If a black hole were to come into sustained existence, assume the smallest one. How long could we stand near it before being unable to escape? And how far is that distance?
R = distance (radius, really) M = Mass of the body G = Universal gravitation constant
We can modify this equation to find for the distance at which you can escape:
r = 2GM/v^2
The answer is largely: it depends on how fast you can go, at the speed of light you can escape from further away, since the pull will increase the closer your are to the "event horizon".
I'm in a car right now (as a passenger ofc) doing this from my phone so not in a situation where I can put together a model, but you should be able to plug in some numbers and estimate a result, just make sure you convert to SI units so you don't accidentally end up 3 orders of magnitude off.
A black hole with the mass of the earth would have a radius of about 2cm, so things less massive than a planet start to get very small, very fast, and you end up fighting quantum effects which become less intuitive.
This is not at all known, as we have no idea what a theory of quantum gravity would look like (which would necessarily enter the game here). We might end up with a black hole remnant, or Hawking radiation might behave differently for microscopic black holes etc.
Source: https://arxiv.org/abs/2004.14192 (There was also a pretty good discussion about it here on HN.)
> would probably be too small to notice without some sort of detection equipment.
What makes you think that?
Theoretically, black holes can have a mass of the tiniest fraction of a gram which would be unimaginabley small. It's my own speculation that you wouldn't be able to detect that with a naked eye.
> you wouldn't be able to detect that with a naked eye.
What if you touched it? No idea what the spacetime would look like near a gram-sized black hole with lots of heavier matter surrounding it but I suppose there would still be pretty severe tidal forces.
Keep in mind that event horizon isn't a shell, just a point at which your future (which is in the singularity) is certain.
Yeah, exactly my thought. Then again, we're silently assuming here that spacetime would pretty much look like one of the vacuum black hole solutions plus some additional matter (our body) near it. That doesn't seem too likely, given that our body is much heavier and can't just be treated as a test particle. OTOH it doesn't seem too likely, either, that the actual spacetime would look completely different: There will surely still be a black hole and an event horizon.
What would it take to get an event horizon on a human scale (a feet or two across?)
A solar mass black hole is stupid dense. But a supermassive black hole is less dense than the earth, and can be less dense than water. That's still an insane amount of mass, but it's not really all that dense.
A human scale black hole would be even denser than a solar mass black hole. It would require over 200 earth masses, though that's still a tiny fraction of a solar mass.
A non spinning black hole is an absolutely perfect sphere, with no "hair". A spinning black hole is flattened, or maybe even a torus, but is still mathematically perfect.
Unless quantum mechanics intervenes in ways nobody has yet figured out.
The singularity occurs at a nominal point at the center (or a ring for a rotating black hole). It has no volume, but all of the mass ends up there, causing divide by zero errors.
Correct. Any mass M taking up a spherical volume of radius less than 2GM/c² (the Schwarzschild radius) will necessarily be a black hole. Black holes are thus the objects in the universe with the highest mass density and, coincidentally, the highest entropy density.
> What would it take to get an event horizon on a human scale (a feet or two across?)
A mass M = Lc²/2G, where L = 1ft for a black hole 2ft across.
https://spacemath.gsfc.nasa.gov/blackh/4Page33.pdf
The calculation in that document is representative: for a solar-mass hole (event horizon radius 2.9km) the tidal forces on a human are 51000x Earth gravity at 100km away!