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?