Mathematicians have shown that extreme black holes are real
In studying the universe, scientists often turn to its most extreme manifestations in an attempt to uncover fundamental laws of nature. As mathematical physicist Carsten Gundlach of the University of Southampton points out, understanding extreme cases is especially important for gaining insight into cosmic processes.
Black holes, some of the most mysterious objects in the universe, have been used for decades to probe the limits of our understanding of gravity, space, and time. According to Einstein's general theory of relativity, matter inside a black hole is compressed so tightly that nothing can escape. This phenomenon is described by Einstein's famous equation:
However, even among black holes, there are extreme cases. Extreme black holes are a special class of objects that achieve the maximum possible charge or rotation speed for their mass. Their unique property is that their surface gravity at the event horizon is zero. This means that objects on the surface of such a black hole experience no gravitational attraction, but the slightest impulse towards the center will cause them to fall inward.
From hypothesis to proof
In 1973, eminent physicists Stephen Hawking, James Bardeen, and Brandon Carter proposed the hypothesis that extreme black holes could not exist in reality. Their assumption was based on an analogy with the third law of thermodynamics and the lack of plausible mechanisms for the formation of such objects. This hypothesis was considered correct for a long time, and in 1986, physicist Werner Israel even published a proof of it.
But two mathematicians, Christoph Koehle of MIT and Ryan Unger of Stanford University and the University of California at Berkeley, have recently overturned this long-standing hypothesis. In their work, they mathematically proved that the known laws of physics do not prohibit the formation of extreme black holes.
For a charged black hole (described by the Reissner-Nordström solution), the extreme state is reached when the condition is satisfied:
Where M – the mass of the black hole, Q – its electric charge, and a – angular momentum per unit mass. This condition means that the energy associated with the charge and rotation of the black hole is exactly equal to its mass-energy. In this case, the event horizon becomes degenerate, and the surface gravity is zero.
Kele and Unger used a complex mathematical model in which an uncharged, non-rotating black hole interacts with a scalar field. They showed that it is possible to increase the charge of a black hole faster than its mass, reaching an extreme state. This process is described by the equation:
Where dQ/dt represents the rate of change of charge, and dM/dt – the rate of change of the mass of a black hole. If the charge increases faster than the mass, the black hole can reach an extreme state in a finite time.
Michalis Dafermos, a mathematician at Princeton University, called their proof “beautiful, technically innovative, and physically astonishing.” The discovery hints at a potentially more diverse universe where extreme black holes could exist in astrophysical reality.
Future research directions
The proof of the possibility of the existence of extreme black holes opens up new horizons in theoretical physics. It allows us to explore the limits of applicability of general relativity and quantum mechanics in extreme conditions, which may lead to new discoveries at the intersection of these fundamental theories.
However, despite the theoretical possibility, observing extreme black holes remains a difficult task. Black holes with a noticeable electric charge have never been observed in nature, and detecting rapidly rotating black holes requires further development of observational techniques.
Kele and Unger plan to extend their work by investigating the possibility of spinning black holes reaching an extreme state. This requires solving more complex equations, such as the Kerr metric:
This metric describes the spacetime around a rotating black hole, where r – radial coordinate, θ – polar angle, φ – azimuth angle, t – time coordinate, M – the mass of the black hole, and a – rotation parameter. The Kerr metric allows one to study complex effects that occur near rapidly rotating black holes, including the dragging of inertial reference frames and the formation of an ergosphere.
Gaurav Khanna of the University of Rhode Island points out that while Einstein initially thought black holes were too weird to be real, we now know that the universe is teeming with them. By analogy, we shouldn't rule out the possibility of extreme black holes by limiting “nature's creativity.”
The study of extreme black holes demonstrates how mathematics and theoretical physics can push the boundaries of our understanding of the universe. While the existence of extreme black holes in nature remains an open question, the very possibility of their existence forces us to rethink our understanding of space, time, and matter in the most extreme conditions the cosmos has to offer.
Elena Giorgi, a mathematician at Columbia University, calls the discovery “a beautiful example of mathematics coming back to physics.” Indeed, Kaehle and Unger's work shows how abstract mathematical research can lead to profound physical insights that potentially change our understanding of the fundamental structure of the universe.
In conclusion, although we are still a long way from detecting extreme black holes in the real world, the very possibility of their existence opens up exciting prospects for future research. It reminds us that even in seemingly well-studied areas of physics, there is always room for unexpected discoveries that can revolutionize our understanding of the cosmos.
Links from the article
An article about How Black Holes Behave on the Brink of Gravitational Collapse.
The final disproof of the third law of thermodynamics for black holes.
All this and much more — TG “Mathematics is not for everyone”
