The event horizon of a black hole with a mass of half a billion tons, similar to the mass of a kilometer-size asteroid, has the radius of a proton. Such a black hole evaporates by Hawking radiation on a timescale ten times longer than the age of the Universe.
A mini black hole of this mass could have been produced in the infant universe, about 10 to the power of -23 seconds (ten septillionths of a second) after the Big Bang. As exotic as it sounds, such primordial black holes might not be particularly rare. In fact, they could account for dark matter which is nearly six times more abundant in the cosmic mass budget than ordinary matter.
In a new paper, I show that embedding such a black hole in a typical astrophysical environment would not result in a significant accretion of matter onto it because of quantum mechanics. Previous extensive papers by astrophysicists overestimated the accretion of matter by primordial black holes. They treated the background matter as a continuous fluid. I argue that the accretion onto mini black holes with an event horizon smaller than the size of an atom is suppressed by the quantum mechanical description of matter. Whereas a mini black hole with a mass of 4 billion tons could absorb a proton which is ten times smaller, the tiny event horizon has a lower probability of absorbing an electron with a widespread wave function. The likelihood for the probability-cloud of the electron to overlap with the mini event horizon is much smaller. As a result, the mini black hole can be endowed temporarily with the positive electric charge of the proton, while the electron occupies a bound state around it, making for a `black hole atom’.
The electric binding of the electron to the charged black hole is enhanced by gravity. As a result, the black hole atom is more compact than the hydrogen atom. The binding energy is increased beyond the standard Rydberg scale for hydrogen. This scale is named after the Swedish physicist Johannes Rydberg who derived it empirically in 1888 before it was explained theoretically by the Danish physicist and Nobel laureate Niels Bohr in 1913. Because of the overlap between the electron wave function and the black hole horizon, the bound state of the black hole atom has a finite half-life. I calculated this time to be of order a picosecond (a trillionth of a second) for a black hole mass of 4 billion tons. As short as this timescale sounds, the accretion of one electron-proton pair over this timescale leads to a negligible addition of mass to the black hole even if this process continues steadily for the entire age of the Universe, 13.8 billion years.
Fifty years ago, Stephen Hawking predicted that black holes with a mass of billions of tons shine brightly. The Hawking radiation temperature exceeds the binding energy of the `black hole atom’. In addition, the Hawking luminosity exceed the ratio between the binding energy and the half-life of the transient bound state. As a result, I concluded in my paper that the outward flux of energetic photons and electron-positron pairs would suppress the accretion of matter onto such black holes altogether in rarefied astrophysical environments.
A few days ago, I mentioned my preliminary thoughts on this subject in a brief article. In response, I had received an email from the distinguished physicist, Professor Paul Davies, who in his early career made seminal contributions to quantum field theory in curved spacetime. Paul wrote:
“Hi Avi.
I read this article with interest. In 1972 I attended a summer school in Erice and I recall discussing this very problem with John Wheeler. Wheeler told me he had a graduate student, Jacob Bekenstein, working on it. Well, as we know, that work led to the Bekenstein-Hawking area theorem. I have no idea whether Bekenstein also calculated the capture rate from the ground state of the electron-black hole system, and he is of course now deceased. Wheeler told me that Bekenstein had discovered that a free electron was reluctant to be swallowed because if its Compton wavelength exceeded the black hole radius it tended to scatter instead. But a bound electron is a different matter. I am curious to know whether you or a student have written anything on it. I was planning to set it as an undergraduate project.
All best,
Paul”
My response was short:
“Dear Paul,
Great minds think alike. I was not aware of the Bekenstein-Wheeler history.
Yes, I wrote the paper linked here and submitted it for publication in `The Astrophysical Journal Letters’.
With warm regards,
Avi”
Gladly, scientific insights are resilient to the passage of time. I was delighted to learn that Paul Davies, John Wheeler and Jacob Bekenstein pondered about similar ideas to mine. Their thoughts were never documented and so I was unaware of them 52 years later. As of now, my preprint is being blocked temporarily from being posted on the arXiv online server by moderators. They are not convinced that the content is of wide interest because my paper is too short to their taste. The previous papers which calculated incorrectly accretion onto mini black holes were posted instantly on the arXiv because they were long enough to the taste of the moderators and included plots from numerical calculations. But who cares about arXiv moderators or any other gatekeepers. As the physicist and Nobel laureate Richard Feynman noted in the title of a short book, there is nothing better than the pleasure of finding things out.
ABOUT THE AUTHOR
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.
