Classical black holes could be likened to being the ultimate prisons. A researcher could hypothetically visit one, checking in to study quantum-gravity effects near its center, but this journey is a one-way ticket, offering no opportunity to check out.
However, continuing with the prison metaphor, imagine trying to fit a plump prisoner into a prison that is much smaller than the prisoner’s body size. Even if the prison officer is successful at placing a small part of the prisoner’s body behind bars, the remainder might still be able to run away.
In the case of a black hole, the prison walls are the event horizon. Karl Schwarzschild derived in 1916 the radius of the event horizon to be 2GM/c^2, where G is Newton’s constant, c is the speed of light and M is the mass of the (non-spinning) black hole. The Schwarzschild radius equals the Bohr radius of the hydrogen atom for a black hole mass of 36 trillion tons. This is comparable to the mass of the Chicxulub impactor that killed the non-avian dinosaurs on Earth 66 million years ago.
A primordial black hole of this mass could have been produced in the early Universe. The Schwarzschild radius equals the radius of the proton for a black hole mass of 580 million tons. This happens to be a few times above the mass of a primordial black hole that evaporates by Hawking radiation on a timescale comparable to the age of the Universe. Smaller primordial black holes would have disappeared by now.
Black holes tend to grow in mass by accreting matter from their astrophysical environment. For black holes more massive than the Sun, the horizon is large enough to be considered as a mouth that absorbs many atoms at once and so the background matter can be approximated as a continuous fluid. But in the regime of primordial black holes of asteroid masses, this assumption is no longer valid. These black holes feast on one atom, one proton or one electron at a time, because their mouth is smaller than the quantum-mechanical size of these particles.
Consider a situation where a hydrogen atom is attracted gravitationally to a primordial black hole. In the naïve perception of the atom as a point particle, it would freely fall into the black hole horizon over a short time. However, if the horizon is smaller than the size of the atom, then this `plump prisoner’ will not be captured by the prison walls, and most of its body will remain outside the prison. Even if the primordial black hole is massive enough to absorb the proton, the electron cloud could remain outside the resulting black hole which just acquired a positive electric charge. The usual electric force that binds the electron to the proton, will be enhanced by gravity.
At distances much larger than the horizon, both electric and gravitational forces decline with the distance squared. Hence, the Bohr radius of the atom will shrink by the enhancement factor in the binding force. The gravitational and electric forces are of equal magnitude for a black hole mass of 4,000 million tons. Gravity wins for larger masses.
The wave function of the bound electron would be modified close to the event horizon, where the quantum-mechanical Dirac equation must be solved in the background metric of a charged black hole. The overlap of the electron wave function with the volume of the event horizon sets a finite lifetime for the electron to stay in a bound state outside the horizon. After this lifetime, the electron will join the proton inside the horizon and the black hole will become electrically neutral again, with the addition of a hydrogen atom to its mass.
In the quantum world, there is a finite probability per unit time for a plump prisoner to be captured by a small prison. The quantum transition to the final state of capture resembles the possibility of quantum tunneling through a barrier. From the perspective of classical physics known to Homer, Sisyphus in the Iliad had to bring the massive boulder to the top of the hill in order for it to overcome gravity and get it to the other side. But quantum-mechanically, Sisyphus could have just waited long enough for the boulder to show up on the other side of the hill, because there is no energy difference between the states of the boulder on the two sides of the hill. The rate of quantum tunneling depends on overlap of the tail of the quantum-mechanical wave function of the boulder with the other side of the hill.
For the same reason, a bound electron can be captured by an atomic nucleus even without combining forces with a central black hole. This process is called electron capture, in which the proton-rich nucleus of a neutral atom absorbs an electron from an inner atomic shell. The rate by which this happens is related to the overlap of the electron wave function with the volume of the atomic nucleus.
Similarly, the accretion of protons and electrons by primordial black holes in the asteroid mass range, involves subtle quantum-mechanical effects. Altogether, if we ever witness such a black hole in the solar system, it could serve as a sandbox for testing quantum-gravitational physics on a subatomic scale.
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s – Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011-2020). He is a former member of the President’s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of the textbook “Life in the Cosmos”, both published in 2021. His new book, titled “Interstellar”, was published in August 2023.