Similar to the quantised description of ordinary atoms consisting of protons and neutrons, a new type of atom with microscopic black holes as their nuclei could exist. In ordinary atoms electrons orbit the nucleus due to their electric attraction to the oppositely charged protons in the nucleus, while quantum mechanics tells us that these electrons are quantised to certain energy levels depending on their distance from the nucleus.
The same principle can be applied to the gravitational force. The quantisation of gravity can be approximately described by applying the ideas of Bohr’s atom to Newtonian gravity. Another way would be solving the Schrödinger equation by replacing the electromagnetic potential energy with the Newtonian gravitational potential energy and then assuming the mass of the nucleus to be equal to that of a black hole as given by the Schwarzschild radius. The resulting equation for the nth radius of a mass, m and a black hole of radius, R is given by
r(n) = 2ħ²n²÷c²m²R
Hence, the mass’s orbital radius is inversely proportional to its own mass squared and the radius of the black hole. For large black holes, there is no mass small enough for the initial quantised radii to be observed as they all lie within the event horizon. One could increase n but, just like in ordinary atoms this would result in a continuous energy spectrum and hide the discrete nature of gravity. The energy levels E(n) in terms of the black holes mass M are given as follows
E(n) = -G²M²m³÷2ħ²n²
Hence, the quantisation of gravity is not evident on large scales, where the effects of gravity are mostly considered.
One can calculate the radius a black hole would need in order for all of these radii to be outside of the event horizon by setting the the radius of the horizon equal to the radius of the first energy level of a mass m and solving for it. The result is equal to
R∗ = √2ħ÷cm
Assuming the most natural and simplistic scenario by setting the mass m equal to the mass of an electron, the radius R∗ is equal to 6.7*10^-13 meters which is smaller than an atom yet bigger than a nucleus. This describes a scenario in which the ground state of the electrons is actually on the event horizon of the black hole. By considering a smaller black hole radius, the radius of the ground state is increased. For example, considering the black hole radius to be the size of a nucleus would result in an electron ground state radius on the order of Ångstroms or the size of an ordinary atom.
The mass of such an atom would be equal to 10^11 kg which is about the mass of the entire human population. It is also approximately the mass of a primordial black hole with an evaporation time equal to the age of the universe which raises the thought if there is a connection between the two. Perhaps it would be possible to observe the effect of such black hole atoms evaporating.
In principle it should be possible not only for electrons, but also any other particle to orbit the black hole. The black hole acts as a filtering mechanism which divides each particle into its distinct orbit. Heavier particles lie closer or within the horizon while lighter particles orbit further outside and may leak out of the horizon. This depends on the size of the black hole.
These black hole atoms fit the description of the observed dark matter in the universe due to their high mass and weak interaction with their surrounding other than their gravitational interaction. Moreover, it should be possible to detect them. Just like in ordinary atoms, when electrons jump from one energy level to the next they emit or absorb radiation equal to the change in the energy states.
With electrons orbiting around the black holes, perhaps entire molecules could be composed out of these black hole atoms.