# Would it be possible to have a small black hole dissipate as it 'sinks' into the earth?

I recently watched this Kurzgesagt video exploring the question of what would happen if you had a black hole the size of a coin.

To summarize the video:

A black hole with the MASS of a coin would have a radius 10^-30m and would decay by Hawking Radiation in 10^-23 seconds with the explosive force of 450 terajoules.

A black hole with the SIZE of a coin would have a mass comparable to Earth and become a dominant gravitational force in the Earth-moon system, passing into the Earth as the Earth begins to orbit it. It wouldn't 'fall' directly inwards, but slightly and then carve out rings of material as the Earth orbits around their shared center of gravity. (2:04 in video). Earth would later collapse into a disk of hot rock. Length of dissipation via Hawking Radiation is not specified.

I want to know if it's possible for the black hole to be created, begin to 'sink' into and consume the planet, only to dissipate before it really begins to destroy the planet.

I guess the outcome I'm looking for is a big, scarily deep hole in the ground with a diameter of several kilometers.

I think the main thing to consider here is would Hawking Radiation be able to overcome the additional mass that the black hole is taking on?

I'm more interested in the hard science behind the formation of the hole than the stability of the hole itself.

• That tiny little black hole which decays "in 10^-23 seconds with the explosive force of 450 terajoules"... That's about 100 kilotons, about 6 times the energy of the Hiroshima bomb. Oct 8, 2017 at 0:18
• A black hole that is large enough to suck up more than a few atoms will be large enough to almost never evaporate. Even the earth mass black hole is too small to suck up much mass. The black hole is an energy source, either from hawking or form the few atoms that get sucked up blasting out energy. Oct 8, 2017 at 23:36

Yes a black hole of a certain size could be capable of falling someway into the earth and then detonating. Unfortunately the size of black hole sufficient to reach a depth of even 2km would be sufficiently large to create an explosion of such magnitude that a big hole would not be an adequate description.

By my calculations roughly 5x10^22 Joules would be released. The largest US Hydrogen bomb at Bikini atoll released ~ 6.3 *10^16 Joules. So very roughly the power of 1 million of the largest H bombs would be available to excavate this "hole" starting off at 2km below the surface. This is roughly of the same order of magnitude as the asteroid impact that destroyed the dinosaurs.

The problem can be split into 3 parts

1) The time to detonation
If we take the ‘detonation’ depth as 2000m how long will it take a black hole to fall this far? s=at^2 where s = distance, a = acceleration and t = time If the black hole starts off at rest on the surface and we assume no friction, it wil take the black hole rougly 20 seconds to fall 2000m.

2) How big is a black hole that will live for 20 sec?
t = (5120*pi*G^2*M^3)/(h*c^4)

Where

t = the life time of the black hole
pi = 3.142
G = The gravitational constant 6.67408×10^−11
M = The mass of the black hole
h = The reduced planck constant (h-bar) 1.055x10^-34
c = The speed of light 3.00×10^8 m/s

Assuming t = 20 sec and substituting all of the standard constants and rearranging gives A black hole mass of around 620,000 kg

3) Can this black hole gain sufficient mass in its short life time to sustain itself?
Could a black hole weighing only 620 tons accumulate more than 620 tons of material when falling for 20 seconds? Considering the gravitational pull and the diameter of a black hole of 620 tons would be miniscule I suggest that it could not replenish even a small fraction of its own weight within 20 seconds and that therefore at around 2000 metres down it would explode. All 620 tons would be released as energy. By E=mc^2 that’s around 5x10^22 Joules (thats a big hole).

Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of gamma rays