Would a micro black hole keep oscillating around the center of Earth forever?

Question

If a micro black hole was dropped from the surface of the Earth, and that it was too small to absorb particles and eat the Earth, it would fall until emerging from the other side, then fall again and so on...

One would think that it may however collide with particles on the way, which would slow it down and make it ultimately settle in the center of the Earth.

How large would be such an effect, if existing at all? How long would a black hole take before settling depending on its mass?

Edit: “Micro” here means only means that it is too small to feed on Earth's particles, so it could still be very massive at human scales. A $10^{12}\,$kg black hole would still be smaller than a proton, after all.

Also, we are ignoring how such a black hole could have appeared at rest at the surface of the Earth in the first place.

Answer 1

We will use the value of $10^{12}\,\text{kg}$, proposed in OP, for black hole initial mass for estimates.

This value is large enough that the evaporation time through Hawking radiation for such a black hole is longer than the current age on Universe, moreover, the accretion rates if such a black hole is placed inside the Earth would be larger than the loss of mass through Hawking radiation by at least an order of magnitude, so such a black hole would be gradually consuming Earth, with its mass growing exponentially.

Characteristic time for such growth through accretion could be estimated through the Eddington luminosity limit: $$ \tau_\text{E} = \frac{\eta}{4\pi} \, \frac{\sigma_\text{T} c}{ G m_p} \simeq 2.6\times10^7 \,\text{yr}. $$ Here $\sigma_\text{T}$ is Thompson scattering cross-section, $m_p$ is the mass of a proton and $\eta$ is the efficiency of conversion of accreting mass into radiation, which we assume to be about $5\,\%$.

Of course, mass absorption would slow our black hole down, but the main mechanism which would be responsible for the dampening of black hole oscillations is the dynamical friction (a.k.a. gravitational drag), because our black hole could lose its velocity not only through direct absorption of mass/energy but also through long-range gravitational interaction with the surrounding medium.

Characteristic decay timescale of such process is shorter (for oscillations which would reach the surface of the Earth) and could be estimated as $$ \tau_\text{d}=\frac{v^3}{9\pi G^2 M \rho \ln \Lambda} \simeq 2.5\times 10^6 \times\left(\frac{M}{10^{12}\,\text{kg}} \right)^{-1}\left(\frac{v}{8 \, \text{km/s}}\right)^3 \, \text{yr} , $$ where $M$ is the black hole mass, $v$ is its average velocity (for which we assume value of $8\,\text{km/s}$), $\rho$ is an average density of the material through which the black hole is moving ($\rho\simeq 5 \, \text{g/cm}^3$), and $ \ln \Lambda$ is the gravitational Coulomb logarithm. We see that the dampening effect from the dynamical friction is proportional to the black hole mass (decay timescales are shorter for larger black hole), and is inversely proportional to the cube of black hole velocity, so once the black hole lost most of its velocity and is oscillating around the core, the dampening is greatly increased and the decay timescales would be much shorter.

Within a several million years our black hole would be oscillating with much smaller amplitude deeply in the Earth's core where it will be growing doubling in mass every dozen million years. And by the time the black hole accretion rates become large enough for the accompanying radiation to heat up the Earth surface to make it uninhabitable, several dozen million years would pass.

Answer 2

A $10^{12}$ kg Black Hole would be putting out a constant 356 megawatts of Hawking radiation in all directions and it would have an Event Horizon of only 1.4 femtometers (smaller than an atomic nucleus) No particles would even manage to get close to this thing, let alone absorbed by it.

However, it's not going to evaporate any time soon either. By the equation: $t = 5120 M^3 \pi G^2 / (\hbar c ^4)$ we can see it's life time will be $2.6 \times 10^{12}$ years, longer than the history of the Earth and quite a bit longer than it's expected future.

Now this bubble of hawking radiation will produce interesting effects as it vaporizes a pocket of matter inside the earth as it passes back and forth. Some of the light emitted by the black hole will be reflected back into it and in an asymmetric way. The matter behind the black hole will have had more time to be heated by the megawatts of radiation than the matter in front of it. This means that it will emit more radiation back into the black hole and thus increase it's speed through the momentum of the photons (similar to a light sail.) It won't be a large effect considering the small target of 1 fm$^2$ squared, but it will add up over time giving the black hole more and more momentum with each pass until it finally reaches escape velocity and leaves the Earth.