How small would the earth have to be squashed so that it would become a black hole?
[Physics] How small the earth should be for it to become a black hole
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In the LHC, we are talking about mini black holes of mass around $10^{-24}kg$, so when you talk about $10^{15}-10^{20}kg$ you talk about something in the range from the mass of Deimos (the smallest moon of Mars) up to $1/100$ the mass of the Moon. So we are talking about something really big.
The Schwarzschild radius of such a black hole (using the $10^{20}$ value) would be
$$R_s=\frac{2GM}{c^2}=1.46\times 10^{-7}m=0.146\mu m$$
We can consider that radius to be a measure of the cross section that we can use to calculate the rate that the BH accretes mass. So, the accretion would be a type of Bondi accretion (spherical accretion) that would give an accretion rate
$$\dot{M}=\sigma\rho u=(4\pi R_s^2)\rho_{earth} u,$$
where $u$ is a typical velocity, which in our case would be the speed of sound and $\rho_{earth}$ is the average density of the earth interior. The speed of sound in the interior of the earth can be evaluated to be on average something like
$$c_s^2=\frac{GM_e}{3R_e}.$$
So, the accretion rate is
$$\dot{M}=\frac{4\pi}{\sqrt{3}}\frac{G^2M_{BH}^2}{c^4}\sqrt{\frac{GM_e}{R_e}}.$$
That is an order of magnitude estimation that gives something like $\dot{M}=1.7\times10^{-6}kg/s$. If we take that at face value, it would take something like $10^{23}$ years for the BH to accrete $10^{24}kg$. If we factor in the change in radius of the BH, that time is probably much smaller, but even then it would be something much larger than the age of the universe.
But that is not the whole picture. One should take also in to account the possibility of having a smaller accretion rate due to the Eddington limit. As the matter accretes to the BH it gets hotter since the gravitational potential energy is transformed to thermal energy (virial theorem). The matter then radiates with some characteristic luminosity. The radiation excerpts some back-force on the matter that is accreting lowering the accretion rate. In this case I don't thing that this particular effect plays any part in the evolution of the BH.
An interesting and horrifying possibility for your book could be that of a tiny black hole with negligible mass relative to the mass of the Earth. It would silently sink into the ground, completely unnoticed. It would make damped oscillations around the center of the Earth until eventually staying within the nucleus. There it would stay unnoticed, slowly growing like a parasite. The Earth radius would start to diminish, and thus the crust would have to adapt by means of earthquakes - very weak at first, but of increasing frequency and strength. Eventually, chains of volcanoes would appear along giant fault lines, heating and poisoning the atmosphere, bringing total obliteration. After that, the Earth would continue shrinking until all that was left would be a tiny black hole with little more than one Earth mass in the place where our planet was. And the Moon would stay there as a horrified witness of the catastrophe.
Nothing on Earth would be able to stop the process and save us. Nothing except Chuck Norris.
Note: Classical GR black holes are fully determined with just three values: mass, electric charge and angular momentum (No-hair theorem). If the charge is large enough in the black hole of your novel, then the hero might be able to confine it by using strong magnetic fields.
Best Answer
You can define a Schwarzschild Radius based solely on total mass, i.e. $R_s = \frac{2GM}{c^2}$. If you plug in the mass of the earth, the radius is about 9 mm --- which is how small you would have to compress it to make a black-hole.