
M = \frac{4}{3} \pi r^3 \rho
$$
$$
\rho = \frac{3}{8 \pi}\frac{c^2}{G r^2}
$$
Which can be as small as you like by having $r$ large enough.
So, assuming that the average density of a person is approximately that of water, I calculate that if I achieved a radius of approximately $4 \times 10^8 km$ then I would become a black hole. If I was centred at the Sun then I would extend into the asteroid belt. Of course, people are not usually spherical though if you get this large then it is probably a good approximation. There may also be some health issues and practical issues with obtaining sufficient food, oxygen, etc but they are off-topic in this group.
Am I right, is a black hole with the same average density as water possible?
Apart from the biological and nutrient issues, what others might I face? I guess that I will suffer severe problems long before I reach this size. Would I collapse and become a star (quite literally)?
Alternatively, if an extremely large number of people gathered in a huge group hug, could they become a black hole?
Update
Thanks to the comments from kleingordon and dmitry-brant, my main question is answered. This leaves:
-
Is my calculation right? (To 1 significant digit)
-
Would I become a star or face some other calamity before becoming a black hole?
Best Answer
As you suggest, long before you got that large gravitation would become dominant. One of the early what ifs is about what you get if you take a mole (ie $6\times 10^{23}$) of moles (small animals) which results in something a little larger than the Moon, and the answer is not very pretty (at least not for the moles).
This is indeed why stars happen: if you get very large collections of matter, they collapse in on themselves gravitationally, fusion starts in their cores (you are mostly carbon, which would fuse I presume). I don't know if anyone has done the sums describing the formation of stars whose precursors were very large humans: I suspect not. But this would happen long before you got large enough to reach your Schwarzschild radius.
In fact, long before that you would hit another problem: surface to volume and temperature. A grovel over the internet says that humans dissipate about $100\,\mathrm{W}$, so if the average human has a mass of $60\,\mathrm{kg}$, then this is about $1.6\,\mathrm{W/kg}$ or (assuming humans are about as dense as water) about $1.6\,\mathrm{kW/m^3}$. If you assume humans are spherical then you get surface flux, I think, of about $550 r\,\mathrm{W/m^3}$, where $r$ is the radius of the human (the units are right here: r has the dimensions of length so the flux comes out as $\mathrm{W/m^2}$). And since humans are approximately black bodies, this corresponds to a surface temperature of
$$\left(\frac{550r}{\sigma}\right)^{\frac{1}{4}}$$
So this is going like $r^{1/4}$ which smells right to me. Plugging in some numbers, a $100\,\mathrm{m}$ radius human is at around $1000\,\mathrm{K}$. So you're dead well before you get that big.
So no, you can't eat yourself into a black hole: you die of heat exhaustion before gravity even becomes significant.