Stellar-mass black holes are obviously dangerous. Their massive gravitational field and tidal forces threaten any planet or star system that gets too close to them.

Microscopic black holes are also dangerous, but not because of gravity. They emit huge amounts of hawking radiation before evaporating completely.

Is there an optimal size of black-hole that is "safest"? One where there is a non-lethal dose of radiation, but the tidal forces will not rip you apart?

What would be the "safest" size for a black hole on the surface of the Earth (or at least, one that will kill us the slowest)?

What would be the safest size if it was orbiting Earth (or vice versa)? How about the size for one that simply passes through the solar system?

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    $\begingroup$ A stellar mass black hole is safer than a star of the same mass. There’s less radiation to worry about (unless it has a dense accretion disk), and no risk of a nova or supernova. Its gravitational effect is exactly the same as the star’s would be, unless you get close enough to it that you would be inside it if it was a star, at which point you would already be dead. Obviously it would kill us if it was on the surface of the Earth, but so would a star. $\endgroup$ – Mike Scott Aug 6 '20 at 15:41
  • $\begingroup$ Larger black holes are actually safer than smaller ones since the gravitation gradient is smaller $\endgroup$ – DKNguyen Aug 6 '20 at 18:51
  • $\begingroup$ Remember that by most definitions of black hole, our entire universe is one. And it is moderately safe to live in, at least from my viewpoint. en.wikipedia.org/wiki/Black_hole_cosmology $\endgroup$ – user79911 Nov 18 '20 at 16:32

The thing you have to realize about Hawking radiation is that for mosr black holes, it's not very powerful. The power scales inversely with the square of the mass of the black hole: $$P=\frac{\hbar c^6}{15360\pi GM^2}$$ The upshot of this is that a black hole of $\sim10^{6}\;\text{kg}$ would be only one-millionth as luminous as the Sun, or, according to Wolfram Alpha, comparable in power to a hurricane. This black hole would live for 1.4 minutes; if you chose a black hole that would be stable on geologic timescales - say, one million years - then the emission would be even less intense.

As a rule of thumb, if your black hole is stable over human timescales, it won't produce dangerous amounts of Hawking radiation, and a black hole is only dangerous in the very last moments of its life. You'd have to reach exceedingly small masses for it to be problematic at all. The lower limit, then, is bounded not by Hawking radiation but by how long the black hole has to live. For instance, if we want a black hole to live for 1000 years, then its mass will be $\sim7\times10^8\;\text{kg}$, and it would produce about $2\times10^{-12}L_{\odot}$. In outer space, it would be virtually undetectable, either from its radiation or its gravitational effects.

Now, the tidal forces from a black hole of mass $M$ at a distance $R$ are no different than the tidal forces from any other body of mass $M$ at a distance $R$. We only say that black holes have strong tidal forces because they are so compact, and therefore you can get quite close while remaining outside them. In other words, the tidal forces 10 km away from the center of a black hole are no stronger than the tidal forces 10 km away from any other object of the same mass smaller than 10 km.

From the above, we can imagine that a black hole on the surface of Earth that survives for about 1000 years can fall into the mass range of $\sim10^9\;\text{kg}$ (below which it will evaporate) and $\sim10^{14}\;\text{kg}$ (above which the black hole begins to have a gravitational pull significant in comparison to Earth's within a few hundred meters of it. Outside a few kilometers, the tidal forces are no stronger than that of the Moon. For that black hole of $10^9$ kg, we could go within 1000 feet before the tidal forces became that strong.

A black hole within the Solar System can have a mass comparable to that of, say, a massive moon before it begins to have a gravitational or tidal impact, depending on where it is. If it's no closer than the Oort cloud, it could be of planetary mass and still pose no threat in terms of gravitational disruption; if it's closer than that, perhaps being comparable to a high-mass moon could lead to problems.

  • $\begingroup$ This is a great answer, thank you so much. I never thought about it this way. Since scientific notation is very difficult for me to comprehend, I found something to compare against: Mount Everest likely has a mass of 10^14 kg. But also, since you taught me that the Hawking radiation is only dangerous when the black hole dies, this gives me the idea that a small but eternal black hole could be created, through the proper maintenance of feeding it mass at exactly the correct rate. $\endgroup$ – cowlinator Aug 6 '20 at 16:03
  • $\begingroup$ @cowlinator with the right size black hole, you could keep it exposed to space and have its mass fluctuate very little because of the ambient temperature and particle density. At least for a good bit of cosmic time. $\endgroup$ – BMF Aug 7 '20 at 16:40

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